Lewin - Some Compositional Uses of Projective Geometry

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Perspectives of New Music

Some Compositional Uses of Projective GeometryAuthor(s): David LewinSource: Perspectives of New Music, Vol. 42, No. 2 (Summer, 2004), pp. 12-63Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/25164553

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2/53

Some Compositional Uses

of

Projective Geometry

Le?7ta

David Lewin

Part

I: Introduction

to the

Projective

Plane;

Diatonic Modal

Examples

Imagine

a

school

that has

seven

students:

Ann, Bill, Carol,

Dan,

Eve,

Frank, and Gladys. Suppose the school offers seven courses: Latin,

Math,

Neurology, Psychology,

Quantum mechanics, Russian,

and

Span

ish.

Suppose

the enrollment of

students

in

courses

is

given by

Example

1.

On

the

example,

each

asterisk

indicates

that

the

student whose

initial

appears

to

the left

is

taking

the

course

whose

initial

appears

above.

Thus

Ann

is

taking

Math,

Neurology,

and

Quantum mechanics;

the students

enrolled

in

Spanish

are

Carol, Frank,

and

Gladys;

and

so

forth.

http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/53

ProjectiveGeometry

13

A

~B~

~C

~D

~E

~F~

~G

M

N

Q

R

EXAMPLE

1

Example

1

has the

following

properties:

1.

Any

two

students

have

just

one course

in

common.

2.

Any

two

courses

have

just

one

student

in

common.

3.

Every

course

has

at

least

three

students;

every

student is

taking

at

least three

courses.

4. It is

not

the

case

that

every

student

is

taking

every

course.

The

properties

manifest

a

structure

called

a

"projective

plane."

To

appreciate

the

geometric metaphor,

we can

replace

the word "student"

by

the

word

"point,"

the word

"course"

by

the word

"line,"

and

the

notion

of

a

student's

taking

a course

by

the notions

of

a

point's "lying

on"

a

line and

(equivalently)

of

the line's

"passing through"

the

point.

The properties above then translate as follows:

1.

Any

two

points

lie

on

just

one

line.

2.

Any

two

lines

pass

through just

one

point.

3.

Every

line

passes

through

at

least three

points;

every

point

lies

on

at

least

three

lines.

4. It is

not

the

case

that

every

point

lies

on

every

line.



4/53

14

Perspectives

of

New Music

As

translated,

the

properties

suggest

things

we

apprehend

about

"plane geometry."

The

things

have

to

do in

particular

with incidence

relations

of

points

and lines.

Only

property

2

departs

from

Euclidean

intuition:

in

Euclidean

plane

geometxy^parallel

lines

do

not

pass

through

any

common

point. Projective

geometry,

unlike Euclidean

geometry

(or

"affine

geometry")

does

not

recognize

the

notion

of

parallel

lines.

To

turn

the

Euclidean

plane

into

a

projective plane

one can

adjoin

a

"line

at

infinity,"

comprising

formal

"points"

where

parallel

lines

meet.

In

the

Euclidean

plane

one

fixes

a

reference

line;

in

a

standard

Cartesian

coordinate

system

this could be the horizontal X axis. For each number a

between 0 inclusive

and

180

exclusive,

stipulate

a

corresponding

"point

at

infinity."

Each line in the Euclidean

plane

will

be

tilted

with

respect

to

the reference

line

at some

angle

a

degrees;

the

given

line

meets

the

line

at-infinity

at

point

a.

So

a

line

perpendicular

to

the

reference

line

meets

the

line-at-infinity

at

point

90.

A line

parallel

to

the

reference

line

meets

the

line-at-infinity

at

point

0. More

generally,

two

parallel

lines will both

be

tilted

the

same

amount

with

respect

to

the

reference

line;

if

the

com

mon tilt is a degrees, then the two linesmeet at infinite-point a. The

extended

structure

is called the "Real

Projective

Plane." The

reader

can

verify

that it satisfies

properties

1

through

4

above.1

A

projective plane

is

highly

structured,

and

that

structure

can

be musi

cally suggestive

in

various

contexts.

As

a

first

illustration,

let

us

return to

Example

1

and

now

read the

seven

"students"

as

the

seven

white-note

pitch

classes.

The

seven

"courses" then

become

a

family

of

seven

privi

leged

Chords,

the

Chords

L

through

S

displayed

in

Example

2.

The four

properties now read:

1.

Any

two notes

belong

to

just

one

Chord.

2.

Any

two

Chords

have

just

one common

tone.

3.

Every

Chord has

at

least

three

notes;

every

note

belongs

to

at

least

three Chords.

4. It

is

not

the

case

that

every

note

belongs

to

every

Chord.

L M

N

P

Q

R

S

EXAMPLE

2



5/53

Projective Geometry

15

EXAMPLE

3

Indeed

in

this

particular

geometry

every

Chord

has

exactly

three

notes,

and

every

note

belongs

to

exactly

three

Chords.

Chords

L, M,

and

N,

taken

in

conjunction,

suggest

Phrygian modality;

Chord

P,

when

suitably

linearized,

is

appropriate

for

executing

a

melodic

Phrygian

cadence.

Obviously

I

constructed

the

example

with these features in

mind/ear.

Later

on,

I

shall show how

I

did

that.

Meanwhile

I

shall

present

a

Lehrst?ck

that

demonstrates

how

various

aspects

of

the

geometry

can

be

projected compositionally.

After

setting

up

the seven-note seven-chord

projective plane

as in

Examples

1 and

2,

to

build

a

species

of

"Phrygian

mode,"

I

decided

to

write

a

three-part

vocal

piece

where

every

three-note

verticality

would be

a

Chord.

I

decided

to

base this

piece

on a

cantus

firmus,

and had the idea

of

com

posing

the

cantus

as

a

twenty-one-note

tune

comprising

concatenated

linearizations

of the

seven

three-note

Chords.

Example

3 shows the

can

tus,

shaped

to

have

a

"Phrygian"

character

(and

somewhat

adjusted

as

the

composition

went

into later

stages).

As

shown

by

the brackets and

letters at the bottom of the

example,

the first three notes of the cantus

linearize

Chord

M,

the

next

three

linearize Chord

L,

and

so

on;

the

melodic cadence

is

achieved

through

a

linearized form of Chord

P.

Imagining

a

vocal

piece,

I

looked for

a

suitable

text.

I

tried

to

find

a

stylistically appropriate

twenty-one-syllable

text

that could be

sung

to

the

tune

of the

cantus

firmus.

Example

4,

setting

three

seven-syllable

verses

of

text,

seemed

apt. (I

modified

my

original

cantus

firmus

slightly,

the

better

to

accommodate

this

text.)

Ben

-

e

-

die

-

tus

qui

ven

-

it

in

no

-

mi

-

ne

Do

-

mi

-

ni,

in

no

-

mi

-

ne

Do

-

mi

-

ni.

EXAMPLE

4



6/53

16

Perspectives

of

New

Music

I wanted to harmonize each note of the cantus firmuswith each of the

three

Chords

to

which

it

belongs. Example

5a

sketches the

idea.

Wsym

bolizes

the

note

of

the

cantus

firmus;

the lines

on

Example

5

a

indicate

the three Chords

containing

W,

that

is

{W,

XI,

X2],

{W,

Yl,

Y2\,

and

[W,Z1,Z2\.

a)

b)

c)

/W\ ,WV /Wl

W2

XI Yl Zl XI-Yl-Zl XI-Yl-W3

III

III II

XI Yl 72 X2?Y2-22 X2-Y2-W3

EXAMPLE

5

It is

not

hard

to

see

that

the

seven

"points"

of

Example

5a

must

be

the

seven

distinct

notes

of the

mode.2

One

naturally

hopes

to

be

able

to

arrange

matters

as

in

Example

5b,

so

that

{XI,

Yl,

Zl]

and

{X2,

T2,

Z2\

will

be

Chords

(i.e.,

"lines"

of the

geometry), projected

as

vocal lines in

the

two

accompanying

voices.

But

this

is

not

possible.

The line deter

mined

by

XI

and

Yl will

indeed contain

one

of

the

Z-points,

but that

same

Z-point

(not

the

other

one)

will also be the

third

point

on

the

X2

72 line.3

This feature

of

the

system suggests

the

compositional algorithm

sketched

by Example

5c.

Here Wl

symbolizes

the

present

note

of the

cantus

firmus,

and

W2

symbolizes

the

next

note

of

the

cantus.

(The

can

tus was

constructed

so

as

not to

have

repeated

notes.)

If

W3

is the

third

note

of

the Chord

containing

Wl

and

W2

(the

third

point

on

the

line

containing

Wl and

W2),

then the

X-notes

and

T-notes

can

be

arranged

so

that

X1-Y1-W3

is

a

geometric

line

(i.e.,

a

Chord),

and

X2-Y2-W3

is

also a geometric line (a Chord). Indeed, this arrangement isunique, up

to

contrapuntal

inversion of "voice

1"

and "voice

2,"

and/or

exchange

of

notes

Xn

with

notes

Tw.4

The basic

algorithm

of

Example

5

c

produced

the

compositional

study

of

Example

6.

Other

features

of

the

projective

geometry suggest

other

compositional

possibilities.

Consider for

instance the fact

that

every

pair

of distinct

notes

belongs

to

exactly

one

Chord,

to

which

exactly

one

other

note

also

belongs.

This

enables

one to

add

a

third

part automatically,

to

any two-part



7/53

Projective

Geometry

17

nit,

ui

ve

-

-

-

nit,

n

m

mi

-

-

ni_

Do-mi

ni,_Do

-

mi

-

ni,

_

Do

- -

mi

-

ni,.

in_

no

-

-

mi

-

- -

ne inno

-

mi

-

ne

_

EXAMPLE

6



8/53

18

Perspectives

of

New Music

example

7

diatonic

counterpoint

that does

not

contain

a

vertical

octave

or

unison.

Example

7

illustrates,

adding

a

third

part

to

an

example

from

Zarlino.5

The example illustrates his "third mode"; given our particular Chord

vocabulary,

it is

of

course

natural

to

select

a

two-part

model

from

one or

another E-final mode. The exercise could be made

more

elegant

if

it did

not

cling

so

doggedly

to

the

notion

that

every

three-note

verticality

had

to

be

a

Chord

of the

geometric

system;

protocols

for non-Chordal verti

calities

in

three

voices

could

of

course

be

worked

out.

Another

resource

for

composition

in

our

geometric

mode

is

"modula

tion"

to

secondary

modes. This

can

be

accomplished

in

several

ways. First,

the Chords of

Example

2

can

all

be

transposed by

a

fixed number of semi



9/53

Projective

Geometry

19

a) b)

L'

W

N'

P'

?r

R' S' V

M' N' P'

Q

R'

S'

(M)

(Q)

(R)

(Q)

(P)

EXAMPLE

8

tones,

introducing

ficta

as

appropriate.

An

appropriate

interval

of

transpo

sition is

five

semitones

(Example

8a):

that

is

because

the

two

harmonic

triads

of

themode

are

related

by

T5;

also the

transposition

introduces

only

one ficta tone. The a-minor triad serves as

pivot:

Chord V of the trans

posed

mode

coincides with

Chord

M

of

the

original

mode.Example

8b

shows

the

original

mode

transposed

by

T2.

This introduces ficta

tones

F|

and

C|.

Chord

5'of the

transposed

mode

coincides with Chord

Q

of the

original

mode.

Example

8c

shows

a

particularly

idiomatic

pitch-class

inver

sion of

the

original

mode,

inversion-about-D. No

ficta

tones

are

intro

duced,

and

there

are

many

pivot-Chords:

P'and

il'coincide

with R and P

respectively;

Q

'coincides with

Q.

The

mode

of

Example

8c could be taken

in itself as a sort of

"Ionian/C-major"

structure.

That

suggests

another

sort

of

modulation

one can

make,

from the

original

mode

of

Example

2.

Namely,

one can

modulate

to

other diatonic

modes,

modes

given

by

other choices of

Chords,

to

realize

a

projective

plane

with the

seven

white

notes

as

"points."

One

can

easily

construct

large

numbers of such modes.

For

that

purpose,

the

abstract

configura

tion of

Example

9 is useful.

First

one

selects three basic

Chords,

which

one

might

think

of

as

"pri

mary." These are the Chords corresponding to lines 123, 345, and 561

on

the

figure.

The

primary

Chords

must

be

selected

so

that

any

two

of

them

have

exactly

one common

tone.

C-, G-,

and

F-major

triads could

not

be selected

as

the

three

Chords,

because the G

and the F triads

have

no

common

tone.

One

could,

however,

select

{C,

E,

G}, {F,A,

C},

and

{G, F,

B}.

Arranging

the

notes

to

suit

the

configuration

of

Example

9,

one

could take

C

=

1,

G

=

3,

F

=

5,

E

=

2,

B

=

4,

A

=

6. Then

the

remain

ing

note,

D,

would be

number 7 in

the

example.



10/53

20

Perspectives

of

New Music

If

123, 345,

and

561

are

all

formal

"lines"

of

the

geometry,

then

so are

174,

376, 572,

and 246.

example

9

The choice of the "basic triangle" on Example 9 exhausts six notes of

the

mode.

The seventh

note

appears

at

the

center

of the

triangle.

It

must

be

the

case

that

174, 376,

and 572

are

all formal

"lines" of

the

geometry;

the

corresponding

notes

will then be

formal Chords

of

the

system.6

Finally,

it

must

be the

case

that 246 is

a

formal

"line";

the

corresponding

notes

then

form the

seventh

Chord.7

(a) (b)

(c)

"Ionian/Cmaj";

"Dorian";

primary "Mixolydian";

primary

Chords

CEG,

Chords

DFA, GBD,

primary

Chords

GBD,

FAC, GBF;

other

EGA;

other

Chords

DFA, GAC;

other

Chords

BCD,

DGA,

CDE, ABC, FGC,

Chords

CDE, EFG,

DEF,

EAB.

BEF.

EAB,

BCF.

EXAMPLE

10



11/53

Projective Geometry

21

Example 10a shows themode generated by the "primary Chords" {C,

E,

G},

{F,A, C},

and

{G,

F,

B},

as

begim

above.

This

turns

out

to

be the

"Ionian/C-maj"

mode

encountered earlier in

Example

8c;

it

was

the

inversion

of

our

original

"Phrygian"

mode about

D.

Example

10b shows

a

"Dorian"

mode

generated by

taking

{D, F,

A},

{G, B, D},

and

{E,

G,

A}

as

primary

Chords.

Example

10b

inverts about D

to

yield

Example

10c,

a

"Mixolydian"

mode with

primary

Chords

{G, B, D},

{D,

F,

A},

and

{G,

A,

C}.

Chords

DFA,

GBD,

and

CDE

serve as

pivots

between

our

Dorian

and

Mixolydian modes;

FGC

pivots

as a

Chord

for

both

our

Phrygian

and

our

Dorian mode.

There

is

of

course

little

reason,

other than

historicist

sentimentality,

to

assign

special priority

to

harmonic

triads

as

Chords,

"primary"

or

other

wise,

in

constructing

any

formal

diatonic mode

(i.e.,

geometry).

Further

techniques

of

"modulation"

are

available.

Some mathemati

cally

idiomatic

ones

will

be

discussed

later,

"collineations"

in

particular.

* *

In

any

projective

plane,

there

are

many

ways

of

choosing

a

structure

almost

like that of

Example

9,

a

structure

consisting

of

seven

points

{1,2,

3, 4,

5,

6, 7}

in

which

123, 345, 561,

174, 376,

and

572

are

all

formal

"lines" of

the

geometry.

In

general,

246 need

not

be

a

formal

line.

We

shall

call

any

such

structure

a

"reference

quadrangle."

A

generic

reference

quadrangle

may

be

constructed

as

follows.

Given

any

projective plane

(whether

a

seven-point

plane

or

some

other),

select

"1" and "3" as any two distinct

points.

(Any line contains at least three

distinct

points.)

Since

not

all

points

of

the

plane

lie

on one

line,

select

"5"

as

any

point

not

on

the

line 13.

The

lines

13, 35,

and 15

are

all distinct.

(If

any

two

coincided,

then

points

1,3,

and 5 would all

be

collinear,

and

point

5

would lie

on

line

13,

contrary

to

construction.)

So,

in

particular,

lines

13 and 15

are

dis

tinct. Let

L

be

a

third line

passing

through

point

1.

(Every point

lies

on

at

least

three

distinct

lines.)

Take "4"

to

be the

point

where

line

L

meets

line 35. Point 4 isnot the same as point 3, since Lis distinct from line 13.

Point

4

is

not

the

same as

point

5,

since

L

is

distinct from

line

15.

Line L

contains

at

least three

distinct

points,

including

points

1

and 4.

Take "7"

to

be

a

third

point

on

line L.

Then 7

cannot

lie

on

any

of

the

other lines

so

far constructed

(13,

15,

or

35).

If

7

were on

line

13,

then

line

17,

which is

L,

would

be the

same

as

line 13?but L

was

taken

to

be

distinct from

line 13. If

7

were on

line

15,

then

line

17,

which

is

L,

would be

the

same as

line

15?but

L

was

taken

to

be

distinct

from

line



12/53

22

Perspectives

of

New Music

Jl

=

[p2,p3, ql, ri} Kl

-

[pi, ri, s2, s3)

J2

=

[pi,

p3, q2,

r2\

K2

=

[p2,

r2, si,

s3}

J3

=

[plyp2,

q3,

r3]

K3

=

[p3,

r3,

si,

s2]

Ll

=

[pi, ql,

si,

t)

Ml

=

[q2, q3,

ri,

si)

L2

=

[p2,

q2,

s2, t)

M2

=

[ql,

q3,

r2,

s2\

L3

=

[p3, q3,

s3,

t]

M3

=

[ql,

q2,

r3,

s3)

N=[rl,r2,r3,t\

EXAMPLE

11

15. If 7

were on

line

35,

7 would be the

point

where

L

meets

line

35?

but that

point

is

point

4,

and

7

was

chosen

to

be

distinct

from

4.

Take

"2"

to

be the

point

where line 57

meets

line

13. Point

2

is

dis

tinct from

point

3: if2

=

3,

then line 25

=

line

35,

and

point

7 would lie

on

line 35?which it doesn't. Point 2 is distinct from

point

1:

if 2

=

1,

then line 25

=

line

15,

and

point

7 would lie

on

line 15?which

it

doesn't.

So

1, 2,

and 3

are

different

points

of line 13.

In like

manner,

taking

"6"

to

be the

point

where line 37

meets

line

15,

we see

that

1,

6,

and 5

are

distinct

points

of

line

15.

In

sum,

points

1

through

7

are

all

distinct; 123, 345,

561,

275,

471,

and

673

are

all dis

tinct lines.

In

the

general

projective

plane,

it

need

not

be the

case

that

points

2, 4,

and 6

are

collinear.

Part

II:

Some Other

Projective

Planes,

with

Musical Applications

A

generic thirteen-point projective plane

can

be

conceptualized

as

fol

lows. Let the

"points"

of

the

system

be

symbolized

as

pi, p2, p3, ql, q2,

q3,

rl, r2,

r3,

si,

s2,

s3,

and

t.

Then take

as

the thirteen "lines"

the

sets

of

points

J1,J2,

etc.

given by Example

11.

The given thirteenpoints, togetherwith the stipulated thirteen lines,form

a

projective plane.

The reader

may

check

the four

required properties:

1.

Any

two

points

lie

on

just

one

line.

2.

Any

two

lines

pass

through

just

one

point.

3.

Every

line

passes

through

at

least three

points;

every

point

lies

on

at

least

three lines.



13/53

Projective Geometry

23

4. It is

not

the

case

that every point lies

on

every line.

In the

thirteen-point

plane,

every

point

lies

on

exactly

four lines and

every

line

passes

through

exactly

four

points.

The

thirteen-point plane

can

be used

to

structure

"chromatic"

modes,

if

one

adjoins

a

special

symbol

$

to

the

twelve

chromatic

pitch

classes.

The

special

symbol

could

be realized

musically

as

silence,

or as a non

pitched

event,

or as a

wild card.

Example

12

shows the

structure

of

one

such

mode.

pl=B,p2

=

G,p3=?, ql=D,

#2=Bt,

q3

=

G$,

ri=C, r2=E,

r3-Cf,

?i=Et,?2=Ff,.tf

=A,?

=

$.

/i={G,F,D,C}

tfi?{B,C,Ff,A}

J2

=

{B,

F,

Bt,

E}

K2

=

{G,

E,

Et,

A}

J3

=

{B,G,

Gf, Cf}

K3

-

{F,

Cf,

Et, Ff}

LI -

{B,

D,

Et,

$}

Ml =

{Bt,Gt,

C,

Et}

L2

=

{G,

Bt,

F|,

$}

M2

-

{D,

Gf,

E,

Ff}

L3

=

{F,

Gf,

A, $}

M3

=

{D,

Bt, C|,

A}

tf={C,E,Cf,$}

EXAMPLE

12

To

construct

the

mode of

Example

12

I

proceeded

as

follows.

I

decided

to

make the

special

symbol

$

the

point

t

of

Example

11. I

then

considered

the

four

lines

of

Example

11

that

would contain the

special

symbol

t

=

$.

These four

formal lines would be realized

in

the musical

mode

by

formal

Chords

thatwould each

comprise

three

"genuine

notes"

plus

the

special

effect.

I

decided

to structure

the mode

so

that those four

Chords

would be

transposed

or

inverted

forms,

each of

any

other.

I

spe

cificallydecided thatLI

=

[pi, ql, si, t)would be {B,D, Et, $}, that L2

=

[p2, q2,

s2,

t)

would be

{Ff,

G,

Bt,

$},

that L3

=

[p3,

q3,

s3,

t)

would be

{F,

Gf,

A,

$},

and

that

N

-

[rl,

r2,

r3,

t)

would be

{C,

Cf,

E,

$}.

Thus

the three

notes

B, D,

and

Et

would

be

assigned

in

some

order

as

the

points

pi,

ql,

and si

of

Example

11;

likewise the

three

notes

Ff,

G,

and

Bt

would

be

assigned

in

some

order

as

the

points

p2, q2,

and s2

of

Example

11,

and

so

forth.

Playing

around with

various

possibilities

for

those

assignments,

considering

the other

Chords

that

Example

11

would



14/53

24

Perspectives

of

New Music

generate in each case, I arrived

at

themode of Example 12. Evidently

an

enormous

number of such modes

can

be

constructed.

(t-1-1)-1-p3-s3v

rl

r

-s

/-qJv pi-p

I-r3-(q3-q3-q3-q3)

rl-p

I-P3-(ql?ql-ql-ql)-J--ql

r3-ql-$3'

s2-rl-qV

si

EXAMPLE

13

Example

13 demonstrates

an

algorithm

for

harmonizing

certain

types

of

cantus

firmus.

The

example

supposes

the

succession

t,

q2, q3

as a

seg

ment of such a cantus firmus.All vertically aligned sets of four events in

the

example

are

Chords of the

generic

system.

At the

left,

the

cantus

fir

mus

event

tis harmonized

by

the four vertical Chords that contain it:

{t,

rl,

r2, r3}, {t,

si,

pi,

ql],

{t,

q3, p3,

s3],

and

(moving

onwards)

{t,

p2, q2,

s2).

The

voice

leading

can

be

arranged

so

that the three

accompanying

voices

of the first three chords all form formal lines with

q2,

the

next

event

of the

cantus

firmus. That

is?as

indicated

by

the horizontal and

diagonal

lines

in

Example

13?[rl,

si,

q3, q2)

is

a

formal

line;

so

is

\r2,

pi, p3, q2}y

and

so

is

[r3,

ql,

s3,

q2\. q2

now

takes

over as

cantus

firmus

event,

and it is first armonized

by

the

(unique)

chord that contains both

q2

and

t.

The

process

then

repeats,

aiming

at

q3,

the

next event

of

the

cantus

firmus.

Evidently,

in

this

arrangement,

it is

necessary

for

q3

not to

be

p2

or

s2?i.e.,

q3

must not

lie

on

the

line

defined

by

t

and

q2.

And

of

course

t

and

q2

must

be

distinct.

The

cantus

is thus

constrained

as

follows:

it

must

contain

no

immediately

repeated

events,

and

no

three consecutive

events

may

be collinear.

In

Example

13, the three initial lines of "alto," "tenor," and "bass"

voices,

under

t

in the

cantus

firmus,

become

verticalities when

q2

takes

over

the

cantus

role:

the

alto's initial

line

rl-sl-q3-q2

becomes the last

four-point

verticality

shown

in

the

example;

the tenor's

initial

line

r2-pl

p3-q2

becomes the fifth four-note

verticality,

and

so

forth. This

is

a

fea

ture

of the

algorithm.



15/53

Projective

Geometry

25

* *

*

The

thirteen-point

plane

can

also be

applied

to

other musical

phenom

ena.

For

instance,

there

are

exactly

thirteen

of what

Larry Polansky

calls

"ternary

contours."8

These

are

illustrated

in

Example

14. The

thirteen

point

plane

will

organize

these

contours

into

mathematical

"lines,"

each

of

which

is

a

group

of

four

contours.

Example

11 will

help

the

reader

experiment

with

various modes

of

such

grouping, assigning

the

symbols

ply p2,

etc.

in

various

ways

to

the thirteen

symbols

of

Example

14.

EXAMPLE

14

* * *

There

is

a

twenty-one-point projective plane;

every

line therein has

exactly five points, and every point lies on exactly five lines. A generic

structure

for

the

plane

can

be

obtained

by

algebraic

techniques

to

be dis

cussed later. The

geometric

structure

could be

applied,

in various

"modes,"

to

the

family

of

twenty-one

diatonic unordered

dyads.

There is also

a

thirty-one-point projective plane;

every

line therein has

exactly

six

points,

and

every

point

lies

on

exactly

six

lines. The

structure

might

be of

interest

to

those who

are

interested in the

thirty-one-tone

scale;

the

geometry

would

single

out

a

privileged

family

of

thirty-one

hexachords

as

formal "lines" of the

system.

The

thirty-one privileged

hexachords

could be chosen

in

a

vast

variety

of

different

"modes."

Part

III: Algebraic

Structure

of the

Projective

Plane

There is

a

common

algebraic

structure

underlying

all

projective planes,

a

structure

which is

very

helpful

for

constructing

a

"generic" specimen

of

a



16/53

26

Perspectives

of

New

Music

given plane, along the lines of Example 11 earlier. To grasp suchmatters,

one

must

first

nderstand the

notion

of

an

algebraic

"skew-field."

By

that

term,

a

mathematician

understands

a

set

F

of

objects

x,

y,

etc.,

upon

which

two

binary

combinations

are

defined.

The

combinations

are

usually

denoted

using

conventional

symbols

for

addition

and

multiplica

tion:

thus

objects

x

and

ymay

be

combined

so as

to

yield

a

formal

"sum"

x

+

y,

and

a

formal

"product"

xy.

In

order

to

constitute

a

"skew-field,"

the

system

must

satisfy requirements

1, 2,

and

3

following.

1. Pis

a

commutative

group

under addition.

The

identity

element is

denoted

by

0;

the additive

inverse

of

element

a;

is

denoted

"-#".

2.

The

non-zero

elements

of F

form

a

group

under

multiplication.

The

identity

element of the

group

is

denoted

by

1;

the

multiplicative

inverse of

a

non-zero

#is denoted

by

"aT1"

or

by

"1/V\

3.

Multiplication

distributes

over

addition,

according

to

the

laws

x(y

+

z)

=

xy

+

xz\ (x +y)z

=

xz

+yz.

If

multiplication

is

commutative,

the

skew-field

is

called

a

"field."

The

real

numbers

form

a

field.

So

do the rational

numbers?those

which

can

be

expressed

as

quotients

of

integers.

So

do

the numbers

of

form

a

+

b(Jl),

where

a

and

b

are

rational

numbers

and

Jl

is

the

(irrational)

square

root

of

2.

The

multiplicative

inverse

of

a non-zero a +

b(

Jl

)

is

(a

-b(j2))/(aa-

2bb)?the

denominator

cannot

vanish because

a2

cannot

possibly equal

2e2,

a

and b

being

rational and

not

both

zero.9

The integersmod 2 form a field. Those are the integers 0 and 1with

all

arithmetic reduced modulo

2?e.g.,

1

+

1

=

0. The

integers

mod 3

are

also

a

field,

that is the

symbols

0,

1,

and

2

with

arithmetic

reduced

mod

ulo

3:

2

+

1

=

0,

2 2

=

1

(4

is

one

more

than

some

multiple

of

3),

and

so

forth.

The

integers

mod 5

are a

field,

that is

the

symbols

0,1, 2, 3,

and

4,

with

all arithmetic reduced

modulo

5.

(4

3,

for

instance,

=

2,

since

12

is

two

more

than

some

multiple

of

5.)

In

general,

for

any

prime

number

p,

the

integers

mod

p

form

a

field.

There

are

other

finitefields

as

well. For

instance,

there is

a

field

of four

elements which

one can

denote

as

0,1,

a,

and

a

+

1. In

this

field,

anything

added

to

itself

produces

0,

and

a2

equals

a +

1.

These

properties

enable

one

to

carry

out

the

necessary arithmetic,

to

verify

that

the

structure

is

a

field. For

instance,

a(a

+

1)

=

a2

+

al

=

(a

+

I)

+

a

=

I

+

a

+

a

=

1.

Then

(a

+

l)(a

+

1)

=

a(a+

1)

+

l(a+

1)

=

(I)

+

(a

+

1)

=

1

+

1

+

a

=

a.

The

cardinality

of the field

just

studied,

four,

is

not

a

prime

number,

but it

is

a

power

of

a

prime.

In

general,

for

any

prime

number

p

and

any

positive integer

?,

there will be

a

field

of

cardinality

p^.

Furthermore, any



17/53

Projective Geometry

27

finite fieldmust be one of those. Indeed, any finite skew-field must be

one

of

those,

since

a

remarkable theorem shows

that

any

finite skew-field

must

be

a

field.

* * *

Now

we are

ready

to

construct

a

generic projective plane.

Let

F

be

some

skew-field

(any

skew-field).

Consider the

family

of all

ordered

triples

formed from members

bl, b2,

and b3 of F.

We

stipulate

that

two

such

triples,

say

and

,

are

"left-equivalent"

if

there is

some non-zero

number k in

F

such that bl'

=

k

-

bl,

b2'

=

k

-

b2,

and b3'

=

k b3. This

in

indeed

an

equivalence

relation

among

the

triples.

The relation is

reflexive

(bj-

1

bj,forj

=

1, 2,

3);

it is

symmetric

(if

bj'=

k

-

bj,

then

bj

=

(l/k)

bjy,

the relation is also

transitive

(if

bj"

=

h

bj',

then

bj"

=

(hk)

bj).

The

points

of the F

Projective

Plane

are

labeled

by

the

non-zero

left

equivalence-classes

of

F-triples.

That

is,

is excluded

as a

"point"

of

the

geometry,

but

every

other

F-triple

represents

a

point;

among

those

triples

and

represent

the

same

point

if

and

only

if

there

is

some

non-zero

?in F such

that

cj

=

k-

?/for

each

y

=

1,2,3.

We shall label the

formal lines

of the

F

Projective

Plane

by

the

non

zero

right equivalence-classes

of

F-triples,

which

we

shall write

using

square

brackets:

the

triple

[XI, X2, X3]

is

r^i-equivalent

to

the

triple

[XI

',

T, X_H

if there is

some non-zero

k in

_Fsuch that

Xj'

=

Xj

k,

for

each

j

=

1,

2,

3.

Every

non-zero

triple

represents

a

line; among

those

triples [XI, X2, X3] and [Tl, T2, T3] represent the same line ifand

only

if there is

some non-zero

k in _Fsuch that

Yj

=

Xj-

k,

for each

j

=

1,2,3.

The

point

b=

lies

on

the line

X=

[XI,

X2,

X3]

if

and

only

if

he sixnumbers

atisfy

he

equation (bl)(Xl)

+

(b2)(X2)

+

(b3)(X3)

=

0 in the skew-field

F.

One

sees

that if b'

is

left-equivalent

to

b,

and/or

right-equivalent

to

X,

the numbers

?y'and Xj'wi? satisfy

the

same

equa

tion. Hence the relation

"? lies

on

_?" is

well-defined for

points

and lines.

The known

algebraic

behavior of such

equations

in

skew-fields

guaran

tees that the "points" and "lines" of the "F Projective Plane," as defined

above,

do indeed constitute

a

projective

plane.

That is:

1.

Any

two

points

lie

on

just

one

line.

2.

Any

two

lines

pass

through just

one

point.

3.

Every

line

passes

through

at

least

three

points;

every

point

lies

on

at

least three lines.

4. It isnot the case that every point lies on every line.



18/53

28

Perspectives

of

New

Music

The Property of Desargues (discussed in note 1) also obtains. Remark

ably

enough,

every

projective plane

(i.e.,

every

aggregation

of

formal

points

and

formal

lines

satisfying

the

four

properties

above

and the

Prop

erty

of

Desargues)

can

be

labeled

by

numbers from

some

F

Projective

Plane

so as

to

realize the

algebraic

structure

described

above.

That

is,

every

Desarguesian) Projective

Plane

"is"

in

fact

an

F

Projective

Plane,

for

some

suitable F.

If

the

Pis

finite

(and

hence

a

field)

we

can

count

the

number of

points

in

the

F

Projective

Plane.

Say

the

cardinality

of Pis

q,

where

q

is

a

power

of

some

prime

p

(as

earlier).

The

number

of

F-triples

is

then

q3,

and the

number of

non-zero

F-triples

is

q3

-

1.

Of

those,

each

is

equivalent

to

q

1

others,

since each

triple

can

be

multiplied

by

any

of

the

q

-

1

non-zero

members of F

to

form

an

equivalent

triple.

The

number

of

non-zero

equivalence

classes

is then

q*

-

1

divided

by

q

-

1.

That

number is

equal

to

1

+

q

+

q2,

which

will

be the

number of

points

in

the

corresponding

projective plane.

If

#

=

2,l

+

#

+

#2

=

l+2

+

4

=

7;

the

mod-2

projective

plane

is

our

seven-point

"diatonic"

plane.

If

#=3,

1

+

^

+

^

=

1

+

3

+

9

=

13;

themod-3

projective plane

is our

thirteen-point

"chromatic"

or

"ternary

contour"

plane.

If

q

=

4,

we

obtain

the

twenty-one-point

plane;

if

q

=

5,

we

obtain

the

thirty-one-point

plane.

Six is

not

a

prime

power;

there is

no

six-element

(skew-)field,

so

there is

no

forty-three-point

pro

jective

plane.

But

there

is

a

seven-element

field,

so

there

is

a

fifty-seven

point

projective

plane. Eight

and nine

are

each

prime

powers,

so

there

is

a

seventy-three-point

plane,

and

a

ninety-one-point plane.

Part

IV:

Applications?Mathematical

and

Musical?of Part

III

to

the

Twenty-One-Point

Plane

21

=

1+4+16;

the

twenty-one-point

projective plane

"is"

in

fact the

F4

Projective

Plane,

where F4 is

the four-element

field

discussed in

Part III.

F4

=

(0,1,?,#+1};

anything

added

to

itself

is

0;

a +

1

=

02.

One

verifies

that

0(0

+

1)

=

(0

+

1)0

=

03

=

1;

(0

+

l)2

=

0;

(0

+

l)3

=

1.

1

+

(0

+

1)

[= 1 + 1 + 0]

=

a, and so forth.

Given

a

twenty-one-point projective

plane,

we

shall

now see

how it

manifests

the

structure

of

the F4

Projective

Plane.

Using

the

method

dis

cussed

at

the

end of

Part

I

earlier,

we

select

any

"reference

quadrangle"

in

the

plane;

let its

points

and

lines be

denoted

as

in

Example

15:

points

pi,

p2,

p3,

q,

rl,

r2,

and

r3\

lines

Jl,

J2,

J3,

Ml, M2,

and

M3,

each line

containing

inter

alia

the

points specified

on

the

figure.

Now,

using

numbers

from

F4,

we

label

pi

=

,

p2

=

,

p3

=

,

and

q

=

. The

algebraic

structure

of the F4 Plane



19/53

Projective

Geometry

29

Jl=[p2,p3,rl,

J2

=

[p3,pl,r2,

J3

=

[pl,p2,r3,

Ml

=

[pl,

q,

rl,

M2

=

[p2, q, r2,

M3

=

[p3,

q,

r3,

EXAMPLE

15

tells

us

what

the

line coordinates for

Jl

will

be. Both

p2

and

p3

have

first

coordinate

equal

to

zero.

That

is,

both

are

points

b

=

satisfy

ing the equation bl

-

0. Thus, both points satisfythe equation (bl)(Xl)

+

(b2)(X2)

+

(b3)(X3)

=

0,

where

XI

=

1,

X2

=

0,

and

X3

=

0. We

know,

from

the

algebraic

structure

developed

in Part

III,

that

the

coordinates

for

Jl

will

then

be

[1,

0, 0]:

Jlis

that line

containing exactly

such

points

b

as

satisfy

bl =0

(more

explicitly,

bl

1,

plus

b2

0,

plus

b3

0

equals

0).

Similarly,

J2

=

[0,1,

0];

J2

contains

just

those

points

b

satisfying

b2

=

0.

And

J3

=

[0, 0, 1];

J3

contains

just

those

points

b

satisfying

b3

?

0.

Example

16

updates Example

15

to

display

the

new

information.

pl

-

,

pl=,p3

=

q-

/Z-[1,0,0]

J2

=[0,1,0]

J3

=

[0,0,1]

Ml=[pl,q,rl,.

M2

=

[ply

q,

rl,

.

M3

=

[p3y

}

r3,

.

EXAMPLE

16

[pl,p3,rl,..

[p3,pl,rl,..

[pl

l(W-O)

.K*2-0)

p2yv3y...

}(b3=0)

Next

we

shall find

the line

coordinates

for

Ml

in

F4.

The

lineMl

=

[XI,

X2,

X3]

passes

through

the

points pl

=

and

q

=

.

Accordingly,

we

must

have

1 XI

plus

0

X2

plus

0

X3

equal

to

0

(since



20/53

30

Perspectives

of

New

Music

pi lies onMl), and also 1 XI plus 1 X2 plus 1 X3 equal to 0 (since q

lies

on

Ml).

That

is,

we

must

have

XI

=

0,

XI

+

X2

+

X3

=

0. Or: XI

=

0,

X2

=

X3.

Accordingly

Ml

=

[0,

1,

l].10

In similar

fashion,

we see

thatM2

=

[1, 0, 1],

and

that

M3

=

[1,

1, 0].

The

new

information

is

displayed

in

Example

17.

pi

=

,

p2

=

,

p3

=

,

q

=

/i-

[1,0,0]

/2

=

[0,1,0]

J3

=

[0,0,1]

{p2,p3,rl,.

[p3,pl,r2,.

\pl,p2,r3,.

?fi-[0,1,1]:

{?I,

ft ri,.

?I?

-

[1, 0,1]:

{?2,

ftr2,.

M3-[l,l,0]:f^,ftr3,.

(M-0)

(?2=0)

(W-0)

(b2

=

b3)

(bl=b3)

(bl=b2)

EXAMPLE 17

Now

we

shall find the

point

coordinates

for

point

rl.

Suppose

those

coordinates

are

. Since rl lies

on

line/i

=

[1,

0,

0],

we

have

bl

=

0. Since

rl

lies

on

lineMl

=

[0,

1,

1],

we

have b2

+

b3

=

0,

whence b2

=

b3

(in F4).

In

sum:

bl

=

0;

b2

=

?3.

ri is

then labeled

by

the

triple

.

Similarly,

r2

is

labeled

by

the

triple

,

and

r3 is

labeled

by

the

triple

.

Example

18

updates

Example

17

with

the

new

information.

In the

particular

field

F4,

1 + 1 = 0.

Therefore,

in the

particular

sys

tem

we are now

considering,

the

sum

of rl's

three coordinates

is

zero.

So

is the

sum

of r2's

three

coordinates,

and

so

is the

sum

of

r3,s

three

coor

dinates. That

is,

the

points

rl, r2,

and

r3 all

satisfy

the

equation

bl

+

b2

+

b3

=

0.

(This

would

not

be

the

case

in

a

field where

things

added

to

themselves

were

not

zero.)

Hence

the three

r-points

all lie

on

the line

L

=

[1,

1, 1].

Example

19

updates Example

18 with the

new

information.



21/53

Projective

Geometry

31

pl

=

,

p2

=

,

p3

=

,

q

=

rl

=

,

r2

=

,

r3

=

Jl

=[1,0,0]:

[p2,p3,rl,.

J2

=

[0,1,0]:

[p3,pl,

r2,

.

J3

=

[0,0,1]:

[pl,p2,r3,.

Ml

=[0,1,1]:

[pl,q,rl,.

M2

=

[1,0,1]:

[P2,q,r2,.

M3 =

[1,1,0]:

[p3,q,r3,.

(bl=0)

(b2=0)

(b3

=

0)

(b2

=

b3)

(bl=b3)

(bl=b2)

EXAMPLE

18

Example

19,

we

now

see,

realizes

a

seven-point plane

embedded

within

the

twenty-one-point

plane.

That

is

because the two-element field

F2

=

{0, 1}

is

a

subfield

of F4. The

seven

points

and

seven

lines of

Example 19 are exactly those points and lineswhose coordinates can be

written

using

only

numbers from F2.

Since

our

original

choice of refer

ence

quadrangle

was

arbitrary,

we can

pause

to

note

the

interesting

fact:

any

choice

of reference

quadrangle

in

the

twenty-one-point projective plane

realizes the

seven-point

plane.

The visual

image

of

Example

19,

on

the

Euclidean

plane

of

the

page,

begins

to

be

deceptive

now?for

instance

it does

not

show

us

that

rl, r2,

and r3

are

collinear.11

So

we

shall continue onwards

from

Example

19

purely algebraically, adjoining

other

points

and

lines

of

the

twenty-one

point plane using number-triples

of F4. The

coordinates

of

our new

points

and lines

will

all

involve numbers of

F4 that

are

not

numbers of

F2;

thus

they

will all involve

the

number

a

of F4.

Example

20 lists the

twenty-one

points

of the

plane,

identifying

them

with

twenty-one

non-equivalent

number-triples

from F4. And

Example

21

lists the

twenty-one

lines

of the

plane, giving

their line coordinates

in

square brackets;

Example

21

also

shows

which

points

of

Example

20 lie

on

which of

those

lines.



22/53

32

Perspectives of

New Music

pi

=

,

p2

=

,

p3

=

,

q

=

ri

?

,

r2

=

,

r3

=

Jl=[l,0,0]:{p2,p3,rl,..

J2

=

[Q,l,0]:{p3,pl,r2,..

J3

=

[0,0,1]:

[pl,p2,r3,.

.

L-[l,

1,1]:

[ri, r2,r3,...

Ml=[0,l,l]:[pl,q,rl,..

M2

=

[l,Q,l]:{p2,q,r2,..

M3

=

[l,l,0]:{p3,q,r3,..

.}(W-0)

.}(f?-0)

.}(f??0)

}(M

+

*2

+

*3-0)

. }(t?-*3)

,

}(M-*3)

.

)(W-t?)

EXAMPLE

19

?i=,

ri=l>)

si

=,

ii'

=

,

tl

=,

fi'

=


23/53

Projective Geometry

33

Jl

=[1,0,0]

J2

=

[0,1,0]

/5

=

[0,0,1]

?

=

[1,1,1]:

Ml

=[0,1,1]

Ml-

[1,0,1]

M3

=

[1,1,0]

M

=[0,0,1]

#2

=

[0,0,1]

N3

=

[0,1,0]

ATi'

-[0,*

+

l,l]

N2'

=

[0 +1,0,1]

AT3'

=

[0

+

1,1,0]

?-[^,1,1]

K2-[1,*>1]

K3-[l,l,*]

Kl'

=[0

+

1,1,1]

Kl' =[1,0 + 1,1]

K3'

=[1,1,0

+

1]

G

=

[l,0,0

+

1]:

#=[1,0

+

1,0]:

[p2yP3ytlySlySV\

[plyp3yrlyslysi]

[pl,pl,r3,s3,s3'}

[rly

rly r3y

u,

v)

[plyqyTlytlyt?]

[pl,

qy

rly tly

ti

}

[p3y

y

r3y 3y

3'

)

[Ply

Sly

t?

yt3yU)

[flySlyt? yt3yV)

[p3y

3y ly

t?

y

U)

[plySl'

ytlyt3' yV)

[fly

S?

y

lyt3'

y

U)

[p3yS3'

ytlyti yV)

[rlySlyS3y

i'

yt? }

[Tly

ly

S3'

y

i'

y

3'

}

[r3ySl'

y

?

yti yti

[rly

S?

y

3'

y

ly

t3\

[rlyS3ySly tly t3\

[r3,

Sly

Sly

tly

tl\

[sly

i

y

3'

y

} u)

[Sly

S3y

S?

y

,

V)

(arithmetic nF4(\))

(W-0)

(?2-0)

(?5=0)

(bl+bl

+

b3

=

0)

(bl

=

b3)

(bl=b3)

(bl-bl)

(b3-a-bl)

(b3?a-bl)

(bl-a-bl)

(b3-(a

+

l)-bl)

(b3

=

(a

+

l)-

bl)

(bl=(a

+

l)-bl)

(a-bl-bl

+

b3)

(a-bl-bl+bJ)

(a-b3

=

bl+bl)

((a

+

l)bl=bl

+

b3)

((a + l)bl

=

bl+b3)

((a

+

l)b3-bl+b2>)

(bl+a(bl)

+

(a

+

l)b3=0)

(bl+(a

+

l)bl

+

a(b3)

=

0)

EXAMPLE

21

points listed in Example 20. One strategy?not the only one ?is to

begin

with

a

reference

quadrangle.

Example

22

assigns

to

Example

19

the values

pl

=

{CE},

pl

=

{FA},

p3

=

{GB},

r3

=

{EG},

rl

=

{DA},

r2

-

{BFU

=

{AC}.

EXAMPLE

22



24/53

34

Perspectives

of

New Music

These assignments give

a

"C-major"

sort

of reference

to

thequadrangle?a

feature evident

in

Example

23a,

where the bracketed

segments

are

collinear.

pi

r3

pi

rl

p3

rl

pi

pi

q

rl

r3

q

p3 p2

q

r2 r3

rl

p3

rl

pi

EXAMPLE 23

Example

23b

projects

the

same

formal

"lines"

as

verticalities,

indicat

ing

another

textural

possibility.

No

rhythm

is

indicated.

I

have

not

exer

cised

any

special

control

over

the

voice

leading

here,

simply

trying

to

project

vaguely Stravinskyish spacings

of the

dyad-triples.

Example

23

illustrates how the reference

quadrangle

of

Example

22

yields

a

seven-point subplane

of the

twenty-one-point plane. In the

present

context,

it is natural

to

arrive

at

the full

plane

by

"diminuting"

Example

23a. To

see

what that

means,

consider the first three

dyads

of

Example

23a. These

dyads,

playing

the roles of

pi,

r3,

and

p2,

all lie

on

line

J3

of the

full

plane,

but

they

do

not

constitute the

entire

line

J3

of

that

plane.

The

full line

J3,

as we see

in

Example

21,

contains

not

only

the

points

pi,

r3,

and

r2,

but

also the

points

s3 and

s3',

points

not

appearing

on

the reference

quadrangle.

We

can

naturally

imagine

dyads

s3 and

s3',

in

this

context,

appearing

"in

passing"

to

elaborate the

pro

gression pi,

r3,

p2

of

Example

23a. The

first

phrase

of

Example

24a

materializes

that

notion.

In

that

phrase,

the

progression pi,

r3,

p2,

which

began

Example

23a,

is

diminuted

by

transitional

dyads

{CG}

and

{EB}

that

appear

on

the off

beat

eighths.

To subsume those

dyads

onto

line

J3,

we

assign

them the

point-values

s3

and

s3',

the other

points

of line

J3.

The first

ive

dyads

of

Example

24a will then

project

line

J3

in

its

entirety.

In

like

spirit,

the

second five

dyads

of

Example

24a take the

progression

p2,

rl,

p3

from



25/53

Projective

Geometry

35

pi

r3

p2 p2

rl

p3 p3

r2

pi pi

q

ri r3

q

p3

p2

q

r2 r2

r3

rl

s3 s3'

si si'

s2 s2'

tl tl'

t3 t3' t2

t2'

v u

EXAMPLE

24

Example

23a,

and diminute

it

by

inserting

transitional

dyads

{FG}

and

{CD}

on

the

off-beat

eighths;

the

transitional

dyads

are

then

assigned

as

points

si and

si',

so

the

phrase

as a

whole

projects

line

Jl

=

{p2,

p3,

rl,

si,

si'}

in

its

entirety.

Example

24a

proceeds

in

the

same

spirit through

out.

It

turns

out

(in

this

plane )

that

every

remaining point

of the

plane

will be referenced exactly once, by diminuting Example 23a in such a

way.

One

can

see

that

by

inspecting

Example

21;

there

one

sees

that the

lines

Jl,

J2, J3,

L, Ml, M2,

and M3

collectively

reference

point

u,

point

v,

each

?-point,

each s'

-point,

each

i-point,

and each t'

-point

exactly

once.

And those

are

the lines

which

appear

in

part

on

Example

23a,

as

progres

sions

to

be

"diminuted."

Example

24b

realizes the

five-dyad

lines

as

verticalities,

elaborating

the

three-dyad

verticalities of

Example

23b.

The

new

dyads

are

shown with

filled-in

noteheads.

Orchestration

and/or

other

compositional

means

(dynamics,

order of

entrance

and

exit,

amount

of

sustained

time,

etc.)

could

be used

to

project

the

idea that the

open-notehead

parts

of each

sonority

is

more

in

the

nature

of

a

Zentralklang,

while

the filled-in

noteheads

project

Akzidentien}2

On the

other

hand,

a

composer

might

not

choose

to

project

explicitly

the

"referentiality"

of

Example

23b,

treating

that

only

as

a

work

ing

method

to

arrive

at

the full

twenty-one-point

structure.



26/53

36

Perspectives

of

New

Music

DEFGABEFGABFGAB

CCCCCCDDDDDEEEE

si'

pi

t3' s3

q

tT t2' s2' t3 rl t3

u

r3

v

s3'

G

A B A B B

F F F G G A

si

p2

r2 s2

p3

t2

EXAMPLE

25

Example

25 shows

the

assignment

of

dyads

to

points,

that

eventuates

from the diminutions of

Example

24. In

working

out

those

diminutions,

it is

helpful

of

course to

keep

a

running log

of

Example

25

as

it

develops,

to

be

sure

that each

dyad

is

used,

and used

only

once.

Now

that

we

have

assigned

a

dyad

to

each formal

point

of

Example

20,

we

automatically

know

the formal lines of

Example

21?in

particular

the

fourteen lines of that example not yet involved in the constructions of

Example

24.

t3 si

u

tT

pi

v

t3'

tl

pi

si' rl

si

s3 t3' tT

EXAMPLE

26

Example

26a

projects

lines

Nl,

Nlf,

and

Kl

of

Example

21 into

the

musical

texture

of

Examples

23a and

24a,

loosening

up

the

rhythm

and

"bowing"

a

bit.

Example

26b

imagines

a more

flexible

texture

than

any

so

far

presented;

in that

texture

it

projects

lines

J3,

G,

and

if

as

indicated.

The interested reader, taking hints from this example, will quickly dis



27/53

Projective

Geometry

37

cover

many

other

textures

inwhich

to

project formal

lines

of the plane,

and

many

other

musical

resources

for

linking

the lines

together.

Many,

many

other "modes"

are

of

course

available for

assigning

dia

tonic

dyads

to

the formal

points

of

Example

20. The

choice of

our

"C

major"

mode

was

only

to

facilitate

hearing

for

the

first

such

mode

to

be

studied.

Within

a

composition,

one

could

"modulate"

between

different

such

modes,

particularly

if

they

have formal

lines

in

common,

with

which

to

"pivot."

Or

one

could

transpose

a

mode via

any

number of

semitones,

introducing

new

tones.

Example

24a,

using

a

ficta

F|

within

our

mode,

suggests

one

such

possibility.

Finally

one

can use

mathematical transfor

mations that

are

particularly

characteristic for

projective planes.

Part

V: Collineations

in the

Projective

Plane

It is

time

now

to

bring

such

transformations

into the

picture.

A

collinea

tion

is

a

function f that

permutes

the

points

of

a

projective plane

among

themselves in such fashion

that,

whenever

points

p,

q,

and r are

collinear,

so are

?(p),

?(q),

and

f(r).

Loosely

speaking,

a

collineation

is

an

operation

on

points

that

"preserves

lines."

If

point

p

varies

along

line

L,

then

f(L),

the

locus of the

points f(p),

will

itself be

a

line.

f(Z)

may

or

may

not

be

the

same

line

as

L.

Obviously,

collineations

are

particularly

idiomatic

sorts

of

transforma

tions

to

consider,

given

the

points

of

a

projective plane.

The

collineations

form

a

group

of

operations.13

Here is the basic theorem concerning collineations: letpl, p2, p3, and

p4

be

points,

no

three of which

are

collinear;

let

pi, p2',

p3',

and

p4'

also

be

such

a

quartet

of

points;

then there exists

a

collineation

f

such

that

f(pl) =pl',

f(p2)

=

pi, f(p3)

=

p3',

and

?(p4)

=

p4'.

In

this

context

the

p'

-points

may

all

be distinct from the

^-points,

or

various of

the

p'

-

points

may

be the

same as

various

of the

^-points;

the

theorem obtains

in

any

case.14

As

one

intuits from

the

theorem,

there

are a

"large

number" of

col

lineations

on a

given plane. To sharpen that intuition, letus examine col

lineations

on

the

seven-point plane.

For that

purpose

Example

27

essentially

reproduces

an

earlier

sche

matic for the

plane.

Points

1,

3,

and 5 of the

example

are

non-collinear;

we

can

take them

as

the

generic

"pl,

p2,

and

p3"

of

the

theorem.

We

are

also

to

consider

a

generic

"?>4"

not

lying

on

line

13,

or on

line

35,

or

on

line

51.

Point 7 is

the

only possible

such

p4,

for this

plane.

Given

points

pi,

pi,

p3',

and

p4'

as

stipulated,

let

us

find

a

collineation f

as

in the

theorem.

We

know that

any

such

f

must

satisfyf(l)

=

pl ', f(3)

=

pi, f(5)



28/53

38

Perspectives

of

New

Music

A

seven-point plane:

123, 345,

and

561

are

all formal "lines" of the

geometry;

so are

174, 376, 572,

and 246

example

27

=

p3', f(7)

=

p4'. Subject

t

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Lewin - Some Compositional Uses of Projective Geometry

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