Lambert, Interval Cycles, Spectrum

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Society for Music Theory

Interval Cycles as Compositional Resources in the Music of Charles IvesAuthor(s): J. Philip LambertSource: Music Theory Spectrum, Vol. 12, No. 1 (Spring, 1990), pp. 43-82Published by: {oupl} on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/746146Accessed: 23-09-2015 16:49 UTC

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I n t e r v a l

y c l e s

s

ompositional

esources

in

t h e

u s i c

o f

h a r l e s

v e s

J.

Philip

Lambert

I n t e r v a l

y c l e s

s

ompositional

esources

in

t h e

u s i c

o f

h a r l e s

v e s

J.

Philip

Lambert

I n t e r v a l

y c l e s

s

ompositional

esources

in

t h e

u s i c

o f

h a r l e s

v e s

J.

Philip

Lambert

Students

of

the music

of

CharlesIves

over the

past

20

years

have made

convincing

arguments

or

elevating

his musical

quo-

tations to

a

level of structure ar above that

of

the musicalsur-

face. Dennis

Marshall

provided

a seminal definitionof the

is-

sue,

asking

whether borrowed materials are

part

of the

surface-level "manner"or are more

integrated

nto the "sub-

stance"

of a

composition,1

and

analysts

have

since affirmed hat

Ives's

quotations

are more than tidbitsof musical Americana

added to enhance the nationalistic lavor.2Recently, J. Peter

Burkholderhas identified

and defined

specific echniques,

such

as

"modeling"

and

"paraphrasing,"

hat Ives

employs

n incor-

porating

borrowed

material,

supporting

the assertion

that

'Dennis

Marshall,

"Charles Ives's

Quotations:

Manner or

Substance?"

Perspectives f

New Music

6/2

(1968),

45-56;

repr.

n

Perspectives

n American

Composers,

ed.

Benjamin

Boretz and EdwardT. Cone

(New

York:

Norton,

1971),

13-24.

Henry

and

Sidney

Cowell address

he

same ssues n Charles ves

and His

Music

(1955;

reissued

with additional

material,

1969;

repr.

New

York:

OxfordUniversityPress, 1975), 147.

2Some

representative

studies are Gordon

Cyr,

"Intervallic

Structural

Ele-

ments

n Ives's Fourth

Symphony,"

Perspectives f

New Music

9/2-10/1

(1971),

291-303;

Christopher

Ballantine,

"Charles ves and

the

Meaning

of

Quotation

in

Music,"

Musical

Quarterly

5

(1979),

167-184;

Stuart

Feder,

"Decoration

Day:

A

Boyhood Memory

of

Charles

Ives,"

Musical

Quarterly

6

(1980),

234-

261.

For a

typicalopposing

view,

see Kurt

Stone,

"Ives'sFourth

Symphony:

A

Review,"

Musical

Quarterly

2

(1966),

1-16.

Students

of

the music

of

CharlesIves

over the

past

20

years

have made

convincing

arguments

or

elevating

his musical

quo-

tations to

a


of

the musicalsur-

face. Dennis

Marshall

provided


is-

sue,

asking


part

of the


integrated

nto the "sub-

stance"

of a

composition,1

and

analysts

have

since affirmed hat

Ives's

quotations




and defined

specific echniques,

such

as

"modeling"

and

"paraphrasing,"

hat Ives

employs

n incor-

porating

borrowed

material,

supporting

the assertion

that

'Dennis

Marshall,

"Charles Ives's

Quotations:

Manner or

Substance?"

Perspectives f

New Music

6/2

(1968),

45-56;

repr.

n

Perspectives

n American

Composers,

ed.

Benjamin


(New

York:

Norton,

1971),

13-24.

Henry

and

Sidney

Cowell address

he


and His

Music

(1955;

reissued

with additional

material,

1969;

repr.

New

York:


2Some

representative

studies are Gordon

Cyr,

"Intervallic

Structural

Ele-

ments

n Ives's Fourth

Symphony,"

Perspectives f

New Music

9/2-10/1

(1971),

291-303;

Christopher

Ballantine,

"Charles ves and

the

Meaning

of

Quotation

in

Music,"

Musical

Quarterly

5

(1979),

167-184;

Stuart

Feder,

"Decoration

Day:

A

Boyhood Memory

of

Charles

Ives,"

Musical

Quarterly

6

(1980),

234-

261.

For a

typicalopposing

view,

see Kurt

Stone,

"Ives'sFourth

Symphony:

A

Review,"

Musical

Quarterly

2

(1966),

1-16.

Students

of

the music

of

CharlesIves

over the

past

20

years

have made

convincing

arguments

or

elevating

his musical

quo-

tations to

a


of

the musicalsur-

face. Dennis

Marshall

provided


is-

sue,

asking


part

of the


integrated

nto the "sub-

stance"

of a

composition,1

and

analysts

have

since affirmed hat

Ives's

quotations




and defined

specific echniques,

such

as

"modeling"

and

"paraphrasing,"

hat Ives

employs

n incor-

porating

borrowed

material,

supporting

the assertion

that

'Dennis

Marshall,

"Charles Ives's

Quotations:

Manner or

Substance?"

Perspectives f

New Music

6/2

(1968),

45-56;

repr.

n

Perspectives

n American

Composers,

ed.

Benjamin


(New

York:

Norton,

1971),

13-24.

Henry

and

Sidney

Cowell address

he


and His

Music

(1955;

reissued

with additional

material,

1969;

repr.

New

York:


2Some

representative

studies are Gordon

Cyr,

"Intervallic

Structural

Ele-

ments

n Ives's Fourth

Symphony,"

Perspectives f

New Music

9/2-10/1

(1971),

291-303;

Christopher

Ballantine,

"Charles ves and

the

Meaning

of

Quotation

in

Music,"

Musical

Quarterly

5

(1979),

167-184;

Stuart

Feder,

"Decoration

Day:

A

Boyhood Memory

of

Charles

Ives,"

Musical

Quarterly

6

(1980),

234-

261.

For a

typicalopposing

view,

see Kurt

Stone,

"Ives'sFourth

Symphony:

A

Review,"

Musical

Quarterly

2

(1966),

1-16.

"Ives's

reworking

of

existing

music

is

the

single

most central

technique

in his

process

of

creation."3From

this

perspective,

much of Ives's music

responds favorably

to close

analytical

scrutiny,

and

the

incorporation

and

integration

of

quotations

s

revealed to be a vital

organizing

orce in

a

musical

language

breakingaway

from

tonality.4

While Ives

may

have been

a

supremepractitioner

f musical

borrowing,

however,

he was also a

composer

of

complex

nter-

ests who refused to be confined to a single compositionalpos-

ture.

His methods of

achieving

musical

unity

without

tonality

may

have relied

heavily,

and

successfully,

on

"modeling"pro-

cedures concentratedon

existing

deas,

but

they

also embraced

attempts

at

developing

a more abstract

anguage,

ndependent

of

any

structural ramework hat borrowedmaterial

mightpro-

vide.

Works without

musical

quotations

may

receive some uni-

fying

contributions rom

textual or other extramusical

actors,

but

theymight

also

pursue

a

structural

purity

born of relation-

3J.

Peter

Burkholder,

"

'Quotation'

and

Emulation:

Charles

ves's

Uses

of

His

Models,"

Musical

Quarterly

1

(1985),

20.

See also

Burkholder,

"

'Quota-

tion'

and

Paraphrase

n

Ives's

Second

Symphony,"Nineteenth-Century

Music

11

(1987),

3-25.

4Robert

Morgan

discusses

Ives's

quotations

n this

ight

in

"Rewriting

Mu-

sic

History:

Second

Thoughts

on Ives

and

Varese,"

Musical Newsletter3/1

(January1973),

8-12.

"Ives's

reworking

of

existing

music

is

the

single

most central

technique

in his

process

of

creation."3From

this

perspective,


responds favorably

to close

analytical

scrutiny,

and

the

incorporation

and

integration

of

quotations

s


organizing

orce in

a

musical

language

breakingaway

from

tonality.4

While Ives

may

have been

a

supremepractitioner

f musical

borrowing,

however,

he was also a

composer

of

complex

nter-


ture.

His methods of

achieving

musical

unity

without

tonality

may

have relied

heavily,

and

successfully,

on

"modeling"pro-


existing

deas,

but

they

also embraced

attempts

at

developing

a more abstract

anguage,

ndependent

of

any


mightpro-

vide.

Works without

musical

quotations

may

receive some uni-

fying

contributions rom


actors,

but

theymight

also

pursue

a

structural

purity

born of relation-

3J.

Peter

Burkholder,

"

'Quotation'

and

Emulation:

Charles

ves's

Uses

of

His

Models,"

Musical

Quarterly

1

(1985),

20.

See also

Burkholder,

"

'Quota-

tion'

and

Paraphrase

n

Ives's

Second


Music

11

(1987),

3-25.

4Robert

Morgan

discusses

Ives's

quotations

n this

ight

in

"Rewriting

Mu-

sic

History:

Second

Thoughts

on Ives

and

Varese,"


(January1973),

8-12.

"Ives's

reworking

of

existing

music

is

the

single

most central

technique

in his

process

of

creation."3From

this

perspective,


responds favorably

to close

analytical

scrutiny,

and

the

incorporation

and

integration

of

quotations

s


organizing

orce in

a

musical

language

breakingaway

from

tonality.4

While Ives

may

have been

a

supremepractitioner

f musical

borrowing,

however,

he was also a

composer

of

complex

nter-


ture.

His methods of

achieving

musical

unity

without

tonality

may

have relied

heavily,

and

successfully,

on

"modeling"pro-


existing

deas,

but

they

also embraced

attempts

at

developing

a more abstract

anguage,

ndependent

of

any


mightpro-

vide.

Works without

musical

quotations

may

receive some uni-

fying

contributions rom


actors,

but

theymight

also

pursue

a

structural

purity

born of relation-

3J.

Peter

Burkholder,

"

'Quotation'

and

Emulation:

Charles

ves's

Uses

of

His

Models,"

Musical

Quarterly

1

(1985),

20.

See also

Burkholder,

"

'Quota-

tion'

and

Paraphrase

n

Ives's

Second


Music

11

(1987),

3-25.

4Robert

Morgan

discusses

Ives's

quotations

n this

ight

in

"Rewriting

Mu-

sic

History:

Second

Thoughts

on Ives

and

Varese,"


(January1973),

8-12.

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44

Music

Theory Spectrum

4

Music

Theory Spectrum

4

Music

Theory Spectrum

ships

defined

contextually

within

the

available resources.

While a single broadlydefined process may characterize he

creative

evolution of an

Ives

composition,

the

organizational

problems posed

by

the

individual

approaches

and solutions n

variousworks

are

numerousand

diverse.5

Indeed,

it is in

the

works without

quotations

where

Ives's

compositional

objectives

are most

overt,

his search or

organi-

zationalalternativesmost

consistently

apparent.

These works

generallyseparate

hemselvesfrom

the main

body

of his

music,

distinguished by

their

searches

for

components

of

abstract

structureand by their attention to technical detail. Ives gives

many

of them

titles such

as

"Study"

or

"Exercise,"

making

clear that his

compositional purpose

is to work

out technical

problems.6

He also refers

to works of

this

type

as "memos in

notes,"

a

phrase

that

alludes to their

unrefined tate as

well

as

the

very private

nature of

his

perspective

on them.7 These

pieces

are,

in

fact,

"experiments"

n

musical

organization-

musical

ncarnationsof

ideas about

structure hat

may

receive

theirformulation

only

in

the courseof

composing

he work. Al-

togetherIves'sexperimentalworks form a distinctive ubset of

his

music,

serving

as a

forum for the isolation

and

exploration

of

specific

echnical ssues.8

5Burkholder eels that

Ives's

"working

out

of technical

problems

and his

creationof musical

analogues

o

texts and to

programmatic

onceptions

ollow

the same

pattern

as his

elaborationof

borrowed

material,"

and that

the

process

common to these activities would be

the basis of a

"unifiedview

of

Ives's

ap-

proach

o

composition"

"

'Quotation'

and

Emulation,"20,

n.

36).

Testing

or

this

hyphothesis

will come

as each

compositional

area s

subjected

o

thorough

investigation.

6For

example,

the

piano

pieces

that fall in this

category

are entitled"Stud-

ies."

Similarly,

the

song Soliloquy

is

entitled

"A

Study

in 7ths and Other

Things."

Also

typical

s the

subtitle to

Chromdtimelodtune:

Ear-Study

aural

& mental

exercise )."

7Charles

E.

Ives, Memos,

edited and with

appendicesby

John

Kirkpatrick

(New

York:

Norton,

1972),

64.

8Ives's

memoirs describe an

active influenceof his

father on these

experi-

mental

attitudes,

although

the

reliability

of

Ives's accounthas been

questioned

ships

defined

contextually

within

the



creative

evolution of an

Ives

composition,

the

organizational

problems posed

by

the

individual

approaches

and solutions n

variousworks

are

numerousand

diverse.5

Indeed,

it is in

the

works without

quotations

where

Ives's

compositional

objectives

are most

overt,

his search or

organi-


consistently

apparent.

These works

generallyseparate

hemselvesfrom

the main

body

of his

music,

distinguished by

their

searches

for

components

of

abstract


many

of them

titles such

as

"Study"

or

"Exercise,"

making

clear that his


is to work

out technical

problems.6

He also refers

to works of

this

type

as "memos in

notes,"

a

phrase

that

alludes to their

unrefined tate as

well

as

the

very private

nature of

his

perspective

on them.7 These

pieces

are,

in

fact,

"experiments"

n

musical

organization-

musical

ncarnationsof

ideas about

structure hat

may

receive

theirformulation

only

in

the courseof

composing

he work. Al-


his

music,

serving

as a


and

exploration

of

specific

echnical ssues.8


Ives's

"working

out

of technical

problems

and his

creationof musical

analogues

o

texts and to

programmatic

onceptions

ollow

the same

pattern

as his

elaborationof

borrowed

material,"

and that

the

process


the basis of a

"unifiedview

of

Ives's

ap-

proach

o

composition"

"

'Quotation'

and

Emulation,"20,

n.

36).

Testing

or

this

hyphothesis

will come

as each

compositional

area s

subjected

o

thorough

investigation.

6For

example,

the

piano

pieces

that fall in this

category

are entitled"Stud-

ies."

Similarly,

the

song Soliloquy

is

entitled

"A

Study

in 7ths and Other

Things."

Also

typical

s the

subtitle to

Chromdtimelodtune:

Ear-Study

aural

& mental

exercise )."

7Charles

E.

Ives, Memos,

edited and with

appendicesby

John

Kirkpatrick

(New

York:

Norton,

1972),

64.

8Ives's

memoirs describe an


father on these

experi-

mental

attitudes,

although

the

reliability

of


questioned

ships

defined

contextually

within

the



creative

evolution of an

Ives

composition,

the

organizational

problems posed

by

the

individual

approaches

and solutions n

variousworks

are

numerousand

diverse.5

Indeed,

it is in

the

works without

quotations

where

Ives's

compositional

objectives

are most

overt,

his search or

organi-


consistently

apparent.

These works

generallyseparate

hemselvesfrom

the main

body

of his

music,

distinguished by

their

searches

for

components

of

abstract


many

of them

titles such

as

"Study"

or

"Exercise,"

making

clear that his


is to work

out technical

problems.6

He also refers

to works of

this

type

as "memos in

notes,"

a

phrase

that

alludes to their

unrefined tate as

well

as

the

very private

nature of

his

perspective

on them.7 These

pieces

are,

in

fact,

"experiments"

n

musical

organization-

musical

ncarnationsof

ideas about

structure hat

may

receive

theirformulation

only

in

the courseof

composing

he work. Al-


his

music,

serving

as a


and

exploration

of

specific

echnical ssues.8


Ives's

"working

out

of technical

problems

and his

creationof musical

analogues

o

texts and to

programmatic

onceptions

ollow

the same

pattern

as his

elaborationof

borrowed

material,"

and that

the

process


the basis of a

"unifiedview

of

Ives's

ap-

proach

o

composition"

"

'Quotation'

and

Emulation,"20,

n.

36).

Testing

or

this

hyphothesis

will come

as each

compositional

area s

subjected

o

thorough

investigation.

6For

example,

the

piano

pieces

that fall in this

category

are entitled"Stud-

ies."

Similarly,

the

song Soliloquy

is

entitled

"A

Study

in 7ths and Other

Things."

Also

typical

s the

subtitle to

Chromdtimelodtune:

Ear-Study

aural

& mental

exercise )."

7Charles

E.

Ives, Memos,

edited and with

appendicesby

John

Kirkpatrick

(New

York:

Norton,

1972),

64.

8Ives's

memoirs describe an


father on these

experi-

mental

attitudes,

although

the

reliability

of


questioned

Central

to the issues under

investigation

n Ives's

experi-

mentation is his focus on intervallicconstructionsas primary

structural

components.

Even in his earliest

compositional

ef-

forts,

the interval

provided

he means

for

constructing

northo-

dox but

orderly

melodic

and harmonic

structures,

often based

on standard

types

of formations

employing

atypical

units of

combination.

Ives

recalls,

for

example,

his father's

suggestion

that

"If one can use

chords

of

3rds

and make

them mean

some-

thing, why

not chordsof 4ths?"9 ves

took these

and similar

suggestions

o

heart,

leading

to his well-known

experiments

n

"quartal"harmony, such as the piano accompaniment o the

song

"The

Cage,"

and

other instances

of intervallic satura-

tion.10

Analysts

have also

noted more

complex

approaches

o

intervallic

tructure,

such

as a

system

of

permutations

f a cer-

tain intervallic

profilel

and

a

process

of

shifting

ocusfromone

interval

or

group

of intervalsto

another over the course of

a

by Maynard

Solomon,

"Charles ves:

Some

Questions

of

Veracity,"

Journal

of

the American

MusicologicalSociety

40

(1987),

443-470.

Solomon

speculates

that Ives

exaggerated

his father's nfluence

as

part

of a

mourningprocess

that

included

an idealizationof his father'smusical nnovationsand a realization

of

some of his father's musical

aspirations.

Solomon

discusses

mportant

ssues

about

the Ives

biography,

but his views remainas

questions

that are left

unan-

swered;

any

substantiationof his thesis will come

only

after extensive

non-

speculativescrutiny

of the availableevidence.

The conventionalview of Ives's debt to his father's

experimental

attitudes

is

described

by

J. Peter

Burkholder,

Charles

ves: The Ideas

behind he Music

(New

Haven

and

London:

Yale

UniversityPress, 1985),

45-50. The author

makesa useful distinctionbetween

"experimental"

nd

"concert"

music,

em-

phasizing

the

private

nature of the former as

opposed

to an attitude

toward

concert

works that

encourages

revision

and

refinement,

ostensibly eading

o a

public presentation.

9Ives,Memos,

140.

10Soliloquy,

he

"Study

n

7ths,"

is another

example.

"Cyr,

"Intervallic

Structural

Elements

in Ives's Fourth

Symphony."

Cyr's

observations

establish

a

link between

the intervallic

constructions

and

the

structure

of

many

of the tunes

quoted

in

the

symphony.

Central

to the issues under

investigation

n Ives's

experi-


structural

components.


compositional

ef-

forts,

the interval

provided

he means

for

constructing

northo-

dox but

orderly

melodic

and harmonic

structures,

often based

on standard

types

of formations

employing

atypical

units of

combination.

Ives

recalls,

for

example,

his father's

suggestion

that

"If one can use

chords

of

3rds

and make

them mean

some-

thing, why


took these

and similar

suggestions

o

heart,

leading

to his well-known

experiments

n


song

"The

Cage,"

and

other instances


tion.10

Analysts

have also

noted more

complex

approaches

o

intervallic

tructure,

such

as a

system

of

permutations

f a cer-

tain intervallic

profilel

and

a

process

of

shifting

ocusfromone

interval

or

group

of intervalsto


a

by Maynard

Solomon,

"Charles ves:

Some

Questions

of

Veracity,"

Journal

of

the American


40

(1987),

443-470.

Solomon

speculates

that Ives

exaggerated


as

part

of a

mourningprocess

that

included


of


aspirations.

Solomon

discusses

mportant

ssues

about

the Ives

biography,


questions

that are left

unan-

swered;

any


only

after extensive

non-

speculativescrutiny



experimental

attitudes

is

described

by

J. Peter

Burkholder,

Charles

ves: The Ideas

behind he Music

(New

Haven

and

London:

Yale


45-50. The author


"experimental"

nd

"concert"

music,

em-

phasizing

the

private


opposed

to an attitude

toward

concert

works that

encourages

revision

and

refinement,

ostensibly eading

o a


9Ives,Memos,

140.

10Soliloquy,

he

"Study

n

7ths,"

is another

example.

"Cyr,

"Intervallic

Structural

Elements

in Ives's Fourth

Symphony."

Cyr's

observations

establish

a

link between

the intervallic

constructions

and

the

structure

of

many

of the tunes

quoted

in

the

symphony.

Central

to the issues under

investigation

n Ives's

experi-


structural

components.


compositional

ef-

forts,

the interval

provided

he means

for

constructing

northo-

dox but

orderly

melodic

and harmonic

structures,

often based

on standard

types

of formations

employing

atypical

units of

combination.

Ives

recalls,

for

example,

his father's

suggestion

that

"If one can use

chords

of

3rds

and make

them mean

some-

thing, why


took these

and similar

suggestions

o

heart,

leading

to his well-known

experiments

n


song

"The

Cage,"

and

other instances


tion.10

Analysts

have also

noted more

complex

approaches

o

intervallic

tructure,

such

as a

system

of

permutations

f a cer-

tain intervallic

profilel

and

a

process

of

shifting

ocusfromone

interval

or

group

of intervalsto


a

by Maynard

Solomon,

"Charles ves:

Some

Questions

of

Veracity,"

Journal

of

the American


40

(1987),

443-470.

Solomon

speculates

that Ives

exaggerated


as

part

of a

mourningprocess

that

included


of


aspirations.

Solomon

discusses

mportant

ssues

about

the Ives

biography,


questions

that are left

unan-

swered;

any


only

after extensive

non-

speculativescrutiny



experimental

attitudes

is

described

by

J. Peter

Burkholder,

Charles

ves: The Ideas

behind he Music

(New

Haven

and

London:

Yale


45-50. The author


"experimental"

nd

"concert"

music,

em-

phasizing

the

private


opposed

to an attitude

toward

concert

works that

encourages

revision

and

refinement,

ostensibly eading

o a


9Ives,Memos,

140.

10Soliloquy,

he

"Study

n

7ths,"

is another

example.

"Cyr,

"Intervallic

Structural

Elements

in Ives's Fourth

Symphony."

Cyr's

observations

establish

a

link between

the intervallic

constructions

and

the

structure

of

many

of the tunes

quoted

in

the

symphony.






Interval

ycles

as

Compositional

esources

45nterval

ycles

as

Compositional

esources

45nterval

ycles

as

Compositional

esources

45

work.12n

maintaining

an active

nterest n alternatives

o struc-

tural conventions, Ives took the interval as his primaryre-

source,

and the evolution

of his

experimentation

centers

in

large part

on this

element

of

musical

structure.

This

orientation,

combined

with an

ever-present

oncern

or

maximizingpitch-class

variety,

develops finally

nto

an aware-

ness

of

the size

and characterof

repetitive

ntervallic

tructures,

viewed

as

cycles,

and their

potential

for

compositional

xploita-

tion.

Many

of his

experiments,

including

those

exhibiting

higher degrees

of

technical

sophistication,

are centered on

in-

tervalcyclesas contributors o acontextuallydefinedharmonic

language

and a unified

catalogue

of

developmental

devices.

Ives thus allies

himself,

spiritually,

with his

contemporaries:

Berg's

"master

array"

of

interval

cycles,

as described

by

George

Perle,

and the role of

cyclic

structures

in

early

Stravinsky

as

explained

by

Elliott

Antokoletz,

demonstrate

similar

approaches

o intervallicconstruction.13 s an alterna-

tive

to traditional

compositional

resources,

the interval

cycles

12Nors

S.

Josephson,

"Charles Ives: Intervallische Permutationen

im

Spatwerk,"

Zeitschrift ur

Musiktheorie /2

(1978),

27-33.

Josephson's

study

encompasses

works of various

types

from all

periods

of Ives's

compositional

career.

13George

Perle,

"Berg's

Master

Array

of the Interval

Cycles,"

Musical

Quarterly

3

(1977),

1-30;

Elliott

Antokoletz,

"Interval

Cycles

n

Stravinsky's

Early

Ballets,"

Journal

of

the

American


34

(1986),

578-

614;

Marianne

Kielian-Gilbert,

"Relationships

of

Symmetrical

Pitch-Class

Sets and

Stravinsky's

Metaphor

of

Polarity," Perspectivesof

New Music 21

(1982-83), 209-240; MenachemZur, "TonalAmbiguitiesas a Constructive

Force

in

the

Language

of

Stravinsky,"

Musical

Quarterly

8

(1982),

516-526.

Cyclic

ntervallic

repetitions

are also central

to

the studies of Bart6k

by

Erno

Lendvai,

The

Workshop of

Bart6k and

Kodaly

(Budapest:

Editio

Musica,

1983)

and

by

Elliott

Antokoletz,

TheMusic

of

Bela Bart6k:A

Studyof

Tonality

and

Progression

n

Twentieth-Century

Music

(Berkeley: University

of

Califor-

nia

Press,

1984).

Another

composer

from this

period

who workedwith nterval

cycles

is Karol

Szymanowski

1882-1937);

see Ann K.

McNamee,

"Bitonality,

Mode,

and Interval in the Music of Karol

Szymanowski,"

Journal

of

Music

Theory

29

(1985),

61-84.

work.12n

maintaining

an active


o struc-


source,

and the evolution

of his

experimentation

centers

in

large part

on this

element

of

musical

structure.

This

orientation,

combined

with an

ever-present

oncern

or


variety,

develops finally

nto

an aware-

ness

of

the size

and characterof

repetitive

ntervallic

tructures,

viewed

as

cycles,

and their

potential

for

compositional

xploita-

tion.

Many

of his

experiments,

including

those

exhibiting

higher degrees

of

technical

sophistication,

are centered on

in-


language

and a unified

catalogue

of

developmental

devices.

Ives thus allies

himself,

spiritually,

with his

contemporaries:

Berg's

"master

array"

of

interval

cycles,

as described

by

George

Perle,

and the role of

cyclic

structures

in

early

Stravinsky

as

explained

by

Elliott

Antokoletz,

demonstrate

similar

approaches


tive

to traditional

compositional

resources,

the interval

cycles

12Nors

S.

Josephson,


im

Spatwerk,"

Zeitschrift ur

Musiktheorie /2

(1978),

27-33.

Josephson's

study

encompasses

works of various

types

from all

periods

of Ives's

compositional

career.

13George

Perle,

"Berg's

Master

Array

of the Interval

Cycles,"

Musical

Quarterly

3

(1977),

1-30;

Elliott

Antokoletz,

"Interval

Cycles

n

Stravinsky's

Early

Ballets,"

Journal

of

the

American


34

(1986),

578-

614;

Marianne

Kielian-Gilbert,

"Relationships

of

Symmetrical

Pitch-Class

Sets and

Stravinsky's

Metaphor

of


New Music 21


Force

in

the

Language

of

Stravinsky,"

Musical

Quarterly

8

(1982),

516-526.

Cyclic

ntervallic

repetitions

are also central

to


by

Erno

Lendvai,

The

Workshop of

Bart6k and

Kodaly

(Budapest:

Editio

Musica,

1983)

and

by

Elliott

Antokoletz,

TheMusic

of

Bela Bart6k:A

Studyof

Tonality

and

Progression

n

Twentieth-Century

Music


of

Califor-

nia

Press,

1984).

Another

composer

from this

period


cycles

is Karol

Szymanowski

1882-1937);

see Ann K.

McNamee,

"Bitonality,

Mode,


Szymanowski,"

Journal

of

Music

Theory

29

(1985),

61-84.

work.12n

maintaining

an active


o struc-


source,

and the evolution

of his

experimentation

centers

in

large part

on this

element

of

musical

structure.

This

orientation,

combined

with an

ever-present

oncern

or


variety,

develops finally

nto

an aware-

ness

of

the size

and characterof

repetitive

ntervallic

tructures,

viewed

as

cycles,

and their

potential

for

compositional

xploita-

tion.

Many

of his

experiments,

including

those

exhibiting

higher degrees

of

technical

sophistication,

are centered on

in-


language

and a unified

catalogue

of

developmental

devices.

Ives thus allies

himself,

spiritually,

with his

contemporaries:

Berg's

"master

array"

of

interval

cycles,

as described

by

George

Perle,

and the role of

cyclic

structures

in

early

Stravinsky

as

explained

by

Elliott

Antokoletz,

demonstrate

similar

approaches


tive

to traditional

compositional

resources,

the interval

cycles

12Nors

S.

Josephson,


im

Spatwerk,"

Zeitschrift ur

Musiktheorie /2

(1978),

27-33.

Josephson's

study

encompasses

works of various

types

from all

periods

of Ives's

compositional

career.

13George

Perle,

"Berg's

Master

Array

of the Interval

Cycles,"

Musical

Quarterly

3

(1977),

1-30;

Elliott

Antokoletz,

"Interval

Cycles

n

Stravinsky's

Early

Ballets,"

Journal

of

the

American


34

(1986),

578-

614;

Marianne

Kielian-Gilbert,

"Relationships

of

Symmetrical

Pitch-Class

Sets and

Stravinsky's

Metaphor

of


New Music 21


Force

in

the

Language

of

Stravinsky,"

Musical

Quarterly

8

(1982),

516-526.

Cyclic

ntervallic

repetitions

are also central

to


by

Erno

Lendvai,

The

Workshop of

Bart6k and

Kodaly

(Budapest:

Editio

Musica,

1983)

and

by

Elliott

Antokoletz,

TheMusic

of

Bela Bart6k:A

Studyof

Tonality

and

Progression

n

Twentieth-Century

Music


of

Califor-

nia

Press,

1984).

Another

composer

from this

period


cycles

is Karol

Szymanowski

1882-1937);

see Ann K.

McNamee,

"Bitonality,

Mode,


Szymanowski,"

Journal

of

Music

Theory

29

(1985),

61-84.

form a viable

system

of

internally

defined

pitch

relationships

and suggestfertile means for musicaldevelopmentand trans-

formation.

The

description

of

Ives's

incorporation

of interval

cycles

in

the

following

pages

citesevidencefrom musicwritten

at various

stages

in his

composing period, primarily

ncluding

works that

are of the

experimental

ype, though

not exclusive

to this cate-

gory.

The discussion

centers first on

cyclic repetitions

of

single

intervals,

and

then

on

alternatingrepetitions

of

two

different

intervals,

or combination

ycles.

Each

topic

includes

definitions

of terminologyand establishmentof analyticalmethodology.A

finalareaof discussion llustrates

particular

ompositional

ap-

plications

of

cycles, generally

n

larger

musicalcontexts.

SINGLE-INTERVAL

YCLES. linear

pitch-class

(pc) presenta-

tion translates

to a

segment

notated

as

integers(C

=

0) sepa-

rated

by

commas within

angled

brackets.14 he

adjacent

nter-

vals,

or ordered

pitch-class

intervals,

form the INT

of a

segment,

notated as

integers (1-11) separated

by

dashes and

enclosedinangledbrackets.15 xample1illustratesheapplica-

tion

of

these notations to

a

violin line from Ives's

Largo

Riso-

luto No.

1

(1906).16

An INT

containing

exclusive

repetitions

of

14The erm

segment

s used here as defined

n Robert D.

Morris,

Composi-

tion

with

Pitch

Classes:

A

Theoryof Compositional

Design (New

Haven

and

London:

Yale

University

Press,

1987),

37,

64.

Morris defines

segments

of

pitches

(pseg)

and

pitch

classes

(pcseg).

A

segment

s ordered

by

definition.

15Anordered

pc

interval,

or directed nterval n Milton

Babbitt's erminol-

ogy, is calculatedby subtracting mod 12) the firstpc from the second. See

Morris,

62

and John

Rahn,

Basic Atonal

Theory New

York:

Longman,

1980),

25-27.

The term INT

is defined n

Morris,

107.

'6Dates

of

composition

are those

given

by

John

Kirkpatrick

n The

New

Grove

Dictionaryof

Music and

Musicians,

6th

ed.,

s.v.

"Ives,

Charles

E."

These

dates are

primarily

based

on

evidence

from

Ives's scoresand

memoirs,

although

the

reliability

of these sources

has been

questioned

by

Solomon

("Questions

of

Veracity").

Until

Solomon's

questions

and

speculations

are

subjected

to further

analysis,

Kirkpatrick's

ates

represent

he most

accurate

informationavailable.

form a viable

system

of

internally

defined

pitch

relationships


formation.

The

description

of

Ives's

incorporation

of interval

cycles

in

the

following

pages


at various

stages

in his


ncluding

works that

are of the

experimental

ype, though

not exclusive

to this cate-

gory.

The discussion

centers first on

cyclic repetitions

of

single

intervals,

and

then

on


of

two

different

intervals,

or combination

ycles.

Each

topic

includes

definitions



particular

ompositional

ap-

plications

of

cycles, generally

n

larger

musicalcontexts.

SINGLE-INTERVAL

YCLES. linear

pitch-class

(pc) presenta-

tion translates

to a

segment

notated

as

integers(C

=

0) sepa-

rated

by

commas within

angled

brackets.14 he

adjacent

nter-

vals,

or ordered

pitch-class

intervals,

form the INT

of a

segment,

notated as


by

dashes and


tion

of

these notations to

a


Largo

Riso-

luto No.

1

(1906).16

An INT

containing

exclusive

repetitions

of

14The erm

segment


n Robert D.

Morris,

Composi-

tion

with

Pitch

Classes:

A


Design (New

Haven

and

London:

Yale

University

Press,

1987),

37,

64.

Morris defines

segments

of

pitches

(pseg)

and

pitch

classes

(pcseg).

A

segment

s ordered

by

definition.

15Anordered

pc

interval,


Babbitt's erminol-


Morris,

62

and John

Rahn,

Basic Atonal

Theory New

York:

Longman,

1980),

25-27.

The term INT

is defined n

Morris,

107.

'6Dates

of

composition

are those

given

by

John

Kirkpatrick

n The

New

Grove

Dictionaryof

Music and

Musicians,

6th

ed.,

s.v.

"Ives,

Charles

E."

These

dates are

primarily

based

on

evidence

from

Ives's scoresand

memoirs,

although

the

reliability

of these sources

has been

questioned

by

Solomon

("Questions

of

Veracity").

Until

Solomon's

questions

and

speculations

are

subjected

to further

analysis,

Kirkpatrick's

ates

represent

he most

accurate


form a viable

system

of

internally

defined

pitch

relationships


formation.

The

description

of

Ives's

incorporation

of interval

cycles

in

the

following

pages


at various

stages

in his


ncluding

works that

are of the

experimental

ype, though

not exclusive

to this cate-

gory.

The discussion

centers first on

cyclic repetitions

of

single

intervals,

and

then

on


of

two

different

intervals,

or combination

ycles.

Each

topic

includes

definitions



particular

ompositional

ap-

plications

of

cycles, generally

n

larger

musicalcontexts.

SINGLE-INTERVAL

YCLES. linear

pitch-class

(pc) presenta-

tion translates

to a

segment

notated

as

integers(C

=

0) sepa-

rated

by

commas within

angled

brackets.14 he

adjacent

nter-

vals,

or ordered

pitch-class

intervals,

form the INT

of a

segment,

notated as


by

dashes and


tion

of

these notations to

a


Largo

Riso-

luto No.

1

(1906).16

An INT

containing

exclusive

repetitions

of

14The erm

segment


n Robert D.

Morris,

Composi-

tion

with

Pitch

Classes:

A


Design (New

Haven

and

London:

Yale

University

Press,

1987),

37,

64.

Morris defines

segments

of

pitches

(pseg)

and

pitch

classes

(pcseg).

A

segment

s ordered

by

definition.

15Anordered

pc

interval,


Babbitt's erminol-


Morris,

62

and John

Rahn,

Basic Atonal

Theory New

York:

Longman,

1980),

25-27.

The term INT

is defined n

Morris,

107.

'6Dates

of

composition

are those

given

by

John

Kirkpatrick

n The

New

Grove

Dictionaryof

Music and

Musicians,

6th

ed.,

s.v.

"Ives,

Charles

E."

These

dates are

primarily

based

on

evidence

from

Ives's scoresand

memoirs,

although

the

reliability

of these sources

has been

questioned

by

Solomon

("Questions

of

Veracity").

Until

Solomon's

questions

and

speculations

are

subjected

to further

analysis,

Kirkpatrick's

ates

represent

he most

accurate







46 Music

Theory

Spectrum

6 Music

Theory

Spectrum

6 Music

Theory

Spectrum

Example

1.

Largo

Risoluto No.

1,

mm.

24-25,

first violin.

xample

1.

Largo

Risoluto No.

1,

mm.

24-25,

first violin.

xample

1.

Largo

Risoluto No.

1,

mm.

24-25,

first violin.

3

A I4

-I-

>I

,>

3

A I4

-I-

>I

,>

3

A I4

-I-

>I

,>

I

J

r

^

)-

^

6.-

I

,^J

r

^

)-

^

6.-

I

,^J

r

^

)-

^

6.-

I

,^

2555

Y

L

I i

3

3I

pc <4,

9, 2,

7,

0, 5, 10,

3, 8, 1,

6,

11>

<4,...

INT

<5-5-5-5-5-5-5-5-5-5-5>

a

single

interval

generates

a

segment

that is

cyclic

n that

it re-

turnsto its

point

of

origin;

the

repetition

of interval5

in Exam-

ple

1is aninterval-5

ycle

that

generates

he returnof

pc

4 atthe

downbeat

of m.

25.17

The numberof

pcs generatedby

the

cyclic

repetition

of a

given

interval

s the

cardinality,

r

CARD

of

the

cycle,

which totals

12 in

Example

1,

a

completion

of the

aggre-

gate.

The familiar

cyclicpitch-class

tructures

have CARDs of

12

(interval-1,

-5,

-7,

or -11

cycles),

6

(interval-2

or

-10),

4

(interval-3

or

-9),

3

(interval-4

or

-8),

or 2

(interval-6).

The or-

derof

presentation

of

the

pcs generated

by

a

cycle

may

be indi-

cated

by

italicized

integers representing

order

position

(op),

extending

from0 for the initial

pc

to CARD-1forthe last.

Ives's interest in

cyclic

structures

originates

with the

in-

stancesof chromaticand whole-tone scales

appearing

n

music

from various

stages

of his life.

In his

Memos,

he describes

an

early,

father-influenced xercise

in

which he

played

the chro-

matic

scale

with octave

displacements,

creating

"wide

umps

n

the

counterpoint

and lines."18 ves'schromatic cale

presenta-

Notes in brackets in musical excerpts are correctionsof misprints n the

published

score,

confirmed

through comparison

with

the

original

manuscript

(housed

in the

Ives Collection

at

Yale).

17Morris efines a

cycle (pcyc

and

pccyc)

more

generally

to

apply

to

any

reiterative

segment,

regardless

of whether an intervallic

repetition

occurs

in

the

INT

(pp.

37,

65).

The

resulting

ntervallic

uccession s the

cyclic

INT,

or

CINT(pp.

40,

107).

18Memos,

44

(and

musical

example). Kirkpatrick

ecalls that

"Ives told

George

Roberts

[one

of

his

copyists]

that

his father

had him do

chromatic

scales

with each

interval

a

minor 9th"

(Memos,

44,

n.

5).

Y

L

I i

3

3I

pc <4,

9, 2,

7,

0, 5, 10,

3, 8, 1,

6,

11>

<4,...

INT

<5-5-5-5-5-5-5-5-5-5-5>

a

single

interval

generates

a

segment

that is

cyclic

n that

it re-

turnsto its

point

of

origin;

the

repetition

of interval5

in Exam-

ple

1is aninterval-5

ycle

that

generates

he returnof

pc

4 atthe

downbeat

of m.

25.17

The numberof

pcs generatedby

the

cyclic

repetition

of a

given

interval

s the

cardinality,

r

CARD

of

the

cycle,

which totals

12 in

Example

1,

a

completion

of the

aggre-

gate.

The familiar

cyclicpitch-class

tructures

have CARDs of

12

(interval-1,

-5,

-7,

or -11

cycles),

6

(interval-2

or

-10),

4

(interval-3

or

-9),

3

(interval-4

or

-8),

or 2

(interval-6).

The or-

derof

presentation

of

the

pcs generated

by

a

cycle

may

be indi-

cated

by

italicized


order

position

(op),

extending


pc


Ives's interest in

cyclic

structures

originates

with the

in-


appearing

n

music

from various

stages

of his life.

In his

Memos,

he describes

an

early,


in

which he

played

the chro-

matic

scale

with octave

displacements,

creating

"wide

umps

n

the

counterpoint


presenta-


published

score,

confirmed

through comparison

with

the

original

manuscript

(housed

in the

Ives Collection

at

Yale).

17Morris efines a

cycle (pcyc

and

pccyc)

more

generally

to

apply

to

any

reiterative

segment,

regardless


repetition

occurs

in

the

INT

(pp.

37,

65).

The

resulting

ntervallic

uccession s the

cyclic

INT,

or

CINT(pp.

40,

107).

18Memos,

44

(and

musical


ecalls that

"Ives told

George

Roberts

[one

of

his

copyists]

that

his father

had him do

chromatic

scales

with each

interval

a

minor 9th"

(Memos,

44,

n.

5).

Y

L

I i

3

3I

pc <4,

9, 2,

7,

0, 5, 10,

3, 8, 1,

6,

11>

<4,...

INT

<5-5-5-5-5-5-5-5-5-5-5>

a

single

interval

generates

a

segment

that is

cyclic

n that

it re-

turnsto its

point

of

origin;

the

repetition

of interval5

in Exam-

ple

1is aninterval-5

ycle

that

generates

he returnof

pc

4 atthe

downbeat

of m.

25.17

The numberof

pcs generatedby

the

cyclic

repetition

of a

given

interval

s the

cardinality,

r

CARD

of

the

cycle,

which totals

12 in

Example

1,

a

completion

of the

aggre-

gate.

The familiar

cyclicpitch-class

tructures

have CARDs of

12

(interval-1,

-5,

-7,

or -11

cycles),

6

(interval-2

or

-10),

4

(interval-3

or

-9),

3

(interval-4

or

-8),

or 2

(interval-6).

The or-

derof

presentation

of

the

pcs generated

by

a

cycle

may

be indi-

cated

by

italicized


order

position

(op),

extending


pc


Ives's interest in

cyclic

structures

originates

with the

in-


appearing

n

music

from various

stages

of his life.

In his

Memos,

he describes

an

early,


in

which he

played

the chro-

matic

scale

with octave

displacements,

creating

"wide

umps

n

the

counterpoint


presenta-


published

score,

confirmed

through comparison

with

the

original

manuscript

(housed

in the

Ives Collection

at

Yale).

17Morris efines a

cycle (pcyc

and

pccyc)

more

generally

to

apply

to

any

reiterative

segment,

regardless


repetition

occurs

in

the

INT

(pp.

37,

65).

The

resulting

ntervallic

uccession s the

cyclic

INT,

or

CINT(pp.

40,

107).

18Memos,

44

(and

musical


ecalls that

"Ives told

George

Roberts

[one

of

his

copyists]

that

his father

had him do

chromatic

scales

with each

interval

a

minor 9th"

(Memos,

44,

n.

5).

tions often take this

form,

bringing

the

scale,

and

thus the

interval-1 or -11 cycle, to its completion in a melodic setting

that either reverses directions

frequently,

producing

linear

an-

gularity,

or

moves

in the

same direction to

cover a

large regis-

tral

span.

The

early

sketch shown

in

Example

2a

employs

the

"wide

jumps" technique

in a melodic line with direction

changes, establishing

a

displacement

pattern

in the first mea-

sure that is

duplicated

a

fourth

higher

in the second.19

Example

2b

gives

a

portion

of

the

piano part

to the

song

Soliloquy

that

employs

the unidirectional

approach,

stating

scale

segments

first with interval 1 (pitch interval 13), and then with interval

11.

The second

arpeggio

(mm. 4-5)

presents

six

pitch-classes

that

are not stated in the

arpeggio

of the

previous

two

mea-

sures,

leaving

only

pc

0

absent from the

upper

voice

of the

passage.20

Ives's

best-known

incorporations

of the

whole-tone

scale

are in the Finales

to the

Second

String

Quartet

(1907-13)

and

Fourth

Symphony

(1909-16).

Both works conclude

with

thick,

layered

textures

of

repeated figures

above reiterated

whole-

tone scales, producing an arrival point of stability that repre-

sents

a kind of resolution of the

many

musical

and extramusical

conflicts

that

have

previously

been

prevalent.21

The

beginning

19The ketch

appears

n Charles Ives's hand

in

George

Ives's

Copybook.

See

John

Kirkpatrick,

A

TemporaryMimeographedCatalogue

of

the Music

Manuscripts

and Related Materials

of

CharlesEdward

Ives 1874-1954

(New

Haven:

Library

of the Yale

University

School of

Music,

1960),

214,

Cat.

No.

7A2.

The

pagination

of the

Copybook

s

Kirkpatrick's.

The

catalogue p.

214)

liststhe probabledatesfor Ives'ssketchings n the Copybook

as 1890-93.

This

scale

setting

reappears

on

p. [71]

of the

Copybook

in a

short

organ

work

Kirkpatrick

alls

"Burlesque

Postlude"

Kirkpatrick,

Catalogue,

19,

Cat.

No.

7C6).

Ives uses the scale

in

canon,

preceded

by

chromatic

ines

in

contrary

mo-

tion.

20Similar

echniques appear

n Over he Pavements

1906-13),

mm.

81-92,

piano,

and

in four works named

by Kirkpatrick

n

Memos,

44,

n.

5.

21Ives's

itle for the finale of the

Quartet

s "The

Callof the

Mountains,"

indicating

that the

"4 men"

personified

by

the

instruments

have

concluded

their "Discussions"of the firstmovement

and

"Arguments"

f the second

and


form,

bringing

the

scale,

and

thus the



frequently,

producing

linear

an-

gularity,

or

moves

in the

same direction to

cover a

large regis-

tral

span.

The

early

sketch shown

in

Example

2a

employs

the

"wide

jumps" technique



a

displacement

pattern

in the first mea-

sure that is

duplicated

a

fourth

higher

in the second.19

Example

2b

gives

a

portion

of

the

piano part

to the

song

Soliloquy

that

employs

the unidirectional

approach,

stating

scale

segments


11.

The second

arpeggio

(mm. 4-5)

presents

six

pitch-classes

that


arpeggio

of the

previous

two

mea-

sures,

leaving

only

pc

0

absent from the

upper

voice

of the

passage.20

Ives's

best-known

incorporations

of the

whole-tone

scale

are in the Finales

to the

Second

String

Quartet

(1907-13)

and

Fourth

Symphony

(1909-16).

Both works conclude

with

thick,

layered

textures

of

repeated figures

above reiterated

whole-


sents


many

musical

and extramusical

conflicts

that

have

previously

been

prevalent.21

The

beginning

19The ketch

appears


in

George

Ives's

Copybook.

See

John

Kirkpatrick,

A


of

the Music

Manuscripts


of

CharlesEdward

Ives 1874-1954

(New

Haven:

Library

of the Yale

University

School of

Music,

1960),

214,

Cat.

No.

7A2.

The

pagination

of the

Copybook

s

Kirkpatrick's.

The

catalogue p.

214)


as 1890-93.

This

scale

setting

reappears

on

p. [71]

of the

Copybook

in a

short

organ

work

Kirkpatrick

alls

"Burlesque

Postlude"

Kirkpatrick,

Catalogue,

19,

Cat.

No.

7C6).

Ives uses the scale

in

canon,

preceded

by

chromatic

ines

in

contrary

mo-

tion.

20Similar

echniques appear

n Over he Pavements

1906-13),

mm.

81-92,

piano,

and

in four works named

by Kirkpatrick

n

Memos,

44,

n.

5.

21Ives's


Quartet

s "The

Callof the

Mountains,"

indicating

that the

"4 men"

personified

by

the

instruments

have

concluded


and

"Arguments"

f the second

and


form,

bringing

the

scale,

and

thus the



frequently,

producing

linear

an-

gularity,

or

moves

in the

same direction to

cover a

large regis-

tral

span.

The

early

sketch shown

in

Example

2a

employs

the

"wide

jumps" technique



a

displacement

pattern

in the first mea-

sure that is

duplicated

a

fourth

higher

in the second.19

Example

2b

gives

a

portion

of

the

piano part

to the

song

Soliloquy

that

employs

the unidirectional

approach,

stating

scale

segments


11.

The second

arpeggio

(mm. 4-5)

presents

six

pitch-classes

that


arpeggio

of the

previous

two

mea-

sures,

leaving

only

pc

0

absent from the

upper

voice

of the

passage.20

Ives's

best-known

incorporations

of the

whole-tone

scale

are in the Finales

to the

Second

String

Quartet

(1907-13)

and

Fourth

Symphony

(1909-16).

Both works conclude

with

thick,

layered

textures

of

repeated figures

above reiterated

whole-


sents


many

musical

and extramusical

conflicts

that

have

previously

been

prevalent.21

The

beginning

19The ketch

appears


in

George

Ives's

Copybook.

See

John

Kirkpatrick,

A


of

the Music

Manuscripts


of

CharlesEdward

Ives 1874-1954

(New

Haven:

Library

of the Yale

University

School of

Music,

1960),

214,

Cat.

No.

7A2.

The

pagination

of the

Copybook

s

Kirkpatrick's.

The

catalogue p.

214)


as 1890-93.

This

scale

setting

reappears

on

p. [71]

of the

Copybook

in a

short

organ

work

Kirkpatrick

alls

"Burlesque

Postlude"

Kirkpatrick,

Catalogue,

19,

Cat.

No.

7C6).

Ives uses the scale

in

canon,

preceded

by

chromatic

ines

in

contrary

mo-

tion.

20Similar

echniques appear

n Over he Pavements

1906-13),

mm.

81-92,

piano,

and

in four works named

by Kirkpatrick

n

Memos,

44,

n.

5.

21Ives's


Quartet

s "The

Callof the

Mountains,"

indicating

that the

"4 men"

personified

by

the

instruments

have

concluded


and

"Arguments"

f the second

and






Interval

ycles

as

Compositional

esources 47nterval

ycles

as

Compositional

esources 47nterval

ycles

as

Compositional

esources 47

Example

2. "Wide

jumps"

reatmentof the chromatic

cale.

a. GeorgeIves'scopybook, p. [68].

pc<0,

1,

2, 3, 4, 5, 6,

7, 8,

9

10, 11,

1,

0>

b.

Soliloquy,

mm.

2-5,

piano.

A

A

4

A1

18

18

18

18

"'

1>

IL

'

i\if

^^

^

Example

2. "Wide

jumps"


cale.


pc<0,

1,

2, 3, 4, 5, 6,

7, 8,

9

10, 11,

1,

0>

b.

Soliloquy,

mm.

2-5,

piano.

A

A

4

A1

18

18

18

18

"'

1>

IL

'

i\if

^^

^

Example

2. "Wide

jumps"


cale.


pc<0,

1,

2, 3, 4, 5, 6,

7, 8,

9

10, 11,

1,

0>

b.

Soliloquy,

mm.

2-5,

piano.

A

A

4

A1

18

18

18

18

"'

1>

IL

'

i\if

^^

^

upper-

voice

v

pc:

<1, 2,

3,

4,

5>

INT: < 1--

1-

1

-

1

>

upper-

voice

v

pc:

<1, 2,

3,

4,

5>

INT: < 1--

1-

1

-

1

>

upper-

voice

v

pc:

<1, 2,

3,

4,

5>

INT: < 1--

1-

1

-

1

>

va-

- J

loco

8va

<

11,

10,

9,

8,

7,

6>

< 11-11-11-11 -

11>

va-

- J

loco

8va

<

11,

10,

9,

8,

7,

6>

< 11-11-11-11 -

11>

va-

- J

loco

8va

<

11,

10,

9,

8,

7,

6>

< 11-11-11-11 -

11>

of

this

section of the

Quartet,

shown n

Example

3,

is anchored

on

the

repetition

of the

scale in

the

cello,

forming

a

seven-beat

ostinatothat is

noncoincidentalwith

the

one-measure

pattern

in

the

viola,

the ten-beat

ostinato in

the second

violin,

and the

fragment

of the tune

"Bethany"

n

the first

violin,

all

of

which

support

a tonal

center of D

(see

brackets n

Ex.

3).

The absence

of

rhythmic

variation in

the

whole-tone

presentation

in

the

cello, the lack of intervallicvariety nthe cello'srepeatedmate-

rial,

and

the avoidance

of metric

correlation with

the other

have now walked

"up

the

mountainside o view

the firmament "

Kirkpatrick,

Catalogue,

60).

Ives describes

he

Finale to the

Symphony

as "an

apotheosis

of

the

preceding

content,

in terms

that

have

something

to do

with the

reality

of

existence and its

religious

experience."

See CharlesE.

Ives,

"TheFourth

Sym-

phony

for

Large

Orchestra,"

New

Music

Quarterly

/2

(January

1929): [ii].

of

this

section of the

Quartet,

shown n

Example

3,

is anchored

on

the

repetition

of the

scale in

the

cello,

forming

a

seven-beat

ostinatothat is

noncoincidentalwith

the

one-measure

pattern

in

the

viola,

the ten-beat

ostinato in

the second

violin,

and the

fragment

of the tune

"Bethany"

n

the first

violin,

all

of

which

support

a tonal

center of D

(see

brackets n

Ex.

3).

The absence

of

rhythmic

variation in

the

whole-tone

presentation

in

the


rial,

and

the avoidance

of metric

correlation with

the other

have now walked

"up

the

mountainside o view

the firmament "

Kirkpatrick,

Catalogue,

60).

Ives describes

he

Finale to the

Symphony

as "an

apotheosis

of

the

preceding

content,

in terms

that

have

something

to do

with the

reality

of

existence and its

religious

experience."

See CharlesE.

Ives,

"TheFourth

Sym-

phony

for

Large

Orchestra,"

New

Music

Quarterly

/2

(January

1929): [ii].

of

this

section of the

Quartet,

shown n

Example

3,

is anchored

on

the

repetition

of the

scale in

the

cello,

forming

a

seven-beat

ostinatothat is

noncoincidentalwith

the

one-measure

pattern

in

the

viola,

the ten-beat

ostinato in

the second

violin,

and the

fragment

of the tune

"Bethany"

n

the first

violin,

all

of

which

support

a tonal

center of D

(see

brackets n

Ex.

3).

The absence

of

rhythmic

variation in

the

whole-tone

presentation

in

the


rial,

and

the avoidance

of metric

correlation with

the other

have now walked

"up

the

mountainside o view

the firmament "

Kirkpatrick,

Catalogue,

60).

Ives describes

he

Finale to the

Symphony

as "an

apotheosis

of

the

preceding

content,

in terms

that

have

something

to do

with the

reality

of

existence and its

religious

experience."

See CharlesE.

Ives,

"TheFourth

Sym-

phony

for

Large

Orchestra,"

New

Music

Quarterly

/2

(January

1929): [ii].

parts

contribute

to a

minimizationof

temporal

qualities, sup-

portingthe projectionof an ethereal, spiritually ranscendent

quality

or the

conclusion

to the work.22

Two basic

features of

the

chromaticand

whole-tone scales

are

relevantto Ives's

pitch

structures n

general.

First,

thecom-

plete uniformity

of interval

sizes in

the scales

s a

desirablechar-

acteristic hat

Ives

exploits

to

highlight

heeffects of

repetition,

as at the

end of the

Second

Quartet,

and to

create a

consistency

of

pitch

structure rom

intervallic

aturation.

Applied

to other

intervals,

repetition

may generate

pitch

materials hat are

less

attractiveowing to their tonal connotations and low cardinal-

ity;

this

may

be

true of

the

repetitions

of

intervals3 and 4 or

their

nverses,

and of

interval

6.

However,

intervals

5

and 7

pro-

vide

rich

resources for

harmonic

saturation

and

have been

widely

used to

ensure

nontraditional

constructionaluniform-

ity.23

ves

views his

song

"The

Cage,"

for

example,

as "a

study

of

how chordsof

4thsand

5ths

may

throw

melodies

away

rom a

set

tonality,"

noting

that "To

make music in

no

particular

key

has a

nice name

nowadays-'atonality.'

"24

The secondinfluential eature of the scales,as foranyof the

cycles,

is

their

insurance of

pitch-class

variety,

or

non-

repetition.

Ives's concern

for avoidance of

pc repetition

allies

22Theseactors

demonstrate he

"spatial"

qualities

of Ives'smusic

observed

by

Robert

Morgan,

"Spatial

Form

n

Ives,"

in An Ives

Celebration:

apers

and

Panels

of

theIves Centennial

Festival-Conference,

d. H.

Wiley

Hitchcockand

Vivian

Perlis

(Urbana:

University

of Illinois

Press,

1977),

145-158.

Morgan

cites, amongother factors, "fragmentation" nd "thesimultaneouscombina-

tion of two

or more

independent,

though

related

musicalcontinuities"

as

con-

tributors o

spatial

effects,

both of

which would

apply

to this

passage

in the

Quartet.

23This

s the

intent of

Schoenberg,

for

example,

in the

chapter

entitled

"Chords Constructed

in

Fourths,"

in

Theory

of Harmony,

trans.

Roy

E.

Carter

(Berkeley:

University

of California

Press,

1978),

399-410. See

also

"Chords

by

Fourths,"

in Vincent

Persichetti,

Twentieth-Century

armony

(New

York:

Norton,

1961),

93-108.

24Ives,Memos,

56.

parts

contribute

to a

minimizationof

temporal

qualities, sup-


quality

or the

conclusion

to the work.22

Two basic

features of

the

chromaticand

whole-tone scales

are

relevantto Ives's

pitch

structures n

general.

First,

thecom-

plete uniformity

of interval

sizes in

the scales

s a

desirablechar-

acteristic hat

Ives

exploits

to

highlight

heeffects of

repetition,

as at the

end of the

Second

Quartet,

and to

create a

consistency

of

pitch

structure rom

intervallic

aturation.

Applied

to other

intervals,

repetition

may generate

pitch

materials hat are

less


ity;

this

may

be

true of

the

repetitions

of

intervals3 and 4 or

their

nverses,

and of

interval

6.

However,

intervals

5

and 7

pro-

vide

rich

resources for

harmonic

saturation

and

have been

widely

used to

ensure

nontraditional


ity.23

ves

views his

song

"The

Cage,"

for

example,

as "a

study

of

how chordsof

4thsand

5ths

may

throw

melodies

away

rom a

set

tonality,"

noting

that "To

make music in

no

particular

key

has a

nice name


"24


cycles,

is

their

insurance of

pitch-class

variety,

or

non-

repetition.

Ives's concern

for avoidance of

pc repetition

allies

22Theseactors

demonstrate he

"spatial"

qualities

of Ives'smusic

observed

by

Robert

Morgan,

"Spatial

Form

n

Ives,"

in An Ives

Celebration:

apers

and

Panels

of

theIves Centennial


d. H.

Wiley

Hitchcockand

Vivian

Perlis

(Urbana:

University

of Illinois

Press,

1977),

145-158.

Morgan


tion of two

or more

independent,

though

related


as

con-

tributors o

spatial

effects,

both of

which would

apply

to this

passage

in the

Quartet.

23This

s the

intent of

Schoenberg,

for

example,

in the

chapter

entitled

"Chords Constructed

in

Fourths,"

in

Theory

of Harmony,

trans.

Roy

E.

Carter

(Berkeley:

University

of California

Press,

1978),

399-410. See

also

"Chords

by

Fourths,"

in Vincent

Persichetti,

Twentieth-Century

armony

(New

York:

Norton,

1961),

93-108.

24Ives,Memos,

56.

parts

contribute

to a

minimizationof

temporal

qualities, sup-


quality

or the

conclusion

to the work.22

Two basic

features of

the

chromaticand

whole-tone scales

are

relevantto Ives's

pitch

structures n

general.

First,

thecom-

plete uniformity

of interval

sizes in

the scales

s a

desirablechar-

acteristic hat

Ives

exploits

to

highlight

heeffects of

repetition,

as at the

end of the

Second

Quartet,

and to

create a

consistency

of

pitch

structure rom

intervallic

aturation.

Applied

to other

intervals,

repetition

may generate

pitch

materials hat are

less


ity;

this

may

be

true of

the

repetitions

of

intervals3 and 4 or

their

nverses,

and of

interval

6.

However,

intervals

5

and 7

pro-

vide

rich

resources for

harmonic

saturation

and

have been

widely

used to

ensure

nontraditional


ity.23

ves

views his

song

"The

Cage,"

for

example,

as "a

study

of

how chordsof

4thsand

5ths

may

throw

melodies

away

rom a

set

tonality,"

noting

that "To

make music in

no

particular

key

has a

nice name


"24


cycles,

is

their

insurance of

pitch-class

variety,

or

non-

repetition.

Ives's concern

for avoidance of

pc repetition

allies

22Theseactors

demonstrate he

"spatial"

qualities

of Ives'smusic

observed

by

Robert

Morgan,

"Spatial

Form

n

Ives,"

in An Ives

Celebration:

apers

and

Panels

of

theIves Centennial


d. H.

Wiley

Hitchcockand

Vivian

Perlis

(Urbana:

University

of Illinois

Press,

1977),

145-158.

Morgan


tion of two

or more

independent,

though

related


as

con-

tributors o

spatial

effects,

both of

which would

apply

to this

passage

in the

Quartet.

23This

s the

intent of

Schoenberg,

for

example,

in the

chapter

entitled

"Chords Constructed

in

Fourths,"

in

Theory

of Harmony,

trans.

Roy

E.

Carter

(Berkeley:

University

of California

Press,

1978),

399-410. See

also

"Chords

by

Fourths,"

in Vincent

Persichetti,

Twentieth-Century

armony

(New

York:

Norton,

1961),

93-108.

24Ives,Memos,

56.






48

Music

Theory

Spectrum

8

Music

Theory

Spectrum

8

Music

Theory

Spectrum

Example

3.

Second

String

Quartet,

third

movement,

mm.

123-126.

xample

3.

Second

String

Quartet,

third

movement,

mm.

123-126.

xample

3.

Second

String

Quartet,

third

movement,

mm.

123-126.

Adagio

maestoso

125

8va

..

.-----------

--

- 3 3

,

3

,

Otlt

-a

It9Rt

-e

iff: -&

Adagio

maestoso

125

8va

..

.-----------

--

- 3 3

,

3

,

Otlt

-a

It9Rt

-e

iff: -&

Adagio

maestoso

125

8va

..

.-----------

--

- 3 3

,

3

,

Otlt

-a

It9Rt

-e

iff: -&

v

i

I I

I I

I

I

I

ff

ff

a n_

IXtt

3

-

3

,--_ ---3

---

-

--

3--

--

3 -

ir

i

r

IT

i

ff

I I

F

l-

dJu a I

v

i

I I

I I

I

I

I

ff

ff

a n_

IXtt

3

-

3

,--_ ---3

---

-

--

3--

--

3 -

ir

i

r

IT

i

ff

I I

F

l-

dJu a I

v

i

I I

I I

I

I

I

ff

ff

a n_

IXtt

3

-

3

,--_ ---3

---

-

--

3--

--

3 -

ir

i

r

IT

i

ff

I I

F

l-

dJu a I

'

t?-

F

t

--4

I

I.

r

f4w

-E-

-.

'

t?-

F

t

--4

I

I.

r

f4w

-E-

-.

'

t?-

F

t

--4

I

I.

r

f4w

-E-

-.

him with his friend Carl

Ruggles,

who is said to have advocated

stating

at least "sevenor

eight

differentnotes

[pitchclasses]

na

melody"

before

introducing

a

repetition.25

he

composer

s as-

sured of maximal

pitch-class variety,

without

repetition,

in

chromatic

completions

and in structuresbased

on

intervals5

or

7,

such as the melodic line illustrated n

Example

1.

Methodical

generation

of the

aggregate

s also

possible

in

a

complementa-

tion of the other

cycles,

such

as,

for

example,

a combination

of

pitch

classes

from "odd" and

"even"

whole-tone scales.26

2sHenry

Cowell,

New Musical Resources

(1930;repr.

New York:

Knopf,

1950),

41-42. See

Steven

E.

Gilbert,

"The 'Twelve-Tone

System'

of

Carl

Rug-

gles:

A

Study

of the Evocations or

Piano,"

Journal

of

Music

Theory

14

(1970),

68-91.

26DaveHeadlam defines this

type

of

complementation

s an "extension"

of

an

interval

cycle

in "The Derivation of Rows in

Lulu,"

Perspectives

of

New

Music 24/1

(1985),

203.


Ruggles,


stating

at least "sevenor

eight

differentnotes

[pitchclasses]

na

melody"

before

introducing

a

repetition.25

he

composer

s as-

sured of maximal


without

repetition,

in

chromatic

completions


on

intervals5

or

7,


Example

1.

Methodical

generation

of the

aggregate

s also

possible

in

a

complementa-

tion of the other

cycles,

such

as,

for

example,

a combination

of

pitch

classes

from "odd" and

"even"


2sHenry

Cowell,


(1930;repr.

New York:

Knopf,

1950),

41-42. See

Steven

E.

Gilbert,

"The 'Twelve-Tone

System'

of

Carl

Rug-

gles:

A

Study


Piano,"

Journal

of

Music

Theory

14

(1970),

68-91.


type

of

complementation

s an "extension"

of

an

interval

cycle


Lulu,"

Perspectives

of

New

Music 24/1

(1985),

203.


Ruggles,


stating

at least "sevenor

eight

differentnotes

[pitchclasses]

na

melody"

before

introducing

a

repetition.25

he

composer

s as-

sured of maximal


without

repetition,

in

chromatic

completions


on

intervals5

or

7,


Example

1.

Methodical

generation

of the

aggregate

s also

possible

in

a

complementa-

tion of the other

cycles,

such

as,

for

example,

a combination

of

pitch

classes

from "odd" and

"even"


2sHenry

Cowell,


(1930;repr.

New York:

Knopf,

1950),

41-42. See

Steven

E.

Gilbert,

"The 'Twelve-Tone

System'

of

Carl

Rug-

gles:

A

Study


Piano,"

Journal

of

Music

Theory

14

(1970),

68-91.


type

of

complementation

s an "extension"

of

an

interval

cycle


Lulu,"

Perspectives

of

New

Music 24/1

(1985),

203.

The union

of intervallic

uniformity

and

pitch-classvariety

may

yield

a

plan

of

organization

or

melodic

or

harmonic truc-

tures.

In a

brief sketch that Ives

gives

the title

Song

in

5's,

for

example,

chords of stackedfifths are related

through

associa-

tion

with a

generating

interval-5

cycle.27

The

sketch,

tran-

scribed

n

Example

4,

consists

of

four sonoritiesconnected

by

an

upper

melodic

line that

is itself

a

whole-tone

pentachord,

a

combination reminiscent of the

juxtaposition

of

chords in

fourths

with a

whole-tone

melody

in "The

Cage."

The

pitch

classes

in the lowest voices of chords 1 and 2 connect

to

the

up-

per

voices

of

the

ensuing

chords as continuations

of the

interval-5

cycle

indicated

below the score.

Aggregate

comple-

27Kirkpatrick, atalogue,

226,

No.

7E38.

The

compiler

does

not

suggest

a

date for the sketch. It

appears

within materials

relating

to

the "Thoreau"

movement of the

Concord

Sonata,

which was

composed

around

1910-15.

The union

of intervallic

uniformity

and

pitch-classvariety

may

yield

a

plan

of

organization

or

melodic

or

harmonic truc-

tures.

In a


gives

the title

Song

in

5's,

for

example,


through

associa-

tion

with a

generating

interval-5

cycle.27

The

sketch,

tran-

scribed

n

Example

4,

consists

of


by

an

upper

melodic

line that

is itself

a

whole-tone

pentachord,

a


juxtaposition

of

chords in

fourths

with a

whole-tone

melody

in "The

Cage."

The

pitch

classes


to

the

up-

per

voices

of

the

ensuing


of the

interval-5

cycle

indicated

below the score.

Aggregate

comple-


226,

No.

7E38.

The

compiler

does

not

suggest

a


appears

within materials

relating

to

the "Thoreau"

movement of the

Concord

Sonata,

which was

composed

around

1910-15.

The union

of intervallic

uniformity

and

pitch-classvariety

may

yield

a

plan

of

organization

or

melodic

or

harmonic truc-

tures.

In a


gives

the title

Song

in

5's,

for

example,


through

associa-

tion

with a

generating

interval-5

cycle.27

The

sketch,

tran-

scribed

n

Example

4,

consists

of


by

an

upper

melodic

line that

is itself

a

whole-tone

pentachord,

a


juxtaposition

of

chords in

fourths

with a

whole-tone

melody

in "The

Cage."

The

pitch

classes


to

the

up-

per

voices

of

the

ensuing


of the

interval-5

cycle

indicated

below the score.

Aggregate

comple-


226,

No.

7E38.

The

compiler

does

not

suggest

a


appears

within materials

relating

to

the "Thoreau"

movement of the

Concord

Sonata,

which was

composed

around

1910-15.

I

ff

I

ff

I

ff






Interval

ycles

as

Compositional

esources

49

nterval

ycles

as

Compositional

esources

49

nterval

ycles

as

Compositional

esources

49

Example

4.

Song

in

5's.

chord

1 2

Example

4.

Song

in

5's.

chord

1 2

Example

4.

Song

in

5's.

chord

1 2 3

444

n 7

d

3

'

[7]

b

I:

6

n 7

d

3

'

[7]

b

I:

6

n 7

d

3

'

[7]

b

I:

6

chord: 1 2 3

pc<9,

2,

7,

0,

5

5,

10,

3,

8,

1

1,6, 11,

4>

INT

<5

-

5

-5

-5 . . .

4

tion is

achieved

in

chord

3,

followed

in the final

sonority

by

a

repetition

of

pitch

classes from the latter

part

of the

cycle.

In

the

early

choral work

Psalm

54

(1894?),

a bidimensional

projection

of

repetitive

ntervallic tructures

xploits

the

circu-

larity nherent in the materials,achievinga moreextensive ex-

position

of

cyclic

deas than n similar

passages

rom the

Finales

of

the Second

String

Quartet

and Fourth

Symphony.

Example

5

is

a reduction in

short score

of the

first

of

the

seven

verses,

which exhibits

a

pattern

of

half-note chords n the

lower voices

embellished

by

added notes in the

upper

voices.

This same

ar-

rangement

is used in the

setting

of verse

2

(mm. 7-13)

and

again

n

the

setting

of

the

final

verse

(mm.

44-57)

with the

roles

exchanged

(half-note

chords in the

upper

voices,

and so

forth).28

The

upperembellishments circled

n Ex.

5)

decorate

a

lower-rangeprojection

of

augmented

riadsrooted

on notes

of

the

descending

whole-tone

scale,

completing

an

octave

de-

scent at the downbeat of m. 4. The

projection

of the triads hen

reverses direction and shifts to the

complementary

whole-tone

28Verses

through

6

are set in a

contrasting

contrapuntal tyle

employing

double canon.

chord: 1 2 3

pc<9,

2,

7,

0,

5

5,

10,

3,

8,

1

1,6, 11,

4>

INT

<5

-

5

-5

-5 . . .

4

tion is

achieved

in

chord

3,

followed

in the final

sonority

by

a

repetition

of

pitch


part

of the

cycle.

In

the

early

choral work

Psalm

54

(1894?),

a bidimensional

projection

of

repetitive


xploits

the

circu-


position

of

cyclic

deas than n similar

passages

rom the

Finales

of

the Second

String

Quartet

and Fourth

Symphony.

Example

5

is

a reduction in

short score

of the

first

of

the

seven

verses,

which exhibits

a

pattern

of


lower voices

embellished

by

added notes in the

upper

voices.

This same

ar-

rangement

is used in the

setting

of verse

2

(mm. 7-13)

and

again

n

the

setting

of

the

final

verse

(mm.

44-57)

with the

roles

exchanged

(half-note

chords in the

upper

voices,

and so

forth).28

The


n Ex.

5)

decorate

a


of

augmented

riadsrooted

on notes

of

the

descending

whole-tone

scale,

completing

an

octave

de-


projection

of the triads hen


complementary

whole-tone

28Verses

through

6

are set in a

contrasting

contrapuntal tyle

employing

double canon.

chord: 1 2 3

pc<9,

2,

7,

0,

5

5,

10,

3,

8,

1

1,6, 11,

4>

INT

<5

-

5

-5

-5 . . .

4

tion is

achieved

in

chord

3,

followed

in the final

sonority

by

a

repetition

of

pitch


part

of the

cycle.

In

the

early

choral work

Psalm

54

(1894?),

a bidimensional

projection

of

repetitive


xploits

the

circu-


position

of

cyclic

deas than n similar

passages

rom the

Finales

of

the Second

String

Quartet

and Fourth

Symphony.

Example

5

is

a reduction in

short score

of the

first

of

the

seven

verses,

which exhibits

a

pattern

of


lower voices

embellished

by

added notes in the

upper

voices.

This same

ar-

rangement

is used in the

setting

of verse

2

(mm. 7-13)

and

again

n

the

setting

of

the

final

verse

(mm.

44-57)

with the

roles

exchanged

(half-note

chords in the

upper

voices,

and so

forth).28

The


n Ex.

5)

decorate

a


of

augmented

riadsrooted

on notes

of

the

descending

whole-tone

scale,

completing

an

octave

de-


projection

of the triads hen


complementary

whole-tone

28Verses

through

6

are set in a

contrasting

contrapuntal tyle

employing

double canon.

scale,

eventually

stating

an

augmented

triad on

each

of the 12

possibleroots.29

Because each

pc representation

of an

interval-4

ycle

is

sym-

metrically

ituated within one of the whole-tone

scales,

the lin-

ear whole-tone

presentationmay

be

viewed as

a

gradual

"un-

folding"

of

alternate members

of

the

interval-4

cycles.

Correspondingly,

he

augmented

triads built above

each scale

step

alternate

appearances

as notated below the score

n

Exam-

ple

5:

cycles

I and III

alternate

n

the initial descent

(mm. 1-4)

and

cycles

II and IV

alternate n the

subsequent

ascent. Each

cycleexhibitsacompleterotationof its verticalarrangemento

exhaust

the

possible

augmented-triad

oots and thus the

aggre-

gate. Cycle

I,

for

instance,

moves from

pc

<0,4,8>

in m. 1 to

<8,4,0>

in

m.

2

and to

<4,8,0>

in

m.

3. The

passage

thus

be-

comes saturated with

cyclic

formations from the horizontal

whole-tone

cycles (interval-2

or

-10)

and their

support

of verti-

cal interval-4

cycles

undergoing

regularpatterns

of rotation.

The

excerpt

from Psalm54 in

Example

5

is

typical

n

that the

pure

whole-tone

saturation n the lower

voices

provides

a

cyclic

framework or a processof embellishment n the uppervoices

that introduces

pitch

classes from outside

the

prevailing

har-

monies. The

cycle

thus

serves

a

role that

might

be fulfilled

by

a

diatonic

scale

or

scale

segment

in a tonal

context.

This

ap-

proach

is also evident in

Ives's music as

an

embellishment

of

repetitive

ntervallic tructures hat do not

reach

cycliccomple-

tion,

includingpassages

based on

seconds,

fourths,

or

fifths as

controllersof

linear motion.30

29The

assage

is described

similarly

n H.

Wiley

Hitchcock,

Ives: A

Survey

of

the Music

(London:

Oxford

University

Press, 1977;

repr.

New York: Insti-

tute for

Studies in American

Music,

1983),

29-31. The chords could also be

viewed as

resulting

from

three whole-tone scales

moving

in

parallel

major

thirds.

30See,

or

example,

the

embellishmentof an

interval-5

equence

n the

first

movement

of

the Second

String

Quartet,

mm.

28-33,

firstviolin.

scale,

eventually

stating

an

augmented

triad on

each

of the 12

possibleroots.29

Because each

pc representation

of an

interval-4

ycle

is

sym-

metrically


scales,

the lin-

ear whole-tone

presentationmay

be

viewed as

a

gradual

"un-

folding"

of

alternate members

of

the

interval-4

cycles.

Correspondingly,

he

augmented

triads built above

each scale

step

alternate

appearances


n

Exam-

ple

5:

cycles

I and III

alternate

n

the initial descent

(mm. 1-4)

and

cycles

II and IV

alternate n the

subsequent

ascent. Each


exhaust

the

possible

augmented-triad

oots and thus the

aggre-

gate. Cycle

I,

for

instance,

moves from

pc

<0,4,8>

in m. 1 to

<8,4,0>

in

m.

2

and to

<4,8,0>

in

m.

3. The

passage

thus

be-


cyclic


whole-tone

cycles (interval-2

or

-10)

and their

support

of verti-

cal interval-4

cycles

undergoing

regularpatterns

of rotation.

The

excerpt

from Psalm54 in

Example

5

is

typical

n

that the

pure

whole-tone


voices

provides

a

cyclic


that introduces

pitch


the

prevailing

har-

monies. The

cycle

thus

serves

a

role that

might

be fulfilled

by

a

diatonic

scale

or

scale

segment

in a tonal

context.

This

ap-

proach

is also evident in

Ives's music as

an

embellishment

of

repetitive


reach

cycliccomple-

tion,

includingpassages

based on

seconds,

fourths,

or

fifths as

controllersof

linear motion.30

29The

assage

is described

similarly

n H.

Wiley

Hitchcock,

Ives: A

Survey

of

the Music

(London:

Oxford

University

Press, 1977;

repr.

New York: Insti-

tute for

Studies in American

Music,

1983),


viewed as

resulting

from


moving

in

parallel

major

thirds.

30See,

or

example,

the

embellishmentof an

interval-5

equence

n the

first

movement

of

the Second

String

Quartet,

mm.

28-33,

firstviolin.

scale,

eventually

stating

an

augmented

triad on

each

of the 12

possibleroots.29

Because each

pc representation

of an

interval-4

ycle

is

sym-

metrically


scales,

the lin-

ear whole-tone

presentationmay

be

viewed as

a

gradual

"un-

folding"

of

alternate members

of

the

interval-4

cycles.

Correspondingly,

he

augmented

triads built above

each scale

step

alternate

appearances


n

Exam-

ple

5:

cycles

I and III

alternate

n

the initial descent

(mm. 1-4)

and

cycles

II and IV

alternate n the

subsequent

ascent. Each


exhaust

the

possible

augmented-triad

oots and thus the

aggre-

gate. Cycle

I,

for

instance,

moves from

pc

<0,4,8>

in m. 1 to

<8,4,0>

in

m.

2

and to

<4,8,0>

in

m.

3. The

passage

thus

be-


cyclic


whole-tone

cycles (interval-2

or

-10)

and their

support

of verti-

cal interval-4

cycles

undergoing

regularpatterns

of rotation.

The

excerpt

from Psalm54 in

Example

5

is

typical

n

that the

pure

whole-tone


voices

provides

a

cyclic


that introduces

pitch


the

prevailing

har-

monies. The

cycle

thus

serves

a

role that

might

be fulfilled

by

a

diatonic

scale

or

scale

segment

in a tonal

context.

This

ap-

proach

is also evident in

Ives's music as

an

embellishment

of

repetitive


reach

cycliccomple-

tion,

includingpassages

based on

seconds,

fourths,

or

fifths as

controllersof

linear motion.30

29The

assage

is described

similarly

n H.

Wiley

Hitchcock,

Ives: A

Survey

of

the Music

(London:

Oxford

University

Press, 1977;

repr.

New York: Insti-

tute for

Studies in American

Music,

1983),


viewed as

resulting

from


moving

in

parallel

major

thirds.

30See,

or

example,

the

embellishmentof an

interval-5

equence

n the

first

movement

of

the Second

String

Quartet,

mm.

28-33,

firstviolin.






50

Music

Theory

Spectrum

0

Music

Theory

Spectrum

0

Music

Theory

Spectrum

Example

5. Psalm

54,

mm. 1-6.

xample

5. Psalm

54,

mm. 1-6.

xample

5. Psalm

54,

mm. 1-6.

Save

me,

[Largo

maestoso]

Save

me,

[Largo

maestoso]

Save

me,

[Largo

maestoso]

O

God,God,God,

I

III

I

III

I

III

name,

and

judge

me

by

I

III

I

III

I

III

name,

and

judge

me

by

I

III

I

III

I

III

name,

and

judge

me

by

thy

strength,

hy

strength,

hy

strength,

I

II

IV

II

IV

II

IV

II

IV

II

IV

II

IV

II

IV

II

IV

II

IV

I.

<0,

4,

8>

.

<0,

4,

8>

.

<0,

4,

8>

IV.

<3,

7,

11>

V.

<3,

7,

11>

V.

<3,

7,

11>

Embellishment

techniques

also

help

interpret passages

where,

unlike

the Psalm

54

excerpt,

the

underlying

ycle

is less

explicitly

stated

within

the

texture.

An

elaboration

of an

interval-11

cycle,

realized

as a

descending

chromatic

cale,

oc-

curs

in the

passage

from

Ives's

Study

No.

20

(1908?)

given

in

Example 6a, spanningeight barsand reachingcompletionin

the

bass

register

four

octaves

below

its initial

pitch.

The

chro-

matic descent

begins

with the

C#6

in beat

1 of

m.

6,

stated

as the

top

note

in an instance

of 3-5

[0,1,6].31

Thistrichord

appears

31Set-class

abels

throughout

his

article

are those

of Allen Forte.

See

The

Structure

of

Atonal

Music

(New

Haven

and London:

Yale

University

Press,

1973), Appendix

1(179-181).

Embellishment

techniques

also

help

interpret passages

where,

unlike

the Psalm

54

excerpt,

the

underlying

ycle

is less

explicitly

stated

within

the

texture.

An

elaboration

of an

interval-11

cycle,

realized

as a

descending

chromatic

cale,

oc-

curs

in the

passage

from

Ives's

Study

No.

20

(1908?)

given

in


the

bass

register

four

octaves

below

its initial

pitch.

The

chro-

matic descent

begins

with the

C#6

in beat

1 of

m.

6,

stated

as the

top

note

in an instance

of 3-5

[0,1,6].31

Thistrichord

appears

31Set-class

abels

throughout

his

article

are those

of Allen Forte.

See

The

Structure

of

Atonal

Music

(New

Haven

and London:

Yale

University

Press,

1973), Appendix

1(179-181).

Embellishment

techniques

also

help

interpret passages

where,

unlike

the Psalm

54

excerpt,

the

underlying

ycle

is less

explicitly

stated

within

the

texture.

An

elaboration

of an

interval-11

cycle,

realized

as a

descending

chromatic

cale,

oc-

curs

in the

passage

from

Ives's

Study

No.

20

(1908?)

given

in


the

bass

register

four

octaves

below

its initial

pitch.

The

chro-

matic descent

begins

with the

C#6

in beat

1 of

m.

6,

stated

as the

top

note

in an instance

of 3-5

[0,1,6].31

Thistrichord

appears

31Set-class

abels

throughout

his

article

are those

of Allen Forte.

See

The

Structure

of

Atonal

Music

(New

Haven

and London:

Yale

University

Press,

1973), Appendix

1(179-181).

beneath

several

of

the scale

steps,

combining

he

intervals

of a

fourth

or fifth

plus

a

tritone

to

accompany

he

participants

n

the

chromatic

descent.

The

circled

notes

in

Example

6a

follow

the

descent

of the

scale

down

a

fifth

in the

right

hand

to

F#5

in m. 8.

Singularly

absent from the scalarunfoldingis pc 9, whichoccursas the

bass

grace

note

in

mm.

6-8

and

as the

pedal

starting

n m.

1 of

the

Study.

Example

6bsummarizes

he

descent

plus

the

accom-

panying

3-5s,

incorporating

he

missingpc

9

in

its

appropriate

position

and

actual

register.

When

the

line

transfers

o

the

left

hand

in m.

9,

two

scale

tones

arestated

simultaneously

s

part

of a 3-5

and are

subsequently

reiterated

n

the

bass

of m.

10

(the

reiteration

s enclosed

in brackets

n Ex.

6b).

The

remain-

beneath

several

of

the scale

steps,

combining

he

intervals

of a

fourth

or fifth

plus

a

tritone

to

accompany

he

participants

n

the

chromatic

descent.

The

circled

notes

in

Example

6a

follow

the

descent

of the

scale

down

a

fifth

in the

right

hand

to

F#5

in m. 8.

Singularly


bass

grace

note

in

mm.

6-8

and

as the

pedal

starting

n m.

1 of

the

Study.

Example

6bsummarizes

he

descent

plus

the

accom-

panying

3-5s,

incorporating

he

missingpc

9

in

its

appropriate

position

and

actual

register.

When

the

line

transfers

o

the

left

hand

in m.

9,

two

scale

tones

arestated

simultaneously

s

part

of a 3-5

and are

subsequently

reiterated

n

the

bass

of m.

10

(the

reiteration

s enclosed

in brackets

n Ex.

6b).

The

remain-

beneath

several

of

the scale

steps,

combining

he

intervals

of a

fourth

or fifth

plus

a

tritone

to

accompany

he

participants

n

the

chromatic

descent.

The

circled

notes

in

Example

6a

follow

the

descent

of the

scale

down

a

fifth

in the

right

hand

to

F#5

in m. 8.

Singularly


bass

grace

note

in

mm.

6-8

and

as the

pedal

starting

n m.

1 of

the

Study.

Example

6bsummarizes

he

descent

plus

the

accom-

panying

3-5s,

incorporating

he

missingpc

9

in

its

appropriate

position

and

actual

register.

When

the

line

transfers

o

the

left

hand

in m.

9,

two

scale

tones

arestated

simultaneously

s

part

of a 3-5

and are

subsequently

reiterated

n

the

bass

of m.

10

(the

reiteration

s enclosed

in brackets

n Ex.

6b).

The

remain-

byyy

thyhyhy

II.

<1,

5,

9>

III.

<2,

6, 10>I.

<1,

5,

9>

III.

<2,

6, 10>I.

<1,

5,

9>

III.

<2,

6, 10>






Interval

ycles

as

Compositional

esources

51

nterval

ycles

as

Compositional

esources

51

nterval

ycles

as

Compositional

esources

51

Example

6.

Study

No.

20,

mm. 6-14.

a.

tem~

.

tmpo

pnmo

06

1

2 13

4i

J

-'

-TI ,J ,~

^^

n

i

I I

i

,

l^^r

-

,^1

"

dlJ '^n^

rK

^n

Example

6.

Study

No.

20,

mm. 6-14.

a.

tem~

.

tmpo

pnmo

06

1

2 13

4i

J

-'

-TI ,J ,~

^^

n

i

I I

i

,

l^^r

-

,^1

"

dlJ '^n^

rK

^n

Example

6.

Study

No.

20,

mm. 6-14.

a.

tem~

.

tmpo

pnmo

06

1

2 13

4i

J

-'

-TI ,J ,~

^^

n

i

I I

i

,

l^^r

-

,^1

"

dlJ '^n^

rK

^n

derof the descent isaccomplishedby successive ranspositions

of

a

three-beat

pattern

n the left

handthat

ncludes

his

reitera-

tion

and

comes to rest

on the

3-5;

the

firststatementof the

pat-

tern is

m. 10

through

the first

beat

of m.

11,

bracketed

below

the

score

in

Example

6a. Since each three-beat

unit

is

trans-

posed

down a half

step,

the descent continueswith each new

transposition

evel,

as

highlightedby

the

stemmed

notes

in

Ex-

ample

6b. After

arriving

at

C#2

on the third beat

of m.

13,

the

pattern

returns

to its

original

pitch

level,

with mm.

14-17

re-

peatingmm. 10-13 exactly.

The

adjacent

intervals

forming

the 3-5s beneath the scale

tones,

as notatedin

Example

6b,

reverse their

positions

n the

course

of

the

presentation

rom

7

above

6

at the firsttwo scale

tones

to

6 above

7

in

m. 7

and in mm.

9-13,

portraying

he final

left-hand trichordsas mirrorsof

the first wo

in

the

right

hand.

These

trichords,

which also include chromatic intervals be-

tween the

outer

notes,

provide

a consistent character or the


of

a

three-beat

pattern

n the left

handthat

ncludes

his

reitera-

tion

and

comes to rest

on the

3-5;

the


pat-

tern is

m. 10

through

the first

beat

of m.

11,

bracketed

below

the

score

in

Example


unit

is

trans-

posed

down a half

step,


transposition

evel,

as

highlightedby

the

stemmed

notes

in

Ex-

ample

6b. After

arriving

at

C#2

on the third beat

of m.

13,

the

pattern

returns

to its

original

pitch

level,

with mm.

14-17

re-


The

adjacent

intervals

forming


tones,

as notatedin

Example

6b,

reverse their

positions

n the

course

of

the

presentation

rom

7

above

6


tones

to

6 above

7

in

m. 7

and in mm.

9-13,

portraying

he final


the first wo

in

the

right

hand.

These

trichords,


tween the

outer

notes,

provide



of

a

three-beat

pattern

n the left

handthat

ncludes

his

reitera-

tion

and

comes to rest

on the

3-5;

the


pat-

tern is

m. 10

through

the first

beat

of m.

11,

bracketed

below

the

score

in

Example


unit

is

trans-

posed

down a half

step,


transposition

evel,

as

highlightedby

the

stemmed

notes

in

Ex-

ample

6b. After

arriving

at

C#2

on the third beat

of m.

13,

the

pattern

returns

to its

original

pitch

level,

with mm.

14-17

re-


The

adjacent

intervals

forming


tones,

as notatedin

Example

6b,

reverse their

positions

n the

course

of

the

presentation

rom

7

above

6


tones

to

6 above

7

in

m. 7

and in mm.

9-13,

portraying

he final


the first wo

in

the

right

hand.

These

trichords,


tween the

outer

notes,

provide


steps in the chromatic ine, contributing o the associationof

pitches

that are

registrally eparated

as

part

of a

unified state-

ment of the

cyclic

pitch

source.

Other

embellishment

procedures

alter a source not

through

addition

of notes

surrounding

he

cyclic unfolding

but

through

rearrangement

f the

pitch

classes

of the source alone.

A

cycli-

cally generated

pc

set

is

thus used

as an

unordered

collection

that

provides

material or a melodic

or

harmonic

etting.

With

cycles

of

cardinality

12,

the

source

provides

a convenient

point

of departure or aggregateconstructions; ves typicallyretains

features

of the source so that the

origins

of the

material

are

im-

mediately

apparent.

A

principal

heme in

the Robert

Browning

Overture

1908-12) employs

a

displacement

of a

single pitch

class

within the

generation

of an interval-7

cycle,

producing

an

aggregate

ordering

and an INT

of

<7-7-2-7-7-7-7-7-7-11-8>.

One

transposition

of

this

theme,

for

example,

is an alteration

of

<3,10,5,0,7,2,9,4,11,6,1,8>,

a

strict

repetition

of

interval

7,

to


pitches

that are


as

part

of a

unified state-

ment of the

cyclic

pitch

source.

Other

embellishment

procedures

alter a source not

through

addition

of notes

surrounding

he

cyclic unfolding

but

through

rearrangement

f the

pitch

classes


A

cycli-

cally generated

pc

set

is

thus used

as an

unordered

collection

that

provides


or

harmonic

etting.

With

cycles

of

cardinality

12,

the

source

provides

a convenient

point


features


origins

of the

material

are

im-

mediately

apparent.

A

principal

heme in

the Robert

Browning

Overture

1908-12) employs

a

displacement

of a

single pitch

class

within the

generation

of an interval-7

cycle,

producing

an

aggregate

ordering

and an INT

of

<7-7-2-7-7-7-7-7-7-11-8>.

One

transposition

of

this

theme,

for

example,

is an alteration

of

<3,10,5,0,7,2,9,4,11,6,1,8>,

a

strict

repetition

of

interval

7,

to


pitches

that are


as

part

of a

unified state-

ment of the

cyclic

pitch

source.

Other

embellishment

procedures

alter a source not

through

addition

of notes

surrounding

he

cyclic unfolding

but

through

rearrangement

f the

pitch

classes


A

cycli-

cally generated

pc

set

is

thus used

as an

unordered

collection

that

provides


or

harmonic

etting.

With

cycles

of

cardinality

12,

the

source

provides

a convenient

point


features


origins

of the

material

are

im-

mediately

apparent.

A

principal

heme in

the Robert

Browning

Overture

1908-12) employs

a

displacement

of a

single pitch

class

within the

generation

of an interval-7

cycle,

producing

an

aggregate

ordering

and an INT

of

<7-7-2-7-7-7-7-7-7-11-8>.

One

transposition

of

this

theme,

for

example,

is an alteration

of

<3,10,5,0,7,2,9,4,11,6,1,8>,

a

strict

repetition

of

interval

7,

to






52 Music

Theory

Spectrum

2 Music

Theory

Spectrum

2 Music

Theory

Spectrum

Example

7.

Three-Page

Sonata,

derivation

of left

hand,

mm.77-79.

a.

source

interval-5

op

0

1

2 3

4 5 6 7

8

9

10 11

cycle

p

--

cycl:0

?

b"o

to

t0

f

lo

o

pc

9 2

7

0 5 10 3

8 1

6

11

4

b.

left

pc

9

2

7

0

5

6

1

4

10 8 3 8 11

hand,

mm 77 3 r

3-

78 ---3 r-- 79 ----- 3-m. - 7 3

.-

79

77-79

-

I

K

I

,

.

r

INT

<7-5--5

-

5--1-7-3-

--

10-7-5-3>

produce

<3,10,5,7,2,9,4,11,6,1,0,8>,

accomplished

by

mov-

ing

pc

0 from

op

3 to

op

10.32

A

pitch-class

succession

n Ives's

Three-Page

Sonata

(1905)

employsa moreextensivereorderingn anaggregateconstruc-

tion

while still

retaining

a

display

of elements

of its intervallic

source.

Example

7a

displays

he interval-5

ycle

that serves as

a

point

of

departure

or the

aggregate

of

Example

7b

(the

final

pc

2 of m. 79

begins

a

repetition

of the

pitch-class

material).33

f-

ter

five

pitch

classes

drawn

without variation rom

the

source,

the line

presents

pc

<6,1>,

or a reversal

of source

ops

8 and

9,

followed

by pc

4,

which s the final

note in the

source,

and then

pc

<10,8,3,8>,

a

rearrangement

of

ops

5, 6,

and

7. The INT

notatedbelowExample7billustrates hepresenceof interval5

and

ts inverse

7

within

a

sequence

that also

includes ntervals

1,

3, 6,

and

10. Within

this intervallic

variety,

ntervals

5 and

7 re-

32This ccurs

n the

cello,

mm. 50-52. The

first

presentation

of this

theme

(bassoon

and

trombone,

mm.

46-49)

has some

pitch

naccuracies

hat

prevent

aggregatecompletion,

probably

he results

of

copying

or calculation

rrors.

33In

he

original,

this line is doubled

an octave

lower.

Example

7.

Three-Page

Sonata,

derivation

of left

hand,

mm.77-79.

a.

source

interval-5

op

0

1

2 3

4 5 6 7

8

9

10 11

cycle

p

--

cycl:0

?

b"o

to

t0

f

lo

o

pc

9 2

7

0 5 10 3

8 1

6

11

4

b.

left

pc

9

2

7

0

5

6

1

4

10 8 3 8 11

hand,

mm 77 3 r

3-

78 ---3 r-- 79 ----- 3-m. - 7 3

.-

79

77-79

-

I

K

I

,

.

r

INT

<7-5--5

-

5--1-7-3-

--

10-7-5-3>

produce

<3,10,5,7,2,9,4,11,6,1,0,8>,

accomplished

by

mov-

ing

pc

0 from

op

3 to

op

10.32

A

pitch-class

succession

n Ives's

Three-Page

Sonata

(1905)


tion

while still

retaining

a

display

of elements

of its intervallic

source.

Example

7a

displays

he interval-5

ycle

that serves as

a

point

of

departure

or the

aggregate

of

Example

7b

(the

final

pc

2 of m. 79

begins

a

repetition

of the

pitch-class

material).33

f-

ter

five

pitch

classes

drawn


the

source,

the line

presents

pc

<6,1>,

or a reversal

of source

ops

8 and

9,

followed

by pc

4,

which s the final

note in the

source,

and then

pc

<10,8,3,8>,

a

rearrangement

of

ops

5, 6,

and

7. The INT


and

ts inverse

7

within

a

sequence

that also

includes ntervals

1,

3, 6,

and

10. Within

this intervallic

variety,

ntervals

5 and

7 re-

32This ccurs

n the

cello,

mm. 50-52. The

first

presentation

of this

theme

(bassoon

and

trombone,

mm.

46-49)

has some

pitch

naccuracies

hat

prevent


probably

he results

of

copying

or calculation

rrors.

33In

he

original,


an octave

lower.

Example

7.

Three-Page

Sonata,

derivation

of left

hand,

mm.77-79.

a.

source

interval-5

op

0

1

2 3

4 5 6 7

8

9

10 11

cycle

p

--

cycl:0

?

b"o

to

t0

f

lo

o

pc

9 2

7

0 5 10 3

8 1

6

11

4

b.

left

pc

9

2

7

0

5

6

1

4

10 8 3 8 11

hand,

mm 77 3 r

3-

78 ---3 r-- 79 ----- 3-m. - 7 3

.-

79

77-79

-

I

K

I

,

.

r

INT

<7-5--5

-

5--1-7-3-

--

10-7-5-3>

produce

<3,10,5,7,2,9,4,11,6,1,0,8>,

accomplished

by

mov-

ing

pc

0 from

op

3 to

op

10.32

A

pitch-class

succession

n Ives's

Three-Page

Sonata

(1905)


tion

while still

retaining

a

display

of elements

of its intervallic

source.

Example

7a

displays

he interval-5

ycle

that serves as

a

point

of

departure

or the

aggregate

of

Example

7b

(the

final

pc

2 of m. 79

begins

a

repetition

of the

pitch-class

material).33

f-

ter

five

pitch

classes

drawn


the

source,

the line

presents

pc

<6,1>,

or a reversal

of source

ops

8 and

9,

followed

by pc

4,

which s the final

note in the

source,

and then

pc

<10,8,3,8>,

a

rearrangement

of

ops

5, 6,

and

7. The INT


and

ts inverse

7

within

a

sequence

that also

includes ntervals

1,

3, 6,

and

10. Within

this intervallic

variety,

ntervals

5 and

7 re-

32This ccurs

n the

cello,

mm. 50-52. The

first

presentation

of this

theme

(bassoon

and

trombone,

mm.

46-49)

has some

pitch

naccuracies

hat

prevent


probably

he results

of

copying

or calculation

rrors.

33In

he

original,


an octave

lower.

ceive

emphasis

at

the

beginning

of m. 79

from the

repetition

of

pc 8, matching he repetitionof pc 2 in m. 77; this reinforcesa

less

direct connection

with

the source

cycle

in

the latter

part

of

the

pattern.

Ives's most extensive

and most

systematically

omprehen-

sive

employment

of

single-interval

cycles

occurs

in the

early

choral

work

Psalm

24

(1894?).34

The

setting

of each

of

the

ten

versesis based

on a

mirroring

f

inversionally omplementary

cycles

that makes

possible

a

gradual egistral

xpansion

or con-

traction,

simulating

a

registral"wedge"

betweenlines

moving

in contrarymotion. Typically,a mirroring f a givencyclewill

provide

the

pitch-class

source

material,

or

"model,"

for the

verse,

while the

actual

pitch

realization

may employ

octave

dis-

placements

o obscure

a literal

registral

nactment

of the

wedge

shape.

In the

pitch-class

model,

the

gradual

inear

changespro-

duce

gradations

of vertical nterval

sizes

and a consistent

sym-

metrical

axis,

while

octave variations

on the

pattern

n the mu-

sical

settings disrupt

the vertical intervallic

regularity

and

mobilize

the axisof

symmetry.

Verse 1 of the Psalm,forexample,is basedon themirroring

of

intervals

1 and 11 outlined

n

Figure

1. This chromatic

wedge

expands

the vertical

distances

between

voices in

increments

of

two from

0 to 24.

Ives's realization

of the

model,

however,

em-

ploys

the "wide

jumps" echnique

(see Example

2),

so that

the

cyclic

model

determines

only

the

pitch-class

content

of the

voices

involved,

not

a

pitch-specific

egistral

pattern.

Example

8

is

a two-stave

reduction

of verse

1,

illustrating

he octave-

displaced

cycles

of intervals

1 and 11

in the outer

voices,

sup-

ported by chromatic ines in the innervoices. In the firsttwo

measures,

the octave

displacements

are

themselves

mirrored:

both

soprano

and

bass shiftto outer octaves

at the

firstchord

of

m.

2,

emphasizing

the

word "Lord's."

Because

both voices

34As s the

case

with Psalm

54,

Ives

probably

worked on

Psalm 24

with his

father

around

1893-94,

though

no

specific

recollection

of a collaboration

ap-

pears

in Memos or elsewhere.

(See

Memos,

47.)

ceive

emphasis

at

the

beginning

of m. 79

from the

repetition

of


less

direct connection

with

the source

cycle

in

the latter

part

of

the

pattern.


and most

systematically

omprehen-

sive

employment

of

single-interval

cycles

occurs

in the

early

choral

work

Psalm

24

(1894?).34

The

setting

of each

of

the

ten

versesis based

on a

mirroring

f


cycles

that makes

possible

a

gradual egistral

xpansion

or con-

traction,

simulating

a

registral"wedge"

betweenlines

moving


provide

the

pitch-class

source

material,

or

"model,"

for the

verse,

while the

actual

pitch

realization

may employ

octave

dis-

placements

o obscure

a literal

registral

nactment

of the

wedge

shape.

In the

pitch-class

model,

the

gradual

inear

changespro-

duce

gradations

of vertical nterval

sizes

and a consistent

sym-

metrical

axis,

while

octave variations

on the

pattern

n the mu-

sical

settings disrupt


regularity

and

mobilize

the axisof

symmetry.


of

intervals

1 and 11 outlined

n

Figure

1. This chromatic

wedge

expands

the vertical

distances

between

voices in

increments

of

two from

0 to 24.

Ives's realization

of the

model,

however,

em-

ploys

the "wide

jumps" echnique

(see Example

2),

so that

the

cyclic

model

determines

only

the

pitch-class

content

of the

voices

involved,

not

a

pitch-specific

egistral

pattern.

Example

8

is

a two-stave

reduction

of verse

1,

illustrating

he octave-

displaced

cycles

of intervals

1 and 11

in the outer

voices,

sup-


measures,

the octave

displacements

are

themselves

mirrored:

both

soprano

and


at the

firstchord

of

m.

2,

emphasizing

the

word "Lord's."

Because

both voices

34As s the

case

with Psalm

54,

Ives

probably

worked on

Psalm 24

with his

father

around

1893-94,

though

no

specific

recollection

of a collaboration

ap-

pears


(See

Memos,

47.)

ceive

emphasis

at

the

beginning

of m. 79

from the

repetition

of


less

direct connection

with

the source

cycle

in

the latter

part

of

the

pattern.


and most

systematically

omprehen-

sive

employment

of

single-interval

cycles

occurs

in the

early

choral

work

Psalm

24

(1894?).34

The

setting

of each

of

the

ten

versesis based

on a

mirroring

f


cycles

that makes

possible

a

gradual egistral

xpansion

or con-

traction,

simulating

a

registral"wedge"

betweenlines

moving


provide

the

pitch-class

source

material,

or

"model,"

for the

verse,

while the

actual

pitch

realization

may employ

octave

dis-

placements

o obscure

a literal

registral

nactment

of the

wedge

shape.

In the

pitch-class

model,

the

gradual

inear

changespro-

duce

gradations

of vertical nterval

sizes

and a consistent

sym-

metrical

axis,

while

octave variations

on the

pattern

n the mu-

sical

settings disrupt


regularity

and

mobilize

the axisof

symmetry.


of

intervals

1 and 11 outlined

n

Figure

1. This chromatic

wedge

expands

the vertical

distances

between

voices in

increments

of

two from

0 to 24.

Ives's realization

of the

model,

however,

em-

ploys

the "wide

jumps" echnique

(see Example

2),

so that

the

cyclic

model

determines

only

the

pitch-class

content

of the

voices

involved,

not

a

pitch-specific

egistral

pattern.

Example

8

is

a two-stave

reduction

of verse

1,

illustrating

he octave-

displaced

cycles

of intervals

1 and 11

in the outer

voices,

sup-


measures,

the octave

displacements

are

themselves

mirrored:

both

soprano

and


at the

firstchord

of

m.

2,

emphasizing

the

word "Lord's."

Because

both voices

34As s the

case

with Psalm

54,

Ives

probably

worked on

Psalm 24

with his

father

around

1893-94,

though

no

specific

recollection

of a collaboration

ap-

pears


(See

Memos,

47.)






Interval

ycles

as

Compositional

esources

53

nterval

ycles

as

Compositional

esources

53

nterval

ycles

as

Compositional

esources

53

Example

8.

Psalm

24,

verse

1,

two-stavereduction.

The

earth is the Lord's

and

the

ful- ness

there-of,

the world

and

they

that dwell there

-

in.

1i

2 3

4

6

^

^

..

-

F^^r

Example

8.

Psalm

24,

verse

1,

two-stavereduction.

The

earth is the Lord's

and

the

ful- ness

there-of,

the world

and

they

that dwell there

-

in.

1i

2 3

4

6

^

^

..

-

F^^r

Example

8.

Psalm

24,

verse

1,

two-stavereduction.

The

earth is the Lord's

and

the

ful- ness

there-of,

the world

and

they

that dwell there

-

in.

1i

2 3

4

6

^

^

..

-

F^^r

Figure 1. Pitch-class model of Psalm 24, verse 1 (mm. 1-6).igure 1. Pitch-class model of Psalm 24, verse 1 (mm. 1-6).igure 1. Pitch-class model of Psalm 24, verse 1 (mm. 1-6).

12

3 45

6 789

2

3 45

6 789

2

3 45

6 789

0

11

10 9

1

10 9

1

10 9

8

7

6

5

4

3

7

6

5

4

3

7

6

5

4

3

0 2 4

6 8 10

12

14 16 182 4

6 8 10

12

14 16 182 4

6 8 10

12

14 16 18

10

11

2

1

20

22

10

11

2

1

20

22

10

11

2

1

20

22

0

0

0

0

0

0

4444

shift,

the

symmetrical

axis remains stable

(at C4).

Subse-

quently,

in

m.

3,

the

soprano

employs

the octave

leap

while the

bass continues its

half-step

descent,

avoiding

pitches

in the

lower

part

of the bass

register;

this shiftsthe axis

upward

(to

F#4)

for the first two

beats of

m. 3.

The unmirrored

hifts con-

tinue in

the remainder

of

the

verse,

continuallymoving

the axis

andunpredictably arying he sizes of the vertical ntervals.

Similar

proceduresgovern

a

mirroring

f

intervals

2 and 10

in

verse

2,

initiating

a

verse-by-verse

pattern

of

progressive

n-

creases

in

the interval sizes of the

cyclic

model that culminates

with a

cycle

of interval 7

mirrored

by

5

beginning

in

verse

7

(mm.

35-44).

This is followed

by

a

rapid

reduction

n interval

sizes in the final two

verses,

with several

cycles

used

in

each

verse,

concluding

with an interval-1

source

in the final

phrase

shift,

the

symmetrical

axis remains stable

(at C4).

Subse-

quently,

in

m.

3,

the

soprano

employs

the octave

leap

while the

bass continues its

half-step

descent,

avoiding

pitches

in the

lower

part

of the bass

register;

this shiftsthe axis

upward

(to

F#4)

for the first two

beats of

m. 3.

The unmirrored

hifts con-

tinue in

the remainder

of

the

verse,

continuallymoving

the axis


Similar

proceduresgovern

a

mirroring

f

intervals

2 and 10

in

verse

2,

initiating

a

verse-by-verse

pattern

of

progressive

n-

creases

in


cyclic


with a

cycle

of interval 7

mirrored

by

5

beginning

in

verse

7

(mm.

35-44).

This is followed

by

a

rapid

reduction

n interval


verses,

with several

cycles

used

in

each

verse,

concluding

with an interval-1

source

in the final

phrase

shift,

the

symmetrical

axis remains stable

(at C4).

Subse-

quently,

in

m.

3,

the

soprano

employs

the octave

leap

while the

bass continues its

half-step

descent,

avoiding

pitches

in the

lower

part

of the bass

register;

this shiftsthe axis

upward

(to

F#4)

for the first two

beats of

m. 3.

The unmirrored

hifts con-

tinue in

the remainder

of

the

verse,

continuallymoving

the axis


Similar

proceduresgovern

a

mirroring

f

intervals

2 and 10

in

verse

2,

initiating

a

verse-by-verse

pattern

of

progressive

n-

creases

in


cyclic


with a

cycle

of interval 7

mirrored

by

5

beginning

in

verse

7

(mm.

35-44).

This is followed

by

a

rapid

reduction

n interval


verses,

with several

cycles

used

in

each

verse,

concluding

with an interval-1

source

in the final

phrase

(mm.

53-57,

not mirrored n the bass

voice).

Thus the overall

structure

of Psalm

24

is based on

a

wedge-like

expansion

and

contraction

of

interval sizes that

is reflected

n

the

cyclic

pitch-

class

model of

each

verse.35

Ives's interest in

maximizingpitch-classvariety

s

naturally

relevant

to a

systematicexploration

of

cycles

such

as Psalm

24.

Intervals1and 2 are sourcesfor verses1 and2, intervals5 and7

control

verses 5 and

7,

respectively,

and interval

3,

despite

a

low

cardinality (4),

is used

(with

pc

repetitions)

in verse

3.

Rather than

employing

nterval

4

(CARD

3)

in verse

4 and in-

(mm.

53-57,


voice).

Thus the overall

structure

of Psalm

24

is based on

a

wedge-like

expansion

and

contraction

of

interval sizes that

is reflected

n

the

cyclic

pitch-

class

model of

each

verse.35

Ives's interest in


s

naturally

relevant

to a


of

cycles

such

as Psalm

24.


control

verses 5 and

7,

respectively,

and interval

3,

despite

a

low

cardinality (4),

is used

(with

pc

repetitions)

in verse

3.

Rather than

employing

nterval

4

(CARD

3)

in verse

4 and in-

(mm.

53-57,


voice).

Thus the overall

structure

of Psalm

24

is based on

a

wedge-like

expansion

and

contraction

of

interval sizes that

is reflected

n

the

cyclic

pitch-

class

model of

each

verse.35

Ives's interest in


s

naturally

relevant

to a


of

cycles

such

as Psalm

24.


control

verses 5 and

7,

respectively,

and interval

3,

despite

a

low

cardinality (4),

is used

(with

pc

repetitions)

in verse

3.

Rather than

employing

nterval

4

(CARD

3)

in verse

4 and in-

35Hitchcock escribes

Psalm24

similarly

n Ives:A

Surveyof

the

Music,

31-

32.


Psalm24

similarly

n Ives:A

Surveyof

the

Music,

31-

32.


Psalm24

similarly

n Ives:A

Surveyof

the

Music,

31-

32.

int.

1

cycle:

int.

11

cycle:

vertical

distance:

int.

1

cycle:

int.

11

cycle:

vertical

distance:

int.

1

cycle:

int.

11

cycle:

vertical

distance:






54 Music

Theory

Spectrum

4 Music

Theory

Spectrum

4 Music

Theory

Spectrum

terval 6

(CARD

2)

in verse

6,

however,

Ives introduces

varia-

tions in the scheme that promote greater pitch-classvariety.

The

pitch-class

models for verses

4 and

6

are based

on alterna-

tions

of intervals:verse

4

alternates

ntervals3

and

4

(mirrored

by

intervals 9 and

8),

and verse 6 alternates

ntervals5

and 6

(mirroredby

7 and

6).

Cyclic

repetitions

of this

type,

essentially

combinations

of

two

single-interval

ycles, represent

a fruitful

area

for further

nquiry

nto the

more

sophisticated

xperimen-

tation of Ives's later

years.

COMBINATIONYCLES. he idea of a "combination cycle," or

a

cyclic

alternation

of two

intervals,

presents

itself

early

in

Ives's works

as,

for

example,

a

combination

of

major

and mi-

nor thirds

(intervals

4 and

3).

In addition

to their

role in the

pitch-classwedge

material

of verse

4 of Psalm

24,

these

inter-

vals combine

to form chords

of

"stacked

hirds"n

organ

works

from the

1890s,

ncluding

an interlude

or a church

ervice36

nd

organ

parts

rom the cantata

The

Celestial

Country

1898-99).37

These

structures

realize a

suggestion

from Ives's

father,

re-

calledby Ives in his Memos: "If two majoror minor3rds can

make

up

a

chord,

why

not more?"38While

Ives's chords

exhibit

different distributions

of the

two

intervals,

some,

such

as the

sonorities

n the Introduction

o No. 7 in The Celestial

Country,

employ

a strict alternation

between the two

intervals

o

gener-

ate a

large, regularly

tructured hord

without

pitch-class

dupli-

cations.39 ves's

exploration

of other intervallic

alternations

raises

questions

about the

gamut

of

possibilities

or these

types

36Transcribed

n

Henry

and

Sidney

Cowell,

Charles

ves and

His

Music,

35.

Kirkpatrick

Catalogue,

107) gives

the

probable

datefor the

interlude

as 1892.

37See,

or

example,

the Prelude

before No.

2 and the Introduction

o

No.

7.

38Ives,

Memos,

47.

See

also

related

recollections,

33 and 120.

39Another nstance

occurs

in Psalm

90,

m. 2. The

stacked

thirds

of

m. 2

recur

hroughout

he

work to serve

as

part

of a "harmonic

eitmotif"

system,

as

described

by

Donald

Grantham,

"A Harmonic

'Leitmotif'

System

in Ives's

Psalm

90,"

In

TheoryOnly

5/2

(1979),

3-14.

terval 6

(CARD

2)

in verse

6,

however,

Ives introduces

varia-


The

pitch-class

models for verses

4 and

6

are based

on alterna-

tions

of intervals:verse

4

alternates

ntervals3

and

4

(mirrored

by

intervals 9 and

8),


ntervals5

and 6

(mirroredby

7 and

6).

Cyclic

repetitions

of this

type,

essentially

combinations

of

two

single-interval

ycles, represent

a fruitful

area

for further

nquiry

nto the

more

sophisticated

xperimen-


years.


a

cyclic

alternation

of two

intervals,

presents

itself

early

in

Ives's works

as,

for

example,

a

combination

of

major

and mi-

nor thirds

(intervals

4 and

3).

In addition

to their

role in the

pitch-classwedge

material

of verse

4 of Psalm

24,

these

inter-

vals combine

to form chords

of

"stacked

hirds"n

organ

works

from the

1890s,

ncluding

an interlude

or a church

ervice36

nd

organ

parts

rom the cantata

The

Celestial

Country

1898-99).37

These

structures

realize a

suggestion

from Ives's

father,

re-


make

up

a

chord,

why

not more?"38While

Ives's chords

exhibit


of the

two

intervals,

some,

such

as the

sonorities

n the Introduction


Country,

employ


between the two

intervals

o

gener-

ate a

large, regularly

tructured hord

without

pitch-class

dupli-

cations.39 ves's

exploration


alternations

raises

questions

about the

gamut

of

possibilities

or these

types

36Transcribed

n

Henry

and

Sidney

Cowell,

Charles

ves and

His

Music,

35.

Kirkpatrick

Catalogue,

107) gives

the

probable

datefor the

interlude

as 1892.

37See,

or

example,

the Prelude

before No.


o

No.

7.

38Ives,

Memos,

47.

See

also

related

recollections,

33 and 120.

39Another nstance

occurs

in Psalm

90,

m. 2. The

stacked

thirds

of

m. 2

recur

hroughout

he

work to serve

as

part

of a "harmonic

eitmotif"

system,

as

described

by

Donald

Grantham,

"A Harmonic

'Leitmotif'

System

in Ives's

Psalm

90,"

In

TheoryOnly

5/2

(1979),

3-14.

terval 6

(CARD

2)

in verse

6,

however,

Ives introduces

varia-


The

pitch-class

models for verses

4 and

6

are based

on alterna-

tions

of intervals:verse

4

alternates

ntervals3

and

4

(mirrored

by

intervals 9 and

8),


ntervals5

and 6

(mirroredby

7 and

6).

Cyclic

repetitions

of this

type,

essentially

combinations

of

two

single-interval

ycles, represent

a fruitful

area

for further

nquiry

nto the

more

sophisticated

xperimen-


years.


a

cyclic

alternation

of two

intervals,

presents

itself

early

in

Ives's works

as,

for

example,

a

combination

of

major

and mi-

nor thirds

(intervals

4 and

3).

In addition

to their

role in the

pitch-classwedge

material

of verse

4 of Psalm

24,

these

inter-

vals combine

to form chords

of

"stacked

hirds"n

organ

works

from the

1890s,

ncluding

an interlude

or a church

ervice36

nd

organ

parts

rom the cantata

The

Celestial

Country

1898-99).37

These

structures

realize a

suggestion

from Ives's

father,

re-


make

up

a

chord,

why

not more?"38While

Ives's chords

exhibit


of the

two

intervals,

some,

such

as the

sonorities

n the Introduction


Country,

employ


between the two

intervals

o

gener-

ate a

large, regularly

tructured hord

without

pitch-class

dupli-

cations.39 ves's

exploration


alternations

raises

questions

about the

gamut

of

possibilities

or these

types

36Transcribed

n

Henry

and

Sidney

Cowell,

Charles

ves and

His

Music,

35.

Kirkpatrick

Catalogue,

107) gives

the

probable

datefor the

interlude

as 1892.

37See,

or

example,

the Prelude

before No.


o

No.

7.

38Ives,

Memos,

47.

See

also

related

recollections,

33 and 120.

39Another nstance

occurs

in Psalm

90,

m. 2. The

stacked

thirds

of

m. 2

recur

hroughout

he

work to serve

as

part

of a "harmonic

eitmotif"

system,

as

described

by

Donald

Grantham,

"A Harmonic

'Leitmotif'

System

in Ives's

Psalm

90,"

In

TheoryOnly

5/2

(1979),

3-14.

of

structures,

their

ability

to

generate pitch-class

variety,

and

theircontributionso Ives'scompositional anguage.

The

alternation

of

intervals

4

and 3

in

Figure

2,

a

"4/3"

com-

bination

cycle,

can be viewed

as a combination

of

an "A set"

of

pitch

classes

in

the

even-numbered

order

positions

with a

"B

set"

in the

odd-numbered

positions.

Both the

A

and B

sets

in

this

case

are

generated

by

cycles

of interval

7,

which s the

sum

(mod

12)

of the two

alternating

ntervals.

Conceptually,

he

A

and

B sets

constitute

an

"overlay"

of a

cycle

with itself

ata

par-

ticular

nterval:

he 4/3

cycle

overlays

wo interval-7

ycles

at

a

distanceof interval4.40ViewingIves's ntervallic lternations s

combination

cycles

assumes

a

transpositional

quivalence

be-

tween

the combined

cycles,

and

encompasses

any possible

wo-

interval

alternation.41

n

general

terms,

x +

y

=

n

(mod

12),

where

x and

y

are

any

two

alternating

ntervals

overlaying

ycles

of interval

n.

The

possible

values

of x and

y

for

a

given

n are

defined

by

the

operator

cycles

TnI

(see

Table

1).

For

example,

where

n

=

7,

as in

Figure

2,

the

possible

values

of

x and

y

are

indicated

by

T7I

cycles:

x/y

=

0/7, 7/0,

1/6, 6/1,2/5, 5/2,

3/4, 4/3, 8/11,

11/8,

9/10,

or

10/9.

A

simple

exchange

of the

x/y

values,

as in

converting

9/10

to

10/9,

has

only

a subtle

effect

on the

presentation

of the

two

al-

4This

approach

bears

similarities

o that

of Elliott

Antokoletz,

in The

Mu-

sic

ofBela

Bart6k,

in which

nterval-1

cycles

are

combined

n order

to

explain

symmetrical

pitch

constructions,

hough

his

study

does not

extend

the

concept

to

encompass

combinations

of other

cycles.

Along

these

same

lines,

the

com-

positionalsystemof GeorgePerle outlined n Twelve-ToneTonalityBerkeley:

University

of California

Press,

1977)

combines

cycles

of

every

interval,

but is

primarily

concerned

with

inverse-related

cycles

and their

compositional

applications.

41The

oncept

may

be

naturally

extended

to

encompass

more

thantwo

in-

tervals

n

alternation,

thus

possibly

nvolving

a combination

of several

cycles.

Ives

rarely

uses

such a

sequence,

and the

present

discussion

s

limited

to

two

alternating

ntervals.

Morris's

oncept

of

a

cyclic

INT

would

apply

o

cyclicrep-

etitions

of

any type (Composition

WithPitch

Classes,40,

107).

of

structures,

their

ability

to


variety,

and


The

alternation

of

intervals

4

and 3

in

Figure

2,

a

"4/3"

com-

bination

cycle,

can be viewed

as a combination

of

an "A set"

of

pitch

classes

in

the

even-numbered

order

positions

with a

"B

set"

in the

odd-numbered

positions.

Both the

A

and B

sets

in

this

case

are

generated

by

cycles

of interval

7,

which s the

sum

(mod

12)

of the two

alternating

ntervals.

Conceptually,

he

A

and

B sets

constitute

an

"overlay"

of a

cycle

with itself

ata

par-

ticular

nterval:

he 4/3

cycle

overlays

wo interval-7

ycles

at

a


combination

cycles

assumes

a

transpositional

quivalence

be-

tween

the combined

cycles,

and

encompasses

any possible

wo-

interval

alternation.41

n

general

terms,

x +

y

=

n

(mod

12),

where

x and

y

are

any

two

alternating

ntervals

overlaying

ycles

of interval

n.

The

possible

values

of x and

y

for

a

given

n are

defined

by

the

operator

cycles

TnI

(see

Table

1).

For

example,

where

n

=

7,

as in

Figure

2,

the

possible

values

of

x and

y

are

indicated

by

T7I

cycles:

x/y

=

0/7, 7/0,

1/6, 6/1,2/5, 5/2,

3/4, 4/3, 8/11,

11/8,

9/10,

or

10/9.

A

simple

exchange

of the

x/y

values,

as in

converting

9/10

to

10/9,

has

only

a subtle

effect

on the

presentation

of the

two

al-

4This

approach

bears

similarities

o that

of Elliott

Antokoletz,

in The

Mu-

sic

ofBela

Bart6k,

in which

nterval-1

cycles

are

combined

n order

to

explain

symmetrical

pitch

constructions,

hough

his

study

does not

extend

the

concept

to

encompass

combinations

of other

cycles.

Along

these

same

lines,

the

com-


University

of California

Press,

1977)

combines

cycles

of

every

interval,

but is

primarily

concerned

with

inverse-related

cycles

and their

compositional

applications.

41The

oncept

may

be

naturally

extended

to

encompass

more

thantwo

in-

tervals

n

alternation,

thus

possibly

nvolving

a combination

of several

cycles.

Ives

rarely

uses

such a

sequence,

and the

present

discussion

s

limited

to

two

alternating

ntervals.

Morris's

oncept

of

a

cyclic

INT

would

apply

o

cyclicrep-

etitions

of


WithPitch

Classes,40,

107).

of

structures,

their

ability

to


variety,

and


The

alternation

of

intervals

4

and 3

in

Figure

2,

a

"4/3"

com-

bination

cycle,

can be viewed

as a combination

of

an "A set"

of

pitch

classes

in

the

even-numbered

order

positions

with a

"B

set"

in the

odd-numbered

positions.

Both the

A

and B

sets

in

this

case

are

generated

by

cycles

of interval

7,

which s the

sum

(mod

12)

of the two

alternating

ntervals.

Conceptually,

he

A

and

B sets

constitute

an

"overlay"

of a

cycle

with itself

ata

par-

ticular

nterval:

he 4/3

cycle

overlays

wo interval-7

ycles

at

a


combination

cycles

assumes

a

transpositional

quivalence

be-

tween

the combined

cycles,

and

encompasses

any possible

wo-

interval

alternation.41

n

general

terms,

x +

y

=

n

(mod

12),

where

x and

y

are

any

two

alternating

ntervals

overlaying

ycles

of interval

n.

The

possible

values

of x and

y

for

a

given

n are

defined

by

the

operator

cycles

TnI

(see

Table

1).

For

example,

where

n

=

7,

as in

Figure

2,

the

possible

values

of

x and

y

are

indicated

by

T7I

cycles:

x/y

=

0/7, 7/0,

1/6, 6/1,2/5, 5/2,

3/4, 4/3, 8/11,

11/8,

9/10,

or

10/9.

A

simple

exchange

of the

x/y

values,

as in

converting

9/10

to

10/9,

has

only

a subtle

effect

on the

presentation

of the

two

al-

4This

approach

bears

similarities

o that

of Elliott

Antokoletz,

in The

Mu-

sic

ofBela

Bart6k,

in which

nterval-1

cycles

are

combined

n order

to

explain

symmetrical

pitch

constructions,

hough

his

study

does not

extend

the

concept

to

encompass

combinations

of other

cycles.

Along

these

same

lines,

the

com-


University

of California

Press,

1977)

combines

cycles

of

every

interval,

but is

primarily

concerned

with

inverse-related

cycles

and their

compositional

applications.

41The

oncept

may

be

naturally

extended

to

encompass

more

thantwo

in-

tervals

n

alternation,

thus

possibly

nvolving

a combination

of several

cycles.

Ives

rarely

uses

such a

sequence,

and the

present

discussion

s

limited

to

two

alternating

ntervals.

Morris's

oncept

of

a

cyclic

INT

would

apply

o

cyclicrep-

etitions

of


WithPitch

Classes,40,

107).






Interval

ycles

as

Compositional

esources

55

nterval

ycles

as

Compositional

esources

55

nterval

ycles

as

Compositional

esources

55

Figure

2. 4/3 combination

cycle.

igure

2. 4/3 combination

cycle.

igure

2. 4/3 combination

cycle.

op:

0

B set:

(pc)

A set:

0

(pc)

I

op:

0

B set:

(pc)

A set:

0

(pc)

I

op:

0

B set:

(pc)

A set:

0

(pc)

I

1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19

20

21 22

23

4 11 6

1

8 3 10

5 0 7

2

9

7 2 9 4 11

6

1

8 3 10

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19

20

21 22

23

4 11 6

1

8 3 10

5 0 7

2

9

7 2 9 4 11

6

1

8 3 10

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19

20

21 22

23

4 11 6

1

8 3 10

5 0 7

2

9

7 2 9 4 11

6

1

8 3 10

5

PCL

=

8CL

=

8CL

=

8

first

repetition

irst

repetition

irst

repetition

ternating

intervals,

but

may

more

significantly

nfluence as-

pects

of

the

pitch-class

succession.

Where n is

even,

the

presence

of a

singleton

cycle

indicates

that

x

=

y,

expressing

a

single-interval ycle

as an alternation

of two identical

ntervals.

If

n

=

10,

for

example, x/y

can be 5/5

or 11/11 n addition o

0/10,

1/9,

andso

on,

because 5 and11are

the

singletons

of

the

T1oI

cycles.

A subdivision

of

this

type

within

an

interval-5

cycle

(x/y

=

5/5)

is

apparent

n

Example

1,

where Ives highlights he underlyingnterval-10cycle through

accentuationand

registral

association.

Though

a

combination

nvolving

a CARD-12

cycle

will con-

tinue

to

op

23,

Ives

is

typically

most concernedwith the number

of

pcs prior

to

repetition,

the

"pitch-class

ength"

(PCL)

of a

combination

cycle.

The PCL of the 4/3

cycle

illustrated

n

Fig-

ure 2

is

8,

because

eight pcs (op 0-7)

are stated

prior

o the first

repetition:pc

4 at

op

8

is

a

repetition

of

the

pc

at

op

1.

Thus,

in a

numbering

of order

positions

that

begins

with

0,

the

op

of

the

firstrepeated pc will correspond o the value of the PCL.

The value of the PCLis a

function

of

the CARD

of

the over-

laid


and of

the content

of

the

A

and

B

sets.

For

combinations

of

cycles

of CARD

12,

the

pitch

classes

of A

and B will

always

be

the

same,

allowing

a

variety

of

possible

PCL

values. Overlaid

cycles

with

cardinalities maller

han

12,

however,

exhibit

either

complete equivalence

of

pitch-class

content

between the A and B

sets or

complete

nonequivalence.

ternating

intervals,

but

may

more

significantly

nfluence as-

pects

of

the

pitch-class

succession.

Where n is

even,

the

presence

of a

singleton

cycle

indicates

that

x

=

y,

expressing

a


as an alternation

of two identical

ntervals.

If

n

=

10,

for

example, x/y

can be 5/5


0/10,

1/9,

andso

on,

because 5 and11are

the

singletons

of

the

T1oI

cycles.

A subdivision

of

this

type

within

an

interval-5

cycle

(x/y

=

5/5)

is

apparent

n

Example

1,


accentuationand

registral

association.

Though

a

combination

nvolving

a CARD-12

cycle

will con-

tinue

to

op

23,

Ives

is

typically


of

pcs prior

to

repetition,

the

"pitch-class

ength"

(PCL)

of a

combination

cycle.

The PCL of the 4/3

cycle

illustrated

n

Fig-

ure 2

is

8,

because

eight pcs (op 0-7)

are stated

prior

o the first

repetition:pc

4 at

op

8

is

a

repetition

of

the

pc

at

op

1.

Thus,

in a

numbering

of order

positions

that

begins

with

0,

the

op

of

the



function

of

the CARD

of

the over-

laid


and of

the content

of

the

A

and

B

sets.

For

combinations

of

cycles

of CARD

12,

the

pitch

classes

of A

and B will

always

be

the

same,

allowing

a

variety

of

possible

PCL

values. Overlaid

cycles

with


han

12,

however,

exhibit

either


of

pitch-class

content

between the A and B

sets or

complete

nonequivalence.

ternating

intervals,

but

may

more

significantly

nfluence as-

pects

of

the

pitch-class

succession.

Where n is

even,

the

presence

of a

singleton

cycle

indicates

that

x

=

y,

expressing

a


as an alternation

of two identical

ntervals.

If

n

=

10,

for

example, x/y

can be 5/5


0/10,

1/9,

andso

on,

because 5 and11are

the

singletons

of

the

T1oI

cycles.

A subdivision

of

this

type

within

an

interval-5

cycle

(x/y

=

5/5)

is

apparent

n

Example

1,


accentuationand

registral

association.

Though

a

combination

nvolving

a CARD-12

cycle

will con-

tinue

to

op

23,

Ives

is

typically


of

pcs prior

to

repetition,

the

"pitch-class

ength"

(PCL)

of a

combination

cycle.

The PCL of the 4/3

cycle

illustrated

n

Fig-

ure 2

is

8,

because

eight pcs (op 0-7)

are stated

prior

o the first

repetition:pc

4 at

op

8

is

a

repetition

of

the

pc

at

op

1.

Thus,

in a

numbering

of order

positions

that

begins

with

0,

the

op

of

the



function

of

the CARD

of

the over-

laid


and of

the content

of

the

A

and

B

sets.

For

combinations

of

cycles

of CARD

12,

the

pitch

classes

of A

and B will

always

be

the

same,

allowing

a

variety

of

possible

PCL

values. Overlaid

cycles

with


han

12,

however,

exhibit

either


of

pitch-class

content

between the A and B

sets or

complete

nonequivalence.

The combinationwill be

termed "reiterative"where the

pitch-

class content of A and

B

is

the

same,

or

"nonreiterative"where

the

pitch-class

content of

the A and B sets

is different. Mem-

bership

of

any

combination none of these two

categories

s dis-

played

by

the

x/y

values: if x and

y

are

multiples

mod

12)

of

n,

thecombination s

reiterative,

and

f x

and

y

are not

multiples

of

n the combination s

nonreiterative.

The four

CARD-12

cycles

(n

=

1, 5, 7,

11)

generate only

reiterative

combinations

be-

cause allpossible x/yvalues aremultiples mod 12)of the n val-

ues,

while the other

cycles may

or

may

not

generate

reiterative

combinations.

For

example,

where

n

=

2,

reiterative

ombina-

tions result

when

x/y

=

(multiples

of

2)

0/2, 2/0,

4/10,

10/4,

6/8,

and

8/6,

but

nonreiterative

combinations

result when

x/y

=

(non-multiples)

1/1, 3/11,

11/3,

5/9,

9/5,

and 7/7. In the

caseof interval-2

ycles

(and

nterval-10),

a nonreiterative om-

bination

produces

aggregate completion

from

combining

the

odd

and even whole-tone collections.

Prediction of the PCL in nonreiterativecombinationsis

achieved

by doubling

the

CARD

of the

n-cycle:

PCL

=

CARD(n)

x

2. That

is,

the numberof

pitch

classes in both

A

and

B

together

totals the number

of

unique

pitch

classes n the

segment.

PCL

prediction

or

reiterativecombinations

ntails a

more extensive calculation

procedure.Everypossible

combina-

tion of a

cycle

with

itself,

and

thus

every

PCL

value,

is

displayed

in a

comparison

of a

constant

A

set

with

each rotation

of



pitch-


B

is

the

same,

or


the

pitch-class

content of

the A and B sets

is different. Mem-

bership

of

any


categories

s dis-

played

by

the

x/y

values: if x and

y

are

multiples

mod

12)

of

n,

thecombination s

reiterative,

and

f x

and

y

are not

multiples

of

n the combination s

nonreiterative.

The four

CARD-12

cycles

(n

=

1, 5, 7,

11)

generate only

reiterative

combinations

be-


ues,

while the other

cycles may

or

may

not

generate

reiterative

combinations.

For

example,

where

n

=

2,

reiterative

ombina-

tions result

when

x/y

=

(multiples

of

2)

0/2, 2/0,

4/10,

10/4,

6/8,

and

8/6,

but

nonreiterative

combinations

result when

x/y

=

(non-multiples)

1/1, 3/11,

11/3,

5/9,

9/5,

and 7/7. In the

caseof interval-2

ycles

(and

nterval-10),


bination

produces


from

combining

the

odd



achieved

by doubling

the

CARD

of the

n-cycle:

PCL

=

CARD(n)

x

2. That

is,

the numberof

pitch

classes in both

A

and

B

together

totals the number

of

unique

pitch

classes n the

segment.

PCL

prediction

or


ntails a



combina-

tion of a

cycle

with

itself,

and

thus

every

PCL

value,

is

displayed

in a

comparison

of a

constant

A

set

with

each rotation

of



pitch-


B

is

the

same,

or


the

pitch-class

content of

the A and B sets

is different. Mem-

bership

of

any


categories

s dis-

played

by

the

x/y

values: if x and

y

are

multiples

mod

12)

of

n,

thecombination s

reiterative,

and

f x

and

y

are not

multiples

of

n the combination s

nonreiterative.

The four

CARD-12

cycles

(n

=

1, 5, 7,

11)

generate only

reiterative

combinations

be-


ues,

while the other

cycles may

or

may

not

generate

reiterative

combinations.

For

example,

where

n

=

2,

reiterative

ombina-

tions result

when

x/y

=

(multiples

of

2)

0/2, 2/0,

4/10,

10/4,

6/8,

and

8/6,

but

nonreiterative

combinations

result when

x/y

=

(non-multiples)

1/1, 3/11,

11/3,

5/9,

9/5,

and 7/7. In the

caseof interval-2

ycles

(and

nterval-10),


bination

produces


from

combining

the

odd



achieved

by doubling

the

CARD

of the

n-cycle:

PCL

=

CARD(n)

x

2. That

is,

the numberof

pitch

classes in both

A

and

B

together

totals the number

of

unique

pitch

classes n the

segment.

PCL

prediction

or


ntails a



combina-

tion of a

cycle

with

itself,

and

thus

every

PCL

value,

is

displayed

in a

comparison

of a

constant

A

set

with

each rotation

of






56

Music

Theory

Spectrum

6

Music

Theory

Spectrum

6

Music

Theory

Spectrum

Table 1.

Values for

x/y

and

PCL

n combination

ycles.

[x/y,PCL] [etc.]

n R=0

R= 1 R=2 R=3 R=4 R=

1

0/1,

1

1/0,

2

2/11,4 3/10,6 4/9,8

5/E

2

0/2,

1

2/0,

2

4/10,4 6/8,

6

8/6,

5 10/,

*1/1,12 3/11,12 5/9,12 7/7,12

9/5,12

11/

3

0/3,

1

3/0,

2

6/9,

4

9/6,

3

*1/2,

8

4/11,8 7/8,

8

10/5,8

*2/1,

8

5/10,8

8/7,

8

11/4,8

4

0/4,

1

4/0,

2

8/8,

3

*1/3,

6

5/11,6 9/7,

6

*2/2,

6

6/10,6 10/6,6

*3/1,

6

7/9,

6

11/5,6

5

0/5,

1

5/0,

2

10/7,4 3/2,

6

8/9,

8 1/'

6

0/6,

1

6/0,

2

*1/5,

4

7/11,4

*2/4,

4

8/10,4

*3/3,

4

9/9,

4

*4/2,

4

10/8,

4

*5/1,

4

11/7,

4

7

0/7,

1

7/0,

2

2/5,

4

9/10,

6

4/3,

8 11/

8

0/8,

1

8/0,

2

4/4,

3

*1/7,

6

9/11,6 5/3,

6

*2/6,

6

10/10,6 6/2,

6

*3/5,

6

11/9,

6

7/1,

6

9

0/9,

1

9/0,

2

6/3,

4

3/6,

3

*1/8,

8

10/11,8 7/2,

8

4/5,

8

*2/7,

8

11/10,8 8/1,

8

5/4,

8

10

0/10,1 10/0,

2

8/2,

4

6/4,

6

4/6,

5

2/

*1/9,12

11/11,12

9/1,12 7/3,12

5/5,12

3/

11

0/11,1

11/0,

2

10/1,

4

9/2,

6

8/3,

8

7/

Table 1.

Values for

x/y

and

PCL

n combination

ycles.

[x/y,PCL] [etc.]

n R=0

R= 1 R=2 R=3 R=4 R=

1

0/1,

1

1/0,

2

2/11,4 3/10,6 4/9,8

5/E

2

0/2,

1

2/0,

2

4/10,4 6/8,

6

8/6,

5 10/,

*1/1,12 3/11,12 5/9,12 7/7,12

9/5,12

11/

3

0/3,

1

3/0,

2

6/9,

4

9/6,

3

*1/2,

8

4/11,8 7/8,

8

10/5,8

*2/1,

8

5/10,8

8/7,

8

11/4,8

4

0/4,

1

4/0,

2

8/8,

3

*1/3,

6

5/11,6 9/7,

6

*2/2,

6

6/10,6 10/6,6

*3/1,

6

7/9,

6

11/5,6

5

0/5,

1

5/0,

2

10/7,4 3/2,

6

8/9,

8 1/'

6

0/6,

1

6/0,

2

*1/5,

4

7/11,4

*2/4,

4

8/10,4

*3/3,

4

9/9,

4

*4/2,

4

10/8,

4

*5/1,

4

11/7,

4

7

0/7,

1

7/0,

2

2/5,

4

9/10,

6

4/3,

8 11/

8

0/8,

1

8/0,

2

4/4,

3

*1/7,

6

9/11,6 5/3,

6

*2/6,

6

10/10,6 6/2,

6

*3/5,

6

11/9,

6

7/1,

6

9

0/9,

1

9/0,

2

6/3,

4

3/6,

3

*1/8,

8

10/11,8 7/2,

8

4/5,

8

*2/7,

8

11/10,8 8/1,

8

5/4,

8

10

0/10,1 10/0,

2

8/2,

4

6/4,

6

4/6,

5

2/

*1/9,12

11/11,12

9/1,12 7/3,12

5/5,12

3/

11

0/11,1

11/0,

2

10/1,

4

9/2,

6

8/3,

8

7/

Table 1.

Values for

x/y

and

PCL

n combination

ycles.

[x/y,PCL] [etc.]

n R=0

R= 1 R=2 R=3 R=4 R=

1

0/1,

1

1/0,

2

2/11,4 3/10,6 4/9,8

5/E

2

0/2,

1

2/0,

2

4/10,4 6/8,

6

8/6,

5 10/,

*1/1,12 3/11,12 5/9,12 7/7,12

9/5,12

11/

3

0/3,

1

3/0,

2

6/9,

4

9/6,

3

*1/2,

8

4/11,8 7/8,

8

10/5,8

*2/1,

8

5/10,8

8/7,

8

11/4,8

4

0/4,

1

4/0,

2

8/8,

3

*1/3,

6

5/11,6 9/7,

6

*2/2,

6

6/10,6 10/6,6

*3/1,

6

7/9,

6

11/5,6

5

0/5,

1

5/0,

2

10/7,4 3/2,

6

8/9,

8 1/'

6

0/6,

1

6/0,

2

*1/5,

4

7/11,4

*2/4,

4

8/10,4

*3/3,

4

9/9,

4

*4/2,

4

10/8,

4

*5/1,

4

11/7,

4

7

0/7,

1

7/0,

2

2/5,

4

9/10,

6

4/3,

8 11/

8

0/8,

1

8/0,

2

4/4,

3

*1/7,

6

9/11,6 5/3,

6

*2/6,

6

10/10,6 6/2,

6

*3/5,

6

11/9,

6

7/1,

6

9

0/9,

1

9/0,

2

6/3,

4

3/6,

3

*1/8,

8

10/11,8 7/2,

8

4/5,

8

*2/7,

8

11/10,8 8/1,

8

5/4,

8

10

0/10,1 10/0,

2

8/2,

4

6/4,

6

4/6,

5

2/

*1/9,12

11/11,12

9/1,12 7/3,12

5/5,12

3/

11

0/11,1

11/0,

2

10/1,

4

9/2,

6

8/3,

8

7/

= 5

3,10

4,3

3,12

= 5

3,10

4,3

3,12

= 5

3,10

4,3

3,12

R=6

6/7,12

R=6

6/7,12

R=6

6/7,12

R=7

7/6,11

R=7

7/6,11

R=7

7/6,11

4,10

6/11,12 11/6,11

,10

6/11,12 11/6,11

,10

6/11,12 11/6,11

8,10 6/1,12

8,

3

7,12

4,10

6/5,12

8,10 6/1,12

8,

3

7,12

4,10

6/5,12

8,10 6/1,12

8,

3

7,12

4,10

6/5,12

R=8

8/5,9

R=8

8/5,9

R=8

8/5,9

R=9

9/4,7

R=9

9/4,7

R=9

9/4,7

R

=

10 R

=

11

10/3,5 11/2,3

R

=

10 R

=

11

10/3,5 11/2,3

R

=

10 R

=

11

10/3,5 11/2,3

4/1,

9

9/8,

7

2/3,

5

7/10,3/1,

9

9/8,

7

2/3,

5

7/10,3/1,

9

9/8,

7

2/3,

5

7/10,3

1/6,11

8/11,9

/6,11

8/11,9

/6,11

8/11,9 3/4,7 10/9,5 5/2,3/4,7 10/9,5 5/2,3/4,7 10/9,5 5/2,3

5/6,11 4/7,

9

3/8,7 2/9,5

1/10,3

*rows

preceded

by

asterisks ist nonreiterativecombinations

5/6,11 4/7,

9

3/8,7 2/9,5

1/10,3

*rows

preceded

by


5/6,11 4/7,

9

3/8,7 2/9,5

1/10,3

*rows

preceded

by







Interval

ycles

as

Compositional

esources 57

nterval

ycles

as

Compositional

esources 57

nterval

ycles

as

Compositional

esources 57

Table

1

(cont.)

able

1

(cont.)

able

1

(cont.)

TI

Cycles

I

Cycles

I

Cycles

T1I

(0-1)

T2I

(0-2)

T31

(0-3)

T41

(0-4)

T5I

(0-5)

T61

(0-6)

T7I

(0-7)

T8I

(0-8)

TgI

(0-9)

TloI (0-10)

TllI (0-11)

T1I

(0-1)

T2I

(0-2)

T31

(0-3)

T41

(0-4)

T5I

(0-5)

T61

(0-6)

T7I

(0-7)

T8I

(0-8)

TgI

(0-9)

TloI (0-10)

TllI (0-11)

T1I

(0-1)

T2I

(0-2)

T31

(0-3)

T41

(0-4)

T5I

(0-5)

T61

(0-6)

T7I

(0-7)

T8I

(0-8)

TgI

(0-9)

TloI (0-10)

TllI (0-11)

(2-11)

(1)

(1-2)

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

(1-8)

(1-9)

(1-10)

(2-11)

(1)

(1-2)

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

(1-8)

(1-9)

(1-10)

(2-11)

(1)

(1-2)

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

(1-8)

(1-9)

(1-10)

(3-10)

(3-11)

(4-11)

(2)

(2-3)

(2-4)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(3-10)

(3-11)

(4-11)

(2)

(2-3)

(2-4)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(3-10)

(3-11)

(4-11)

(2)

(2-3)

(2-4)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(4-9)

(4-10)

(5-10)

(5-11)

(6-11)

(3)

(3-4)

(3-5)

(3-6)

(3-7)

(3-8)

(4-9)

(4-10)

(5-10)

(5-11)

(6-11)

(3)

(3-4)

(3-5)

(3-6)

(3-7)

(3-8)

(4-9)

(4-10)

(5-10)

(5-11)

(6-11)

(3)

(3-4)

(3-5)

(3-6)

(3-7)

(3-8)

(5-8)

(5-9)

(6-9)

(6-10)

(7-10)

(7-11)

(8-11)

(4)

(4-5)

(4-6)

(4-7)

(5-8)

(5-9)

(6-9)

(6-10)

(7-10)

(7-11)

(8-11)

(4)

(4-5)

(4-6)

(4-7)

(5-8)

(5-9)

(6-9)

(6-10)

(7-10)

(7-11)

(8-11)

(4)

(4-5)

(4-6)

(4-7)

(6-7)

(6-8)

(7-8)

(7-9)

(8-9)

(8-10)

(9-10)

(9-11)

(10-11)

(5)

(5-6)

(6-7)

(6-8)

(7-8)

(7-9)

(8-9)

(8-10)

(9-10)

(9-11)

(10-11)

(5)

(5-6)

(6-7)

(6-8)

(7-8)

(7-9)

(8-9)

(8-10)

(9-10)

(9-11)

(10-11)

(5)

(5-6)

the

(pitch-class

equivalent)

B

set;

CARD

different rotations

are

possible.

The PCL of a

specific

combination, herefore,

s a

result of the

number of

rotations of B with

respect

to A.

The

calculation

procedure

for

the

PCL

begins

with the establish-

ment of

order

position

indicators or

the

A

set, or,

in Morris's

notation,

the order

mapping

of

segment A, OMAA.42

The

OMAA

s

numbered from 0

through

CARD-

1,

as shown be-

low

the A set of

the 3/2

combination

cycle

in

Figure

3a

(n

=

5),

for

comparison

with

OMAB,

the order

mappings

of B with re-

spect

to A.

A

four-step

procedure

calculates he PCL:

1.

Determine

R

=

OMABo.

In other

words,

R is the A set or-

der

position

of

the first

pitch

class of the B

set,

showing

the

numberof times B has

been rotated with

respect

to

A.

2.

Find

R',

defined

as the

mod(CARD)

complement

of R.

R + R' = 0 (mod(CARD)).

3.

Compare

R and

R'.

If R

(and

R')

O

andR

<

R'

(Case

),

the

first

pc repetition

occurs n

the A

set,

repeating

he ini-

tial

pc

of

the B

set.

If R

(and

R')

=

0 or

R

>

R'

(Case

2),

42Morris,

15

(DEF 3.13).

The

OMAA

s a

mapping

based on

the

segment

A such that

OMAAs-os.

the

(pitch-class

equivalent)

B

set;

CARD

different rotations

are

possible.

The PCL of a

specific


s a

result of the

number of

rotations of B with

respect

to A.

The

calculation

procedure

for

the

PCL

begins

with the establish-

ment of

order

position

indicators or

the

A

set, or,

in Morris's

notation,

the order

mapping

of

segment A, OMAA.42

The

OMAA

s

numbered from 0

through

CARD-

1,

as shown be-

low

the A set of

the 3/2

combination

cycle

in

Figure

3a

(n

=

5),

for

comparison

with

OMAB,

the order

mappings

of B with re-

spect

to A.

A

four-step

procedure

calculates he PCL:

1.

Determine

R

=

OMABo.

In other

words,

R is the A set or-

der

position

of

the first

pitch

class of the B

set,

showing

the


been rotated with

respect

to

A.

2.

Find

R',

defined

as the

mod(CARD)

complement

of R.

R + R' = 0 (mod(CARD)).

3.

Compare

R and

R'.

If R

(and

R')

O

andR

<

R'

(Case

),

the

first

pc repetition

occurs n

the A

set,

repeating

he ini-

tial

pc

of

the B

set.

If R

(and

R')

=

0 or

R

>

R'

(Case

2),

42Morris,

15

(DEF 3.13).

The

OMAA

s a

mapping

based on

the

segment

A such that

OMAAs-os.

the

(pitch-class

equivalent)

B

set;

CARD

different rotations

are

possible.

The PCL of a

specific


s a

result of the

number of

rotations of B with

respect

to A.

The

calculation

procedure

for

the

PCL

begins

with the establish-

ment of

order

position

indicators or

the

A

set, or,

in Morris's

notation,

the order

mapping

of

segment A, OMAA.42

The

OMAA

s

numbered from 0

through

CARD-

1,

as shown be-

low

the A set of

the 3/2

combination

cycle

in

Figure

3a

(n

=

5),

for

comparison

with

OMAB,

the order

mappings

of B with re-

spect

to A.

A

four-step

procedure

calculates he PCL:

1.

Determine

R

=

OMABo.

In other

words,

R is the A set or-

der

position

of

the first

pitch

class of the B

set,

showing

the


been rotated with

respect

to

A.

2.

Find

R',

defined

as the

mod(CARD)

complement

of R.

R + R' = 0 (mod(CARD)).

3.

Compare

R and

R'.

If R

(and

R')

O

andR

<

R'

(Case

),

the

first

pc repetition

occurs n

the A

set,

repeating

he ini-

tial

pc

of

the B

set.

If R

(and

R')

=

0 or

R

>

R'

(Case

2),

42Morris,

15

(DEF 3.13).

The

OMAA

s a

mapping

based on

the

segment

A such that

OMAAs-os.

the first

pc repetition

occurs

in

the B

set,

repeating

the ini-

tial

pc

of

the A set.

4.

Casel: PCL

=

R

x

2. Case2: PCL

=

(R'

x

2)

+

1.

Figure

3a

executes these four

steps

in

calculating

he PCL

of

the 3/2 combination.At

step

1,

R =

3,

because

OMABo

s

3:

pc

9,

the

first

pc

of

B,

occursat

op

3 inA. R'

is

9

for

step

2,

because

3 +

9 = 0

(mod

12).

Step

3 determines hat

case 1

applies,

and

step

4 calculates

he PCL as 6. The first

pc repetition

occurs

ust

after the sixth

pc

in the

segment,

at

pc

9

in

the

A

set,

which

first

occurs at the beginningof the B set.

Figure

3b

gives

another rotation

of the

same

pitch-class

ma-

terials with the

four-step

PCL calculation.

The value

of

R,

cal-

culated

at

step

1,

is 9

and its

complement,

R' in

step

2,

is 3.

In

step

3,

case

2

applies,

so the

PCL is calculated o be 7 in

step

4.

The first

pc

repetition

occurs

ust

after

he seventh

pc

in the

seg-

ment,

at

pc

6 in the B

set,

which

firstoccursat the

beginning

of

the

A set.

WhereR =

0,

meaning

he

ordering

of A and B is

identical,

the initialpc repetitionoccurs at its earliestpossible position,

the

beginning

of

the

B

set. The

PCL, therefore,

follows thecase

2

conditions,

where the

first

pitch

class

of A

recurs

in B

to

define the

PCL. At

step

4,

either R or R'

can be inserted

n

the

formula,

because both

values

are

0.

Where n =

0,

meaning

x and

y

are

mod

12

complements,

the calculation

procedure

does not

apply,

since,

technically,

this

indicates an

overlay

of

cycles

of

interval

0. For

example,

x/y

= 5/7

defines the

segment

<0,5>.

The PCL

is 2

where

n = 0, except for x/y = 0/0, where the PCL is 1.

Figure

4

calculatesPCL

values for two

reiterativecombina-

tions of

interval-3

cycles.

TheCARD of this

cycle

s

4,

requiring

referenceto

the mod 4

complement

pairs

isted on line a. In

Fig-

ure

4b,

the

x/y

valuesare

3/0,

producing

an

earlypc duplication

that can

easily

be

observed

by inspection,

or can

be derived

through

the

four-step

calculation

procedure,

as listed to the

right

of

the

segment.

The

value

of

R is

1,

which s smaller han

the first

pc repetition

occurs

in

the B

set,

repeating

the ini-

tial

pc

of

the A set.

4.

Casel: PCL

=

R

x

2. Case2: PCL

=

(R'

x

2)

+

1.

Figure

3a

executes these four

steps

in

calculating

he PCL

of


step

1,

R =

3,

because

OMABo

s

3:

pc

9,

the

first

pc

of

B,

occursat

op

3 inA. R'

is

9

for

step

2,

because

3 +

9 = 0

(mod

12).

Step

3 determines hat

case 1

applies,

and

step

4 calculates


pc repetition

occurs

ust

after the sixth

pc

in the

segment,

at

pc

9

in

the

A

set,

which

first


Figure

3b

gives

another rotation

of the

same

pitch-class

ma-

terials with the

four-step

PCL calculation.

The value

of

R,

cal-

culated

at

step

1,

is 9

and its

complement,

R' in

step

2,

is 3.

In

step

3,

case

2

applies,

so the


step

4.

The first

pc

repetition

occurs

ust

after

he seventh

pc

in the

seg-

ment,

at

pc

6 in the B

set,

which

firstoccursat the

beginning

of

the

A set.

WhereR =

0,

meaning

he

ordering

of A and B is

identical,


the

beginning

of

the

B

set. The

PCL, therefore,

follows thecase

2

conditions,

where the

first

pitch

class

of A

recurs

in B

to

define the

PCL. At

step

4,

either R or R'

can be inserted

n

the

formula,

because both

values

are

0.

Where n =

0,

meaning

x and

y

are

mod

12

complements,

the calculation

procedure

does not

apply,

since,

technically,

this

indicates an

overlay

of

cycles

of

interval

0. For

example,

x/y

= 5/7

defines the

segment

<0,5>.

The PCL

is 2

where


Figure

4

calculatesPCL

values for two

reiterativecombina-

tions of

interval-3

cycles.

TheCARD of this

cycle

s

4,

requiring

referenceto

the mod 4

complement

pairs

isted on line a. In

Fig-

ure

4b,

the

x/y

valuesare

3/0,

producing

an

earlypc duplication

that can

easily

be

observed

by inspection,

or can

be derived

through

the

four-step

calculation

procedure,

as listed to the

right

of

the

segment.

The

value

of

R is

1,

which s smaller han

the first

pc repetition

occurs

in

the B

set,

repeating

the ini-

tial

pc

of

the A set.

4.

Casel: PCL

=

R

x

2. Case2: PCL

=

(R'

x

2)

+

1.

Figure

3a

executes these four

steps

in

calculating

he PCL

of


step

1,

R =

3,

because

OMABo

s

3:

pc

9,

the

first

pc

of

B,

occursat

op

3 inA. R'

is

9

for

step

2,

because

3 +

9 = 0

(mod

12).

Step

3 determines hat

case 1

applies,

and

step

4 calculates


pc repetition

occurs

ust

after the sixth

pc

in the

segment,

at

pc

9

in

the

A

set,

which

first


Figure

3b

gives

another rotation

of the

same

pitch-class

ma-

terials with the

four-step

PCL calculation.

The value

of

R,

cal-

culated

at

step

1,

is 9

and its

complement,

R' in

step

2,

is 3.

In

step

3,

case

2

applies,

so the


step

4.

The first

pc

repetition

occurs

ust

after

he seventh

pc

in the

seg-

ment,

at

pc

6 in the B

set,

which

firstoccursat the

beginning

of

the

A set.

WhereR =

0,

meaning

he

ordering

of A and B is

identical,


the

beginning

of

the

B

set. The

PCL, therefore,

follows thecase

2

conditions,

where the

first

pitch

class

of A

recurs

in B

to

define the

PCL. At

step

4,

either R or R'

can be inserted

n

the

formula,

because both

values

are

0.

Where n =

0,

meaning

x and

y

are

mod

12

complements,

the calculation

procedure

does not

apply,

since,

technically,

this

indicates an

overlay

of

cycles

of

interval

0. For

example,

x/y

= 5/7

defines the

segment

<0,5>.

The PCL

is 2

where


Figure

4

calculatesPCL

values for two

reiterativecombina-

tions of

interval-3

cycles.

TheCARD of this

cycle

s

4,

requiring

referenceto

the mod 4

complement

pairs

isted on line a. In

Fig-

ure

4b,

the

x/y

valuesare

3/0,

producing

an

earlypc duplication

that can

easily

be

observed

by inspection,

or can

be derived

through

the

four-step

calculation

procedure,

as listed to the

right

of

the

segment.

The

value

of

R is

1,

which s smaller han






58

Music

TheorySpectrum

8

Music

TheorySpectrum

8

Music

TheorySpectrum

Figure

3.

PCL

calculations.

a. 3/2 combination

cycle.

Figure

3.

PCL

calculations.

a. 3/2 combination

cycle.

Figure

3.

PCL

calculations.

a. 3/2 combination

cycle.

B set:

A set:

OMAA:

B set:

A set:

OMAA:

B set:

A set:

OMAA:

9 2

7 0 5

10 3

8 1

6

11

4

2

7 0 5

10 3

8 1

6

11

4

2

7 0 5

10 3

8 1

6

11

4

6 11

0 1

6 11

0 1

6 11

0 1

4

9 2 7

0 5

10 3

8

1

2 3 4

5 6 7

8 9

10

11

4

9 2 7

0 5

10 3

8

1

2 3 4

5 6 7

8 9

10

11

4

9 2 7

0 5

10 3

8

1

2 3 4

5 6 7

8 9

10

11

1.

R

2. R'

3. R

4.

PCL

PCL

PCL

1.

R

2. R'

3. R

4.

PCL

PCL

PCL

1.

R

2. R'

3. R

4.

PCL

PCL

PCL

=

OMABo

=

3

=

9

0 and R

<

R' (case 1)

=

Rx2

=

3x2

=

6

=

OMABo

=

3

=

9

0 and R

<

R' (case 1)

=

Rx2

=

3x2

=

6

=

OMABo

=

3

=

9

0 and R

<

R' (case 1)

=

Rx2

=

3x2

=

6

b.

9/8

combination

cycle.

.

9/8

combination

cycle.

.

9/8

combination

cycle.

3 8 1

6 11 4 9

2 7

0

5

10

8 1

6 11 4 9

2 7

0

5

10

8 1

6 11 4 9

2 7

0

5

10

6 11

0

1

6 11

0

1

6 11

0

1

4 9 2 7 0 5 10 3 8 1

2

3

4

5 6 7

8

9

10

11

4 9 2 7 0 5 10 3 8 1

2

3

4

5 6 7

8

9

10

11

4 9 2 7 0 5 10 3 8 1

2

3

4

5 6 7

8

9

10

11

1. R

=

OMABo

=9

2.

R'

=

3

3. R >

R'(case2)

4. PCL =

(R'

x

2)

+

1

PCL

=

(3

x

2)+

1

PCL

=

6+1

PCL = 7

1. R

=

OMABo

=9

2.

R'

=

3

3. R >

R'(case2)

4. PCL =

(R'

x

2)

+

1

PCL

=

(3

x

2)+

1

PCL

=

6+1

PCL = 7

1. R

=

OMABo

=9

2.

R'

=

3

3. R >

R'(case2)

4. PCL =

(R'

x

2)

+

1

PCL

=

(3

x

2)+

1

PCL

=

6+1

PCL = 7

its mod

4

complement,

3,

requiring

he

case-1 PCL calculation

shown at

step

4.

The PCL

is 2:

both the A

and B sets state

only

one

pitch

class

before a

duplication

occurs.In

Figure

4c,

differ-

ent

dispositions

of the

same

A

and

B sets

demonstrate

a

case-2

calculationand a

PCL of 3.

its mod

4

complement,

3,

requiring

he


shown at

step

4.

The PCL

is 2:

both the A

and B sets state

only

one

pitch

class

before a

duplication

occurs.In

Figure

4c,

differ-

ent

dispositions

of the

same

A

and

B sets

demonstrate

a

case-2

calculationand a

PCL of 3.

its mod

4

complement,

3,

requiring

he


shown at

step

4.

The PCL

is 2:

both the A

and B sets state

only

one

pitch

class

before a

duplication

occurs.In

Figure

4c,

differ-

ent

dispositions

of the

same

A

and

B sets

demonstrate

a

case-2

calculationand a

PCL of 3.

Once the

pitch-class

succession

of an

entire

segment

has

been

established,

rotational

operations applied

to the

segment-that

is,

to the combinationof A and

B

sets,

not

just

to

the

B

set alone-can

result in

exchanges

of

x/y

values and

variable

PCLs. The values

of

x and

y

for

any complete segment

Once the

pitch-class

succession

of an

entire

segment

has

been

established,

rotational

operations applied

to the

segment-that

is,


B

sets,

not

just

to

the

B

set alone-can

result in

exchanges

of

x/y

values and

variable

PCLs. The values

of

x and

y

for


Once the

pitch-class

succession

of an

entire

segment

has

been

established,

rotational

operations applied

to the

segment-that

is,


B

sets,

not

just

to

the

B

set alone-can

result in

exchanges

of

x/y

values and

variable

PCLs. The values

of

x and

y

for


B

set:

A set:

OMAA:

B

set:

A set:

OMAA:

B

set:

A set:

OMAA:






Interval

ycles

as

Compositional

esources 59

nterval

ycles

as

Compositional

esources 59

nterval

ycles

as

Compositional

esources 59

Figure

4.

PCL

calculations for smaller

cardinalities.

Figure

4.

PCL


cardinalities.

Figure

4.

PCL


cardinalities.

a. mod

4

complements:

. mod

4

complements:

. mod

4

complements:

0 1

2

0 3 2

0 1

2

0 3 2

0 1

2

0 3 2

3

1

3

1

3

1

b.

B set:

A set: 5

OMAA:

0

b.

B set:

A set: 5

OMAA:

0

b.

B set:

A set: 5

OMAA:

0

c.

B

set:

A set: 5

OMAA:

0

c.

B

set:

A set: 5

OMAA:

0

c.

B

set:

A set: 5

OMAA:

0

8

11 2 5

8

11 2

1

2

3

8

11 2 5

8

11 2

1

2

3

8

11 2 5

8

11 2

1

2

3

2

5

8

8

11

2

1 2

3

2

5

8

8

11

2

1 2

3

2

5

8

8

11

2

1 2

3

1. R

2.

R'

3. R

4. PCL

PCL

PCL

1. R

2.

R'

3. R

4. PCL

PCL

PCL

1. R

2.

R'

3. R

4. PCL

PCL

PCL

1111

1. R

2.

R'

3. R

4.

PCL

PCL

PCL

PCL

1. R

2.

R'

3. R

4.

PCL

PCL

PCL

PCL

1. R

2.

R'

3. R

4.

PCL

PCL

PCL

PCL

OMABo

=

1

3

0 and R

<

R'

(case

1)

Rx2

1x2

2

OMABo

=

1

3

0 and R

<

R'

(case

1)

Rx2

1x2

2

OMABo

=

1

3

0 and R

<

R'

(case

1)

Rx2

1x2

2

OMABo

=

3

1

R'

(case 2)

(R'

x

2)

+ 1

(1

x

2)

+

1

2 + 1

3

OMABo

=

3

1

R'

(case 2)

(R'

x

2)

+ 1

(1

x

2)

+

1

2 + 1

3

OMABo

=

3

1

R'

(case 2)

(R'

x

2)

+ 1

(1

x

2)

+

1

2 + 1

3

beginning

on a memberof the A set

exchange positions

for a

segment

starting

with a B set member. In

Figure

4c,

for exam-

ple, x/y

=

9/6

for the

original segment

<5,2,8,5,11,8,2,11>

and for rotations that

begin

with other membersof the

A

set,

but

x/y

=

6/9

for

a

single

rotation

to

<2,8,5,11,8,2,11,5>

or to

a segment beginningon any other memberof the B set. This

distinction

between

x/y exchanges

in reiterativecombinations

is reflected

by

the PCL.

Obviously,

the PCL is not variable

n

rotations and

x/y exchanges

within

nonreiterative

combina-

tions,

where the PCL is

always

wice

the CARD.

However,

ro-

tations of

reiterativecombinationsexhibit

one

of two

PCL

val-

ues,

with

a

differenceof

1,

depending

on whether

hey

begin

on

a

member of the A set or a

member

of

the

B

set.

The

original

beginning


exchange positions

for a

segment

starting


Figure

4c,

for exam-

ple, x/y

=

9/6

for the

original segment

<5,2,8,5,11,8,2,11>


begin


A

set,

but

x/y

=

6/9

for

a

single

rotation

to

<2,8,5,11,8,2,11,5>

or to


distinction

between

x/y exchanges


is reflected

by

the PCL.

Obviously,


n

rotations and

x/y exchanges

within

nonreiterative

combina-

tions,

where the PCL is

always

wice

the CARD.

However,

ro-

tations of


one

of two

PCL

val-

ues,

with

a

differenceof

1,

depending

on whether

hey

begin

on

a


member

of

the

B

set.

The

original

beginning


exchange positions

for a

segment

starting


Figure

4c,

for exam-

ple, x/y

=

9/6

for the

original segment

<5,2,8,5,11,8,2,11>


begin


A

set,

but

x/y

=

6/9

for

a

single

rotation

to

<2,8,5,11,8,2,11,5>

or to


distinction

between

x/y exchanges


is reflected

by

the PCL.

Obviously,


n

rotations and

x/y exchanges

within

nonreiterative

combina-

tions,

where the PCL is

always

wice

the CARD.

However,

ro-

tations of


one

of two

PCL

val-

ues,

with

a

differenceof

1,

depending

on whether

hey

begin

on

a


member

of

the

B

set.

The

original

segment

in

Figure

4c and

any

rotations to

A

set members ex-

hibit PCLsof

3,

while all rotations o B

set members how PCLs

of 4. In

essence,

an

exchange

of

x/y

values in a

reiterativecom-

bination

cycle

increasesor

decreasesthe PCL

by

one.

Table 1

summarizes

x/y

values

arising

rom combinationsof

interval-ncycles and the resultingPCLs. The possiblevalues

for n are

given by

the

cycles

of

TnI

shown

separately

n

the

ta-

ble. Rows in thechart ist differentcombinations f the same n-

cycles

in both

reiterative

and

nonreiterative ombinations

the

latter ndicated

by

asterisks).

Columns

represent

differentrota-

tions of the B set with

respect

to the

A

set. For reiterativecom-

binations,

R

=

0 in column

1,

indicating

hat the orderof theA

and B sets is

identical,

R

=

1 in column

2,

meaning

B is one

segment

in

Figure

4c and

any

rotations to

A

set members ex-

hibit PCLsof

3,



of 4. In

essence,

an

exchange

of

x/y

values in a

reiterativecom-

bination

cycle

increasesor

decreasesthe PCL

by

one.

Table 1

summarizes

x/y

values

arising

rom combinationsof


for n are

given by

the

cycles

of

TnI

shown

separately

n

the

ta-


cycles

in both

reiterative

and


the

latter ndicated

by

asterisks).

Columns

represent

differentrota-


respect

to the

A


binations,

R

=

0 in column

1,

indicating


and B sets is

identical,

R

=

1 in column

2,

meaning

B is one

segment

in

Figure

4c and

any

rotations to

A

set members ex-

hibit PCLsof

3,



of 4. In

essence,

an

exchange

of

x/y

values in a

reiterativecom-

bination

cycle

increasesor

decreasesthe PCL

by

one.

Table 1

summarizes

x/y

values

arising

rom combinationsof


for n are

given by

the

cycles

of

TnI

shown

separately

n

the

ta-


cycles

in both

reiterative

and


the

latter ndicated

by

asterisks).

Columns

represent

differentrota-


respect

to the

A


binations,

R

=

0 in column

1,

indicating


and B sets is

identical,

R

=

1 in column

2,

meaning

B is one






60 Music

TheorySpectrum

0 Music

TheorySpectrum

0 Music

TheorySpectrum

rotation

of

A,

and

so forth. For

nonreiterative

combinations,

the firstcolumnliststhe combinationswiththe smallestvalues

of

x and their

subsequent readings

after rotations

of B. In

the

rotations of interval-3

cycle

combinations,

for

example,

the

first

nonreiterative

x/y

combination s

1/2,

listed

at the

begin-

ning

of the row that

includesrotationsof B

to

produce

4/11,

7/8,

and 10/5.

A

thirdrow then

completes

the

listings

or

n

=

3,

be-

ginning

with

x/y

=

2/1,

the

next smallest availablevalue

of

x,

and

including

B set rotations to

generate

5/10, 8/7,

and

11/4.

The

chart omits values for n

=

0,

where

x/y

are mod-12 com-

plement pairswith PCLsof 1 (x/y = 0/0)or2 (x/y = 1/11,2/10,

3/9,

4/8, 5/7,

6/6).

With

regard

to

pitch-class

content,

the rows in the chartas-

sociated with each value of

n

represent

he different

pitch-class

collections

that are

possible

from the

indicated

combinations.

The

number of

rows

for

each value of

n

corresponds

o the

number

of

differentcollectionsthat can be

generated:

wo rows

are listed

for

n

=

2 or 10 because there are two whole-tone

pitch-class

collections,

three rows are listed for n

=

3 or 9 be-

cause there are three interval-3or -9 collections (diminished

seventh

chords),

and so forth. In

every

case,

combinations

n

the

first

(reiterative)

row can

generate

CARD(n)

unique

pitch

classes,

because

A

and

B

are

pitch-class quivalent,

while com-

binations

n

any subsequent nonreiterative)

ows can

generate

(CARD(n)

x

2)

unique pitch

classes,

because

A

and

B

are not

equivalent.

Assuming,

for

purposes

of

illustration,

that the

pitch

classes

of the A

set are the same

for

each

row,

the

pitch

classes

of

the

B set are increased

by

one in successive rows.

Where n = 3, forexample,a pitch-class epresentationof row

1 could read as follows:

rotation

of

A,

and

so forth. For

nonreiterative

combinations,


of

x and their

subsequent readings

after rotations

of B. In

the


cycle

combinations,

for

example,

the

first

nonreiterative

x/y

combination s

1/2,

listed

at the

begin-

ning

of the row that


to

produce

4/11,

7/8,

and 10/5.

A

thirdrow then

completes

the

listings

or

n

=

3,

be-

ginning

with

x/y

=

2/1,

the


of

x,

and

including

B set rotations to

generate

5/10, 8/7,

and

11/4.

The


=

0,

where

x/y

are mod-12 com-


3/9,

4/8, 5/7,

6/6).

With

regard

to

pitch-class

content,



n

represent

he different

pitch-class

collections

that are

possible

from the

indicated

combinations.

The

number of

rows

for

each value of

n

corresponds

o the

number

of


generated:

wo rows

are listed

for

n

=


pitch-class

collections,


=

3 or 9 be-


seventh

chords),

and so forth. In

every

case,

combinations

n

the

first

(reiterative)

row can

generate

CARD(n)

unique

pitch

classes,

because

A

and

B

are


while com-

binations

n


ows can

generate

(CARD(n)

x

2)

unique pitch

classes,

because

A

and

B

are not

equivalent.

Assuming,

for

purposes

of

illustration,

that the

pitch

classes

of the A

set are the same

for

each

row,

the

pitch

classes

of

the

B set are increased

by




rotation

of

A,

and

so forth. For

nonreiterative

combinations,


of

x and their

subsequent readings

after rotations

of B. In

the


cycle

combinations,

for

example,

the

first

nonreiterative

x/y

combination s

1/2,

listed

at the

begin-

ning

of the row that


to

produce

4/11,

7/8,

and 10/5.

A

thirdrow then

completes

the

listings

or

n

=

3,

be-

ginning

with

x/y

=

2/1,

the


of

x,

and

including

B set rotations to

generate

5/10, 8/7,

and

11/4.

The


=

0,

where

x/y

are mod-12 com-


3/9,

4/8, 5/7,

6/6).

With

regard

to

pitch-class

content,



n

represent

he different

pitch-class

collections

that are

possible

from the

indicated

combinations.

The

number of

rows

for

each value of

n

corresponds

o the

number

of


generated:

wo rows

are listed

for

n

=


pitch-class

collections,


=

3 or 9 be-


seventh

chords),

and so forth. In

every

case,

combinations

n

the

first

(reiterative)

row can

generate

CARD(n)

unique

pitch

classes,

because

A

and

B

are


while com-

binations

n


ows can

generate

(CARD(n)

x

2)

unique pitch

classes,

because

A

and

B

are not

equivalent.

Assuming,

for

purposes

of

illustration,

that the

pitch

classes

of the A

set are the same

for

each

row,

the

pitch

classes

of

the

B set are increased

by




B

set:

A

set:

x/y

=

B

set:

A

set:

x/y

=

B

set:

A

set:

x/y

=

R=0

0369

0369

0/3

R=0

0369

0369

0/3

R=0

0369

0369

0/3

R=1

R=2

3690

6903

0369 0369

3/0

6/9

R=1

R=2

3690

6903

0369 0369

3/0

6/9

R=1

R=2

3690

6903

0369 0369

3/0

6/9

R=3

9036

0369

9/6

R=3

9036

0369

9/6

R=3

9036

0369

9/6

Row 2

could

then

appear

as

a

combination

of

the same

A set

witha B set of pc values that are one greater han the previous

row. The

B

set at R

=

0

would

be

<1,4,7,10>,

rotated as

fol-

lows:

Row 2

could

then

appear

as

a

combination

of

the same

A set


row. The

B

set at R

=

0

would

be

<1,4,7,10>,

rotated as

fol-

lows:

Row 2

could

then

appear

as

a

combination

of

the same

A set


row. The

B

set at R

=

0

would

be

<1,4,7,10>,

rotated as

fol-

lows:

B

set:

A set:

x/

=

B

set:

A set:

x/

=

B

set:

A set:

x/

=

R=0

1 4 7

10

0369

1/2

R=0

1 4 7

10

0369

1/2

R=0

1 4 7

10

0369

1/2

R=1

4 7

10

1

0369

4/11

R=1

4 7

10

1

0369

4/11

R=1

4 7

10

1

0369

4/11

R=2

7

10 1

4

03

69

7/8

R=2

7

10 1

4

03

69

7/8

R=2

7

10 1

4

03

69

7/8

R=3

10 1

4 7

0

369

10/5

R=3

10 1

4 7

0

369

10/5

R=3

10 1

4 7

0

369

10/5

Similarly,row 3 couldbeginwith the B set <2,5,8,11>, or val-

ues one

greater

than

<1,4,7,10>,

rotated to

generate

the re-

maining

x/y

combinations.

Complementary

n values contain nversevalues

of

x/y:

every

x/y

in

Table

1

corresponds

o another

x/y

that is its mod 12

in-

verse.

These inverse

pairs

exhibit he same

R

values

and do not

differ

in PCL.

For

example,

n values of 5 and

7

generate

the

inverse

x/y

values

of

1/4 and

11/8,

both

generated

from

R

=

5

and

exhibiting

a

PCL of

10.

Because

interval 6 is

its own in-

verse, all inversepairsfor n = 6 arecontained n the same six

rows of the chart.

Certainly,

if

pitch-class

variety

is

a

primarycompositional

aim,

many

of the combination

cycles

in Table

1

may

be

of lim-

ited

value. Combinations

of

intervals

4, 6,

or

8,

for

example,

cannot

generate

a PCL

greater

than 6.

This does

not

mean,

however,

that

the smallerPCLs

are without

compositional

ap-

plicability.

Indeed,

the

set-class

ypes

of

many

of the combina-

tion

cycles,

regardless

of the

sizeof their

PCLs,

are some

of

the

most common pitch-classstructures n Ives's music, as in the

music of

other

composers

of his era. In the

interval-6

ombina-

tions,

for

example,

the

nonreiterative

istings

on the second

and

sixth

lines

of Table

1

produce

4-9

[0,1,6,7],

a basic

element

of

Ives's

music

at

every point

in his

development.43

Other

familiar

43See

or

example,

the

combinations

of

half-step

related tritones

in

Ives's

Second

String

Quartet

(first

movement,

mm.

27,

37

[vln.

1,

via.],

38

[vln.

2,


ues one

greater

than

<1,4,7,10>,

rotated to

generate

the re-

maining

x/y

combinations.

Complementary


of

x/y:

every

x/y

in

Table

1

corresponds

o another

x/y

that is its mod 12

in-

verse.

These inverse

pairs

exhibit he same

R

values

and do not

differ

in PCL.

For

example,

n values of 5 and

7

generate

the

inverse

x/y

values

of

1/4 and

11/8,

both

generated

from

R

=

5

and

exhibiting

a

PCL of

10.

Because

interval 6 is

its own in-


rows of the chart.

Certainly,

if

pitch-class

variety

is

a


aim,

many

of the combination

cycles

in Table

1

may

be

of lim-

ited

value. Combinations

of

intervals

4, 6,

or

8,

for

example,

cannot

generate

a PCL

greater

than 6.

This does

not

mean,

however,

that

the smallerPCLs

are without

compositional

ap-

plicability.

Indeed,

the

set-class

ypes

of

many

of the combina-

tion

cycles,

regardless

of the

sizeof their

PCLs,

are some

of

the


music of

other

composers

of his era. In the

interval-6

ombina-

tions,

for

example,

the

nonreiterative

istings

on the second

and

sixth

lines

of Table

1

produce

4-9

[0,1,6,7],

a basic

element

of

Ives's

music

at

every point

in his

development.43

Other

familiar

43See

or

example,

the

combinations

of

half-step

related tritones

in

Ives's

Second

String

Quartet

(first

movement,

mm.

27,

37

[vln.

1,

via.],

38

[vln.

2,


ues one

greater

than

<1,4,7,10>,

rotated to

generate

the re-

maining

x/y

combinations.

Complementary


of

x/y:

every

x/y

in

Table

1

corresponds

o another

x/y

that is its mod 12

in-

verse.

These inverse

pairs

exhibit he same

R

values

and do not

differ

in PCL.

For

example,

n values of 5 and

7

generate

the

inverse

x/y

values

of

1/4 and

11/8,

both

generated

from

R

=

5

and

exhibiting

a

PCL of

10.

Because

interval 6 is

its own in-


rows of the chart.

Certainly,

if

pitch-class

variety

is

a


aim,

many

of the combination

cycles

in Table

1

may

be

of lim-

ited

value. Combinations

of

intervals

4, 6,

or

8,

for

example,

cannot

generate

a PCL

greater

than 6.

This does

not

mean,

however,

that

the smallerPCLs

are without

compositional

ap-

plicability.

Indeed,

the

set-class

ypes

of

many

of the combina-

tion

cycles,

regardless

of the

sizeof their

PCLs,

are some

of

the


music of

other

composers

of his era. In the

interval-6

ombina-

tions,

for

example,

the

nonreiterative

istings

on the second

and

sixth

lines

of Table

1

produce

4-9

[0,1,6,7],

a basic

element

of

Ives's

music

at

every point

in his

development.43

Other

familiar

43See

or

example,

the

combinations

of

half-step

related tritones

in

Ives's

Second

String

Quartet

(first

movement,

mm.

27,

37

[vln.

1,

via.],

38

[vln.

2,






Interval

ycles

as

Compositional

esources 61nterval

ycles

as

Compositional

esources 61nterval

ycles

as

Compositional

esources 61

Example

9.

Musical

applications

of combination

ycles.

a. SecondStringQuartet,secondmovement,mm. 17-18, firstviolin.

Example

9.

Musical

applications

of combination

ycles.


Example

9.

Musical

applications

of combination

ycles.


A

17

_

L^-tt- 4,

17

_

L^-tt- 4,

17

_

L^-tt- 4,

W

r[

I

-

I

fr

I

I

PI

m

-

. LI;

I

f*

Le

F- I

rll

r[

I

-

I

fr

I

I

PI

m

-

. LI;

I

f*

Le

F- I

rll

r[

I

-

I

fr

I

I

PI

m

-

. LI;

I

f*

Le

F- I

rll

PCL=

12

CL=

12

CL=

12

I

-

-

I=I

'='l

-

-

I=I

'='l

-

-

I=I

'='l

Bset: 6 11 4 9 2

1 4 9

2 7

A

set: 0

5 10 3 8

1

0

5 10 3

8 1

(n

=

5)

x/y

=

6/11

b. Over the

Pavements,

mm.

18-22,

clarinet

concertpitch).

Bset: 6 11 4 9 2

1 4 9

2 7

A

set: 0

5 10 3 8

1

0

5 10 3

8 1

(n

=

5)

x/y

=

6/11

b. Over the

Pavements,

mm.

18-22,

clarinet

concertpitch).

Bset: 6 11 4 9 2

1 4 9

2 7

A

set: 0

5 10 3 8

1

0

5 10 3

8 1

(n

=

5)

x/y

=

6/11

b. Over the

Pavements,

mm.

18-22,

clarinet

concertpitch).

21

.

22

1

.

22

1

.

22

,

d

L

J

I -

V

I

LJ

-

IL

I

i

i

i'i

I

,

d

L

J

I -

V

I

LJ

-

IL

I

i

i

i'i

I

,

d

L

J

I -

V

I

LJ

-

IL

I

i

i

i'i

I

x/y

=

x/y

=

x/y

=

I I 1

4

/

7

I I 1

4

/

7

I I 1

4

/

7

(n=

11)

n=

11)

n=

11)

Bset:

3 2

1

0

11 10

A set:

11

10

9

8 7

5

[

7I

6

PCL

=9

\

1st

rep.

Bset:

3 2

1

0

11 10

A set:

11

10

9

8 7

5

[

7I

6

PCL

=9

\

1st

rep.

Bset:

3 2

1

0

11 10

A set:

11

10

9

8 7

5

[

7I

6

PCL

=9

\

1st

rep.

structures

in

the table

include the whole-tone subset 4-25

[0,2,6,8]

on the third and

fifth rows of the

interval-6combina-

tions,

and 6-20

[0,1,4,5,8,9],

one of the all-combinatorial

exa-

chords,

on

the second and fourth

rows

of

the interval-4or

-8

combinations.

When he

uses these combinations o

project cyclic

nterval-

lic

repetitions,

however,

Ives

typically

favors the

larger

PCL

cello]).

In TheStructure

f

Atonal

Music,

Forte cites six

examples

of this tetra-

chord n music

of

Webern,

Scriabin,

Stravinsky,

and

Berg, including

he well-

known

extensive

usage

in

Webern's Five

Movements

or String

Quartet,

Op.

5

No. 4

(p.

27).

This set class s

George

Perle's

"y"

cell in his

analysis

of

Op.

5 No.

4 in

Serial

Composition

and

Atonality,

5th ed.

(Berkeley:

University

of

Califor-

nia

Press,

1981),

16-18. Lendvai's

study

of Bart6k

also focuses on this tetra-

chord,

identifying

t as one of

the

repetitive

nterval

"models"common n Bar-

t6k's

pitch

language.

structures

in

the table


[0,2,6,8]

on the third and

fifth rows of the

interval-6combina-

tions,

and 6-20

[0,1,4,5,8,9],


exa-

chords,

on


rows

of

the interval-4or

-8

combinations.

When he


project cyclic

nterval-

lic

repetitions,

however,

Ives

typically

favors the

larger

PCL

cello]).

In TheStructure

f

Atonal

Music,

Forte cites six

examples

of this tetra-

chord n music

of

Webern,

Scriabin,

Stravinsky,

and

Berg, including

he well-

known

extensive

usage

in

Webern's Five

Movements

or String

Quartet,

Op.

5

No. 4

(p.

27).

This set class s

George

Perle's

"y"

cell in his

analysis

of

Op.

5 No.

4 in

Serial

Composition

and

Atonality,

5th ed.

(Berkeley:

University

of

Califor-

nia

Press,

1981),

16-18. Lendvai's

study

of Bart6k


chord,

identifying

t as one of

the

repetitive

nterval


t6k's

pitch

language.

structures

in

the table


[0,2,6,8]

on the third and

fifth rows of the

interval-6combina-

tions,

and 6-20

[0,1,4,5,8,9],


exa-

chords,

on


rows

of

the interval-4or

-8

combinations.

When he


project cyclic

nterval-

lic

repetitions,

however,

Ives

typically

favors the

larger

PCL

cello]).

In TheStructure

f

Atonal

Music,

Forte cites six

examples

of this tetra-

chord n music

of

Webern,

Scriabin,

Stravinsky,

and

Berg, including

he well-

known

extensive

usage

in

Webern's Five

Movements

or String

Quartet,

Op.

5

No. 4

(p.

27).

This set class s

George

Perle's

"y"

cell in his

analysis

of

Op.

5 No.

4 in

Serial

Composition

and

Atonality,

5th ed.

(Berkeley:

University

of

Califor-

nia

Press,

1981),

16-18. Lendvai's

study

of Bart6k


chord,

identifying

t as one of

the

repetitive

nterval


t6k's

pitch

language.

values in reiterative andnonreiterativecombinations.

Among

the

more common

larger

structures s the octatonic

collection,

8-28

[0,1,3,4,6,7,9,10],

derived n nonreiterative

ombinations

of

intervals 3

or

9. For

combinations

of the

CARD-12

cycles,

every possible

PCL value

(1-12)

is available

(see

Tab.

1),

pro-

viding

material or

some

of

Ives's most

frequent

combinations.

In the

violin

line from his

Second

String

Quartet

given

n

Exam-

ple 9a, for example,Ives employs the 6/11 interval-5combina-

tion to

complete

the

aggregate

hroughprojection

of

the maxi-

mal PCL.

The

segment

s

repeatedbeginning

on the fourthbeat

of m.

17,

following

the

arrivalof

the twelfth

pitch

class,

so that

the

cycles

in the A

and B

sets

individually

do not continue

past

their

midpoints.

However,

since A

and B

are

literal

comple-

ments,

the reiteration

might

be viewed as a continuationof the

individual

cycles

with the

relative order

positions exchanged:


Among

the

more common

larger


collection,

8-28

[0,1,3,4,6,7,9,10],


ombinations

of

intervals 3

or

9. For

combinations

of the

CARD-12

cycles,

every possible

PCL value

(1-12)

is available

(see

Tab.

1),

pro-

viding

material or

some

of

Ives's most

frequent

combinations.

In the

violin

line from his

Second

String

Quartet

given

n

Exam-


tion to

complete

the

aggregate

hroughprojection

of

the maxi-

mal PCL.

The

segment

s

repeatedbeginning

on the fourthbeat

of m.

17,

following

the

arrivalof

the twelfth

pitch

class,

so that

the

cycles

in the A

and B

sets

individually

do not continue

past

their

midpoints.

However,

since A

and B

are

literal

comple-

ments,

the reiteration

might


individual

cycles

with the

relative order



Among

the

more common

larger


collection,

8-28

[0,1,3,4,6,7,9,10],


ombinations

of

intervals 3

or

9. For

combinations

of the

CARD-12

cycles,

every possible

PCL value

(1-12)

is available

(see

Tab.

1),

pro-

viding

material or

some

of

Ives's most

frequent

combinations.

In the

violin

line from his

Second

String

Quartet

given

n

Exam-


tion to

complete

the

aggregate

hroughprojection

of

the maxi-

mal PCL.

The

segment

s

repeatedbeginning

on the fourthbeat

of m.

17,

following

the

arrivalof

the twelfth

pitch

class,

so that

the

cycles

in the A

and B

sets

individually

do not continue

past

their

midpoints.

However,

since A

and B

are

literal

comple-

ments,

the reiteration

might


individual

cycles

with the

relative order


In L_

-f

n L_

-f

n L_

-f

l8

8

,

..

20

_p..

8

8

,

..

20

_p..

8

8

,

..

20

_p..






62 Music

TheorySpectrum

2 Music

TheorySpectrum

2 Music

TheorySpectrum

the

pitch

classes

of the initialA

set

complete

an interval-5

ycle

with the pitchclasses of the reiterationof the B set, and vice

versa.

The

clarinet ine

from Ives's

Over

the

Pavements

1906-13)

given

in

Example

9b

projects

a PCL of 9 from

a

combination

of

interval-11

cycles

in

a

4/7 alternation.Similar

o the extraction

of

interval-10

ycles

from the 5/5

cycle

in

Example

1,

the

gener-

ating

interval-11

cycles

in

Example

9b are

registrally

associ-

ated,

in the manner

of

a

"compoundmelody."

In

mm.

18-21,

each note in the combination

cycle

is of three sixteenths'

dura-

tion, and this durationalconsistency s abandonedwiththe ar-

rival of

the

final

element

of

the

PCL,

pc

7 in m. 22. After this

point,

the

durations

are

shortened

and

the

A and B sets

do

not

continue

to

their individual

cyclic

completions.

As with

single-interval ycles,

combination

cycles

may pro-

vide skeletons

for

patterns

of

embellishment

over

larger

musi-

cal

spans.

This can

generate

an

unsystematic type

of

embellishment-as

is the case for the interval-11

cycle

of Ex-

ample

6-or

a framework or

sequential

repetition.

In Ives's

pi-

ano workRough and Ready (1906-07), for example,the ten-

note

sequential pattern

bracketed

above

the

excerpt

in

Example

10

occurs

in six

transpositions

elated

by descending

whole-steps.44

Accents and slurs

highlight

the first and

sixth

sonorities

n

each

sequential

unit,

projecting

an

alternating

de-

scent

of

even

and

odd whole-tone scales

among

the

upper

ac-

cented

notes.

Since

the

overlay

distance

s

3,

the result s the 3/7

combination

cycle,

a nonreiterative

whole-tone combination.

In the sketchesfor his Universe

Symphony

(1911-28),

Ives

begins to catalogue the possibilitiesfor combinationcycles,

44Kirkpatrick,

atalogue,

96

(Cat.

No.

3B16ii).

This

work,

the full

title of

which s

"Rough

and

Ready

et al. and/or the

JumpingFrog,"

is the second

of

the

Five

Take-Offs

or

piano,

as named

by Kirkpatrick.

t is

transcribed

n Al-

bert

Lotto,

"ExperimentalAspects

of the

Completed

Short

Piano

Pieces

of

CharlesIves"

(D.M.A.

thesis, Juilliard,

1978),

112-118.

the

pitch

classes

of the initialA

set

complete

an interval-5

ycle


versa.

The

clarinet ine

from Ives's

Over

the

Pavements

1906-13)

given

in

Example

9b

projects

a PCL of 9 from

a

combination

of

interval-11

cycles

in

a


o the extraction

of

interval-10

ycles

from the 5/5

cycle

in

Example

1,

the

gener-

ating

interval-11

cycles

in

Example

9b are

registrally

associ-

ated,

in the manner

of

a

"compoundmelody."

In

mm.

18-21,


cycle


dura-


rival of

the

final

element

of

the

PCL,

pc


point,

the

durations

are

shortened

and

the

A and B sets

do

not

continue

to

their individual

cyclic

completions.

As with


combination

cycles

may pro-

vide skeletons

for

patterns

of

embellishment

over

larger

musi-

cal

spans.

This can

generate

an

unsystematic type

of

embellishment-as


cycle

of Ex-

ample

6-or

a framework or

sequential

repetition.

In Ives's

pi-


note

sequential pattern

bracketed

above

the

excerpt

in

Example

10

occurs

in six

transpositions

elated

by descending

whole-steps.44

Accents and slurs

highlight

the first and

sixth

sonorities

n

each

sequential

unit,

projecting

an

alternating

de-

scent

of

even

and


among

the

upper

ac-

cented

notes.

Since

the

overlay

distance

s

3,


combination

cycle,

a nonreiterative



Symphony

(1911-28),

Ives


44Kirkpatrick,

atalogue,

96

(Cat.

No.

3B16ii).

This

work,

the full

title of

which s

"Rough

and

Ready

et al. and/or the

JumpingFrog,"

is the second

of

the

Five

Take-Offs

or

piano,

as named

by Kirkpatrick.

t is

transcribed

n Al-

bert

Lotto,


of the

Completed

Short

Piano

Pieces

of

CharlesIves"

(D.M.A.

thesis, Juilliard,

1978),

112-118.

the

pitch

classes

of the initialA

set

complete

an interval-5

ycle


versa.

The

clarinet ine

from Ives's

Over

the

Pavements

1906-13)

given

in

Example

9b

projects

a PCL of 9 from

a

combination

of

interval-11

cycles

in

a


o the extraction

of

interval-10

ycles

from the 5/5

cycle

in

Example

1,

the

gener-

ating

interval-11

cycles

in

Example

9b are

registrally

associ-

ated,

in the manner

of

a

"compoundmelody."

In

mm.

18-21,


cycle


dura-


rival of

the

final

element

of

the

PCL,

pc


point,

the

durations

are

shortened

and

the

A and B sets

do

not

continue

to

their individual

cyclic

completions.

As with


combination

cycles

may pro-

vide skeletons

for

patterns

of

embellishment

over

larger

musi-

cal

spans.

This can

generate

an

unsystematic type

of

embellishment-as


cycle

of Ex-

ample

6-or

a framework or

sequential

repetition.

In Ives's

pi-


note

sequential pattern

bracketed

above

the

excerpt

in

Example

10

occurs

in six

transpositions

elated

by descending

whole-steps.44

Accents and slurs

highlight

the first and

sixth

sonorities

n

each

sequential

unit,

projecting

an

alternating

de-

scent

of

even

and


among

the

upper

ac-

cented

notes.

Since

the

overlay

distance

s

3,


combination

cycle,

a nonreiterative



Symphony

(1911-28),

Ives


44Kirkpatrick,

atalogue,

96

(Cat.

No.

3B16ii).

This

work,

the full

title of

which s

"Rough

and

Ready

et al. and/or the

JumpingFrog,"

is the second

of

the

Five

Take-Offs

or

piano,

as named

by Kirkpatrick.

t is

transcribed

n Al-

bert

Lotto,


of the

Completed

Short

Piano

Pieces

of

CharlesIves"

(D.M.A.

thesis, Juilliard,

1978),

112-118.

amassing pitch

materials

for his

"Universe

...

in

tones."45

Sketchpage 3038, as numberedby Kirkpatrick,istscyclesof

varioussizes

and combinational

distances,

apparently

n

prepa-

ration for the musical

settings

of

some

of these structures

on

sketch

page

3036.46

Example

11

literally

transcribes

3038,

in-

cluding

the

composer's marginal

notations

indicating

nterval

sizes and instrumental

pecifications,

while

excludingonly

ex-

traneous

markings-some

of which

apparently

cross

out

material-and erasures

or otherwise ndistinctnotations.

In the

transcription,only

bracketed

material,

ncluding

clef

signs

and

the numberingof the stavesin the left margin, s not original.

Along

the

left side

of

the

page,

Ives

writes letters

"a," "b,"

"C,"

and "D" to subdivide he texture

nto four

parts,

although

the musical

notations within

the

parts

of

the

page

correspond-

ing

to each letter

do not

obviously

align

as a score:

the

majority

of the

notations are

clearly

the abstract

pitch

resources

on

whicheachof the

correspondingparts

are

to be based.

Two

groups

of ideas on 3038 are

not abstract

cyclic

struc-

tures. In staves

4

through

8,

roughly

the

right

half

of the

page

containsa melodiclinewithsustainedaccompaniment inbass

clef,

with bar lines

interspersed)

that

apparently

continues

from the

end of staves 5 and 6 to the

middle

of

staves

7

and 8.

A

second

area of

compositionalsetting appears

on staves

11-13.

All other materials

on

the

page

are

cyclicpitch

repetitions,

usu-

ally

notated

in whole

notes,

often

circled,

and often

accompa-

nied

by

an indication

of

the interval

or intervals

hat constitute

the

cyclic repetition.

In a few

cases,

Ives also

makesnote

of the

total number

of

unique pitch

classes that

is

generated.

Includedamong

the

cyclic

notations

on

page

3038

are

single

cycles

of intervals

10 and

11

at the

beginning

of staves7

and

8.

Ives labels

these,

respectively,

"all MIN

7" and

"all

MAJ

7,"

45Memos, 106;

Kirkpatrick,

Catalogue,

27

(Cat.

No.

1A9).

A

"Facsimile/Transcription"

f

the

sketches

s in

preparation

by

Peer-Southern.

46These

numbers are the

photostat

negative

numbers

given

to the

right

of

each

entry

in

Kirkpatrick'sCatalogue.

amassing pitch

materials

for his

"Universe

...

in

tones."45


varioussizes

and combinational

distances,

apparently

n

prepa-


settings

of

some

of these structures

on

sketch

page

3036.46

Example

11

literally

transcribes

3038,

in-

cluding

the

composer's marginal

notations

indicating

nterval


pecifications,

while

excludingonly

ex-

traneous

markings-some

of which

apparently

cross

out



In the

transcription,only

bracketed

material,

ncluding

clef

signs

and


Along

the

left side

of

the

page,

Ives

writes letters

"a," "b,"

"C,"


nto four

parts,

although

the musical

notations within

the

parts

of

the

page

correspond-

ing

to each letter

do not

obviously

align

as a score:

the

majority

of the

notations are

clearly

the abstract

pitch

resources

on

whicheachof the

correspondingparts

are

to be based.

Two

groups


not abstract

cyclic

struc-

tures. In staves

4

through

8,

roughly

the

right

half

of the

page


clef,

with bar lines

interspersed)

that

apparently

continues

from the


middle

of

staves

7

and 8.

A

second

area of


on staves

11-13.

All other materials

on

the

page

are

cyclicpitch

repetitions,

usu-

ally

notated

in whole

notes,

often

circled,

and often

accompa-

nied

by

an indication

of

the interval

or intervals

hat constitute

the

cyclic repetition.

In a few

cases,

Ives also

makesnote

of the

total number

of

unique pitch

classes that

is

generated.

Includedamong

the

cyclic

notations

on

page

3038

are

single

cycles

of intervals

10 and

11

at the

beginning

of staves7

and

8.

Ives labels

these,

respectively,

"all MIN

7" and

"all

MAJ

7,"

45Memos, 106;

Kirkpatrick,

Catalogue,

27

(Cat.

No.

1A9).

A


f

the

sketches

s in

preparation

by

Peer-Southern.

46These

numbers are the

photostat

negative

numbers

given

to the

right

of

each

entry

in


amassing pitch

materials

for his

"Universe

...

in

tones."45


varioussizes

and combinational

distances,

apparently

n

prepa-


settings

of

some

of these structures

on

sketch

page

3036.46

Example

11

literally

transcribes

3038,

in-

cluding

the

composer's marginal

notations

indicating

nterval


pecifications,

while

excludingonly

ex-

traneous

markings-some

of which

apparently

cross

out



In the

transcription,only

bracketed

material,

ncluding

clef

signs

and


Along

the

left side

of

the

page,

Ives

writes letters

"a," "b,"

"C,"


nto four

parts,

although

the musical

notations within

the

parts

of

the

page

correspond-

ing

to each letter

do not

obviously

align

as a score:

the

majority

of the

notations are

clearly

the abstract

pitch

resources

on

whicheachof the

correspondingparts

are

to be based.

Two

groups


not abstract

cyclic

struc-

tures. In staves

4

through

8,

roughly

the

right

half

of the

page


clef,

with bar lines

interspersed)

that

apparently

continues

from the


middle

of

staves

7

and 8.

A

second

area of


on staves

11-13.

All other materials

on

the

page

are

cyclicpitch

repetitions,

usu-

ally

notated

in whole

notes,

often

circled,

and often

accompa-

nied

by

an indication

of

the interval

or intervals

hat constitute

the

cyclic repetition.

In a few

cases,

Ives also

makesnote

of the

total number

of

unique pitch

classes that

is

generated.

Includedamong

the

cyclic

notations

on

page

3038

are

single

cycles

of intervals

10 and

11

at the

beginning

of staves7

and

8.

Ives labels

these,

respectively,

"all MIN

7" and

"all

MAJ

7,"

45Memos, 106;

Kirkpatrick,

Catalogue,

27

(Cat.

No.

1A9).

A


f

the

sketches

s in

preparation

by

Peer-Southern.

46These

numbers are the

photostat

negative

numbers

given

to the

right

of

each

entry

in







Interval

ycles

as

Compositional

esources 63

nterval

ycles

as

Compositional

esources 63

nterval

ycles

as

Compositional

esources 63

Example

10.

Rough

and

Ready,

mm.

8-15,

right

hand.

sequential

unit:

1

2

3

4

5

6

A8

Et.Br^JrTr

I

^ T

1

r

rrT^rrl Tr^

r

1^2

A

A^^

15

8

A A

A

13A

A

14

A

J"

'"-"

-K

L

Ti

PIMal

6"~~~~~~~~~~~~~~~~~~~~~~~~~~~

ta

Example

10.

Rough

and

Ready,

mm.

8-15,

right

hand.

sequential

unit:

1

2

3

4

5

6

A8

Et.Br^JrTr

I

^ T

1

r

rrT^rrl Tr^

r

1^2

A

A^^

15

8

A A

A

13A

A

14

A

J"

'"-"

-K

L

Ti

PIMal

6"~~~~~~~~~~~~~~~~~~~~~~~~~~~

ta

Example

10.

Rough

and

Ready,

mm.

8-15,

right

hand.

sequential

unit:

1

2

3

4

5

6

A8

Et.Br^JrTr

I

^ T

1

r

rrT^rrl Tr^

r

1^2

A

A^^

15

8

A A

A

13A

A

14

A

J"

'"-"

-K

L

Ti

PIMal

6"~~~~~~~~~~~~~~~~~~~~~~~~~~~

ta

__-i

@

-

I

|=I

'lEWl

W

iU;;

I

I

II;

_-i

@

-

I

|=I

'lEWl

W

iU;;

I

I

II;

_-i

@

-

I

|=I

'lEWl

W

iU;;

I

I

II;

x/y= 3 / 7/y= 3 / 7/y= 3 / 7

1111

9

7

6

PCL = 12CL = 12CL = 12

and allows octave

transpositions

n the

latter

out of

necessity

(see

Ex.

11).

On

staff

8,

immediately following

the

major-

seventh

sequence,

he

begins

a

sequence

of

decreasing

nterval

sizes that states

only

intervals

7, 6,

and

5.47

The combination

cycles

on the

page

display

many

of the

possibilities

for

higher

PCLvalues fromcombiningcyclesof intervals1,2, 3, 9, and11.

Figure

5 summarizes he structureof

all

the

cyclic

ormations

n

Example

11,

giving

their

x/y,

n,

and PCL values. Parts a and b

of the

figure

illustrate combination

cycles

found in

the

upper

half of

the

page,

parts

e

and f show

structures

on staves 11 and

12,

near

the

right

margin,

and

parts g through correspond

o

materials

on

the bottom two

staves,

in order from

left

to

right.

The

cycles

of

intervals 10 and 11 on

staves

7

and 8 are listed as

10/10 and 11/11 n

parts

c and d of

Figure

5,

viewed

as

overlays

of interval-8andinterval-10cycles, respectively.

47Ives uses

sequences

of

descending

intervals,

similar to Fritz Henrich

Klein's

"Pyramidenakkord,"

on

several

occasions. An

early example

appears

in his father's

Copybook, p. [165] (Kirkpatrick

Catalogue

No.

7E77).

See

also

Tone Roads No.

1,

downbeat of

m.

12.

See

Fritz Heinrich

Klein,

"Die Grenze

der

Halbtonwelt,"

Die Musik 17/4

(January 1925),

284.

and allows octave

transpositions

n the

latter

out of

necessity

(see

Ex.

11).

On

staff

8,


the

major-

seventh

sequence,

he

begins

a

sequence

of

decreasing

nterval

sizes that states

only

intervals

7, 6,

and

5.47

The combination

cycles

on the

page

display

many

of the

possibilities

for

higher


Figure


all

the

cyclic

ormations

n

Example

11,

giving

their

x/y,

n,


of the

figure


cycles

found in

the

upper

half of

the

page,

parts

e

and f show

structures

on staves 11 and

12,

near

the

right

margin,

and


o

materials

on

the bottom two

staves,

in order from

left

to

right.

The

cycles

of


staves

7

and 8 are listed as

10/10 and 11/11 n

parts

c and d of

Figure

5,

viewed

as

overlays


47Ives uses

sequences

of

descending

intervals,


Klein's

"Pyramidenakkord,"

on

several

occasions. An

early example

appears

in his father's


Catalogue

No.

7E77).

See

also

Tone Roads No.

1,

downbeat of

m.

12.

See

Fritz Heinrich

Klein,

"Die Grenze

der

Halbtonwelt,"

Die Musik 17/4

(January 1925),

284.

and allows octave

transpositions

n the

latter

out of

necessity

(see

Ex.

11).

On

staff

8,


the

major-

seventh

sequence,

he

begins

a

sequence

of

decreasing

nterval

sizes that states

only

intervals

7, 6,

and

5.47

The combination

cycles

on the

page

display

many

of the

possibilities

for

higher


Figure


all

the

cyclic

ormations

n

Example

11,

giving

their

x/y,

n,


of the

figure


cycles

found in

the

upper

half of

the

page,

parts

e

and f show

structures

on staves 11 and

12,

near

the

right

margin,

and


o

materials

on

the bottom two

staves,

in order from

left

to

right.

The

cycles

of


staves

7

and 8 are listed as

10/10 and 11/11 n

parts

c and d of

Figure

5,

viewed

as

overlays


47Ives uses

sequences

of

descending

intervals,


Klein's

"Pyramidenakkord,"

on

several

occasions. An

early example

appears

in his father's


Catalogue

No.

7E77).

See

also

Tone Roads No.

1,

downbeat of

m.

12.

See

Fritz Heinrich

Klein,

"Die Grenze

der

Halbtonwelt,"

Die Musik 17/4

(January 1925),

284.

The

9/4

cycle

on staves 1

and 2

(Fig. 5a),

the

5/8 combination

labeled

by

Ives "5E"on staves 11

and 12

(Fig. 5e),

and

the 6/7

structureon staves 15 and

16,

near the

center

(Fig. 5i)

all

com-

bine

cycles

of interval 1 to

project

different PCL

values.

Ives

indicates,

with the

number

"12,"

that

the latter

generates

a

completeaggregate roman alternationof a "MIN5 [sic]"and

"Perfect5th."48

The

other two

interval-1

ombinations

annot,

of

course,

complete

the

aggregate

without

pc repetitions,

and

Ives continues

the

intervallic

sequences only

to

the final

notes

in

the PCLs.

On staves 1

and

2,

the

9/4

combination

tops

after

the seventh

element

(see

Fig. 5a),

avoiding

he

continuation o

pc

0,

which would

reiterate

the

pitch

class on which

the se-

quence begins.

The

5/8 combination

on

staves 11 and 12

(with

registral shift)

stops

after the

tenth element

(see

Fig.

5e)

to

avoidreiteratingpc 5, firstpresentedas the secondelement.

Interval-11

combinationson

the

page,

displayingx/y

values

that are

complementary

o

those for

interval

1,

present

two ad-

ditional PCL

possibilities.

The 4/7

combination at

the

right

margin

of

staves

11 and 12

(Fig. 5f)

continues

past

the end of

The

9/4

cycle

on staves 1

and 2

(Fig. 5a),

the

5/8 combination

labeled

by


and 12

(Fig. 5e),

and

the 6/7


16,

near the

center

(Fig. 5i)

all

com-

bine

cycles

of interval 1 to

project

different PCL

values.

Ives

indicates,

with the

number

"12,"

that

the latter

generates

a


"Perfect5th."48

The

other two

interval-1

ombinations

annot,

of

course,

complete

the

aggregate

without

pc repetitions,

and

Ives continues

the

intervallic

sequences only

to

the final

notes

in

the PCLs.

On staves 1

and

2,

the

9/4

combination

tops

after

the seventh

element

(see

Fig. 5a),

avoiding

he

continuation o

pc

0,

which would

reiterate

the

pitch

class on which

the se-

quence begins.

The

5/8 combination

on

staves 11 and 12

(with

registral shift)

stops

after the

tenth element

(see

Fig.

5e)

to


Interval-11

combinationson

the

page,

displayingx/y

values

that are

complementary

o

those for

interval

1,

present

two ad-

ditional PCL

possibilities.

The 4/7

combination at

the

right

margin

of

staves

11 and 12

(Fig. 5f)

continues

past

the end of

The

9/4

cycle

on staves 1

and 2

(Fig. 5a),

the

5/8 combination

labeled

by


and 12

(Fig. 5e),

and

the 6/7


16,

near the

center

(Fig. 5i)

all

com-

bine

cycles

of interval 1 to

project

different PCL

values.

Ives

indicates,

with the

number

"12,"

that

the latter

generates

a


"Perfect5th."48

The

other two

interval-1

ombinations

annot,

of

course,

complete

the

aggregate

without

pc repetitions,

and

Ives continues

the

intervallic

sequences only

to

the final

notes

in

the PCLs.

On staves 1

and

2,

the

9/4

combination

tops

after

the seventh

element

(see

Fig. 5a),

avoiding

he

continuation o

pc

0,

which would

reiterate

the

pitch

class on which

the se-

quence begins.

The

5/8 combination

on

staves 11 and 12

(with

registral shift)

stops

after the

tenth element

(see

Fig.

5e)

to


Interval-11

combinationson

the

page,

displayingx/y

values

that are

complementary

o

those for

interval

1,

present

two ad-

ditional PCL

possibilities.

The 4/7

combination at

the

right

margin

of

staves

11 and 12

(Fig. 5f)

continues

past

the end of

48A

perusal

of

the

marginalia

in

Example

11

will reveal

that Ives occasion-

ally

mislabels the

quality

of

intervals.

48A

perusal

of

the

marginalia

in

Example

11

will reveal

that Ives occasion-

ally

mislabels the

quality

of

intervals.

48A

perusal

of

the

marginalia

in

Example

11

will reveal

that Ives occasion-

ally

mislabels the

quality

of

intervals.

B set:

A set: 0

B set:

A set: 0

B set:

A set: 0

3

1000

8

(n = 10)n = 10)n = 10)

IMM"WMM"WMM"W






64

Music

TheorySpectrum

Example

11.

Universe

Symphony,

sketch

page

q3038,

iteral

ranscription.

64

Music

TheorySpectrum

Example

11.

Universe

Symphony,

sketch

page

q3038,

iteral

ranscription.

64

Music

TheorySpectrum

Example

11.

Universe

Symphony,

sketch

page

q3038,

iteral

ranscription.

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43

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1222

wm2 Mtaj

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m2

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/

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""

Maj3

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4

3

4

Maj3

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4

3

4

Maj3

"

4

3

4

Tuba

-

MIN

uba

'\ MIN5

Cor

Perfect

5

Ba

Tuba

-

MIN

uba

'\ MIN5

Cor

Perfect

5

Ba

Tuba

-

MIN

uba

'\ MIN5

Cor

Perfect

5

Ba

Maj

2

min 2

etc.

Maj

2

min 2

etc.

Maj

2

min 2

etc.

maj

2

min 2

mi

3

maj

2

min 2

mi

3

maj

2

min 2

mi

3

[21

[2]

[4]

[5]

[6]

[7]

[8]

[21

[2]

[4]

[5]

[6]

[7]

[8]

[21

[2]

[4]

[5]

[6]

[7]

[8]

[9]

7

9]

7

9]

7

[10]

[

11

I

[

12]

[ 13]

[ 14]

[

15

1

[10]

[

11

I

[

12]

[ 13]

[ 14]

[

15

1

[10]

[

11

I

[

12]

[ 13]

[ 14]

[

15

1

[

161161161

LV

J

r I

LV

J

r I

LV

J

r I

tF

/

4th mm

5

4th

min 5

tF

/

4th mm

5

4th

min 5

tF

/

4th mm

5

4th

min 5






Interval

ycles

as

Compositional

esources

65nterval

ycles

as

Compositional

esources

65nterval

ycles

as

Compositional

esources

65

Figure

5.

Summary

of

cycles

in

Example

11.

igure

5.

Summary

of

cycles

in

Example

11.

igure

5.

Summary

of

cycles

in

Example

11.

a. Lines 1-2.

pc

0 9 1 10 2

11

3

x/y

=

9/4 n= 1 PCL=7

b. Lines 5-6.

pc

0

5 2 7 4 9

6

11

8

1

10 3

x/y=

5/9 n

=

2 PCL

=

12

c. Lines 7-8.

pc

0 10 8 6

4 2 0

x/y

=

10/10

n

=

8

PCL

=

6

d. Lines 7-8.

pc

0 11 10 9 8 7

6

5 4 3 2 1

x/y=

11/11 n

=

10 PCL

=

12

e. Lines 11-12.

pc

0 5 1 6 2 7 3 8 4 9

x/y=

5

/8

n

=

1 PCL

=

10

the

PCL

at the ninth element to return

o

pc

0,

thus

forming

a

sonority

with the same

top

and bottom note. In the

lower left-

hand

corer

of

the

page,

a

5/6 combination

Fig.

5g) presents

a

PCL of

11,

followed

by

a

repetition

of

pc

6 and the

aggregate-

completing pc 0. Ives draws a line to separatethe 11 members

of the PCL from the final

two

notes,

highlighting

he

nonrepeti-

tive

portion

of the

sequence,

while he also notes

that 12

pitch

classes are

present

in

the entire

formation,

deemphasizing

he

single pitch-classrepetition.

The other

combination

cycles

notated

in

Example

11 and

n-

cluded in

Figure

5

are nonreiterativecombinations

of

other

cy-

cles.

Nonintersecting

nterval-2

cycles

are

overlaidon

staves

5-

a. Lines 1-2.

pc

0 9 1 10 2

11

3

x/y

=

9/4 n= 1 PCL=7

b. Lines 5-6.

pc

0

5 2 7 4 9

6

11

8

1

10 3

x/y=

5/9 n

=

2 PCL

=

12

c. Lines 7-8.

pc

0 10 8 6

4 2 0

x/y

=

10/10

n

=

8

PCL

=

6

d. Lines 7-8.

pc

0 11 10 9 8 7

6

5 4 3 2 1

x/y=

11/11 n

=

10 PCL

=

12

e. Lines 11-12.

pc

0 5 1 6 2 7 3 8 4 9

x/y=

5

/8

n

=

1 PCL

=

10

the

PCL


o

pc

0,

thus

forming

a

sonority

with the same

top


lower left-

hand

corer

of

the

page,

a

5/6 combination

Fig.

5g) presents

a

PCL of

11,

followed

by

a

repetition

of

pc

6 and the

aggregate-



two

notes,

highlighting

he

nonrepeti-

tive

portion

of the

sequence,

while he also notes

that 12

pitch

classes are

present

in

the entire

formation,

deemphasizing

he


The other

combination

cycles

notated

in

Example

11 and

n-

cluded in

Figure

5


of

other

cy-

cles.

Nonintersecting

nterval-2

cycles

are

overlaidon

staves

5-

a. Lines 1-2.

pc

0 9 1 10 2

11

3

x/y

=

9/4 n= 1 PCL=7

b. Lines 5-6.

pc

0

5 2 7 4 9

6

11

8

1

10 3

x/y=

5/9 n

=

2 PCL

=

12

c. Lines 7-8.

pc

0 10 8 6

4 2 0

x/y

=

10/10

n

=

8

PCL

=

6

d. Lines 7-8.

pc

0 11 10 9 8 7

6

5 4 3 2 1

x/y=

11/11 n

=

10 PCL

=

12

e. Lines 11-12.

pc

0 5 1 6 2 7 3 8 4 9

x/y=

5

/8

n

=

1 PCL

=

10

the

PCL


o

pc

0,

thus

forming

a

sonority

with the same

top


lower left-

hand

corer

of

the

page,

a

5/6 combination

Fig.

5g) presents

a

PCL of

11,

followed

by

a

repetition

of

pc

6 and the

aggregate-



two

notes,

highlighting

he

nonrepeti-

tive

portion

of the

sequence,

while he also notes

that 12

pitch

classes are

present

in

the entire

formation,

deemphasizing

he


The other

combination

cycles

notated

in

Example

11 and

n-

cluded in

Figure

5


of

other

cy-

cles.

Nonintersecting

nterval-2

cycles

are

overlaidon

staves

5-

f.

Lines

11-12.

pc

0

4

11 3 10

2

9

1

8

x/y=

4/7 n=11 PCL =9

g.

Lines 15-16.

pc

6

11

5 10

4

9

3

8

2 7

1

6

0

x/y

=

5/6

n

=

11 PCL = 11

h. Lines 15-16.

pc

0 4 9 1

6

10 3 7 0

x/y

=

4/ 5 n

=

9 PCL

=

8

i.

Lines 15-16.

pc

0

6

1

7 2 8 3

9

4 10 5

11

x/y

=

6/7 n

=

1 PCL

=

12

j.

Lines 15-16.

pc

0 2 3

5 6 8 9

11

x/y

=

2 /1

n

=

3 PCL

=

8

6

(Fig. 5b)

in a 5/9

alternationthat extends to its PCL

of 12.

Octatonic collections result from a 4/5

combination

overlaying

interval-9

cycles

inthe center of lines

15-16

(Fig. 5h),

and from

a 2/1 alternationon staff

15

in

the lower

right-hand

orner

of the

page(Fig. 5j) overlaying nterval-3cycles.These combinations

complete

a

comprehensive

accumulation f

larger

PCLs,

rang-

ing

from

a PCL

of 6 for the interval-10

cycle

to

completion

of

the

aggregate

by

three

others,

and

includingevery

value

be-

tween. Table

2 charts

this

accumulation,

including

the two

single-interval

ycles

(Fig.

5c and

5d)

notated as combinations.

Noticeably

absent from the n values

are

cycles

of

intervals

5 and

7;

combinations

of

these

cycles,

of

course,

display

a

pattern

of

f.

Lines

11-12.

pc

0

4

11 3 10

2

9

1

8

x/y=

4/7 n=11 PCL =9

g.

Lines 15-16.

pc

6

11

5 10

4

9

3

8

2 7

1

6

0

x/y

=

5/6

n

=

11 PCL = 11

h. Lines 15-16.

pc

0 4 9 1

6

10 3 7 0

x/y

=

4/ 5 n

=

9 PCL

=

8

i.

Lines 15-16.

pc

0

6

1

7 2 8 3

9

4 10 5

11

x/y

=

6/7 n

=

1 PCL

=

12

j.

Lines 15-16.

pc

0 2 3

5 6 8 9

11

x/y

=

2 /1

n

=

3 PCL

=

8

6

(Fig. 5b)

in a 5/9


of 12.


combination

overlaying

interval-9

cycles


15-16

(Fig. 5h),

and from


15

in

the lower

right-hand

orner

of the


complete

a

comprehensive

accumulation f

larger

PCLs,

rang-

ing

from

a PCL


cycle

to

completion

of

the

aggregate

by

three

others,

and

includingevery

value

be-

tween. Table

2 charts

this

accumulation,

including

the two

single-interval

ycles

(Fig.

5c and

5d)


Noticeably


are

cycles

of

intervals

5 and

7;

combinations

of

these

cycles,

of

course,

display

a

pattern

of

f.

Lines

11-12.

pc

0

4

11 3 10

2

9

1

8

x/y=

4/7 n=11 PCL =9

g.

Lines 15-16.

pc

6

11

5 10

4

9

3

8

2 7

1

6

0

x/y

=

5/6

n

=

11 PCL = 11

h. Lines 15-16.

pc

0 4 9 1

6

10 3 7 0

x/y

=

4/ 5 n

=

9 PCL

=

8

i.

Lines 15-16.

pc

0

6

1

7 2 8 3

9

4 10 5

11

x/y

=

6/7 n

=

1 PCL

=

12

j.

Lines 15-16.

pc

0 2 3

5 6 8 9

11

x/y

=

2 /1

n

=

3 PCL

=

8

6

(Fig. 5b)

in a 5/9


of 12.


combination

overlaying

interval-9

cycles


15-16

(Fig. 5h),

and from


15

in

the lower

right-hand

orner

of the


complete

a

comprehensive

accumulation f

larger

PCLs,

rang-

ing

from

a PCL


cycle

to

completion

of

the

aggregate

by

three

others,

and

includingevery

value

be-

tween. Table

2 charts

this

accumulation,

including

the two

single-interval

ycles

(Fig.

5c and

5d)


Noticeably


are

cycles

of

intervals

5 and

7;

combinations

of

these

cycles,

of

course,

display

a

pattern

of






66

Music

TheorySpectrum

6

Music

TheorySpectrum

6

Music

TheorySpectrum

Table 2.

Summary

of

Figure

5

organized by

PCL.

able 2.

Summary

of

Figure

5

organized by

PCL.

able 2.

Summary

of

Figure

5

organized by

PCL.

PCL

6

PCL

6

PCL

6

x/y

10/10

x/y

10/10

x/y

10/10

7

9/4

1

9/4

1

9/4

1

n

Ex.

11,

lines:

Fig.

5,

part:

8 4-5 c

n

Ex.

11,

lines:

Fig.

5,

part:

8 4-5 c

n

Ex.

11,

lines:

Fig.

5,

part:

8 4-5 c

1-2-2-2 a

sions of the combination

cycles

that

precede

them,

forming

me-

lodic

skeletons

from

the

cyclic

sources. These

sketchings

dem-

onstratea

compositionalapproach

hat carries

hrough

o other

pages

of the

Symphony,

as the combination

cycles

provide

sources

of

pitch-class

material or

subsequent

development

and

transformation.


cycles

that

precede

them,

forming

me-

lodic

skeletons

from

the

cyclic

sources. These

sketchings

dem-

onstratea


hat carries

hrough

o other

pages

of the

Symphony,

as the combination

cycles

provide

sources

of

pitch-class

material or

subsequent

development

and

transformation.


cycles

that

precede

them,

forming

me-

lodic

skeletons

from

the

cyclic

sources. These

sketchings

dem-

onstratea


hat carries

hrough

o other

pages

of the

Symphony,

as the combination

cycles

provide

sources

of

pitch-class

material or

subsequent

development

and

transformation.

8 4/5 9

15-16

2/1

3

15-16

9 4/7 11 11-12

10 5/8 1 11-12

11 5/6 11

15-16

8 4/5 9

15-16

2/1

3

15-16

9 4/7 11 11-12

10 5/8 1 11-12

11 5/6 11

15-16

8 4/5 9

15-16

2/1

3

15-16

9 4/7 11 11-12

10 5/8 1 11-12

11 5/6 11

15-16

12

5/9

11/11

6/7

12

5/9

11/11

6/7

12

5/9

11/11

6/7

2

2

1

2

2

1

2

2

1

5-6

7-8

15-16

5-6

7-8

15-16

5-6

7-8

15-16

h

J

f

e

g

b

d

i

h

J

f

e

g

b

d

i

h

J

f

e

g

b

d

i

PCLdistributionanalogousto that for intervals1 and 11, and

thus offer no

unique

contributions

o

the PCL

summary,

al-

though

they

do

display contrasting possibilities

for

x/y

combinations.

The bottom three staves of

Example

11

contain additional

notations

correlating

to

Ives's

cyclic conception

of the

pitch

structures n

the Universe

Symphony.

Severaloftheseare

cyclic

repetitions

of

different

types, including

an interval-3

ycle plus

two added

notes

(right

margin

of staff

14)

andtwo instances

of a

three-intervalalternation: epetitionsof intervals1-2-3and2-1-

3 form octatonic

subsets at the

right margin

of staff 16.49The

other notes on

these bottom

three

staves,

mostly

notated

with

darkened

note

heads,

are linearizedand

slightly

reorderedver-

49Below he

1-2-3

sequence,

Ives indicates the

alternating

ntervallic

pat-

tern

"Maj

2

/

min

2

/

etc,"

referring

o the

sequence

directly

above

this but on

staff15.


thus offer no

unique

contributions

o

the PCL

summary,

al-

though

they

do


for

x/y

combinations.


Example

11

contain additional

notations

correlating

to

Ives's

cyclic conception

of the

pitch

structures n

the Universe

Symphony.

Severaloftheseare

cyclic

repetitions

of

different

types, including

an interval-3

ycle plus

two added

notes

(right

margin

of staff

14)

andtwo instances

of a


3 form octatonic

subsets at the

right margin

of staff 16.49The

other notes on

these bottom

three

staves,

mostly

notated

with

darkened

note

heads,

are linearizedand

slightly

reorderedver-

49Below he

1-2-3

sequence,

Ives indicates the

alternating

ntervallic

pat-

tern

"Maj

2

/

min

2

/

etc,"

referring

o the

sequence

directly

above

this but on

staff15.


thus offer no

unique

contributions

o

the PCL

summary,

al-

though

they

do


for

x/y

combinations.


Example

11

contain additional

notations

correlating

to

Ives's

cyclic conception

of the

pitch

structures n

the Universe

Symphony.

Severaloftheseare

cyclic

repetitions

of

different

types, including

an interval-3

ycle plus

two added

notes

(right

margin

of staff

14)

andtwo instances

of a


3 form octatonic

subsets at the

right margin

of staff 16.49The

other notes on

these bottom

three

staves,

mostly

notated

with

darkened

note

heads,

are linearizedand

slightly

reorderedver-

49Below he

1-2-3

sequence,

Ives indicates the

alternating

ntervallic

pat-

tern

"Maj

2

/

min

2

/

etc,"

referring

o the

sequence

directly

above

this but on

staff15.

CYCLES AS COMPOSITIONAL

OURCES.The

linearizationson

the bottom staves of the

Universe

Symphonypage

(Ex.

11) rep-

resent

a first

step

toward

a

musicalrealizationof the

cyclic

pitch

structures.The

5/6

cycle

notated in

whole notes at the left mar-

gin

(staves

15 and

16),

for

example,

is

immediately

ollowed

by

a series

of

darkenednote

heads,

withand then without

stems,

that

presents

the

notes

of the

cycle

in

order,

except

that the or-

der

of

the

thirdand

fourthnotes

(pcs

5 and

10)

is

reversed,

and

the

penultimatepc

6,

which s a

duplication

of

the

initialnote of

the

cycle,

is omitted. In

essence, then,

the

whole-note notation

is an abstract

expression

of

the

pitch-class

ource

material hat

is

subsequently

realized,

without

rhythmic

values,

in a

specific

register. Presumably,

each

cycle

notated

on

the

page

is

to

pro-

vide source

material of this

nature

in

the

composition

of

the

Symphony,

in

order that

many

musical

deas within the

depic-

tion of a "Universe in tones"

originate

with

a

predetermined,

systematically

conceived

pitch-class

structure. The

cyclic

sources

determine

a

particular itch-class

uccession,

subject

o

slight

order

variations,

as well as

a

uniform ntervallic ucces-

sion and

a

specificpitch-class ength.50

Ives's methods

of

realizing compositional

sources are

not,

however,

limitedto

the

simple

inearizations hown

here

and n

Examples

1 and

9.

In more

complex

incorporations,

cyclic

structuresserve as sources

for

intermingled

vertical

and

hori-

50Similarly,

Michael

J. Babcock

explores

the

circle of fifths

as a

composi-

tional source for the "Thoreau"

movement

of Ives's ConcordSonata

n "Ives's

'Thoreau':

A

Point of

Order,"

Proceedings f

the American

Society

of

Univer-

sity Composers

9 and 10

(1976),

89-102.


OURCES.The

linearizationson


Universe

Symphonypage

(Ex.

11) rep-

resent

a first

step

toward

a


cyclic

pitch

structures.The

5/6

cycle

notated in


gin

(staves

15 and

16),

for

example,

is

immediately

ollowed

by

a series

of

darkenednote

heads,


stems,

that

presents

the

notes

of the

cycle

in

order,

except

that the or-

der

of

the

thirdand

fourthnotes

(pcs

5 and

10)

is

reversed,

and

the

penultimatepc

6,

which s a

duplication

of

the

initialnote of

the

cycle,

is omitted. In

essence, then,

the

whole-note notation

is an abstract

expression

of

the

pitch-class

ource

material hat

is

subsequently

realized,

without

rhythmic

values,

in a

specific


each

cycle

notated

on

the

page

is

to

pro-

vide source

material of this

nature

in

the

composition

of

the

Symphony,

in

order that

many

musical

deas within the

depic-


originate

with

a

predetermined,

systematically

conceived

pitch-class

structure. The

cyclic

sources

determine

a


uccession,

subject

o

slight

order

variations,

as well as

a


sion and

a


Ives's methods

of


sources are

not,

however,

limitedto

the

simple

inearizations hown

here

and n

Examples

1 and

9.

In more

complex

incorporations,

cyclic


for

intermingled

vertical

and

hori-

50Similarly,

Michael

J. Babcock

explores

the

circle of fifths

as a

composi-


movement


n "Ives's

'Thoreau':

A

Point of

Order,"

Proceedings f

the American

Society

of

Univer-

sity Composers

9 and 10

(1976),

89-102.


OURCES.The

linearizationson


Universe

Symphonypage

(Ex.

11) rep-

resent

a first

step

toward

a


cyclic

pitch

structures.The

5/6

cycle

notated in


gin

(staves

15 and

16),

for

example,

is

immediately

ollowed

by

a series

of

darkenednote

heads,


stems,

that

presents

the

notes

of the

cycle

in

order,

except

that the or-

der

of

the

thirdand

fourthnotes

(pcs

5 and

10)

is

reversed,

and

the

penultimatepc

6,

which s a

duplication

of

the

initialnote of

the

cycle,

is omitted. In

essence, then,

the

whole-note notation

is an abstract

expression

of

the

pitch-class

ource

material hat

is

subsequently

realized,

without

rhythmic

values,

in a

specific


each

cycle

notated

on

the

page

is

to

pro-

vide source

material of this

nature

in

the

composition

of

the

Symphony,

in

order that

many

musical

deas within the

depic-


originate

with

a

predetermined,

systematically

conceived

pitch-class

structure. The

cyclic

sources

determine

a


uccession,

subject

o

slight

order

variations,

as well as

a


sion and

a


Ives's methods

of


sources are

not,

however,

limitedto

the

simple

inearizations hown

here

and n

Examples

1 and

9.

In more

complex

incorporations,

cyclic


for

intermingled

vertical

and

hori-

50Similarly,

Michael

J. Babcock

explores

the

circle of fifths

as a

composi-


movement


n "Ives's

'Thoreau':

A

Point of

Order,"

Proceedings f

the American

Society

of

Univer-

sity Composers

9 and 10

(1976),

89-102.






Interval

ycles

as

Compositional

esources 67

nterval

ycles

as

Compositional

esources 67

nterval

ycles

as

Compositional

esources 67

zontal

ideas,

potentially employing

more extensive distortions

of originalordering.The consistent set-class substructure hat

naturally

arises from an

intervallic

equence

may

be

preserved

and

highlighted,

while the

underlying

ntervallic

epetitionmay

be

to some extent

suppressed.

A "sourceset" of

interval

cycles

must exert

unequivocal

control over

pitch

constructions,

ince

substantial order variations

may

obscure

original

structural

properties,

and,

indeed,

largercycles

are

distinguishable

rom

each other

only

by

their

intervallic

tructure,

not

by

their

pitch-

class content.

Still,

the

boundaries et forth

by

comprehensibil-

ity and fidelity to a source allow for ample compositional

freedom.

The

overall

unityprovidedby

this

type

of structural etermi-

nant

appears

n works

composed

as

early

as

the

choral

Psalms,

including

versesofPsalm24 that follow the

excerptgiven

as Ex-

ample

8. While the

chromaticism

f

verse

1

of this Psalm is dis-

played

in the outer voices

only, cyclic

sources in other verses

influence he

complete

texture,

so that an

entire

passage

s satu-

rated with a

distinctive constructional

principle.

In

verse

2 of

Psalm 24 (mm. 7-11), forexample,the notes of the outer-voice

whole-tone

scale also

determine

the

pitch-class

content of the

inner

voices,

with the

result that the

pervasive

pitch

resource s

indeed

a

"scale,"

in the

traditional

ense,

that defines the har-

monic

language

of

the

passage.

With a small

cardinality,

his

source

is

easily

characterized

by

content,

and

thus

requires

no

restrictionsof

ordering

o

retain ts

integrity;

notes

for

the

alto

and

tenor can be

selected from

any

portion

of

the outer-voice

scale,

without

regard

for their

scalar

ordering.

The

resulting

verticalstructuresare whole-tone subsets formed by various

combinations of even-numberedintervals.

Intervals 3 and 9

similarly

prescribe

the harmonic

anguage

of

verse 3

(mm.

12-

16).

For

cyclic

structures hat

generate

higher

numbersof

pitch

classes,

this

type

of

intervallic

aturation annot

be achieved

by

simple

distributionof

pitch

classes from

the

source

throughout

the

texture.

Because

the

source is

distinguished

primarily

by

zontal

ideas,




naturally

arises from an

intervallic

equence

may

be

preserved

and

highlighted,

while the

underlying

ntervallic

epetitionmay

be

to some extent

suppressed.

A "sourceset" of

interval

cycles

must exert

unequivocal

control over

pitch

constructions,

ince


may

obscure

original

structural

properties,

and,

indeed,

largercycles

are

distinguishable

rom

each other

only

by

their

intervallic

tructure,

not

by

their

pitch-

class content.

Still,

the

boundaries et forth

by

comprehensibil-


freedom.

The

overall

unityprovidedby

this

type


nant

appears

n works

composed

as

early

as

the

choral

Psalms,

including


excerptgiven

as Ex-

ample

8. While the

chromaticism

f

verse

1


played

in the outer voices

only, cyclic


influence he

complete

texture,

so that an

entire

passage

s satu-

rated with a


principle.

In

verse

2 of


whole-tone

scale also

determine

the

pitch-class

content of the

inner

voices,

with the

result that the

pervasive

pitch

resource s

indeed

a

"scale,"

in the

traditional

ense,


monic

language

of

the

passage.

With a small

cardinality,

his

source

is

easily

characterized

by

content,

and

thus

requires

no

restrictionsof

ordering

o

retain ts

integrity;

notes

for

the

alto

and

tenor can be

selected from

any

portion

of

the outer-voice

scale,

without

regard

for their

scalar

ordering.

The

resulting



Intervals 3 and 9

similarly

prescribe

the harmonic

anguage

of

verse 3

(mm.

12-

16).

For

cyclic

structures hat

generate

higher

numbersof

pitch

classes,

this

type

of

intervallic

aturation annot

be achieved

by

simple

distributionof

pitch

classes from

the

source

throughout

the

texture.

Because

the

source is

distinguished

primarily

by

zontal

ideas,




naturally

arises from an

intervallic

equence

may

be

preserved

and

highlighted,

while the

underlying

ntervallic

epetitionmay

be

to some extent

suppressed.

A "sourceset" of

interval

cycles

must exert

unequivocal

control over

pitch

constructions,

ince


may

obscure

original

structural

properties,

and,

indeed,

largercycles

are

distinguishable

rom

each other

only

by

their

intervallic

tructure,

not

by

their

pitch-

class content.

Still,

the

boundaries et forth

by

comprehensibil-


freedom.

The

overall

unityprovidedby

this

type


nant

appears

n works

composed

as

early

as

the

choral

Psalms,

including


excerptgiven

as Ex-

ample

8. While the

chromaticism

f

verse

1


played

in the outer voices

only, cyclic


influence he

complete

texture,

so that an

entire

passage

s satu-

rated with a


principle.

In

verse

2 of


whole-tone

scale also

determine

the

pitch-class

content of the

inner

voices,

with the

result that the

pervasive

pitch

resource s

indeed

a

"scale,"

in the

traditional

ense,


monic

language

of

the

passage.

With a small

cardinality,

his

source

is

easily

characterized

by

content,

and

thus

requires

no

restrictionsof

ordering

o

retain ts

integrity;

notes

for

the

alto

and

tenor can be

selected from

any

portion

of

the outer-voice

scale,

without

regard

for their

scalar

ordering.

The

resulting



Intervals 3 and 9

similarly

prescribe

the harmonic

anguage

of

verse 3

(mm.

12-

16).

For

cyclic

structures hat

generate

higher

numbersof

pitch

classes,

this

type

of

intervallic

aturation annot

be achieved

by

simple

distributionof

pitch

classes from

the

source

throughout

the

texture.

Because

the

source is

distinguished

primarily

by

order-solely by

order in the

case of

aggregatecompletion-

the process changesfrom a retention of pitchclasses from the

source to a

perpetuation

of the

intervallic

adjacencies

of the

source.

Thus,

in verse 5 of Psalm 24

(mm. 22-27),

outer-voice

cycles

of

ascending

and

descending perfect

fourths are

sup-

portedexclusivelyby quartal

verticalities.Even when a

source

may

be less

audibly

amiliar,

as

in

some

combination

ycles,

for

example,

a certain

type

of intervallic structure will be

pre-

scribed,

and this

can be

projected

despite

selected

reorderings

and other distortionsof the source.

The structureof Ives's orchestral"tonepoem" TheFourth

of

July

(1911-13)

is based both on

quotations

of familiar

tunes-as is thatof

many

of his

longer

orchestralworks-and

types

of

compositional

calculations associated more with

the

shorter

experimentalpieces.

In

recalling

his

composition

of

the

work,

Ives writes

of

"a

feeling

of freedom as a

boy

has ... who

wants

to do

anything

he wants to do" while at the same time

working

out

"combinations f tones and

rhythms ery

carefully

by

kind

of

prescriptions,

in

the

way

a

chemical

compound

which makes explosions would be made."51The network of

quotations

s indeed

diverse,

resembling

he sort of

"free

asso-

ciation" Ives seems to

describe,

yet

the choices

of

tunes are

hardly

made at

random,

and,

as

Dennis

Marshallhas demon-

strated,

the tune

"Red,

White,

and Blue"

(RWB)

standsat the

structuralcore of the

movement.52Both the tune

quotations

and the

pitch-rhythm

calculations contribute

to a

program-

matic

depiction

of a civic

celebration,

establishing

a series of

musical

interrelationships

and extramusicalassociations that

helpto portray he multiplicityof the experience.53

51Ives,Memos,

104.

52Marshall,

"Charles Ives's

Quotations,"

54-55.

The

following

analysis

supports

Marshall'sobservation

hat

RWB is

used as

"both a

melodic

and har-

monic source" n the

opening

of

the

work.

53See Mark D.

Nelson,

"Beyond

Mimesis: Transcendentalism nd

Pro-

cesses of

Analogy

in

Charles Ives's The Fourth

of July,"

Perspectives

f

New

Music 22/1-2

(1983-84),

353-384.

order-solely by

order in the

case of



source to a

perpetuation

of the

intervallic

adjacencies

of the

source.

Thus,


(mm. 22-27),

outer-voice

cycles

of

ascending

and

descending perfect

fourths are

sup-



source

may

be less

audibly

amiliar,

as

in

some

combination

ycles,

for

example,

a certain

type


pre-

scribed,

and this

can be

projected

despite

selected

reorderings



of

July

(1911-13)

is based both on

quotations

of familiar

tunes-as is thatof

many

of his

longer

orchestralworks-and

types

of

compositional


the

shorter

experimentalpieces.

In

recalling

his

composition

of

the

work,

Ives writes

of

"a

feeling

of freedom as a

boy

has ... who

wants

to do

anything


working

out


rhythms ery

carefully

by

kind

of

prescriptions,

in

the

way

a

chemical

compound


quotations

s indeed

diverse,

resembling

he sort of

"free

asso-


describe,

yet

the choices

of

tunes are

hardly

made at

random,

and,

as

Dennis

Marshallhas demon-

strated,

the tune

"Red,

White,

and Blue"

(RWB)

standsat the



quotations

and the

pitch-rhythm


to a

program-

matic

depiction

of a civic

celebration,

establishing

a series of

musical

interrelationships



51Ives,Memos,

104.

52Marshall,

"Charles Ives's

Quotations,"

54-55.

The

following

analysis

supports


hat

RWB is

used as

"both a

melodic

and har-

monic source" n the

opening

of

the

work.

53See Mark D.

Nelson,

"Beyond


Pro-

cesses of

Analogy

in


of July,"

Perspectives

f

New

Music 22/1-2

(1983-84),

353-384.

order-solely by

order in the

case of



source to a

perpetuation

of the

intervallic

adjacencies

of the

source.

Thus,


(mm. 22-27),

outer-voice

cycles

of

ascending

and

descending perfect

fourths are

sup-



source

may

be less

audibly

amiliar,

as

in

some

combination

ycles,

for

example,

a certain

type


pre-

scribed,

and this

can be

projected

despite

selected

reorderings



of

July

(1911-13)

is based both on

quotations

of familiar

tunes-as is thatof

many

of his

longer

orchestralworks-and

types

of

compositional


the

shorter

experimentalpieces.

In

recalling

his

composition

of

the

work,

Ives writes

of

"a

feeling

of freedom as a

boy

has ... who

wants

to do

anything


working

out


rhythms ery

carefully

by

kind

of

prescriptions,

in

the

way

a

chemical

compound


quotations

s indeed

diverse,

resembling

he sort of

"free

asso-


describe,

yet

the choices

of

tunes are

hardly

made at

random,

and,

as

Dennis

Marshallhas demon-

strated,

the tune

"Red,

White,

and Blue"

(RWB)

standsat the



quotations

and the

pitch-rhythm


to a

program-

matic

depiction

of a civic

celebration,

establishing

a series of

musical

interrelationships



51Ives,Memos,

104.

52Marshall,

"Charles Ives's

Quotations,"

54-55.

The

following

analysis

supports


hat

RWB is

used as

"both a

melodic

and har-

monic source" n the

opening

of

the

work.

53See Mark D.

Nelson,

"Beyond


Pro-

cesses of

Analogy

in


of July,"

Perspectives

f

New

Music 22/1-2

(1983-84),

353-384.






68 Music

TheorySpectrum

8 Music

TheorySpectrum

8 Music

TheorySpectrum

Evidence of

cyclic compositional

ources s

prominent

rom

the

openingsection of the work. Ivesnotes, inthemarginof an

early

score-sketch,

that

some "chords n

'[The] Cage'

"

repre-

sent the

origins

of the firstsection of TheFourth

of July,

appar-

ently referring

o

the series of

sonoritiesconstructed

of

fourths

and fifths in the

strings

of mm. 8-13.54

The

sustained

notes in

the condensed score of this

passage

given

in

Example

12a form

a

cyclic unity

of

intervals

5 and

7

(chords

a and

c,

respectively,

in the

example) alternating

withwhole-tone sonorities

(chord

b)

in mm.

8-11.55

Then in

mm. 12-13 chords a and c are re-

peatedwithout the interruptionof chordb. The lower melodic

voice

is

mostly

independent

of

the

chords.

The notationsbelow

the score

in the

example

trace chord

a,

beginning

with

the

pc

0

of

the

lower

melodic

line,

through

the first seven

elements

of

the interval-5

cycle, connecting

o

chord

c for

the

cycliccomple-

tion.56Chord

c

is constructed

of

fifths and thus "inverts"

hord

a;

in

effect,

the

cycle

progressesupward hrough

a,

connects

n

the

upper

register

of

both

chords,

and then

progresses

down-

ward

hrough

c. The

intervals

n

chord

b

are

also

inverted

from

2 to 10), but the pitchclassesdo not change:both occurrences

of

b state the five-tone whole-tone

subset,

as illustratedbelow

the

score.

The

lower

melodic voice in

this

passagepresents

he firstes-

sentiallycomplete

statement of the first

phrase

of RWB.

Com-

54Kirkpatrick,

Catalogue,

11. Ives

indicates elsewhere the influence

of

other works on The Fourth

of July,

including

March

and

Overture

1776

(Memos,

83)

and The GeneralSlocum

(Memos, 105).

55Ives's ssociationof quartaland whole-tone structures allsto mindparts

of

Schoenberg's

Kammersymphonie,Op.

9

(for

instance,

mm.

1-3).

Schoen-

berg's usages display

the

voice-leading

connections discussed n

his

Theory

of

Harmony,

406.

56ArthurMaisel describes

these

cyclic

completions

as

"mutually

xclusive

collections"

(that

is,

literal

complements)

in "The Fourth

of July

by

Charles

Ives: Mixed Harmonic Criteria n

a

Twentieth-Century

Classic,"

Theory

and

Practice

6/1

(1981),

3-32. These

collections

assume

considerable

ignificance

n

Maisel's

analysis

of the work.

Evidence of


ources s

prominent

rom

the


early

score-sketch,

that

some "chords n

'[The] Cage'

"

repre-

sent the

origins


of July,

appar-

ently referring

o

the series of


of

fourths

and fifths in the

strings

of mm. 8-13.54

The

sustained

notes in


passage

given

in

Example

12a form

a

cyclic unity

of

intervals

5 and

7

(chords

a and

c,

respectively,

in the



(chord

b)

in mm.

8-11.55

Then in



voice

is

mostly

independent

of

the

chords.

The notationsbelow

the score

in the

example

trace chord

a,

beginning

with

the

pc

0

of

the

lower

melodic

line,

through

the first seven

elements

of

the interval-5

cycle, connecting

o

chord

c for

the

cycliccomple-

tion.56Chord

c

is constructed

of


hord

a;

in

effect,

the

cycle


a,

connects

n

the

upper

register

of

both

chords,

and then

progresses

down-

ward

hrough

c. The

intervals

n

chord

b

are

also

inverted

from


of


subset,

as illustratedbelow

the

score.

The

lower

melodic voice in

this

passagepresents

he firstes-

sentiallycomplete


phrase

of RWB.

Com-

54Kirkpatrick,

Catalogue,

11. Ives


of


of July,

including

March

and

Overture

1776

(Memos,

83)


(Memos, 105).


of

Schoenberg's

Kammersymphonie,Op.

9

(for

instance,

mm.

1-3).

Schoen-


the

voice-leading


his

Theory

of

Harmony,

406.


these

cyclic

completions

as

"mutually

xclusive

collections"

(that

is,

literal

complements)

in "The Fourth

of July

by

Charles


a

Twentieth-Century

Classic,"

Theory

and

Practice

6/1

(1981),

3-32. These

collections

assume

considerable

ignificance

n

Maisel's

analysis

of the work.

Evidence of


ources s

prominent

rom

the


early

score-sketch,

that

some "chords n

'[The] Cage'

"

repre-

sent the

origins


of July,

appar-

ently referring

o

the series of


of

fourths

and fifths in the

strings

of mm. 8-13.54

The

sustained

notes in


passage

given

in

Example

12a form

a

cyclic unity

of

intervals

5 and

7

(chords

a and

c,

respectively,

in the



(chord

b)

in mm.

8-11.55

Then in



voice

is

mostly

independent

of

the

chords.

The notationsbelow

the score

in the

example

trace chord

a,

beginning

with

the

pc

0

of

the

lower

melodic

line,

through

the first seven

elements

of

the interval-5

cycle, connecting

o

chord

c for

the

cycliccomple-

tion.56Chord

c

is constructed

of


hord

a;

in

effect,

the

cycle


a,

connects

n

the

upper

register

of

both

chords,

and then

progresses

down-

ward

hrough

c. The

intervals

n

chord

b

are

also

inverted

from


of


subset,

as illustratedbelow

the

score.

The

lower

melodic voice in

this

passagepresents

he firstes-

sentiallycomplete


phrase

of RWB.

Com-

54Kirkpatrick,

Catalogue,

11. Ives


of


of July,

including

March

and

Overture

1776

(Memos,

83)


(Memos, 105).


of

Schoenberg's

Kammersymphonie,Op.

9

(for

instance,

mm.

1-3).

Schoen-


the

voice-leading


his

Theory

of

Harmony,

406.


these

cyclic

completions

as

"mutually

xclusive

collections"

(that

is,

literal

complements)

in "The Fourth

of July

by

Charles


a

Twentieth-Century

Classic,"

Theory

and

Practice

6/1

(1981),

3-32. These

collections

assume

considerable

ignificance

n

Maisel's

analysis

of the work.

Example

12.

The Fourth

of

July,

mm. 8-13.

a. Thirdand fourthviolins, cello, bass.

Example

12.

The Fourth

of

July,

mm. 8-13.


Example

12.

The Fourth

of

July,

mm. 8-13.


9

I

I

a

I

I

a

I

I

a

\

^

7

i

-^

*

-

d -

i

r

o

r

i

i

-a

d

t

r

i

X

F

-

p

"

F

I

F

6Â

t

__

X

0

X

J

,

J

TL3--i

r

\

^

7

i

-^

*

-

d -

i

r

o

r

i

i

-a

d

t

r

i

X

F

-

p

"

F

I

F

6Â

t

__

X

0

X

J

,

J

TL3--i

r

\

^

7

i

-^

*

-

d -

i

r

o

r

i

i

-a

d

t

r

i

X

F

-

p

"

F

I

F

6Â

t

__

X

0

X

J

,

J

TL3--i

r

I I I I II

1

I

i

b

c

b a c

I I I I II

1

I

i

b

c

b a c

I I I I II

1

I

i

b

c

b a c

int.5:

pc

0 5

10 3

8

1

6 11

4 9 2

7 int.2:

pc

0

2

4

6

8 10

I

I

I

I

I

b

c b

b.

"Red, White,

and

Blue,"

first

phrase.

INT <5

-

2-5>

parison

of the lowest voice from

Example

12a with the

more

familiarversion of the first

phrase

notated

n

Example

12b

con-

firms that the former is

rhythmically

aried,

and

only

the

first

note,

the G

anacrusis,

s

missing.

In

the bars

preceding

m. 8

a

motivic

interplay nvolvingprimarily

he first our

pitch

classes

of

the

tune

(including

the

anacrusis)

orecasts the later

more

complete

versions.

Example

13a

gives

a

condensed

score

of

these first

seven

bars, omitting only

a chromatic

neighboring

figure

to

C#

that occurs n

the second violins

of mm.

4-7. The

third violin

opens

the workwith a statement

of

the

motive in

the

key

of

Ct: G#-C#t-C#-D#-Gt

n m.

1,

extending

to

the

downbeat

of

m.

2,

displays

he

interval uccession

5-2-5

that s

the characteristic

beginning

of

RWB

(see

Ex.

12b). Example

13b isolates each

motivic occurrence.In m. 2 the fourth

violin

answers

with a variant n which

the central

nterval s

replaced

int.5:

pc

0 5

10 3

8

1

6 11

4 9 2

7 int.2:

pc

0

2

4

6

8 10

I

I

I

I

I

b

c b

b.

"Red, White,

and

Blue,"

first

phrase.

INT <5

-

2-5>

parison


Example

12a with the

more


phrase

notated

n

Example

12b

con-


rhythmically

aried,

and

only

the

first

note,

the G

anacrusis,

s

missing.

In

the bars

preceding

m. 8

a

motivic


he first our

pitch

classes

of

the

tune

(including

the

anacrusis)

orecasts the later

more

complete

versions.

Example

13a

gives

a

condensed

score

of

these first

seven

bars, omitting only

a chromatic

neighboring

figure

to

C#

that occurs n

the second violins

of mm.

4-7. The

third violin

opens


of

the

motive in

the

key

of

Ct: G#-C#t-C#-D#-Gt

n m.

1,

extending

to

the

downbeat

of

m.

2,

displays

he

interval uccession

5-2-5

that s

the characteristic

beginning

of

RWB

(see

Ex.

12b). Example

13b isolates each


violin

answers


the central

nterval s

replaced

int.5:

pc

0 5

10 3

8

1

6 11

4 9 2

7 int.2:

pc

0

2

4

6

8 10

I

I

I

I

I

b

c b

b.

"Red, White,

and

Blue,"

first

phrase.

INT <5

-

2-5>

parison


Example

12a with the

more


phrase

notated

n

Example

12b

con-


rhythmically

aried,

and

only

the

first

note,

the G

anacrusis,

s

missing.

In

the bars

preceding

m. 8

a

motivic


he first our

pitch

classes

of

the

tune

(including

the

anacrusis)

orecasts the later

more

complete

versions.

Example

13a

gives

a

condensed

score

of

these first

seven

bars, omitting only

a chromatic

neighboring

figure

to

C#

that occurs n

the second violins

of mm.

4-7. The

third violin

opens


of

the

motive in

the

key

of

Ct: G#-C#t-C#-D#-Gt

n m.

1,

extending

to

the

downbeat

of

m.

2,

displays

he

interval uccession

5-2-5

that s

the characteristic

beginning

of

RWB

(see

Ex.

12b). Example

13b isolates each


violin

answers


the central

nterval s

replaced

l10o

11

.

21

,b, J13),

J

10o

11

.

21

,b, J13),

J

10o

11

.

21

,b, J13),

J






Interval

ycles

as

Compositional

esources 69

nterval

ycles

as

Compositional

esources 69

nterval

ycles

as

Compositional

esources 69

Example

13.

TheFourth

of

July,

mm.

1-7,

strings.

a. condensed

score

Example

13.

TheFourth

of

July,

mm.

1-7,

strings.

a. condensed

score

Example

13.

TheFourth

of

July,

mm.

1-7,

strings.

a. condensed

score

vln.

2

ln.

2

ln.

2

+vln.

1

vln.

1

vln.

1

f2 Hr-3---

3

--345

6

7

yin

l

3

2

cb.

b. motivic

structure

ni

I

IJ

t

lT

l

?.J7

J7

f2 Hr-3---

3

--345

6

7

yin

l

3

2

cb.

b. motivic

structure

ni

I

IJ

t

lT

l

?.J7

J7

f2 Hr-3---

3

--345

6

7

yin

l

3

2

cb.

b. motivic

structure

ni

I

IJ

t

lT

l

?.J7

J7

INT:

<5

- 2 -

5>

<5

-

10

-

5>

NT:

<5

- 2 -

5>

<5

-

10

-

5>

NT:

<5

- 2 -

5>

<5

-

10

-

5>

<5

-

2

-

5>

5

-

2

-

5>

5

-

2

-

5>

with its

inverse,

and in mm.

4-5

an exact

transposition

of the

motive in the bass

implies

the

key

of B.

In addition to its

relationship

o

the

primary

quoted

tune

of

the

work,

the 5-2-5 motive exhibits a

repetitive

intervallic

structure

hat

may

be tied

to

more abstract

pitch

resources.

The

first our

pitch

classes

of

RWB

are situatedwithinthe 5/2 com-

bination

cycle,

an

overlay

of

interval-7

cycles

at a distance

of

interval

5.

Taking

he

most

complete

statement

of

RWB,

the

C-

major

version in mm.

8-12

(plus

the

missing anacrusis),

as a

point

of

departure,

a

full

expression

of the

intervallicalterna-

tion

would

read:

with its

inverse,

and in mm.

4-5

an exact

transposition

of the

motive in the bass

implies

the

key

of B.

In addition to its

relationship

o

the

primary

quoted

tune

of

the

work,


repetitive

intervallic

structure

hat

may

be tied

to

more abstract

pitch

resources.

The

first our

pitch

classes

of

RWB


bination

cycle,

an

overlay

of

interval-7

cycles

at a distance

of

interval

5.

Taking

he

most

complete

statement

of

RWB,

the

C-

major

version in mm.

8-12

(plus

the

missing anacrusis),

as a

point

of

departure,

a

full

expression

of the

intervallicalterna-

tion

would

read:

with its

inverse,

and in mm.

4-5

an exact

transposition

of the

motive in the bass

implies

the

key

of B.

In addition to its

relationship

o

the

primary

quoted

tune

of

the

work,


repetitive

intervallic

structure

hat

may

be tied

to

more abstract

pitch

resources.

The

first our

pitch

classes

of

RWB


bination

cycle,

an

overlay

of

interval-7

cycles

at a distance

of

interval

5.

Taking

he

most

complete

statement

of

RWB,

the

C-

major

version in mm.

8-12

(plus

the

missing anacrusis),

as a

point

of

departure,

a

full

expression

of the

intervallicalterna-

tion

would

read:

B

set:

A set:

[x/y

=

5/2]

B

set:

A set:

[x/y

=

5/2]

B

set:

A set:

[x/y

=

5/2]

0

7

2

9

4

11 6 1

7 2

9

4

11

6 1 8

n=7

0

7

2

9

4

11 6 1

7 2

9

4

11

6 1 8

n=7

0

7

2

9

4

11 6 1

7 2

9

4

11

6 1 8

n=7

8 3 10 5

3 10 5 0

8 3 10 5

3 10 5 0

8 3 10 5

3 10 5 0

This

version

of

the 5/2 combination

begins

with the four

pitch

classes of the

motive

as it

appears

n

C

major,

pc

<7,0,2,7>;

any

four-element

segment beginning

on an

A set

member s a

version of the

5-2-5 motive.

The

small PCL

(3)

is,

of

course,

apparent

as a

pitch-classrepetition

between

the

first

and

fourth

pitch

classes in the motive.

This

version

of

the 5/2 combination

begins

with the four

pitch

classes of the

motive

as it

appears

n

C

major,

pc

<7,0,2,7>;

any

four-element

segment beginning

on an

A set

member s a

version of the

5-2-5 motive.

The

small PCL

(3)

is,

of

course,

apparent

as a


between

the

first

and

fourth

pitch


This

version

of

the 5/2 combination

begins

with the four

pitch

classes of the

motive

as it

appears

n

C

major,

pc

<7,0,2,7>;

any

four-element

segment beginning

on an

A set

member s a

version of the

5-2-5 motive.

The

small PCL

(3)

is,

of

course,

apparent

as a


between

the

first

and

fourth

pitch


va.

vc.

cb.

va.

vc.

cb.

va.

vc.

cb.






70

Music

Theory

Spectrum

0

Music

Theory

Spectrum

0

Music

Theory

Spectrum

While

the 5/2

combination

cycle

cannot

be viewed

as

a

tone-

row-like

source

set

for

this

or

any portion

of The

Fourth

of

July,

it can

provide

a

logical

backdrop

or the

pitch

structure

andthe

integration

of the

quoted

tune.

Consistent

reference

to the

characteristic

ntervals

n the

combination

highlights

he

impor-

tance

of the

repetitive

intervallic

structures

n

the

pitch

lan-

guage

of the

composition.

Indeed,

the

two

complete

motives

from

mm.

1-7

(Ex.

13)

arelinked

as

adjacencies

n

the

source,

in reverse

order:

pc

<6,11,1,6>

(mm.

4-5)

immediately

pre-

cedes

pc

<8,1,3,8>

(mm.

1-2)

in

the 5/2combination:

Bset: 0 7 2 9 4 11 6 1 8 3 10 5

Aset:

7 2

9

4

11

6

1

83

10

50

(mm.

4-5Xmm.

1-2)

Further,

the two

alternating

ntervals

ndividually

are

primary

structural

omponents.

This

is

most

obviously apparent

n

the

passage

immediately

ollowing

the

motivic

nterplay,

shown

in

Example

12a,

as chord

structures

built

from

cycles

of intervals

5

and

2

(and

their

inverses).

In the

firstseven

measures,

each

in-

terval

and

ts

inverse

are

prominent

even

apart

rom

exact

repe-

titions

of

the 5-2-5

motive: the

continuation,

or

example,

of

the

initial

motivic

statement

n the

third

violin

of

m. 2 consists

of

intervals

10

and

7,

and

the

violinssustain

nterval

2

through-

out

mm.

4-7

(two

muted

violins

continue

sustaining

his

inter-

val

through

m.

91).

Whole-tone

relationships

are

also

pro-

jected

by

the

pitch

levels

of the

motivic

statements

in the

opening

measures,

as thesecond

statement

vln.

4,

m.

2)

begins

a whole

stephigher

than

the

initial

motive

(vln.

3,

m.

1),

while

the

version

beginning

n m.

4 is

a whole

step

lowerthanthatof

m. 1.

Cyclicpitch

derivations

also

play

an

integral

role

in In

re con

moto

etal

(1913),

a

work

for

pianoquintet

n

which

experimen-

tationwith

complex

methods

of

organizing

pitch

and

rhythm

reaches

a

sort

of saturation

point.

Amid

a

diversity

of

pitch

structures

derived

through

cyclic

and

other

means,

this

work

is

While

the 5/2

combination

cycle

cannot

be viewed

as

a

tone-

row-like

source

set

for

this

or

any portion

of The

Fourth

of

July,

it can

provide

a

logical

backdrop

or the

pitch

structure

andthe

integration

of the

quoted

tune.

Consistent

reference

to the

characteristic

ntervals

n the

combination

highlights

he

impor-

tance

of the

repetitive

intervallic

structures

n

the

pitch

lan-

guage

of the

composition.

Indeed,

the

two

complete

motives

from

mm.

1-7

(Ex.

13)

arelinked

as

adjacencies

n

the

source,

in reverse

order:

pc

<6,11,1,6>

(mm.

4-5)

immediately

pre-

cedes

pc

<8,1,3,8>

(mm.

1-2)

in

the 5/2combination:

Bset: 0 7 2 9 4 11 6 1 8 3 10 5

Aset:

7 2

9

4

11

6

1

83

10

50

(mm.

4-5Xmm.

1-2)

Further,

the two

alternating

ntervals

ndividually

are

primary

structural

omponents.

This

is

most

obviously apparent

n

the

passage

immediately

ollowing

the

motivic

nterplay,

shown

in

Example

12a,

as chord

structures

built

from

cycles

of intervals

5

and

2

(and

their

inverses).

In the

firstseven

measures,

each

in-

terval

and

ts

inverse

are

prominent

even

apart

rom

exact

repe-

titions

of

the 5-2-5

motive: the

continuation,

or

example,

of

the

initial

motivic

statement

n the

third

violin

of

m. 2 consists

of

intervals

10

and

7,

and

the

violinssustain

nterval

2

through-

out

mm.

4-7

(two

muted

violins

continue

sustaining

his

inter-

val

through

m.

91).

Whole-tone

relationships

are

also

pro-

jected

by

the

pitch

levels

of the

motivic

statements

in the

opening

measures,

as thesecond

statement

vln.

4,

m.

2)

begins

a whole

stephigher

than

the

initial

motive

(vln.

3,

m.

1),

while

the

version

beginning

n m.

4 is

a whole

step

lowerthanthatof

m. 1.

Cyclicpitch

derivations

also

play

an

integral

role

in In

re con

moto

etal

(1913),

a

work

for

pianoquintet

n

which

experimen-

tationwith

complex

methods

of

organizing

pitch

and

rhythm

reaches

a

sort

of saturation

point.

Amid

a

diversity

of

pitch

structures

derived

through

cyclic

and

other

means,

this

work

is

While

the 5/2

combination

cycle

cannot

be viewed

as

a

tone-

row-like

source

set

for

this

or

any portion

of The

Fourth

of

July,

it can

provide

a

logical

backdrop

or the

pitch

structure

andthe

integration

of the

quoted

tune.

Consistent

reference

to the

characteristic

ntervals

n the

combination

highlights

he

impor-

tance

of the

repetitive

intervallic

structures

n

the

pitch

lan-

guage

of the

composition.

Indeed,

the

two

complete

motives

from

mm.

1-7

(Ex.

13)

arelinked

as

adjacencies

n

the

source,

in reverse

order:

pc

<6,11,1,6>

(mm.

4-5)

immediately

pre-

cedes

pc

<8,1,3,8>

(mm.

1-2)

in

the 5/2combination:

Bset: 0 7 2 9 4 11 6 1 8 3 10 5

Aset:

7 2

9

4

11

6

1

83

10

50

(mm.

4-5Xmm.

1-2)

Further,

the two

alternating

ntervals

ndividually

are

primary

structural

omponents.

This

is

most

obviously apparent

n

the

passage

immediately

ollowing

the

motivic

nterplay,

shown

in

Example

12a,

as chord

structures

built

from

cycles

of intervals

5

and

2

(and

their

inverses).

In the

firstseven

measures,

each

in-

terval

and

ts

inverse

are

prominent

even

apart

rom

exact

repe-

titions

of

the 5-2-5

motive: the

continuation,

or

example,

of

the

initial

motivic

statement

n the

third

violin

of

m. 2 consists

of

intervals

10

and

7,

and

the

violinssustain

nterval

2

through-

out

mm.

4-7

(two

muted

violins

continue

sustaining

his

inter-

val

through

m.

91).

Whole-tone

relationships

are

also

pro-

jected

by

the

pitch

levels

of the

motivic

statements

in the

opening

measures,

as thesecond

statement

vln.

4,

m.

2)

begins

a whole

stephigher

than

the

initial

motive

(vln.

3,

m.

1),

while

the

version

beginning

n m.

4 is

a whole

step

lowerthanthatof

m. 1.

Cyclicpitch

derivations

also

play

an

integral

role

in In

re con

moto

etal

(1913),

a

work

for

pianoquintet

n

which

experimen-

tationwith

complex

methods

of

organizing

pitch

and

rhythm

reaches

a

sort

of saturation

point.

Amid

a

diversity

of

pitch

structures

derived

through

cyclic

and

other

means,

this

work

is

structured

ccording

o

repeated

projections

of a number

eries

that

determines

meter

changes,

phrase

lengths,

or

rhythmic

groupings.

In

Memos,

Ives

gives

the numberseriesas "2-3-5-

7-11-7-5-3-2,"

a

symmetrical

arrangement

f the

five

prime

numbers

greater

than

1;

the

various

presentations

of

the

"Prime Series"

(PS)

are

sometimes

altered

by

stopping

after

the

midpoint

or

by

inverting

he

order

to

begin

and

end

on

11,

placing

2 in the

center.

Each

occurrence

of the

PS

in

the

work

can be

viewed

as a

"variation,"

projecting

an

overall

form

of:

introduction,

nitial

presentation

of the

PS,

and

the

variations.

In

commenting

on

the structure

of the

work,

Ives

implies

that

only

the PS

represents

an element of

constancy

n the work,

while the

methods

of

projection

are

constantly

hanging;

he

re-

fers

to

repetitions

of the PS

as

"cycles"

that

"grow,

expand,

ebb,

but

never

literally

repeat."57

Prior

to the

first

presentation

of the

PS,

a

one-measure

n-

troduction

n the

strings

displays

he

cyclic

origins

hat

will

simi-

larly

characterize

pitch

structures

n

other

portions

of the

work.

Constructed

according

o

a

proportional

cheme

of

4:3:2:1

be-

tween

the

note

valuesof

the

four

voices

(top

to

bottom),

this

measuresubdividesinto the three

aggregates

hat are boxed

and numbered

n

Example

14. Each

aggregate

s

derived

roma

source

interval-5

cycle,

following

the

op

labels

placed

beside

each

note

in

the score.

The

initial

aggregate

unfolds

the

pitch

classes

in direct

temporal

succession,

starting

with

ops

0-3

stated

simultaneously,

followed

by

op

4

on

the

next

pitch

change

(vln.

1),

leading

to

ops

5 and

6 stated

together

on

beat

2,

and

continuing

n this

manner

to

op

11

on

beat

3

in

the cello.

The

two

other

aggregates

contain

slight

disorderings

of the

57Memos,

01.

Thisview

of

the form

generally

corresponds

o

that

of John

McLain

Rinehart,

"Ives's

Compositional

dioms:

An

Investigation

f Selected

Short

Compositions

as

Microcosms

of

His Musical

Language"

Ph.D.

disserta-

tion,

Ohio State

University,

1970),

48-61.

See

also Ulrich

Maske,

Charles

ves

in seiner

Kammermusik

ur

drei

bis sechs

Instrumente,

Kolner

Beitrage

zur

Mu-

sikforschung,

vol. 64

(Regensburg:

G.

Bosse,

1971),

121-123.

structured

ccording

o

repeated

projections

of a number

eries

that

determines

meter

changes,

phrase

lengths,

or

rhythmic

groupings.

In

Memos,

Ives

gives


7-11-7-5-3-2,"

a

symmetrical

arrangement

f the

five

prime

numbers

greater

than

1;

the

various

presentations

of

the

"Prime Series"

(PS)

are

sometimes

altered

by

stopping

after

the

midpoint

or

by

inverting

he

order

to

begin

and

end

on

11,

placing

2 in the

center.

Each

occurrence

of the

PS

in

the

work

can be

viewed

as a

"variation,"

projecting

an

overall

form

of:

introduction,

nitial

presentation

of the

PS,

and

the

variations.

In

commenting

on

the structure

of the

work,

Ives

implies

that

only

the PS

represents

an element of

constancy

n the work,

while the

methods

of

projection

are

constantly

hanging;

he

re-

fers

to

repetitions

of the PS

as

"cycles"

that

"grow,

expand,

ebb,

but

never

literally

repeat."57

Prior

to the

first

presentation

of the

PS,

a

one-measure

n-

troduction

n the

strings

displays

he

cyclic

origins

hat

will

simi-

larly

characterize

pitch

structures

n

other

portions

of the

work.

Constructed

according

o

a

proportional

cheme

of

4:3:2:1

be-

tween

the

note

valuesof

the

four

voices

(top

to

bottom),

this


aggregates

hat are boxed

and numbered

n

Example

14. Each

aggregate

s

derived

roma

source

interval-5

cycle,

following

the

op

labels

placed

beside

each

note

in

the score.

The

initial

aggregate

unfolds

the

pitch

classes

in direct

temporal

succession,

starting

with

ops

0-3

stated

simultaneously,

followed

by

op

4

on

the

next

pitch

change

(vln.

1),

leading

to

ops

5 and

6 stated

together

on

beat

2,

and

continuing

n this

manner

to

op

11

on

beat

3

in

the cello.

The

two

other

aggregates

contain

slight

disorderings

of the

57Memos,

01.

Thisview

of

the form

generally

corresponds

o

that

of John

McLain

Rinehart,

"Ives's

Compositional

dioms:

An

Investigation

f Selected

Short

Compositions

as

Microcosms

of

His Musical

Language"

Ph.D.

disserta-

tion,

Ohio State

University,

1970),

48-61.

See

also Ulrich

Maske,

Charles

ves

in seiner

Kammermusik

ur

drei

bis sechs

Instrumente,

Kolner

Beitrage

zur

Mu-

sikforschung,

vol. 64

(Regensburg:

G.

Bosse,

1971),

121-123.

structured

ccording

o

repeated

projections

of a number

eries

that

determines

meter

changes,

phrase

lengths,

or

rhythmic

groupings.

In

Memos,

Ives

gives


7-11-7-5-3-2,"

a

symmetrical

arrangement

f the

five

prime

numbers

greater

than

1;

the

various

presentations

of

the

"Prime Series"

(PS)

are

sometimes

altered

by

stopping

after

the

midpoint

or

by

inverting

he

order

to

begin

and

end

on

11,

placing

2 in the

center.

Each

occurrence

of the

PS

in

the

work

can be

viewed

as a

"variation,"

projecting

an

overall

form

of:

introduction,

nitial

presentation

of the

PS,

and

the

variations.

In

commenting

on

the structure

of the

work,

Ives

implies

that

only

the PS

represents

an element of

constancy

n the work,

while the

methods

of

projection

are

constantly

hanging;

he

re-

fers

to

repetitions

of the PS

as

"cycles"

that

"grow,

expand,

ebb,

but

never

literally

repeat."57

Prior

to the

first

presentation

of the

PS,

a

one-measure

n-

troduction

n the

strings

displays

he

cyclic

origins

hat

will

simi-

larly

characterize

pitch

structures

n

other

portions

of the

work.

Constructed

according

o

a

proportional

cheme

of

4:3:2:1

be-

tween

the

note

valuesof

the

four

voices

(top

to

bottom),

this


aggregates

hat are boxed

and numbered

n

Example

14. Each

aggregate

s

derived

roma

source

interval-5

cycle,

following

the

op

labels

placed

beside

each

note

in

the score.

The

initial

aggregate

unfolds

the

pitch

classes

in direct

temporal

succession,

starting

with

ops

0-3

stated

simultaneously,

followed

by

op

4

on

the

next

pitch

change

(vln.

1),

leading

to

ops

5 and

6 stated

together

on

beat

2,

and

continuing

n this

manner

to

op

11

on

beat

3

in

the cello.

The

two

other

aggregates

contain

slight

disorderings

of the

57Memos,

01.

Thisview

of

the form

generally

corresponds

o

that

of John

McLain

Rinehart,

"Ives's

Compositional

dioms:

An

Investigation

f Selected

Short

Compositions

as

Microcosms

of

His Musical

Language"

Ph.D.

disserta-

tion,

Ohio State

University,

1970),

48-61.

See

also Ulrich

Maske,

Charles

ves

in seiner

Kammermusik

ur

drei

bis sechs

Instrumente,

Kolner

Beitrage

zur

Mu-

sikforschung,

vol. 64

(Regensburg:

G.

Bosse,

1971),

121-123.






Interval

ycles

as

Compositional

esources 71nterval

ycles

as

Compositional

esources 71nterval

ycles

as

Compositional

esources 71

Example

14.

In re con moto et

al,

m. 1.

xample

14.

In re con moto et

al,

m. 1.

xample

14.

In re con moto et

al,

m. 1.

r-., 5 7

.10

-., 5 7

.10

-., 5 7

.10

vln.

3

4

'"'1

2 3

4

5

7

10

3

,,f

A

A

^

A

A

va.

4I

1J

.

J

mf

8

6

3

vln.

3

4

'"'1

2 3

4

5

7

10

3

,,f

A

A

^

A

A

va.

4I

1J

.

J

mf

8

6

3

vln.

3

4

'"'1

2 3

4

5

7

10

3

,,f

A

A

^

A

A

va.

4I

1J

.

J

mf

8

6

3

)

aggregates:ggregates:ggregates:

11

1

1

1

1

1

1...

7

8

2...

3...

op

0 1

2 3 4

5

6 7

8 9 10

11

int. 5

cycle:

pc

1

6

11 4

9

2

7 0 5

10

3 8

op

0 1

2 3 4

5

6 7

8 9 10

11

int. 5

cycle:

pc

1

6

11 4

9

2

7 0 5

10

3 8

op

0 1

2 3 4

5

6 7

8 9 10

11

int. 5

cycle:

pc

1

6

11 4

9

2

7 0 5

10

3 8

source,

so

that,

for

example,

op

7

occurs before

op

6 in

aggre-

gate

2

(cello

and

viola),

and

ops

9 and 10

occur

before

op

8

in

aggregate

3

(vln.

1, viola,

cello).

In its

customary

ompositional

role,

the

cyclic

source

provides

for

structural

unity

of

pitch

combinations

and

control

over

pitch-class

urnover.

The initial

presentation

of the PS

(mm. 2-6), immediately

following

the one-measure

ntroduction,

uses durations o

pro-

ject

the

"reciprocal"

PS

values,

beginning

with

11

and shrink-

ing

to

2

at the

midpoint.

These are measured n

eighth

notes

changing

to

sixteenths above the condensed

score

in

Example

15;

the final

duration

(m.

6)

is

nine rather than

eleven

six-

teenths.

The

pitch

structureof each chord

s based on the

repe-

titions indicated

below the score: chords

"a,

b, h,

i"

display

source,

so

that,

for

example,

op

7

occurs before

op

6 in

aggre-

gate

2

(cello

and

viola),

and

ops

9 and 10

occur

before

op

8

in

aggregate

3

(vln.

1, viola,

cello).

In its

customary

ompositional

role,

the

cyclic

source

provides

for

structural

unity

of

pitch

combinations

and

control

over

pitch-class

urnover.

The initial

presentation

of the PS


following

the one-measure

ntroduction,

uses durations o

pro-

ject

the

"reciprocal"

PS

values,

beginning

with

11

and shrink-

ing

to

2

at the

midpoint.


eighth

notes

changing

to


score

in

Example

15;

the final

duration

(m.

6)

is

nine rather than

eleven

six-

teenths.

The

pitch


s based on the

repe-

titions indicated


"a,

b, h,

i"

display

source,

so

that,

for

example,

op

7

occurs before

op

6 in

aggre-

gate

2

(cello

and

viola),

and

ops

9 and 10

occur

before

op

8

in

aggregate

3

(vln.

1, viola,

cello).

In its

customary

ompositional

role,

the

cyclic

source

provides

for

structural

unity

of

pitch

combinations

and

control

over

pitch-class

urnover.

The initial

presentation

of the PS


following

the one-measure

ntroduction,

uses durations o

pro-

ject

the

"reciprocal"

PS

values,

beginning

with

11

and shrink-

ing

to

2

at the

midpoint.


eighth

notes

changing

to


score

in

Example

15;

the final

duration

(m.

6)

is

nine rather than

eleven

six-

teenths.

The

pitch


s based on the

repe-

titions indicated


"a,

b, h,

i"

display

single-interval

tructures,

and the others

employ

intervallical-

ternations.

Toward the end of m.

2,

and

again

in m.

3,

two

notes of the chords are

altered,

temporarilydisrupting

he in-

tervallic scheme.

Though

the basic chord

structure

changes

with each new

duration,

he

interval

izes

do

not

project

a PS of

their

own,

as

they might

have done

by forming

chords of

elev-

enths, sevenths, fifths, thirds,

and

seconds,

for

example.

How-

ever,

the

registral

span

of the chords effects a

general, unsys-

tematic

contraction-expansion

hat

parallels

he

PS,

reaching

a

narrow

point

at the shortest duration:chord e

spans

two oc-

taves

plus

a

fifth,

in

contrast to chords a

and

b

(three

octaves

plus

a

tritone)

and chord

(four

octaves

plus

a

fifth).

The

size of

vertical

intervals also

reaches

a

maximumnear the

end,

with

single-interval

tructures,

and the others

employ

intervallical-

ternations.


2,

and

again

in m.

3,

two


altered,


he in-

tervallic scheme.

Though

the basic chord

structure

changes

with each new

duration,

he

interval

izes

do

not

project

a PS of

their

own,

as

they might

have done

by forming

chords of

elev-


and

seconds,

for

example.

How-

ever,

the

registral

span


general, unsys-

tematic


hat

parallels

he

PS,

reaching

a

narrow

point


spans

two oc-

taves

plus

a

fifth,

in


and

b

(three

octaves

plus

a

tritone)

and chord

(four

octaves

plus

a

fifth).

The

size of

vertical

intervals also

reaches

a

maximumnear the

end,

with

single-interval

tructures,

and the others

employ

intervallical-

ternations.


2,

and

again

in m.

3,

two


altered,


he in-

tervallic scheme.

Though

the basic chord

structure

changes

with each new

duration,

he

interval

izes

do

not

project

a PS of

their

own,

as

they might

have done

by forming

chords of

elev-


and

seconds,

for

example.

How-

ever,

the

registral

span


general, unsys-

tematic


hat

parallels

he

PS,

reaching

a

narrow

point


spans

two oc-

taves

plus

a

fifth,

in


and

b

(three

octaves

plus

a

tritone)

and chord

(four

octaves

plus

a

fifth).

The

size of

vertical

intervals also

reaches

a

maximumnear the

end,

with






72

Music

Theory

Spectrum

2

Music

Theory

Spectrum

2

Music

Theory

Spectrum

Example

15.

In

re con moto et

al,

mm.

2-6,

condensedscore.

xample

15.

In

re con moto et

al,

mm.

2-6,

condensedscore.

xample

15.

In

re con moto et

al,

mm.

2-6,

condensedscore.

J=

11

=

11

=

11

combination

cycle:

5

combination

cycle:

5

combination

cycle:

5

7

311i

=

2

3

5

311i

=

2

3

5

311i

=

2

3

5

a.

b.

c.

d.

e.

f.

g. h.

i.

5/5

7/7

7/6

5/6 4/5 4/9

5/8

10/10

11/1

a.

b.

c.

d.

e.

f.

g. h.

i.

5/5

7/7

7/6

5/6 4/5 4/9

5/8

10/10

11/1

a.

b.

c.

d.

e.

f.

g. h.

i.

5/5

7/7

7/6

5/6 4/5 4/9

5/8

10/10

11/1

interval

10 in chord h and 11

in

chord

representing

n

increase

over the verticalintervals used in

previous

chords

(intervals

4

through9).

Several

adjacent

sonorities n this initial

presentation

of the

PS

are connected

through

a

cyclic

derivational

process

resem-

bling

the

linkages

between

interval-5

and

-7

structures

n mm.

8-13 of

The Fourth

of July

(Ex.

12a).

First,

the interval-5

ycle

of

chord

a

(Ex. 15) "wraps

around,"

or

connects

at the

top,

to

its

inverse,

the interval-7

cycle

of

chord

b,

as between

mm.

8

and 10

or 12

and 13

of The Fourth

of July.

Then the 7/6

cycle

of

chordc

similarly

wraps

around

o the

5/6

cycle

in chord

d.

Lines

1 and 2 of

Figure

6 summarize hese

connections,

withbrackets

and chord

labels

indicatingpositions

of chords within

the

cy-

cles. Because

two

connected

sonorities exhibit converse

regis-

tral

distributions f

pitch-class

order,

the lower

notes in the mu-

sic are those of the outer

portions

of

the

cycle

as it

is

notated

in

the

figure,

and the

higher

notes

appeartogether

in the center.

In line

1,

chord a

encompasses

the first seven elements

of the

interval-5

cycle

and b

spans

the other five

plus

two

repetitions

interval


in

chord

representing

n

increase


previous

chords

(intervals

4

through9).

Several

adjacent


presentation

of the

PS

are connected

through

a

cyclic

derivational

process

resem-

bling

the

linkages

between

interval-5

and

-7

structures

n mm.

8-13 of

The Fourth

of July

(Ex.

12a).

First,

the interval-5

ycle

of

chord

a

(Ex. 15) "wraps

around,"

or

connects

at the

top,

to

its

inverse,

the interval-7

cycle

of

chord

b,

as between

mm.

8

and 10

or 12

and 13

of The Fourth

of July.

Then the 7/6

cycle

of

chordc

similarly

wraps

around

o the

5/6

cycle

in chord

d.

Lines

1 and 2 of

Figure

6 summarize hese

connections,

withbrackets

and chord

labels

indicatingpositions

of chords within

the

cy-

cles. Because

two

connected


regis-

tral

distributions f

pitch-class

order,

the lower

notes in the mu-


portions

of

the

cycle

as it

is

notated

in

the

figure,

and the

higher

notes

appeartogether

in the center.

In line

1,

chord a

encompasses


of the

interval-5

cycle

and b

spans

the other five

plus

two

repetitions

interval


in

chord

representing

n

increase


previous

chords

(intervals

4

through9).

Several

adjacent


presentation

of the

PS

are connected

through

a

cyclic

derivational

process

resem-

bling

the

linkages

between

interval-5

and

-7

structures

n mm.

8-13 of

The Fourth

of July

(Ex.

12a).

First,

the interval-5

ycle

of

chord

a

(Ex. 15) "wraps

around,"

or

connects

at the

top,

to

its

inverse,

the interval-7

cycle

of

chord

b,

as between

mm.

8

and 10

or 12

and 13

of The Fourth

of July.

Then the 7/6

cycle

of

chordc

similarly

wraps

around

o the

5/6

cycle

in chord

d.

Lines

1 and 2 of

Figure

6 summarize hese

connections,

withbrackets

and chord

labels

indicatingpositions

of chords within

the

cy-

cles. Because

two

connected


regis-

tral

distributions f

pitch-class

order,

the lower

notes in the mu-


portions

of

the

cycle

as it

is

notated

in

the

figure,

and the

higher

notes

appeartogether

in the center.

In line

1,

chord a

encompasses


of the

interval-5

cycle

and b

spans

the other five

plus

two

repetitions

(pc

1 and

6,

enclosed in

parentheses).

Line 2 illustrates

a 7/6

cycle concluding

with its

first

pc repetition(PCL

=

11)

at

pc

6,

which

s the bass

note

common to chords

b,

c,

and d. The

wrap-

around

between

chords

c

and d

includes

a

point

of intersection

at

pcs

8, 3,

and

9,

the

upper

three notes

of

both chords.

Pc

0,

which

would

complete

the

aggregate

on line

2 of

Figure

6,

appears

not in chord d but as the lowest note of the subse-

quent

sonority

and of

every

chordfor the remainder f the

pas-

sage.

This

implied

continuationof the

cycle connecting

hordsc

and d thus

extends a

wraparound

process

that includes

cyclic

linkages

between

a and b and

between c and

d

in the

upper

reg-

isters and common-tone

inkages

between b and c and between

c

and d in

the lower

registers.

With

the

arrivalof the

pc

0 bass

"anchor" n chord

e,

subsequent

chords do

not

wrap

around

but

continue the

low-register

common-tone

connections,

with

some association

between

adjacencies.

Chord

e,

for

example,

exhibits

the 4/5

cycle

notated on line

3 of

Figure

6,

which

con-

nects

in

the bass clef to four common

tones

of chord f.

Thus

f

begins identically

to e on line 3 of the

figure

with

pcs

(pc

1 and

6,

enclosed in

parentheses).

Line 2 illustrates

a 7/6

cycle concluding

with its

first

pc repetition(PCL

=

11)

at

pc

6,

which

s the bass

note

common to chords

b,

c,

and d. The

wrap-

around

between

chords

c

and d

includes

a

point

of intersection

at

pcs

8, 3,

and

9,

the

upper

three notes

of

both chords.

Pc

0,

which

would

complete

the

aggregate

on line

2 of

Figure

6,

appears


quent

sonority

and of

every


pas-

sage.

This

implied

continuationof the

cycle connecting

hordsc

and d thus

extends a

wraparound

process

that includes

cyclic

linkages

between

a and b and

between c and

d

in the

upper

reg-


inkages


c

and d in

the lower

registers.

With

the

arrivalof the

pc

0 bass

"anchor" n chord

e,

subsequent

chords do

not

wrap

around

but

continue the

low-register

common-tone

connections,

with

some association

between

adjacencies.

Chord

e,

for

example,

exhibits

the 4/5

cycle

notated on line

3 of

Figure

6,

which

con-

nects

in


tones

of chord f.

Thus

f

begins identically


figure

with

pcs

(pc

1 and

6,

enclosed in

parentheses).

Line 2 illustrates

a 7/6

cycle concluding

with its

first

pc repetition(PCL

=

11)

at

pc

6,

which

s the bass

note

common to chords

b,

c,

and d. The

wrap-

around

between

chords

c

and d

includes

a

point

of intersection

at

pcs

8, 3,

and

9,

the

upper

three notes

of

both chords.

Pc

0,

which

would

complete

the

aggregate

on line

2 of

Figure

6,

appears


quent

sonority

and of

every


pas-

sage.

This

implied

continuationof the

cycle connecting

hordsc

and d thus

extends a

wraparound

process

that includes

cyclic

linkages

between

a and b and

between c and

d

in the

upper

reg-


inkages


c

and d in

the lower

registers.

With

the

arrivalof the

pc

0 bass

"anchor" n chord

e,

subsequent

chords do

not

wrap

around

but

continue the

low-register

common-tone

connections,

with

some association

between

adjacencies.

Chord

e,

for

example,

exhibits

the 4/5

cycle

notated on line

3 of

Figure

6,

which

con-

nects

in


tones

of chord f.

Thus

f

begins identically


figure

with

pcs

1






Interval

ycles

as

Compositional

esources 73

nterval

ycles

as

Compositional

esources 73

nterval

ycles

as

Compositional

esources 73

Figure

6.

Cyclic relationships

of

simultaneities

in

Example

15.

igure

6.


of

simultaneities

in

Example

15.

igure

6.


of

simultaneities

in

Example

15.

bassassass a.

sopr.

.

sopr.

.

sopr.

1.

int.-5cycle:

pc

1

6

.

int.-5cycle:

pc

1

6

.

int.-5cycle:

pc

1

6

11

4 9 2 7

0 5

I

11

4 9 2 7

0 5

I

11

4 9 2 7

0 5

I

10 3 8

(1)

(6)

0 3 8

(1)

(6)

0 3 8

(1)

(6)

b.

bass

.

bass

.

bass

bass

2. 7/6 comb.

cycle: pc

6 1

bass

2. 7/6 comb.

cycle: pc

6 1

bass

2. 7/6 comb.

cycle: pc

6 1

c.

sopr.

7 2

8

3 91 4

I

c.

sopr.

7 2

8

3 91 4

I

c.

sopr.

7 2

8

3 91 4

I

sopr.opr.opr.

d...

10

5

11

6

[0]

bass

\

10

5

11

6

[0]

bass

\

10

5

11

6

[0]

bass

\

- - - - A'- - - A'- - - A'

bass

-

3. 4/5 comb.

cycle: pc

0' 4

9

1

If.

/

/

f.

(bass clef) /

k

v

bass

-

3. 4/5 comb.

cycle: pc

0' 4

9

1

If.

/

/

f.

(bass clef) /

k

v

bass

-

3. 4/5 comb.

cycle: pc

0' 4

9

1

If.

/

/

f.

(bass clef) /

k

v

e.

sopr.

6 10 3 7

e.

sopr.

6 10 3 7

e.

sopr.

6 10 3 7

/

/

/

/

f.

(top

6

notes)

4. 4/9 comb.

cycle:

pc

9 1

10 2 11

3

/

/

/

/

f.

(top

6

notes)

4. 4/9 comb.

cycle:

pc

9 1

10 2 11

3

/

/

/

/

f.

(top

6

notes)

4. 4/9 comb.

cycle:

pc

9 1

10 2 11

3

<0,4,9,1>,

but

then

shifts

to

line

4,

with

pcs

<9,1>

overlap-

ping

to

become

the

beginning

of a

4/9

alternation n the

upper

register.

Other

portions

of

In

re,

while

by

no means

uniformly

con-

ceived,

maintain ome

degree

of

cyclicunderpinning.

Common

to mostof the variationss the recurrence f a

sonority

Ivescalls

the

"Grit

Chord,"

which

s

frequently

used

to

articulate

he be-

ginning

of a

unit

of

the PS.58 n variation

1,

for

example,

meter

changes

project

the

PS,

so that the

beginning

of

each

measure

signals

a new PS unit.59

Measures

in

this variation hat

do not

58Ives,

Memos,

101.

59The

PS determines he

numberof

beats

per

measure n variation

1,

begin-

ning

with

6

meter

(2 beats), (3

beats),

15

(5 beats),

and

so forth.See

Rinehart,

Ives's

Compositional

Idioms,"

50-51.

<0,4,9,1>,

but

then

shifts

to

line

4,

with

pcs

<9,1>

overlap-

ping

to

become

the

beginning

of a

4/9

alternation n the

upper

register.

Other

portions

of

In

re,

while

by

no means

uniformly

con-

ceived,

maintain ome

degree

of

cyclicunderpinning.

Common


sonority

Ivescalls

the

"Grit

Chord,"

which

s

frequently

used

to

articulate

he be-

ginning

of a

unit

of


1,

for

example,

meter

changes

project

the

PS,

so that the

beginning

of

each

measure

signals

a new PS unit.59

Measures

in

this variation hat

do not

58Ives,

Memos,

101.

59The

PS determines he

numberof

beats

per

measure n variation

1,

begin-

ning

with

6

meter

(2 beats), (3

beats),

15

(5 beats),

and

so forth.See

Rinehart,

Ives's

Compositional

Idioms,"

50-51.

<0,4,9,1>,

but

then

shifts

to

line

4,

with

pcs

<9,1>

overlap-

ping

to

become

the

beginning

of a

4/9

alternation n the

upper

register.

Other

portions

of

In

re,

while

by

no means

uniformly

con-

ceived,

maintain ome

degree

of

cyclicunderpinning.

Common


sonority

Ivescalls

the

"Grit

Chord,"

which

s

frequently

used

to

articulate

he be-

ginning

of a

unit

of


1,

for

example,

meter

changes

project

the

PS,

so that the

beginning

of

each

measure

signals

a new PS unit.59

Measures

in

this variation hat

do not

58Ives,

Memos,

101.

59The

PS determines he

numberof

beats

per

measure n variation

1,

begin-

ning

with

6

meter

(2 beats), (3

beats),

15

(5 beats),

and

so forth.See

Rinehart,

Ives's

Compositional

Idioms,"

50-51.

begin

with the Grit Chord

(GC) begin

with

its

literal

comple-

ment

(GCC).

Example

16

gives

the

opening

of variation

1,

dis-

playing

GC on the first beats

of

mm.

7

and 9 and GCC

on the

firstbeat of m. 8 and on the second beat

c m.

9.

The two

sonorities

appear together

in

this

type

of

pairing

(without

change

in

pc content)

in most

subsequent

appearances,

the

only

possible

variable

being

the

registral

distribution

of the

pitch

classes

of

GCC.

In

keeping

with the

cyclic

nature

of

other structural

spects

of

the

work,

GC is formed from

eight

notes,

in

registral

order

from

low

to

high,

of the

7/6

combination

cycle beginning

on

pc

0.

Upward

stems

in

Figure

7

extract GC

from a

7/6

cycle

that

continues

to

the

point

of

aggregate

completion,

one

element

past

the

repetition

of

pc

0

(PCL

=

11).

GCC

(downward

tems

in

the

figure)

then

contains

the

remainingpitch

classesof

the

begin

with the Grit Chord

(GC) begin

with

its

literal

comple-

ment

(GCC).

Example

16

gives

the

opening

of variation

1,

dis-

playing


of

mm.

7

and 9 and GCC

on the


c m.

9.

The two

sonorities

appear together

in

this

type

of

pairing

(without

change

in

pc content)

in most

subsequent

appearances,

the

only

possible

variable

being

the

registral

distribution

of the

pitch

classes

of

GCC.

In

keeping

with the

cyclic

nature

of

other structural

spects

of

the

work,

GC is formed from

eight

notes,

in

registral

order

from

low

to

high,

of the

7/6

combination

cycle beginning

on

pc

0.

Upward

stems

in

Figure

7

extract GC

from a

7/6

cycle

that

continues

to

the

point

of

aggregate

completion,

one

element

past

the

repetition

of

pc

0

(PCL

=

11).

GCC

(downward

tems

in

the

figure)

then

contains

the

remainingpitch

classesof

the

begin

with the Grit Chord

(GC) begin

with

its

literal

comple-

ment

(GCC).

Example

16

gives

the

opening

of variation

1,

dis-

playing


of

mm.

7

and 9 and GCC

on the


c m.

9.

The two

sonorities

appear together

in

this

type

of

pairing

(without

change

in

pc content)

in most

subsequent

appearances,

the

only

possible

variable

being

the

registral

distribution

of the

pitch

classes

of

GCC.

In

keeping

with the

cyclic

nature

of

other structural

spects

of

the

work,

GC is formed from

eight

notes,

in

registral

order

from

low

to

high,

of the

7/6

combination

cycle beginning

on

pc

0.

Upward

stems

in

Figure

7

extract GC

from a

7/6

cycle

that

continues

to

the

point

of

aggregate

completion,

one

element

past

the

repetition

of

pc

0

(PCL

=

11).

GCC

(downward

tems

in

the

figure)

then

contains

the

remainingpitch

classesof

the

sopr.opr.opr.

m

(






74

Music

TheorySpectrum

4

Music

TheorySpectrum

4

Music

TheorySpectrum

Example

16. In

re

con moto et

al,

variation

1,

mm.

7-9.

xample

16. In

re

con moto et

al,

variation

1,

mm.

7-9.

xample

16. In

re

con moto et

al,

variation

1,

mm.

7-9.

inf

c' b; .

.

v

mf

mf

inf

c' b; .

.

v

mf

mf

inf

c' b; .

.

v

mf

mf

)

mf

[GC]

mf

[GC]

mf

[GC]

[GC,

GCC]

[GC, GCC]

[GC,

GCC]

[GC, GCC]

[GC,

GCC]

[GC, GCC]

GCC]GCC]GCC]

cycle,

including

a

pc

3

that

GC

omits from ts

otherwise

contigu-

ous

extraction,

and

excluding

the redundant

pc

0. The

cyclic

source does

not

determinethe

registralordering

of

GCC.

Portions of the

variationsthat

are

not restatements

of GC

and GCC

may

further

perpetuate

a

connection

with a

cyclic

source

through

consistentrestatementof the

primary

ntervals.

Measure

9,

for

example

(Ex.

16),

can

be subdivided

almost

ex-

clusively

into

tritones

(interval 6), starting

with

those that

are

inherent

in

GC

and

GCC,

and

continuing

through

the

struc-

tures

n the

remainder

of the

measure.Each circled

and abeled

dyad

in

Example

16

highlights

an occurrenceof a tritonebe-

tween

registrally

and/or

emporally

associated

pitches.

Follow-

ing

the

completion

of the

aggregate

rom

GC/GCC,

the

dyads

reiterate

pitch

classes from

the

first

part

of the

measure,

effect-

ing

a redistribution f those same intervals:

dyads

, h,

and

g

are

pitch-class

equivalent

o

a, b,

and

c,

respectively.

Figure

8

plots

each

dyad

on the

7/6

cycle,

including

he

three

reiterations,

he

cycle,

including

a

pc

3

that

GC

omits from ts

otherwise

contigu-

ous

extraction,

and

excluding

the redundant

pc

0. The

cyclic

source does

not

determinethe

registralordering

of

GCC.

Portions of the

variationsthat

are

not restatements

of GC

and GCC

may

further

perpetuate

a

connection

with a

cyclic

source

through


primary

ntervals.

Measure

9,

for

example

(Ex.

16),

can

be subdivided

almost

ex-

clusively

into

tritones


with

those that

are

inherent

in

GC

and

GCC,

and

continuing

through

the

struc-

tures

n the

remainder

of the


and abeled

dyad

in

Example

16

highlights


tween

registrally

and/or

emporally

associated

pitches.

Follow-

ing

the

completion

of the

aggregate

rom

GC/GCC,

the

dyads

reiterate

pitch

classes from

the

first

part

of the

measure,

effect-

ing


dyads

, h,

and

g

are

pitch-class

equivalent

o

a, b,

and

c,

respectively.

Figure

8

plots

each

dyad

on the

7/6

cycle,

including

he

three

reiterations,

he

cycle,

including

a

pc

3

that

GC

omits from ts

otherwise

contigu-

ous

extraction,

and

excluding

the redundant

pc

0. The

cyclic

source does

not

determinethe

registralordering

of

GCC.

Portions of the

variationsthat

are

not restatements

of GC

and GCC

may

further

perpetuate

a

connection

with a

cyclic

source

through


primary

ntervals.

Measure

9,

for

example

(Ex.

16),

can

be subdivided

almost

ex-

clusively

into

tritones


with

those that

are

inherent

in

GC

and

GCC,

and

continuing

through

the

struc-

tures

n the

remainder

of the


and abeled

dyad

in

Example

16

highlights


tween

registrally

and/or

emporally

associated

pitches.

Follow-

ing

the

completion

of the

aggregate

rom

GC/GCC,

the

dyads

reiterate

pitch

classes from

the

first

part

of the

measure,

effect-

ing


dyads

, h,

and

g

are

pitch-class

equivalent

o

a, b,

and

c,

respectively.

Figure

8

plots

each

dyad

on the

7/6

cycle,

including

he

three

reiterations,

he

tritone that occurs

within

GCC

(dyad d),

and

the tritone n

the

second violin

(dyade)

formedfrom

pc

6

of

GCC

and

pc

0 on the

next beat. This

segmentation

amasses five of the

six

available

tritones,

avoidingthroughout

he measure

the

pcs

3 and

9

that

are

inherently

absent

from

GC.

The most

striking

evidence of

interval

cycles

in

Ives's

music

comes in

the

form

of a

compositional

"model" hat seems

con-

tinuously

o have

occupied

his

interest.

His

earliest

nspirations

toward the

structure

of

the model

are

suggestedby

several

ac-

counts in Memos

of

experiments

he

conducted

together

with

his

father, including,

for

example,

recollections

of

chord

suc-

cessions

constructed

from

"3rds all and

over,

then 3rds

and

2nds,

then

3rds and

4ths,

then

3rds

and 4ths and

5ths,

etc."60

Such

patterns

of

gradual

change

in

intervalsize evolved

into

a

model of chord

succession,

the broad outlines of which main-

tain

some

degree

of

uniformity

n a

variety

of musicalcontexts.

Most

generally,

the

model

is

comprised

of

successiveverticali-

ties formed from

single-interval

r

combination

ycles,

with the

sizes

of

the

generating

ntervals

gradually ncreasing

r decreas-

ing

in

established

increments.

The

pattern typicallydisplays

a

symmetrical

tructure

by reversing

tself

following

he arrival t

a

high

or

low

point

of interval size. If the numberof voices in

the chords

remains

constant-including

octave

doublings

for

smaller

cardinalities-the

expansion-contraction rocess

may

be

displayed

as

a

registral

"wedge"

shape

outlined

by

the

verti-

cal

span

of

each

chord. If

the chord

voicings

are

flexible,

how-

ever,

the

process may

occur within sonorities

that exhibit no

significant hanges

nvertical

span, only

internal

changes

n in-

tervallicstructure.

The

prime-number

eries

in In re con moto

et al is a variant

of the

model,

with the

pattern

of

changedetermining

aspects

other than

intervallicstructure.

Of

course,

the first

setting

of

the PS

shown in

Example

15 does

exhibit

a

general

pattern

of

intervallic

change

in

support

of

the durational

pattern.

The

tritone that occurs

within

GCC

(dyad d),

and

the tritone n

the

second violin

(dyade)

formedfrom

pc

6

of

GCC

and

pc

0 on the

next beat. This

segmentation

amasses five of the

six

available

tritones,

avoidingthroughout

he measure

the

pcs

3 and

9

that

are

inherently

absent

from

GC.

The most

striking

evidence of

interval

cycles

in

Ives's

music

comes in

the

form

of a

compositional

"model" hat seems

con-

tinuously

o have

occupied

his

interest.

His

earliest

nspirations

toward the

structure

of

the model

are

suggestedby

several

ac-

counts in Memos

of

experiments

he

conducted

together

with

his

father, including,

for

example,

recollections

of

chord

suc-

cessions

constructed

from

"3rds all and

over,

then 3rds

and

2nds,

then

3rds and

4ths,

then

3rds

and 4ths and

5ths,

etc."60

Such

patterns

of

gradual

change

in


into

a

model of chord

succession,


tain

some

degree

of

uniformity

n a

variety

of musicalcontexts.

Most

generally,

the

model

is

comprised

of


ties formed from

single-interval

r

combination

ycles,

with the

sizes

of

the

generating

ntervals

gradually ncreasing

r decreas-

ing

in

established

increments.

The


a

symmetrical

tructure

by reversing

tself

following

he arrival t

a

high

or

low

point


the chords

remains

constant-including

octave

doublings

for

smaller

cardinalities-the


may

be

displayed

as

a

registral

"wedge"

shape

outlined

by

the

verti-

cal

span

of

each

chord. If

the chord

voicings

are

flexible,

how-

ever,

the

process may


that exhibit no

significant hanges

nvertical

span, only

internal

changes

n in-

tervallicstructure.

The

prime-number

eries

in In re con moto

et al is a variant

of the

model,

with the

pattern

of

changedetermining

aspects

other than


Of

course,

the first

setting

of

the PS

shown in

Example

15 does

exhibit

a

general

pattern

of

intervallic

change

in

support

of

the durational

pattern.

The

tritone that occurs

within

GCC

(dyad d),

and

the tritone n

the

second violin

(dyade)

formedfrom

pc

6

of

GCC

and

pc

0 on the

next beat. This

segmentation

amasses five of the

six

available

tritones,

avoidingthroughout

he measure

the

pcs

3 and

9

that

are

inherently

absent

from

GC.

The most

striking

evidence of

interval

cycles

in

Ives's

music

comes in

the

form

of a

compositional

"model" hat seems

con-

tinuously

o have

occupied

his

interest.

His

earliest

nspirations

toward the

structure

of

the model

are

suggestedby

several

ac-

counts in Memos

of

experiments

he

conducted

together

with

his

father, including,

for

example,

recollections

of

chord

suc-

cessions

constructed

from

"3rds all and

over,

then 3rds

and

2nds,

then

3rds and

4ths,

then

3rds

and 4ths and

5ths,

etc."60

Such

patterns

of

gradual

change

in


into

a

model of chord

succession,


tain

some

degree

of

uniformity

n a

variety

of musicalcontexts.

Most

generally,

the

model

is

comprised

of


ties formed from

single-interval

r

combination

ycles,

with the

sizes

of

the

generating

ntervals

gradually ncreasing

r decreas-

ing

in

established

increments.

The


a

symmetrical

tructure

by reversing

tself

following

he arrival t

a

high

or

low

point


the chords

remains

constant-including

octave

doublings

for

smaller

cardinalities-the


may

be

displayed

as

a

registral

"wedge"

shape

outlined

by

the

verti-

cal

span

of

each

chord. If

the chord

voicings

are

flexible,

how-

ever,

the

process may


that exhibit no

significant hanges

nvertical

span, only

internal

changes

n in-

tervallicstructure.

The

prime-number

eries

in In re con moto

et al is a variant

of the

model,

with the

pattern

of

changedetermining

aspects

other than


Of

course,

the first

setting

of

the PS

shown in

Example

15 does

exhibit

a

general

pattern

of

intervallic

change

in

support

of

the durational

pattern.

The

60Ives, Memos,

120.

0Ives, Memos,

120.

0Ives, Memos,

120.

V-_-

-I

I

-_-

-I

I

-_-

-I

I






Interval

ycles

as

Compositional

esources 75

nterval

ycles

as

Compositional

esources 75

nterval

ycles

as

Compositional

esources 75

Figure

7. 7/6 combination

cycle

(portion),

GC/GCC extraction.

igure

7. 7/6 combination

cycle

(portion),

GC/GCC extraction.

igure

7. 7/6 combination

cycle

(portion),

GC/GCC extraction.

I

I I

I

7 1

8 2 9

I

I I

I

7 1

8 2 9

I

I I

I

7 1

8 2 9

I

(SC 8-29)(SC 8-29)(SC 8-29)

3 10

10

10

10

10

10 4 11 5 011 5 011 5 0

GCC:CC:CC:

Figure

8.

Tritones

within

7/6 combination

cycle.

GC

I

I

I

I

I

Figure

8.

Tritones

within

7/6 combination

cycle.

GC

I

I

I

I

I

Figure

8.

Tritones

within

7/6 combination

cycle.

GC

I

I

I

I

I

a

rI

7/6: 0 7 1

f

a

rI

7/6: 0 7 1

f

a

rI

7/6: 0 7 1

f

b

I

I

8 2

h

b

I

I

8 2

h

b

I

I

8 2

h

9

more standard version mixes

cycles

of

single

intervals with

closelyassociated combinations o intensify he gradualnature

of

the

process.

This is

displayed

n

the

early

choralwork Proces-

sional

("Let

there

be

Light,"

1901)

and n a broader ense

in

the

still earlier

Psalm

24,

the

beginning

of which s shown n Exam-

ple

8.

The

growth process

in the

Psalm,

which s

displayed

n

a

linear,

not

vertical

orm,

is

observable n the nonmodular ums

of

the

x/y

values on which each

verse

is

based,

treating every

intervallic

repetition

as

a

combination

cycle.

The chromatic

scale of verse 1 is sum 2

(x/y

=

1/1;

1

+

1

=

2),

the whole tone

of verse2 is sum 4 (x/y = 2/2;2 + 2 = 4), and so on, compiling

the

following progression

of sums in

the

first

seven verses:


cycles

of

single

intervals with


of

the

process.

This is

displayed

n

the

early

choralwork Proces-

sional

("Let

there

be

Light,"

1901)


in

the

still earlier

Psalm

24,

the

beginning


ple

8.

The

growth process

in the

Psalm,

which s

displayed

n

a

linear,

not

vertical

orm,

is


of

the

x/y


verse

is

based,

treating every

intervallic

repetition

as

a

combination

cycle.

The chromatic


(x/y

=

1/1;

1

+

1

=

2),

the whole tone


the


of sums in

the

first

seven verses:


cycles

of

single

intervals with


of

the

process.

This is

displayed

n

the

early

choralwork Proces-

sional

("Let

there

be

Light,"

1901)


in

the

still earlier

Psalm

24,

the

beginning


ple

8.

The

growth process

in the

Psalm,

which s

displayed

n

a

linear,

not

vertical

orm,

is


of

the

x/y


verse

is

based,

treating every

intervallic

repetition

as

a

combination

cycle.

The chromatic


(x/y

=

1/1;

1

+

1

=

2),

the whole tone


the


of sums in

the

first

seven verses:

1

2

3

4

5 6

72

3

4

5 6

72

3

4

5 6

7

combination

cycle:

1/1 2/2 3/3 4/3

5/5 6/5

7/7

ombination

cycle:

1/1 2/2 3/3 4/3

5/5 6/5

7/7

ombination

cycle:

1/1 2/2 3/3 4/3

5/5 6/5

7/7

2 4

6

7

10 11 14

4

6

7

10 11 14

4

6

7

10 11 14

c

I

3 10

4

g

c

I

3 10

4

g

c

I

3 10

4

g

GCC

I

I

4 11 5

I

I

d

d

GCC

I

I

4 11 5

I

I

d

d

GCC

I

I

4 11 5

I

I

d

d

I

*

I

(SC

4-29)

*repetition

(PCL

=

11)

I

*

I

(SC

4-29)

*repetition

(PCL

=

11)

I

*

I

(SC

4-29)

*repetition

(PCL

=

11)

I

0 6

I

e

I

0 6

I

e

I

0 6

I

e

Versions

of

the

pattern

n other later works

might

add,

for ex-

ample, a 1/2cyclebetween 1/1 and 2/2to fill in the gapbetween

sums 2

and

4,

though,

of

course,

some

potential

fill-ins

for

in-

stance,

6/6)

will have

undesirably

ow

PCL values.

Included

among

ater

settings

of

the model are an

attempt

at

integrating

patterns

of

pitch

and

rhythm

n

Over

he Pavements

(1906-13)61

and

a

programmatic

association

of the model's

"wedge" shape

with a

specific

scenario in Tone Roads No.

3

(1915).62

Nowhere is the

model

more

pervasive,

however,

than

61See the

bassoon,

clarinet,

and

trumpet parts,

mm. 81-92.

(Rinehart,

"Ives's

Compositional

Idioms,"

44-46,

91-93.)

62Ives

escribesthe scenario n

Memos,

64;

the

model

is most noticeable n

mm. 24-26

but is

present

elsewhere in the

piece

in variant orms. Some other

occurrences

of

the model are:

CentralPark in the Dark

(1906),

mm.

1-10 and

subsequent

repetitions

of the ten-bar

string

pattern;

Soliloquy (1907),

mm.

6-

7;

Robert

Browning

Overture

1908-12), linearly

in the

upper

strings,

mm.

Versions

of

the

pattern

n other later works

might

add,

for ex-


sums 2

and

4,

though,

of

course,

some

potential

fill-ins

for

in-

stance,

6/6)

will have

undesirably

ow

PCL values.

Included

among

ater

settings

of

the model are an

attempt

at

integrating

patterns

of

pitch

and

rhythm

n

Over

he Pavements

(1906-13)61

and

a

programmatic

association

of the model's

"wedge" shape

with a

specific


3

(1915).62

Nowhere is the

model

more

pervasive,

however,

than

61See the

bassoon,

clarinet,

and

trumpet parts,

mm. 81-92.

(Rinehart,

"Ives's

Compositional

Idioms,"

44-46,

91-93.)

62Ives


Memos,

64;

the

model


mm. 24-26

but is

present

elsewhere in the

piece


occurrences

of

the model are:


(1906),

mm.

1-10 and

subsequent

repetitions

of the ten-bar

string

pattern;

Soliloquy (1907),

mm.

6-

7;

Robert

Browning

Overture

1908-12), linearly

in the

upper

strings,

mm.

Versions

of

the

pattern

n other later works

might

add,

for ex-


sums 2

and

4,

though,

of

course,

some

potential

fill-ins

for

in-

stance,

6/6)

will have

undesirably

ow

PCL values.

Included

among

ater

settings

of

the model are an

attempt

at

integrating

patterns

of

pitch

and

rhythm

n

Over

he Pavements

(1906-13)61

and

a

programmatic

association

of the model's

"wedge" shape

with a

specific


3

(1915).62

Nowhere is the

model

more

pervasive,

however,

than

61See the

bassoon,

clarinet,

and

trumpet parts,

mm. 81-92.

(Rinehart,

"Ives's

Compositional

Idioms,"

44-46,

91-93.)

62Ives


Memos,

64;

the

model


mm. 24-26

but is

present

elsewhere in the

piece


occurrences

of

the model are:


(1906),

mm.

1-10 and

subsequent

repetitions

of the ten-bar

string

pattern;

Soliloquy (1907),

mm.

6-

7;

Robert

Browning

Overture

1908-12), linearly

in the

upper

strings,

mm.

GC:

7/6:

pc

GC:

7/6:

pc

GC:

7/6:

pc

I

6

verse:erse:erse:

I

sums:ums:ums:






76 Music

Theory

Spectrum

6 Music

Theory

Spectrum

6 Music

Theory

Spectrum

in the

song

On the

Antipodes (1915-23),

which,

according

to

the

composer,

is based on a "chordal

cycle

for

a

symphony."63

The "chordal

cycle"

is an extensive intervallic contraction-

expansion pattern

that

occurs three times in the

song, appar-

ently realizing

ideas that were

initially

conceived for

the

incom-

plete

Universe

Symphony (1911-28).64

In both

the

song

and the

Symphony,

Ives

expounds upon

universal themes related to the

forces of nature and

processes

of

life,

and seems to

regard

the

intervallic model

as a

symbol

of some of these cosmic

powers.

The

subjects

of the

composer's

text

for the

song

are

the

"antip-

odal"

aspects

of

nature,

described as

paradoxical

extremes:

nature is both "relentless" and

"kind,"

nature is man's "en-

emy,"

but also his

"friend,"

and so forth. The text concludes

that nature

is

"nothing

but

atomic cosmic

cycles" revolving

be-

tween

the

many antipodes,

advancing

a

cyclic

view

of natural

evolution that is

appropriately

mirrored

by

the

many cyclic

pitch

constructions

in the musical

setting.65

Ives uses

the "chordal

cycle"

at the

beginning

and end

(the

"antipodes")

of the

song,

and in the center. The first and

sec-

ond

statements,

which

are identical

in

pitch-class

content,

pro-

ject

the structure outlined in

Figure

9,

progressing

from

perfect

fifths

to a semitonal cluster in the center of the

symmetrical pat-

tern and then

returning

to fifths.66The

upper-case

letter

labels,

119-136, 312-330,

384-390;

The Fourth

of July (1911-13),

m. 20. Some of

these are

described n Nachum

Schoffman,

"Serialism n the

Worksof Charles

Ives,"

Tempo

138

(September

1981),

21-32.

63Charles ves, NineteenSongs (Bryn Mawr, Pa.: Merion Music, 1935),

[52].

64See,

for

example, page

q3039

of the

Universe

Symphony

sketches

(Kirkpatrick,

Catalogue,

27).

65For urther

commentary

on the

text,

see

Hitchcock,

Ives:A

Survey

of

the

Music, 18-20,

and Nachum

Schoffman,

"The

Songs

of

Charles

Ives"

(Ph.D.

dissertation,

Hebrew

University

of

Jerusalem,

1977),

233.

66Also,

the final

statement

of

chord

A

is

preceded by

a chord of stacked

interval

11s;

this has

been omitted from

Figure

9

because

it is absent from the

in the

song

On the


which,

according

to

the

composer,


cycle

for

a

symphony."63

The "chordal

cycle"


expansion pattern

that


song, appar-

ently realizing

ideas that were

initially

conceived for

the

incom-

plete

Universe

Symphony (1911-28).64

In both

the

song

and the

Symphony,

Ives

expounds upon



processes

of

life,

and seems to

regard

the

intervallic model

as a

symbol


powers.

The

subjects

of the

composer's

text

for the

song

are

the

"antip-

odal"

aspects

of

nature,

described as

paradoxical

extremes:


"kind,"


emy,"

but also his

"friend,"


that nature

is

"nothing

but

atomic cosmic

cycles" revolving

be-

tween

the

many antipodes,

advancing

a

cyclic

view

of natural

evolution that is

appropriately

mirrored

by

the

many cyclic

pitch

constructions

in the musical

setting.65

Ives uses

the "chordal

cycle"

at the

beginning

and end

(the

"antipodes")

of the

song,


sec-

ond

statements,

which

are identical

in

pitch-class

content,

pro-

ject


Figure

9,

progressing

from

perfect

fifths


symmetrical pat-

tern and then

returning

to fifths.66The

upper-case

letter

labels,

119-136, 312-330,

384-390;

The Fourth

of July (1911-13),

m. 20. Some of

these are

described n Nachum

Schoffman,

"Serialism n the

Worksof Charles

Ives,"

Tempo

138

(September

1981),

21-32.


[52].

64See,

for

example, page

q3039

of the

Universe

Symphony

sketches

(Kirkpatrick,

Catalogue,

27).

65For urther

commentary

on the

text,

see

Hitchcock,

Ives:A

Survey

of

the

Music, 18-20,

and Nachum

Schoffman,

"The

Songs

of

Charles

Ives"

(Ph.D.

dissertation,

Hebrew

University

of

Jerusalem,

1977),

233.

66Also,

the final

statement

of

chord

A

is

preceded by

a chord of stacked

interval

11s;

this has

been omitted from

Figure

9

because


in the

song

On the


which,

according

to

the

composer,


cycle

for

a

symphony."63

The "chordal

cycle"


expansion pattern

that


song, appar-

ently realizing

ideas that were

initially

conceived for

the

incom-

plete

Universe

Symphony (1911-28).64

In both

the

song

and the

Symphony,

Ives

expounds upon



processes

of

life,

and seems to

regard

the

intervallic model

as a

symbol


powers.

The

subjects

of the

composer's

text

for the

song

are

the

"antip-

odal"

aspects

of

nature,

described as

paradoxical

extremes:


"kind,"


emy,"

but also his

"friend,"


that nature

is

"nothing

but

atomic cosmic

cycles" revolving

be-

tween

the

many antipodes,

advancing

a

cyclic

view

of natural

evolution that is

appropriately

mirrored

by

the

many cyclic

pitch

constructions

in the musical

setting.65

Ives uses

the "chordal

cycle"

at the

beginning

and end

(the

"antipodes")

of the

song,


sec-

ond

statements,

which

are identical

in

pitch-class

content,

pro-

ject


Figure

9,

progressing

from

perfect

fifths


symmetrical pat-

tern and then

returning

to fifths.66The

upper-case

letter

labels,

119-136, 312-330,

384-390;

The Fourth

of July (1911-13),

m. 20. Some of

these are

described n Nachum

Schoffman,

"Serialism n the

Worksof Charles

Ives,"

Tempo

138

(September

1981),

21-32.


[52].

64See,

for

example, page

q3039

of the

Universe

Symphony

sketches

(Kirkpatrick,

Catalogue,

27).

65For urther

commentary

on the

text,

see

Hitchcock,

Ives:A

Survey

of

the

Music, 18-20,

and Nachum

Schoffman,

"The

Songs

of

Charles

Ives"

(Ph.D.

dissertation,

Hebrew

University

of

Jerusalem,

1977),

233.

66Also,

the final

statement

of

chord

A

is

preceded by

a chord of stacked

interval

11s;

this has

been omitted from

Figure

9

because


which

generally adopt

the

composer's

markings

in

the

sketches,67

how that a

complete symmetrical

arrangement

s

disruptedwhen chordI (3/1)isreplacedbyL (2/2)in thesecond

half.

This

does

not, however,

disrupt

he

regularprogression

f

sums

from 14 to

2

and

back,

which

skipsonly

8

and 12.68 n-

deed,

the

variety

of

sums

portrays

he

diversity

of

cycles

used;

these

range through

most

of

the

possible

n

values

(omitting

only

0

and

8,

inferable

rom

the

absence of

sums 12 and

8),

and

including

the wide

range

of PCL values

listed on the

bottom

line

of

the

figure.

The intervals hemselvesare

mostly

he famil-

iar structures

ound in other realizations

of the

model,

includ-

ingthe circleof fifths(chordsA andD), cyclesof 7/6 and 6/5(B

and

C),

and

the

half-step

clusterat

the

midpoint

K).

Chords

E

and J are octatonic

collections,

and L is whole-tone.

To main-

tain

the

regularity

of

changes

in

sums,

the succession

even

in-

cludes a less

familiar tructure

with a PCL

of

only

4 at chord

G,

a whole-tone

subset

(4-25

[0,2,6,8]).

The

realizations

of this model at the

beginning

and

center of

the

song

exhibit substantial

variations in the

ranges

of the

chords

and

do

not

consistently

project

the

complete

cycles

of

final

"crystallized"

version of the

pattern

(mm.

28-34)

and is not

part

of the

symmetrically

elated series of sums.

Figure

9 reflects

observations common

to

several

analyses

of this work:

Rinehart,

"Ives's

Compositional

dioms," 71-86;

Domenick

Argento,

"A Di-

gest Analysis

of

Ives's

'On

the

Antipodes,'

"

Student

Musicologists

t Minne-

sota

6

(1975-76),

192-200; Hitchcock,

Ives: A

Survey

of

the

Music,

18-20;

Schoffman, "The Songs of Ives," 209-234; and Schoffman, "Serialism n

Ives,"

28-29.

By

contrast,

the

emphasis

here is

placed

on the

cyclic

natureof

each

chord

construction,

not

merely

on

its intervallic ontent.


210:

q2908, q3048.

Ives

makes a distinction

with

superscripts)

between

separate

uses

of

the same intervalsand

does not use the

letter L.

68The

uccession is

provided

with

even finer

gradations

n

the version

in-

tended for the

Universe

Symphony

by

theinclusionof chordswith

quarter-tone

intervals.

(Kirkpatrick,Catalogue,

27:

q3039.)

which

generally adopt

the

composer's

markings

in

the

sketches,67

how that a


arrangement

s


half.

This

does

not, however,

disrupt

he

regularprogression

f

sums

from 14 to

2

and

back,

which

skipsonly

8

and 12.68 n-

deed,

the

variety

of

sums

portrays

he

diversity

of

cycles

used;

these

range through

most

of

the

possible

n

values

(omitting

only

0

and

8,

inferable

rom

the

absence of

sums 12 and

8),

and

including

the wide

range

of PCL values

listed on the

bottom

line

of

the

figure.


mostly

he famil-

iar structures


of the

model,

includ-


and

C),

and

the

half-step

clusterat

the

midpoint

K).

Chords

E

and J are octatonic

collections,


To main-

tain

the

regularity

of

changes

in

sums,

the succession

even

in-

cludes a less

familiar tructure

with a PCL

of

only

4 at chord

G,

a whole-tone

subset

(4-25

[0,2,6,8]).

The

realizations


beginning

and

center of

the

song

exhibit substantial

variations in the

ranges

of the

chords

and

do

not

consistently

project

the

complete

cycles

of

final

"crystallized"

version of the

pattern

(mm.

28-34)

and is not

part

of the

symmetrically


Figure

9 reflects

observations common

to

several

analyses

of this work:

Rinehart,

"Ives's

Compositional

dioms," 71-86;

Domenick

Argento,

"A Di-

gest Analysis

of

Ives's

'On

the

Antipodes,'

"

Student

Musicologists

t Minne-

sota

6

(1975-76),

192-200; Hitchcock,

Ives: A

Survey

of

the

Music,

18-20;


Ives,"

28-29.

By

contrast,

the

emphasis

here is

placed

on the

cyclic

natureof

each

chord

construction,

not

merely

on



210:

q2908, q3048.

Ives

makes a distinction

with

superscripts)

between

separate

uses

of


does not use the

letter L.

68The

uccession is

provided

with

even finer

gradations

n

the version

in-

tended for the

Universe

Symphony

by


quarter-tone

intervals.


27:

q3039.)

which

generally adopt

the

composer's

markings

in

the

sketches,67

how that a


arrangement

s


half.

This

does

not, however,

disrupt

he

regularprogression

f

sums

from 14 to

2

and

back,

which

skipsonly

8

and 12.68 n-

deed,

the

variety

of

sums

portrays

he

diversity

of

cycles

used;

these

range through

most

of

the

possible

n

values

(omitting

only

0

and

8,

inferable

rom

the

absence of

sums 12 and

8),

and

including

the wide

range

of PCL values

listed on the

bottom

line

of

the

figure.


mostly

he famil-

iar structures


of the

model,

includ-


and

C),

and

the

half-step

clusterat

the

midpoint

K).

Chords

E

and J are octatonic

collections,


To main-

tain

the

regularity

of

changes

in

sums,

the succession

even

in-

cludes a less

familiar tructure

with a PCL

of

only

4 at chord

G,

a whole-tone

subset

(4-25

[0,2,6,8]).

The

realizations


beginning

and

center of

the

song

exhibit substantial

variations in the

ranges

of the

chords

and

do

not

consistently

project

the

complete

cycles

of

final

"crystallized"

version of the

pattern

(mm.

28-34)

and is not

part

of the

symmetrically


Figure

9 reflects

observations common

to

several

analyses

of this work:

Rinehart,

"Ives's

Compositional

dioms," 71-86;

Domenick

Argento,

"A Di-

gest Analysis

of

Ives's

'On

the

Antipodes,'

"

Student

Musicologists

t Minne-

sota

6

(1975-76),

192-200; Hitchcock,

Ives: A

Survey

of

the

Music,

18-20;


Ives,"

28-29.

By

contrast,

the

emphasis

here is

placed

on the

cyclic

natureof

each

chord

construction,

not

merely

on



210:

q2908, q3048.

Ives

makes a distinction

with

superscripts)

between

separate

uses

of


does not use the

letter L.

68The

uccession is

provided

with

even finer

gradations

n

the version

in-

tended for the

Universe

Symphony

by


quarter-tone

intervals.


27:

q3039.)






Interval

ycles

as

Compositional

esources

77nterval

ycles

as

Compositional

esources

77nterval

ycles

as

Compositional

esources

77

Figure

9. On

the

Antipodes,

chordal

cycle.

igure

9. On

the

Antipodes,

chordal

cycle.

igure

9. On

the

Antipodes,

chordal

cycle.

label: A B C D E FG H I

7 6 5 5 4 3 4 2 1

7 7

6

5 5 4 2 3 3

sums: 14 13 11

10

9

7

6

5 4

PCL:

12 11 12 12

8 8

4 6

6


7 6 5 5 4 3 4 2 1

7 7

6

5 5 4 2 3 3

sums: 14 13 11

10

9

7

6

5 4

PCL:

12 11 12 12

8 8

4 6

6


7 6 5 5 4 3 4 2 1

7 7

6

5 5 4 2 3 3

sums: 14 13 11

10

9

7

6

5 4

PCL:

12 11 12 12

8 8

4 6

6

J

2

1

3

8

J

2

1

3

8

J

2

1

3

8

K JL

1

2

2

1 1 2

K JL

1

2

2

1 1 2

K JL

1

2

2

1 1 2

H

2

3

H

2

3

H

2

3

G F E

4 3 4

2

4

5

G F E

4 3 4

2

4

5

G F E

4 3 4

2

4

5

2

34

5

6

7

9

12 8 6 6 4

8 8

2

34

5

6

7

9

12 8 6 6 4

8 8

2

34

5

6

7

9

12 8 6 6 4

8 8

D C B

5 5 6

5 6 7

D C B

5 5 6

5 6 7

D C B

5 5 6

5 6 7

A

7

7

A

7

7

A

7

7

10 11 13 14

12 12 11

12

10 11 13 14

12 12 11

12

10 11 13 14

12 12 11

12

eachintervallic tructure.69ome of the

changes

are motivated

by

registral

concerns: the realizations

generally

outline

contracting-expanding edge shapes,

with the

exception

of the

centralcluster. Also

influencing

nconsistencies

are the choices

of

pitch

classes

in the

upper

voice,

which unfold

the

aggregate

with one

repetition.70

The final realization of the

model,

reproduced

as

Example

17 with chord

labels

added,

maximizes the

registral span

of

each

chord with

pitch-classrepetitions

hat

may

extend

beyond

the PCLs.

Accompanying

a vocal line that asks the climactic

textual

question,71

he

four-hand

piano part presents

chords

structured

according

to the

outline

of

Figure

9,

except

that I

and L

exchange positions.Unique

to this realization s the

pc

0,

which,

as the lowest

voice of

every

chord

(doubled

by

an

op-

tional

organ pedal),

serves as the

point

of

departure

or each

cycle

in

the

pattern.

In

addition,

many

of

the sonorities exhibit

or

imply

an

upward-directed yclic

return o

the

pc-0

point

of

origin,

similar

to the

"wraparounds"

n TheFourth

of July (Ex. 12)

and In re

69The core

of

these

versions

s

reproduced

n

Argento,

"A

Digest Analy-

sis,"

201-203 and

Schoffman,

"Serialism n

Ives,"

28.

70This

bservation s

made

in

Schoffman,

"The

Songs

of

Ives,"

216-217.

71Thevocal line is

divided into two

parts,

both of which

produce aggre-

gates.

See

Hitchcock,

Ives: A

Survey

of

the

Music,

19-20. The

aggregate

n

the

upper

ine is formed

by adjacent

[0,1,4]

trichords.


changes

are motivated

by

registral


generally

outline


with the

exception

of the


influencing

nconsistencies

are the choices

of

pitch

classes

in the

upper

voice,

which unfold

the

aggregate

with one

repetition.70


model,

reproduced

as

Example

17 with chord

labels

added,

maximizes the

registral span

of

each

chord with


hat

may

extend

beyond

the PCLs.

Accompanying


textual

question,71

he

four-hand

piano part presents

chords

structured

according

to the

outline

of

Figure

9,

except

that I

and L



pc

0,

which,

as the lowest

voice of

every

chord

(doubled

by

an

op-

tional

organ pedal),

serves as the

point

of

departure

or each

cycle

in

the

pattern.

In

addition,

many

of


or

imply

an


return o

the

pc-0

point

of

origin,

similar

to the

"wraparounds"

n TheFourth

of July (Ex. 12)

and In re

69The core

of

these

versions

s

reproduced

n

Argento,

"A

Digest Analy-

sis,"

201-203 and

Schoffman,

"Serialism n

Ives,"

28.

70This

bservation s

made

in

Schoffman,

"The

Songs

of

Ives,"

216-217.

71Thevocal line is

divided into two

parts,

both of which

produce aggre-

gates.

See

Hitchcock,

Ives: A

Survey

of

the

Music,

19-20. The

aggregate

n

the

upper

ine is formed

by adjacent

[0,1,4]

trichords.


changes

are motivated

by

registral


generally

outline


with the

exception

of the


influencing

nconsistencies

are the choices

of

pitch

classes

in the

upper

voice,

which unfold

the

aggregate

with one

repetition.70


model,

reproduced

as

Example

17 with chord

labels

added,

maximizes the

registral span

of

each

chord with


hat

may

extend

beyond

the PCLs.

Accompanying


textual

question,71

he

four-hand

piano part presents

chords

structured

according

to the

outline

of

Figure

9,

except

that I

and L



pc

0,

which,

as the lowest

voice of

every

chord

(doubled

by

an

op-

tional

organ pedal),

serves as the

point

of

departure

or each

cycle

in

the

pattern.

In

addition,

many

of


or

imply

an


return o

the

pc-0

point

of

origin,

similar

to the

"wraparounds"

n TheFourth

of July (Ex. 12)

and In re

69The core

of

these

versions

s

reproduced

n

Argento,

"A

Digest Analy-

sis,"

201-203 and

Schoffman,

"Serialism n

Ives,"

28.

70This

bservation s

made

in

Schoffman,

"The

Songs

of

Ives,"

216-217.

71Thevocal line is

divided into two

parts,

both of which

produce aggre-

gates.

See

Hitchcock,

Ives: A

Survey

of

the

Music,

19-20. The

aggregate

n

the

upper

ine is formed

by adjacent

[0,1,4]

trichords.

con moto etal (Ex. 15). ChordsA andD, forexample,are com-

plete

interval-7 and interval-5

cycles

that returnto

pc

0 when

continued one

step

further,

mplying

a

linkage

to

the bass note

of the

ensuing verticality.

The same is true of the 6/5

cycle

of

chord

C:

the final

upper

interval 5

(C#6-F#6,

m.

29 beat

1)

would be followed

by

interval

6

to

return

to

pc

0.

In

other

sonorities the return to

pc

0

occurs within the

chord,

either

as

the

upper

voice

or

as

a

result

of

extensive

repetition

of a small

PCL. Chord B

(7/6)

contains

pc

0 in both

top

and bottom

voices, its

only

pc duplication.

The

arpeggiated

tatementof H

(PCL

=

6)

in m. 30

continues far

beyond

its first

pc repetition

(E

3), cyclingthrough

several

pc

duplications

before

returning

to

pc

0

as the

highest

note. Chords hat

repeat

within

an

octave

and that

may

therefore contain several nstances

of

pc

0 are G

(4/2,

PCL

=

4),

I

(3/1,

PCL

=

6),

J

(1/2,

PCL

=

8),

and L

(2/2,

PCL

=

6);

of

these,

only

G contains

pc

0 in the

top

voice,

al-

though

each includes

pc

0

at last twice in inner

voices.

The re-

maining

chords

are

E

(5/4,

PCL

=

8),

which

cycles pastpc

0,

F

(4/3, PCL =

8),

which

stops

short of a returnto the

starting

point,

and

K,

the

four-octave semitonal cluster that

includes

several nstances of

pc

0,

but with

pc

1

in the

upper

voice.

The

series

of

cycles

based

on

successively

smaller ntervals

that return

to

the same

pitch-class

anchor

projects

the

idea ex-

pressed

in

the text

of

"atomic,

cosmic

cycles"

as

a

"spiral"pro-

gressing

inward from

cycles

of

larger

intervals

to those with


plete


cycles

that returnto

pc

0 when

continued one

step

further,

mplying

a

linkage

to

the bass note

of the



cycle

of

chord

C:

the final

upper

interval 5

(C#6-F#6,

m.

29 beat

1)

would be followed

by

interval

6

to

return

to

pc

0.

In

other


pc

0

occurs within the

chord,

either

as

the

upper

voice

or

as

a

result

of

extensive

repetition

of a small

PCL. Chord B

(7/6)

contains

pc

0 in both

top

and bottom

voices, its

only

pc duplication.

The

arpeggiated

tatementof H

(PCL

=

6)

in m. 30

continues far

beyond

its first

pc repetition

(E

3), cyclingthrough

several

pc

duplications

before

returning

to

pc

0

as the

highest

note. Chords hat

repeat

within

an

octave

and that

may


of

pc

0 are G

(4/2,

PCL

=

4),

I

(3/1,

PCL

=

6),

J

(1/2,

PCL

=

8),

and L

(2/2,

PCL

=

6);

of

these,

only

G contains

pc

0 in the

top

voice,

al-

though

each includes

pc

0


voices.

The re-

maining

chords

are

E

(5/4,

PCL

=

8),

which

cycles pastpc

0,

F

(4/3, PCL =

8),

which

stops


starting

point,

and

K,

the


includes

several nstances of

pc

0,

but with

pc

1

in the

upper

voice.

The

series

of

cycles

based

on

successively

smaller ntervals

that return

to

the same

pitch-class

anchor

projects

the

idea ex-

pressed

in

the text

of

"atomic,

cosmic

cycles"

as

a

"spiral"pro-

gressing

inward from

cycles

of

larger

intervals

to those with


plete


cycles

that returnto

pc

0 when

continued one

step

further,

mplying

a

linkage

to

the bass note

of the



cycle

of

chord

C:

the final

upper

interval 5

(C#6-F#6,

m.

29 beat

1)

would be followed

by

interval

6

to

return

to

pc

0.

In

other


pc

0

occurs within the

chord,

either

as

the

upper

voice

or

as

a

result

of

extensive

repetition

of a small

PCL. Chord B

(7/6)

contains

pc

0 in both

top

and bottom

voices, its

only

pc duplication.

The

arpeggiated

tatementof H

(PCL

=

6)

in m. 30

continues far

beyond

its first

pc repetition

(E

3), cyclingthrough

several

pc

duplications

before

returning

to

pc

0

as the

highest

note. Chords hat

repeat

within

an

octave

and that

may


of

pc

0 are G

(4/2,

PCL

=

4),

I

(3/1,

PCL

=

6),

J

(1/2,

PCL

=

8),

and L

(2/2,

PCL

=

6);

of

these,

only

G contains

pc

0 in the

top

voice,

al-

though

each includes

pc

0


voices.

The re-

maining

chords

are

E

(5/4,

PCL

=

8),

which

cycles pastpc

0,

F

(4/3, PCL =

8),

which

stops


starting

point,

and

K,

the


includes

several nstances of

pc

0,

but with

pc

1

in the

upper

voice.

The

series

of

cycles

based

on

successively

smaller ntervals

that return

to

the same

pitch-class

anchor

projects

the

idea ex-

pressed

in

the text

of

"atomic,

cosmic

cycles"

as

a

"spiral"pro-

gressing

inward from

cycles

of

larger

intervals

to those with






78

Music

Theory

Spectrum

Example

17.

On the

Antipodes,

mm. 28-34.

78

Music

Theory

Spectrum

Example

17.

On the

Antipodes,

mm. 28-34.

78

Music

Theory

Spectrum

Example

17.

On the

Antipodes,

mm. 28-34.

Largo maestoso

28 $h

Largo maestoso

28 $h

Largo maestoso

28 $h

r-3 -

L_-

-3 -

L_-

-3 -

L_-

Man

we ask

you

Is Na

- ture

noth-ing

but a

-

tom

-

ic cos

-

mic

cy

- cles _____

loco"

*

f

I

$n/

E--Ip

i= =

fftt8

I

cresc.

it

-i

'

"

i

4

t"bi

Largo

maestoso

cresc.

(Org.

ad

lib.)

(16'

and

32' only)

Man

we ask

you

Is Na

- ture

noth-ing

but a

-

tom

-

ic cos

-

mic

cy

- cles _____

loco"

*

f

I

$n/

E--Ip

i= =

fftt8

I

cresc.

it

-i

'

"

i

4

t"bi

Largo

maestoso

cresc.

(Org.

ad

lib.)

(16'

and

32' only)

Man

we ask

you

Is Na

- ture

noth-ing

but a

-

tom

-

ic cos

-

mic

cy

- cles _____

loco"

*

f

I

$n/

E--Ip

i= =

fftt8

I

cresc.

it

-i

'

"

i

4

t"bi

Largo

maestoso

cresc.

(Org.

ad

lib.)

(16'

and

32' only)

(Org.Ped.)

_

-

-

[A

B

C D E F

G

(Org.Ped.)

_

-

-

[A

B

C D E F

G

(Org.Ped.)

_

-

-

[A

B

C D E F

G

H

smaller units

of

repetition.

Figure

10 illustrates

this

process,

which

primarily

reflects the

changes

in

interval

sizes-as

reflected

by

the sums-not the sizes

of the PCLs

or the

cardi-

nalities of

complete

combinations. Chord

A,

cycle

7/7,

forms

the

outer

layer

of the

spiral,

and its return

o

pc

0 after a com-

plete

revolution coincides with the

beginning

of the

next mem-

ber

of

the chord

succession,

the

7/6

cycle

of chord

B,

on the

next

inner

layer.

Each

successive

layer corresponds

to each

subsequent

chord,

following

he

process

of

gradual

eduction

n

sums,

so that each member of the chordal

cycle

is

represented

by

one revolutionin the

spiral.

The chords

spiral

"inward"

n

the first

half

of

the

pattern,

and then reverse

direction

o return

to the

outer

layer

at

the

conclusionof

the chord

sequence.

smaller units

of

repetition.

Figure

10 illustrates

this

process,

which

primarily

reflects the

changes

in

interval

sizes-as

reflected

by


of the PCLs

or the

cardi-

nalities of

complete

combinations. Chord

A,

cycle

7/7,

forms

the

outer

layer

of the

spiral,

and its return

o

pc

0 after a com-

plete


beginning

of the

next mem-

ber

of

the chord

succession,

the

7/6

cycle

of chord

B,

on the

next

inner

layer.

Each

successive

layer corresponds

to each

subsequent

chord,

following

he

process

of

gradual

eduction

n

sums,


cycle

is

represented

by


spiral.

The chords

spiral

"inward"

n

the first

half

of

the

pattern,

and then reverse

direction

o return

to the

outer

layer

at

the

conclusionof

the chord

sequence.

smaller units

of

repetition.

Figure

10 illustrates

this

process,

which

primarily

reflects the

changes

in

interval

sizes-as

reflected

by


of the PCLs

or the

cardi-

nalities of

complete

combinations. Chord

A,

cycle

7/7,

forms

the

outer

layer

of the

spiral,

and its return

o

pc

0 after a com-

plete


beginning

of the

next mem-

ber

of

the chord

succession,

the

7/6

cycle

of chord

B,

on the

next

inner

layer.

Each

successive

layer corresponds

to each

subsequent

chord,

following

he

process

of

gradual

eduction

n

sums,


cycle

is

represented

by


spiral.

The chords

spiral

"inward"

n

the first

half

of

the

pattern,

and then reverse

direction

o return

to the

outer

layer

at

the

conclusionof

the chord

sequence.

The

overall formof On the

Antipodes

s further

ied

to the

chordal

cycle

through

a

"composing

out"

process

that

distrib-

utes

the sonorities

nto the texture

muchas the

interval-5

ycle

structures

he introduction

of

In

re con moto

et

al

(Ex.

14).

As

explainedby Argento,

the

passages

between

statements

of the

chord

are

entirely

based

on the structures

of the chords

them-

selves,

so that almost

every aspect

of the

song

is connectedto

the

cyclicpattern.72

n the

two measures

mmediately

ollowing

the

initial

presentation

of the

chords,

for

example,

the

A,

B,

and C sonorities

account

for most

of

the

pitch

materials

of the

72"A

Digest

Analysis,"

198-200.

Only

mm. 14-17 are not derived

from the

chordal

cycle.

The


Antipodes

s further

ied

to the

chordal

cycle

through

a

"composing

out"

process

that

distrib-

utes

the sonorities

nto the texture

muchas the

interval-5

ycle

structures

he introduction

of

In

re con moto

et

al

(Ex.

14).

As


the

passages

between

statements

of the

chord

are

entirely

based

on the structures

of the chords

them-

selves,

so that almost

every aspect

of the

song

is connectedto

the

cyclicpattern.72

n the

two measures

mmediately

ollowing

the

initial

presentation

of the

chords,

for

example,

the

A,

B,

and C sonorities

account

for most

of

the

pitch

materials

of the

72"A

Digest

Analysis,"

198-200.

Only


from the

chordal

cycle.

The


Antipodes

s further

ied

to the

chordal

cycle

through

a

"composing

out"

process

that

distrib-

utes

the sonorities

nto the texture

muchas the

interval-5

ycle

structures

he introduction

of

In

re con moto

et

al

(Ex.

14).

As


the

passages

between

statements

of the

chord

are

entirely

based

on the structures

of the chords

them-

selves,

so that almost

every aspect

of the

song

is connectedto

the

cyclicpattern.72

n the

two measures

mmediately

ollowing

the

initial

presentation

of the

chords,

for

example,

the

A,

B,

and C sonorities

account

for most

of

the

pitch

materials

of the

72"A

Digest

Analysis,"

198-200.

Only


from the

chordal

cycle.

, <29 _f-^ 3----, <29 _f-^ 3----, <29 _f-^ 3----

r--

30

cresc.

I 1 30 I

r--

30

cresc.

I 1 30 I

r--

30

cresc.

I 1 30 I






Interval

ycles

as

Compositional

esources

79nterval

ycles

as

Compositional

esources

79nterval

ycles

as

Compositional

esources

79

Example

17.

(cont'd.)

xample

17.

(cont'd.)

xample

17.

(cont'd.)

L

J

K

J

K

J

K

accompaniment.

Example

18 locates

A andB in

m.

5

and

C

in

m. 6

with

indications

of

order

positions

within the

three

cycles

notated

beneath

the score.

Because

the

registral

placement

of

each

pitch

class

generallycorresponds

to

that

of the

original

chord

sequence,

these

occurrences

project

a kind

of

varied

ar-

peggiation

of

the source

sonorities.

Often,

this

produces

a tri-

chordal

distribution

f the

chords,

highlighting

single

trichord

type;

any

three

adjacentpitch

classes

in the B

and

C chords

of

Example

18,

for

example,

form

a

3-5

[0,1,6]

trichord,

and

these

are

prominently

displayed

n mm.

5 and

6,

observable

as

adjacent ops.

The

remaining

uncircled)

notes

then

fill out

the

accompaniment.

Example

18 locates

A andB in

m.

5

and

C

in

m. 6

with

indications

of

order

positions

within the

three

cycles

notated

beneath

the score.

Because

the

registral

placement

of

each

pitch

class


to

that

of the

original

chord

sequence,

these

occurrences

project

a kind

of

varied

ar-

peggiation

of

the source

sonorities.

Often,

this

produces

a tri-

chordal

distribution

f the

chords,

highlighting

single

trichord

type;

any

three

adjacentpitch

classes

in the B

and

C chords

of

Example

18,

for

example,

form

a

3-5

[0,1,6]

trichord,

and

these

are

prominently

displayed

n mm.

5 and

6,

observable

as

adjacent ops.

The

remaining

uncircled)

notes

then

fill out

the

accompaniment.

Example

18 locates

A andB in

m.

5

and

C

in

m. 6

with

indications

of

order

positions

within the

three

cycles

notated

beneath

the score.

Because

the

registral

placement

of

each

pitch

class


to

that

of the

original

chord

sequence,

these

occurrences

project

a kind

of

varied

ar-

peggiation

of

the source

sonorities.

Often,

this

produces

a tri-

chordal

distribution

f the

chords,

highlighting

single

trichord

type;

any

three

adjacentpitch

classes

in the B

and

C chords

of

Example

18,

for

example,

form

a

3-5

[0,1,6]

trichord,

and

these

are

prominently

displayed

n mm.

5 and

6,

observable

as

adjacent ops.

The

remaining

uncircled)

notes

then

fill out

the

measure

with

pitch-class

reiterations,

often

recalling

the

pri-

mary

ntervals

of

the source

sonority.

Derivations

n the

remainderof

the

song

follow

similar

pro-

cedures,

either

continuing

he

varied

arpeggiations

r

redistrib-

uting

the sourcemore

extensively,

while

continuing

o

uphold

distinctive

features

of

each

structure.

These

sources

appear

n

their

original

order

(Fig.

9),

so that

the

entire

song

projects

an

expanded,

elaborated

macrocosm

of

the

chordal

cycle.73

The

series

of derivations

elaborates

roughly

the first

half

of

the se-

measure

with

pitch-class

reiterations,

often

recalling

the

pri-

mary

ntervals

of

the source

sonority.

Derivations

n the

remainderof

the

song

follow

similar

pro-

cedures,

either

continuing

he

varied

arpeggiations

r

redistrib-

uting

the sourcemore

extensively,

while

continuing

o

uphold

distinctive

features

of

each

structure.

These

sources

appear

n

their

original

order

(Fig.

9),

so that

the

entire

song

projects

an

expanded,

elaborated

macrocosm

of

the

chordal

cycle.73

The

series

of derivations

elaborates

roughly

the first

half

of

the se-

measure

with

pitch-class

reiterations,

often

recalling

the

pri-

mary

ntervals

of

the source

sonority.

Derivations

n the

remainderof

the

song

follow

similar

pro-

cedures,

either

continuing

he

varied

arpeggiations

r

redistrib-

uting

the sourcemore

extensively,

while

continuing

o

uphold

distinctive

features

of

each

structure.

These

sources

appear

n

their

original

order

(Fig.

9),

so that

the

entire

song

projects

an

expanded,

elaborated

macrocosm

of

the

chordal

cycle.73

The

series

of derivations

elaborates

roughly

the first

half

of

the se-

73See

Argento,

"A

Digest Analysis."

3See

Argento,

"A

Digest Analysis."

3See

Argento,

"A

Digest Analysis."






80

Music

TheorySpectrum

0

Music

TheorySpectrum

0

Music

TheorySpectrum

Figure

10.

"Spiral"

f

combination

ycles.

igure

10.

"Spiral"

f

combination

ycles.

igure

10.

"Spiral"

f

combination

ycles.

Example

18.

On

the

Antipodes,

mm.

5-6.

Allegro

r--

3

3

Andante

Example

18.

On

the

Antipodes,

mm.

5-6.

Allegro

r--

3

3

Andante

Example

18.

On

the

Antipodes,

mm.

5-6.

Allegro

r--

3

3

Andante

quence,

chords

A

through

K

plus

a return

of

J,

between

the

statements

of the chordal

cycle

at

the

beginning

and

midpoint,

and then elaborates

the

remainder

of

the

sequence,

chords L

back

to

A,

preceding

the final statement

of

the

chordal

cycle.

(Thereturn o A initiates he finalstatementof thechords.) Cy-

clic

principles

hus

permeate

musical

relationships

n the verti-

cal,

or chordal

dimension,

and in the horizontal

dimension

as

the

"cyclic"palindromic

structureof the chord

pattern,

all

of

which extends

from small to

large

structural

evels.

By

impart-

ing

these

qualities

to

the

song,

Ives

likewise characterizes

na-

ture

as ordered and

logical despitecomplexities

andcontradic-

tions that

seemingly defy explanation.

Below

the

complicated

quence,

chords

A

through

K

plus

a return

of

J,

between

the

statements

of the chordal

cycle

at

the

beginning

and

midpoint,

and then elaborates

the

remainder

of

the

sequence,

chords L

back

to

A,

preceding

the final statement

of

the

chordal

cycle.


clic

principles

hus

permeate

musical

relationships

n the verti-

cal,

or chordal

dimension,


dimension

as

the

"cyclic"palindromic


pattern,

all

of

which extends

from small to

large

structural

evels.

By

impart-

ing

these

qualities

to

the

song,

Ives


na-

ture

as ordered and


andcontradic-

tions that


Below

the

complicated

quence,

chords

A

through

K

plus

a return

of

J,

between

the

statements

of the chordal

cycle

at

the

beginning

and

midpoint,

and then elaborates

the

remainder

of

the

sequence,

chords L

back

to

A,

preceding

the final statement

of

the

chordal

cycle.


clic

principles

hus

permeate

musical

relationships

n the verti-

cal,

or chordal

dimension,


dimension

as

the

"cyclic"palindromic


pattern,

all

of

which extends

from small to

large

structural

evels.

By

impart-

ing

these

qualities

to

the

song,

Ives


na-

ture

as ordered and


andcontradic-

tions that


Below

the

complicated

7/6)

p

0

1

2

3 4 5 6 7 8 9

10

11

/)

pc

7

2

8 3

9 4

10 5

11 6 0 7

/6)

p

0

1

2

3 4 5 6 7 8 9

10

11

/)

pc

7

2

8 3

9 4

10 5

11 6 0 7

/6)

p

0

1

2

3 4 5 6 7 8 9

10

11

/)

pc

7

2

8 3

9 4

10 5

11 6 0 7

op

0 1 2

3

4

5 6

7

8 9

(6/5) C:

pc

1 7 0

6 11 5 10

4

9 3

op

0 1 2

3

4

5 6

7

8 9

(6/5) C:

pc

1 7 0

6 11 5 10

4

9 3

op

0 1 2

3

4

5 6

7

8 9

(6/5) C:

pc

1 7 0

6 11 5 10

4

9 3

10 11

8 2

10 11

8 2

10 11

8 2

surface

filled with

antipodal

contrasts

s

a coherent

design

that

can be

understood as

"nothing

but.

. .

cycles."74

The

association,

in

On

the

Antipodes,

of

cyclicpitch

deriva-

74Quoted

rom

the

last

line of

the text

(see

Ex.

17).

This

interpretation

s

adapted

from

Hitchcock

(Ives:

A

Survey

of

the

Music,

17-18)

and Schoffman

("The

Songs

of

Ives,"

233).

surface

filled with

antipodal

contrasts

s

a coherent

design

that

can be

understood as

"nothing

but.

. .

cycles."74

The

association,

in

On

the

Antipodes,

of

cyclicpitch

deriva-

74Quoted

rom

the

last

line of

the text

(see

Ex.

17).

This

interpretation

s

adapted

from

Hitchcock

(Ives:

A

Survey

of

the

Music,

17-18)

and Schoffman

("The

Songs

of

Ives,"

233).

surface

filled with

antipodal

contrasts

s

a coherent

design

that

can be

understood as

"nothing

but.

. .

cycles."74

The

association,

in

On

the

Antipodes,

of

cyclicpitch

deriva-

74Quoted

rom

the

last

line of

the text

(see

Ex.

17).

This

interpretation

s

adapted

from

Hitchcock

(Ives:

A

Survey

of

the

Music,

17-18)

and Schoffman

("The

Songs

of

Ives,"

233).






Interval

ycles

as

Compositional

esources

81

nterval

ycles

as

Compositional

esources

81

nterval

ycles

as

Compositional

esources

81

tions

with natureand

natural

processes

realizes

a

philosophical

stance

common

to much

of Ives'smusic

and

thought.

Themes

relatedto nature,includingspecificpictorial magesandrefer-

ences

to

abstractnatural

orces,

appear

with

familiar

egularity

in the

texts

and

concepts

of his

compositions

and in the

philo-

sophical

positions

of his

writings,

ultimately

connecting

to

a

line

of Transcendentalist

hinking

n which nature

reflectsa

di-

vine

presence.75

n the same

specific

way

that a work

such as

On

the

Antipodes

displays

an

underlying

unity

of

cyclic

pitch

con-

struction,

his view

recognizes

a

cyclic

unity

n nature

embodied

in

planetary

orbits

andthe

resulting

cyclic passage

of time

over

the courseof a dayor year, aswell as in the "lifecycles"of liv-

ing

organisms.

Ives meant

to

make his most

extensive

state-

ment

on such

natural

laws

in the Universe

Symphony,

which

would

"tracewith tonal

imprints

he ... evolution

of all

life,

in

nature

of

humanity

rom the

great

roots

of life to the

spiritual

eternities."76

The

cyclic

structures

notated

on the

sketch

page

of the

Symphony

shown

in

Example

11 above

wereintended

to

represent

the

"body

of the

earth,"

and

subsequent

musical

re-

alizations

of these

structures

would

then

depict

the

formation

of the "rocks,trees andmountains"out of the initialrawmate-

rials.77

As the

Symphony

evolves,

the seminal

function

of the

cyclicpitch

structures

parallels

a

cyclic

underpinning

or natural

laws.

Implicit

n this

perspective,

and

an

important

aspect

of Ives's

cosmic

musical

metaphor,

is

a

juxtaposition

of the

concepts

of

evolution

and

revolution,

for

the

concept

of a

cyclic

revolution

does not

necessarily imply

a

complete

and

literal

return

to

a

point

of

origin,

exclusive

of

any

evolution

or

growth.

Rather,

a

75Ives's

adherence

to

"Transcendentalist"

eliefs,

as

expressed

in his Es-

says

before

A

Sonata,

recognizes

the

importance

of

nature

along

with other

central

hemes,

though

these do not

necessarily

orm a

coherent

"philosophy."

See

Burkholder,

Charles

ves: TheIdeasBehind the

Music,

20-32.

76Kirkpatrick,

atalogue,

27.

7Memos,

107.

tions

with natureand

natural

processes

realizes

a

philosophical

stance

common

to much

of Ives'smusic

and

thought.

Themes


ences

to

abstractnatural

orces,

appear

with

familiar

egularity

in the

texts

and

concepts

of his

compositions

and in the

philo-

sophical

positions

of his

writings,

ultimately

connecting

to

a

line


hinking

n which nature

reflectsa

di-

vine

presence.75

n the same

specific

way

that a work

such as

On

the

Antipodes

displays

an

underlying

unity

of

cyclic

pitch

con-

struction,

his view

recognizes

a

cyclic

unity

n nature

embodied

in

planetary

orbits

andthe

resulting

cyclic passage

of time

over


ing

organisms.

Ives meant

to

make his most

extensive

state-

ment

on such

natural

laws

in the Universe

Symphony,

which

would

"tracewith tonal

imprints

he ... evolution

of all

life,

in

nature

of

humanity

rom the

great

roots

of life to the

spiritual

eternities."76

The

cyclic

structures

notated

on the

sketch

page

of the

Symphony

shown

in

Example

11 above

wereintended

to

represent

the

"body

of the

earth,"

and

subsequent

musical

re-

alizations

of these

structures

would

then

depict

the

formation


rials.77

As the

Symphony

evolves,

the seminal

function

of the

cyclicpitch

structures

parallels

a

cyclic

underpinning

or natural

laws.

Implicit

n this

perspective,

and

an

important

aspect

of Ives's

cosmic

musical

metaphor,

is

a

juxtaposition

of the

concepts

of

evolution

and

revolution,

for

the

concept

of a

cyclic

revolution

does not

necessarily imply

a

complete

and

literal

return

to

a

point

of

origin,

exclusive

of

any

evolution

or

growth.

Rather,

a

75Ives's

adherence

to

"Transcendentalist"

eliefs,

as

expressed

in his Es-

says

before

A

Sonata,

recognizes

the

importance

of

nature

along

with other

central

hemes,

though

these do not

necessarily

orm a

coherent

"philosophy."

See

Burkholder,

Charles


Music,

20-32.

76Kirkpatrick,

atalogue,

27.

7Memos,

107.

tions

with natureand

natural

processes

realizes

a

philosophical

stance

common

to much

of Ives'smusic

and

thought.

Themes


ences

to

abstractnatural

orces,

appear

with

familiar

egularity

in the

texts

and

concepts

of his

compositions

and in the

philo-

sophical

positions

of his

writings,

ultimately

connecting

to

a

line


hinking

n which nature

reflectsa

di-

vine

presence.75

n the same

specific

way

that a work

such as

On

the

Antipodes

displays

an

underlying

unity

of

cyclic

pitch

con-

struction,

his view

recognizes

a

cyclic

unity

n nature

embodied

in

planetary

orbits

andthe

resulting

cyclic passage

of time

over


ing

organisms.

Ives meant

to

make his most

extensive

state-

ment

on such

natural

laws

in the Universe

Symphony,

which

would

"tracewith tonal

imprints

he ... evolution

of all

life,

in

nature

of

humanity

rom the

great

roots

of life to the

spiritual

eternities."76

The

cyclic

structures

notated

on the

sketch

page

of the

Symphony

shown

in

Example

11 above

wereintended

to

represent

the

"body

of the

earth,"

and

subsequent

musical

re-

alizations

of these

structures

would

then

depict

the

formation


rials.77

As the

Symphony

evolves,

the seminal

function

of the

cyclicpitch

structures

parallels

a

cyclic

underpinning

or natural

laws.

Implicit

n this

perspective,

and

an

important

aspect

of Ives's

cosmic

musical

metaphor,

is

a

juxtaposition

of the

concepts

of

evolution

and

revolution,

for

the

concept

of a

cyclic

revolution

does not

necessarily imply

a

complete

and

literal

return

to

a

point

of

origin,

exclusive

of

any

evolution

or

growth.

Rather,

a

75Ives's

adherence

to

"Transcendentalist"

eliefs,

as

expressed

in his Es-

says

before

A

Sonata,

recognizes

the

importance

of

nature

along

with other

central

hemes,

though

these do not

necessarily

orm a

coherent

"philosophy."

See

Burkholder,

Charles


Music,

20-32.

76Kirkpatrick,

atalogue,

27.

7Memos,

107.

cyclically

conceived

structure

provides

an

underlying

cohesive

framework

withinwhich

nonrepetitive

elements

may grow

and

evolve, just as a time periodsuch as a daycanencompassvast

changes

within

its

cyclic

boundaries.78

These

principles

are

clearly displayed

n In re con

moto

et

al,

where

virtually

every

measure

participates

in a

pervasive

constructional

scheme

basedon the

Prime

Series

and articulated

with the Grit

Chord,

yet

the methods

of

projecting

these

unifying

hreads

are com-

plex

and diverse.

Nature,

according

o

Ives,

despite

its funda-

mental

cyclic

character,

"loves

analogy

and

hates

repetition,"

and

any

musical

reflection

of natural

processes

wouldtherefore

reconcilethe necessityof growthwith the universality f cyclic

return 79

Ultimately,

Ives's

cyclic pitch

derivations

reflect

principles

of

pitch

structure

that

are

richly

attractive

to

a

composer

searching

or nontonal methods

of

organization.

Extramusical

considerations,

such as

points

of action

provided

by

a

program

or

scenario,

might suggest

a structural

ramework

n the

ab-

sence

of

tonality,

but would not

provide

the

inherent

and

natu-

ral

logic

of the tonal

system.

With

cyclic

intervallic

repetitions

the techniques of pitch organization, including methods of

pitch-class

exhaustion,

are

given

a

logical

and natural

mpetus

from the

varioussubdivisionsof the octave

into

equal parts

and

the extension

of these

principles

to

produce

cyclic

combina-

78Audrey

Davidson makes

this

point

in

"Transcendental

Unity

in the

Worksof Charles

Ives,"

American

Quarterly

2

(Spring

1970),

35-44.

79CharlesE. Ives,

Essays before

A Sonata, TheMajority,and OtherWrit-

ings,

ed.

Howard

Boatwright

New

York:

Norton,

1970),

22.

"Repetition"

s

a

frequent

target

of criticism

n Ives's

writings,

especially

when

referring

o the

conventions

of

common-practice

onality.

In

his

essay

on

quarter-tone

har-

mony,

for

example,

he

criticizes

"the

drag

of

repetition

n

manyphases

of

art,"

finding

an

absence of an essential

"organic

low"

(Essays, 115).

The

repetitive

nature

of

many

of his

compositional

experiments,

however,

makes

t clearthat

he does

not

reject

all forms

of

repetition;

rather,

he

objects

to

easy

reliance

on

traditional

musical

materials-repetition

without

nspiration.

cyclically

conceived

structure

provides

an

underlying

cohesive

framework

withinwhich

nonrepetitive

elements

may grow

and


changes

within

its

cyclic

boundaries.78

These

principles

are

clearly displayed

n In re con

moto

et

al,

where

virtually

every

measure

participates

in a

pervasive

constructional

scheme

basedon the

Prime

Series

and articulated

with the Grit

Chord,

yet

the methods

of

projecting

these

unifying

hreads

are com-

plex

and diverse.

Nature,

according

o

Ives,

despite

its funda-

mental

cyclic

character,

"loves

analogy

and

hates

repetition,"

and

any

musical

reflection

of natural

processes

wouldtherefore


return 79

Ultimately,

Ives's

cyclic pitch

derivations

reflect

principles

of

pitch

structure

that

are

richly

attractive

to

a

composer

searching

or nontonal methods

of

organization.

Extramusical

considerations,

such as

points

of action

provided

by

a

program

or

scenario,

might suggest

a structural

ramework

n the

ab-

sence

of

tonality,

but would not

provide

the

inherent

and

natu-

ral

logic

of the tonal

system.

With

cyclic

intervallic

repetitions


pitch-class

exhaustion,

are

given

a

logical

and natural

mpetus

from the


into

equal parts

and

the extension

of these

principles

to

produce

cyclic

combina-

78Audrey

Davidson makes

this

point

in

"Transcendental

Unity

in the

Worksof Charles

Ives,"

American

Quarterly

2

(Spring

1970),

35-44.

79CharlesE. Ives,

Essays before


ings,

ed.

Howard

Boatwright

New

York:

Norton,

1970),

22.

"Repetition"

s

a

frequent

target

of criticism

n Ives's

writings,

especially

when

referring

o the

conventions

of

common-practice

onality.

In

his

essay

on

quarter-tone

har-

mony,

for

example,

he

criticizes

"the

drag

of

repetition

n

manyphases

of

art,"

finding

an


"organic

low"

(Essays, 115).

The

repetitive

nature

of

many

of his

compositional

experiments,

however,

makes

t clearthat

he does

not

reject

all forms

of

repetition;

rather,

he

objects

to

easy

reliance

on

traditional

musical


without

nspiration.

cyclically

conceived

structure

provides

an

underlying

cohesive

framework

withinwhich

nonrepetitive

elements

may grow

and


changes

within

its

cyclic

boundaries.78

These

principles

are

clearly displayed

n In re con

moto

et

al,

where

virtually

every

measure

participates

in a

pervasive

constructional

scheme

basedon the

Prime

Series

and articulated

with the Grit

Chord,

yet

the methods

of

projecting

these

unifying

hreads

are com-

plex

and diverse.

Nature,

according

o

Ives,

despite

its funda-

mental

cyclic

character,

"loves

analogy

and

hates

repetition,"

and

any

musical

reflection

of natural

processes

wouldtherefore


return 79

Ultimately,

Ives's

cyclic pitch

derivations

reflect

principles

of

pitch

structure

that

are

richly

attractive

to

a

composer

searching

or nontonal methods

of

organization.

Extramusical

considerations,

such as

points

of action

provided

by

a

program

or

scenario,

might suggest

a structural

ramework

n the

ab-

sence

of

tonality,

but would not

provide

the

inherent

and

natu-

ral

logic

of the tonal

system.

With

cyclic

intervallic

repetitions


pitch-class

exhaustion,

are

given

a

logical

and natural

mpetus

from the


into

equal parts

and

the extension

of these

principles

to

produce

cyclic

combina-

78Audrey

Davidson makes

this

point

in

"Transcendental

Unity

in the

Worksof Charles

Ives,"

American

Quarterly

2

(Spring

1970),

35-44.

79CharlesE. Ives,

Essays before


ings,

ed.

Howard

Boatwright

New

York:

Norton,

1970),

22.

"Repetition"

s

a

frequent

target

of criticism

n Ives's

writings,

especially

when

referring

o the

conventions

of

common-practice

onality.

In

his

essay

on

quarter-tone

har-

mony,

for

example,

he

criticizes

"the

drag

of

repetition

n

manyphases

of

art,"

finding

an


"organic

low"

(Essays, 115).

The

repetitive

nature

of

many

of his

compositional

experiments,

however,

makes

t clearthat

he does

not

reject

all forms

of

repetition;

rather,

he

objects

to

easy

reliance

on

traditional

musical


without

nspiration.






82 Music

Theory Spectrum

2 Music

Theory Spectrum

2 Music

Theory Spectrum

tions.

Coupled

with

their

analogies

to forces of

nature,

the

cy-

clic

procedures provide

a

fertile area for

experimentation,

and

they

constitute a central

organizing principle

for some of Ives's

most

profound

musical

expressions.

ABSTRACT

In the

"experimental"

music of

Charles

Ives,

interval

patterns

and

particular

ntervallic

combinations often

serve as

primary

structural

forces.

Particularly revalent

are

cyclic

ntervallic

repetitions,

reflect-

tions.

Coupled

with

their

analogies

to forces of

nature,

the

cy-

clic

procedures provide

a

fertile area for

experimentation,

and

they



for some of Ives's

most

profound

musical

expressions.

ABSTRACT

In the

"experimental"

music of

Charles

Ives,

interval

patterns

and

particular

ntervallic

combinations often

serve as

primary

structural

forces.


are

cyclic

ntervallic

repetitions,

reflect-

tions.

Coupled

with

their

analogies

to forces of

nature,

the

cy-

clic

procedures provide

a

fertile area for

experimentation,

and

they



for some of Ives's

most

profound

musical

expressions.

ABSTRACT

In the

"experimental"

music of

Charles

Ives,

interval

patterns

and

particular

ntervallic

combinations often

serve as

primary

structural

forces.


are

cyclic

ntervallic

repetitions,

reflect-

ing

compositional

concerns for coherence

through repetition

and

pitch-class variety.

Ives

experiments

both with the familiar

nterval

cycles and with cycles of two alternating ntervals,or "combination

cycles."

Musical

usages

include

straightforward

yclic presentations

as

well

as

developments

of

cyclically generated

structural frame-

works.

In his

most

sophisticatedcyclicexperiments,including

In re

con moto et

al,

Onthe

Antipodes,

and the Universe

Symphony,cycles

and

cyclic

principles

are

central to

the

musicaland

metaphorical

ub-

stance,

mirroring

pervasive

elements

of

Ives's attitudes

toward art

and nature.

ing

compositional


through repetition

and


Ives

experiments


nterval


cycles."

Musical

usages

include

straightforward

yclic presentations

as

well

as

developments

of


structural frame-

works.

In his

most


In re

con moto et

al,

Onthe

Antipodes,

and the Universe

Symphony,cycles

and

cyclic

principles

are

central to

the

musicaland

metaphorical

ub-

stance,

mirroring

pervasive

elements

of

Ives's attitudes

toward art

and nature.

ing

compositional


through repetition

and


Ives

experiments


nterval


cycles."

Musical

usages

include

straightforward

yclic presentations

as

well

as

developments

of


structural frame-

works.

In his

most


In re

con moto et

al,

Onthe

Antipodes,

and the Universe

Symphony,cycles

and

cyclic

principles

are

central to

the

musicaland

metaphorical

ub-

stance,

mirroring

pervasive

elements

of

Ives's attitudes

toward art

and nature.




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Lambert, Interval Cycles, Spectrum

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