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A Declarative Model of Atonal AnalysisAuthor(s): John RoederSource: Music Perception: An Interdisciplinary Journal, Vol. 6, No. 1 (Fall, 1988), pp. 21-34Published by: University of California PressStable URL: http://www.jstor.org/stable/40285414
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8/18/2019 A Declarative Model of Atonal Analysis
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Music
Perception
Fall
1988,
Vol.
6,
No.
1,
21-34
© 1988 BY THE
REGENTSOF THE
UNIVERSITYOF
CALIFORNIA
A DeclarativeModel of Atonal Analysis
TOHN
ROEDER
University
of
British Columbia
Most
computational
models
of musical
understanding
have
focused on
procedural
aspects
of
analysis,
suggesting techniques
for
parsing,
com-
paring,
and
transforming
various
representations
of a
piece,
or
adapting
discovery
procedures
of
artificially
intelligent
(AI)
inference
systems,
which plan and follow agendas and goals. Much contemporary AI re-
search, however,
also focuses
on
declarative
aspects
of
knowledge,
at-
tempting
to
define data
representations
and
relations that are
commen-
surate
with
human
cognition. Naturally,
musical
analysis
has both
procedural
and declarative
aspects:
the declarative
determines
what
the
form
of the
analysis
is,
and the
procedural
determines how the
analysis
is obtained.
However,
a
predominantly procedural analysis
risks sacri-
ficing
the form
of
musical
understanding
to obtain
efficiency
or
compat-
ibility
with
a
particular computer language.
In
this article
I
argue
that,
for a
significant
body
of
twentieth-century
music,
a
declarative
system
models the structure
of
analytical
understanding
better
than do
existing
procedural programs,
and
I
present
a
functioning
declarative
system
that
infers
complex
musical structures
from
the
elementary
musical
relations
that it identifies.
Characteristics of
Atonal
Analysis
In atonal
music,
n
the broadest ense of the
term,
pitch
is
structured
withoutreference
o a
controllingkey.
The
worksto whichthe term
s
com-
monlyapplied,
composed
by
Schoenberg,
Webern,
and other
composers
n
the first
wo decades
of this
century,
are more
specifically
haracterized
y
extreme
and
rapid
contrasts
of
timbre,
register,
exture,
pitch,
pitch
class,
and durations,and negativelyby the lack of sustainedmelody, regular
pulse,
and
consonance.These
features,
and the lack of
themes,
keys,
and
the
phrase
structures
nd formsassociatedwith more
traditionalmeansof
pitch
organization,
ead
analysts
o hear
the music
n
many
different
ways.
However,
most
of them
agree,
expressly
or
tacitly,upon
certain
undamen-
tal
issues,
ncluding
1)
the nature
of
musical
analysis,
2)
the
natureof mu-
sical
structures,
3)
the nature
of the eventsthat
make
up
those
structures,
and
(4)
the nature
of musical
meaning.
For
the
present
discussionthese
points
of
agreement
warrant
a
brief
summary.
Requests for reprintsmay be sent to John Roeder, School of Music, Universityof British
Columbia,
6361 Memorial
Road,
Vancouver,
British Columbia
V6T
1W5,
Canada.
21
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22
John
Roeder
(1)
Music is an
object
for
contemplation,
not
simply
mmediate
xperi-
ence.
The
analytical
understanding
hat
Milton Babbitt
(1972)
calls re-
flective,
contemplated,
nd total
must
be
based
on a
detailed
knowledge
of
every
event nthe
piece,gained
rom
listening
o the music
verycarefully
and
noting
variousstructural
erceptions
Hasty,
1981).
Such
a
synoptic
understanding
s
manifestly
not
acquired
n
the course
of
a
single
istening,
but rather
by
the coordination
of
multiplehearings
and mental
rehearsals,
in
which the events
of the
piece,
even
temporally
distant
ones,
are learned
in
detail and associated
n various
ways.
(2)
The structural
omponents
f a musical
work are
collections,
or
seg-
ments of associated
events
(Forte,
1973).
An
analysisexpresses
a
seg-
mentation which
attributes
o
every
musical
event
membership
n
at least
one
significant
ollection
of events.These
segmentspossess
a
unitary
alue
in
some
domain,
(Hasty,
1981)
that
is,
their
dentity
and coherencearise
from the
perceivedproperties
of the basic
musicalevents
of the
piece.
So
a
segment
may
be a collection
of events
whose
temporaladjacency
defines
a melodic
ine,
or whose
simultaneity
definesa
chord;
a
segment
may
also
be articulated
y
other
musical
properties,depending
upon
the
style
of the
music
under consideration.
(3)
The
only properties
of musicalevents
that are
significant
o musical
structure
rethose that
define
egments;
nalysts
do
not
ordinarily
onsider
extramusical onnotations
of the music
n
determining
ts
segmentai
truc-
ture. Theproperties itedin most analysesarethose that areperceivedn
the intensive
istening
described
above:
rhythm,
essentially
he attack
time
and duration
of
the
events; timbre,
such as
the instrumental
ype
and
ar-
ticulation;
pitch;
and oudness.
Furthermore,
he relations
amongsegments
derive
from
these
same
perceived
properties
of their
events.
For instance
Babbitt
(1972)
claims
that
the
operations
of
inversion,
ransposition,
nd
retrogression
re familiar
and
rudimentary
otions
which
depend
upon
only
the most uncontrovertible
sic]
and essential
acts of musical
percep-
tion: the
capacity
o
recognize
pitch identity
and
nonidentity,
and interval-
lie value under
transposition
n a semitonal
system.
Similarly,
Hasty
(1981) identifies omesegmentsby the collectiveproperties- suchas pat-
terns
of
intervals,
attacks
ypes,
or contour-
which
they
have
in
common
with other
segments.
The structural
mportance
of each musical
domain
varies rom
composer
o
composer
and between
or even
within
pieces,
but
their
comparative
brevity
and untraditional
onstruction
ompels
the
an-
alyst
to discover
many
relations
among
the events.
(4)
The
meaning
of a musical
event
depends,
according
o Boretz
1970a,
compare
Roads,
1984),
upon
its
multiplicity,
r
multivalence,
of refer-
ence.
That
is,
a
simultaneity
of
multiple
mplications
of the
same
entity,
each
one of
which s
cognitive
and
specifiable,
nd
no
two
of
which arecon-
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A Declarative Model
of
Atonal
Analysis
23
tradictory
. .
[Thereare]
several
perfectly
clear
but distinct
'meanings'
attachable
o
single
events.
The
segments
o which an
event
belongs
thus
provide
a
meaningful
ontext for
that
event.
Similarly,
n
Hasty's
(1981)
modelof atonal musical
processes,
he
meaning
of each event is
dynami-
cally
modified
and
augmented
due to the
continually
changing
relations
that
successive
events manifest
with it and
its
predecessors.
These
four
points
of
agreement
bout atonal
analysis
underlie he
pitch-
class
(pc)
set
analyses
of Allen Forte
(Forte,1972,
1973, 1974, 1985;
see
also
Beach,
1979; Rahn, 1980).
Their
synoptic
comprehensiveness
eces-
sitates
heir
presentation
n modified cores hat
represent emporal,pitch,
and,
to
some
extent,
timbrai
properties
f
every
musical
event.
Circles
and
brackets
on these scores
indicatethe structural
omponents
of
the
work,
whichbelongto segmentsof two basictypes.Aprimary egment s a con-
figuration
hat is isolated
as
a
unit
by
conventional
means,
such as a
rhyth-
mically
distinct
melodic
figure
Forte,1973);
other
examples
of
primary
segments
nclude
a
rest-delimited
melodic
fragment
and a
chord.
A
more
complex
kind
of
segment,
which
Forte calls
composite,
s a
segment
formed
by
segments
or
subsegments
hat are
contiguous
or
that are other-
wise
linked
in some
way.
Forte's
analytical
tatementsassert
pitch-class
and
interval-class
ontent relations
among
segments,demonstrating
n
ef-
fect
a
network
of abstract
elations
amongpitch-class
motives.For exam-
ple,
the same
label
given
to two different
egments
ndicates hat
their
pc
content srelatedby transposition rinversion,and thattheyhavethe same
interval-class
ontent.
Thus
he
meaning
of an
individual ventderives rom
its
membership
n various
segments
n
a
complex
networkof related
seg-
ments.
Procedural
Analyses
of Atonal Music
Despite
the
inherently
relationalnature
of
pc-set analyses,
Forte con-
ceives
of
segmentation
s
a
process.
In
fact
he
attempted
o automate
seg-
mentation
by
meansof a
computerprogram
hat
parses
a score into
seg-
ments
and
classifies
hem
(Forte, 1966).
As its
representation
f
music,
Forte's
program
uses
DARMS,
which
encodes
a
score,
part by part,
as a
continuous
tring
of
alphanumeric
haracters.The
programminganguage,
SNOBOL,
n which the
system
s
realized s
especially
well
suited for such
standard
operationsupon
strings
as
concatenation,
earch,
and
compari-
son.
The
program
mploys
hese
operations
o
analyze
he
string
represen-
tation
of the
musicwithout
regard
o how
they correspond
o human
cog-
nitive
processes.
For
nstance,
he
parser
dentifies
ust
one
type
of
primary
segment,consistingof rest-delimited nstrumentalparts. Secondary eg-
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24
John
Roeder
ments are
produced by combining
these
primary
segments
into
pairs
ac-
cording
to the relative
temporal positions
of their first
and
last
events;
this
arbitrarily
binary
combination
procedure
also
seems motivated
by
the
op-
erations available in
SNOBOL,
rather than
by
the more subtle
analytical
procedures
Forte describes
in
his
other
writings.
The
resulting segments
are
similarly
combined
in
pairs,
and
redundant
results
removed.
Although
this
procedure
is
well
defined,
the
arbitrarily
restrictive
definition
and
represen-
tation of
primary
segments,
and the
arbitrarily
binary
combination
proce-
dure make it overselective
(Alphonce,
1980).
Another formalized
procedure
more
closely
modeled
upon
human
an-
alytical
behavior was
proposed
by
Laske
(1984)
to
produce
a
systematized
set
of
examples
for
newly synthesized
concepts. 1
Like Lenat
and Harris's
(1978) scientific discovery system,
Laske's
system represents
a small set
of
given
concepts
as
frame
structures,
then
plans
and
executes
an
agenda
to
find
significant
relations
among segments.
As evidence
for his
procedural
model,
the
author
cites
a
transcription
of a
student's
analysis
of
Debussy's
Syrinx,
which does seem
to involve
finding
examples
for musical
concepts.
Ironically,
however,
that
analysis
also
points
out a crucial
omission
from
Laske's
description:
the
specific
and
logical
representation
of musical
re-
lations
(Smoliar,
1986).
Without it the
analytical
system
cannot tell
which
concepts
are new.
For
instance,
the student
analyst
describes
a redun-
dancy
of motives
in
the
piece.
Laske claims
that
this
concept
is
newly
cre-
ated; logically, however, it would seem to be prior to and implicit in the
definition
of one of the
given
concepts,
the
basic cell.
Whatever
validity
Laske's
system
may possess
as
a model of musical
discovery,
it,
like Forte's
program,
would benefit
from
a
consistent
and
logical
representation
of
mu-
sical relations
obtained
through
the
painstaking
exploration
of the
cog-
nitive
processes
specific
to the
structuring
of music
(Alphonce,
1980).
Such
representations
have been
proposed
by
Boretz
(1969,
1970b),
who
constructed
an
analytical language
from formal
definitions
of
perceivable
event
relations,
and
by
Rahn
(1979),
who
proposed
a
collection
of
defini-
tions
to describe a hierarchical
analysis
of tonal
music.
Both of these
formal
systems
are declarative ratherthan
procedural,
concerned with the logical
definition of musical
events and relations
rather
than the
process
by
which
they
are
perceived.
However,
recently
developed programming
languages
have made
it
possible
to construct
a declarative
system
which not
only rep-
resents
those musical
properties
and relations
specific
to atonal
analysis,
in
logical
predicates analogous
to the declarative
statements
of Boretz
and
Rahn,
but which can also function
to
produce
a
pc-set
segmentation.
1. Similarly, Hasty (1981) states that in the second step of analysis rulesare devised to
form
a
theory
which
might
account
for these
[structural]
perceptions.
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A Declarative Model
of
Atonal
Analysis
25
The
Structure
of
a
Declarative
Analytical System
Following
the
form
of
atonal
analysis
outlined
above,
the
system
consists
of a collection of
predicates
that describe the formal structure of a
segmen-
tation.
The most
primitive
statements
are
the
facts of a
piece,
which de-
scribe
the
properties
of
every
event
in
sufficient detail to
support analytical
statements.
Representing
a more
complex
level of
musical
understanding
are
predicates
that
express
how events
may
associate
in
various
kinds
of
segments.
Still
higher-level
predicates specify
how
segments
are related in
a
segmentation.
The
system
attributes
meaning
to events
by
identifying
their
membership
in
segments
that have
significant
set-theoretical relations.
Thus the
analytical
understanding
that could be
represented
procedurally
by
the results
of a
segmentation process
is
represented
instead
declaratively
by
the instantiation
of musical relations
among
events
and
segments.2
To
attribute
declarative
meaning
to
a
musical
event
in
a
simple pc-set
analysis,
the
system
needs
information about
just
four
properties
of
the
event:
its
pitch,
the instrument
that
plays
it,
its attack
time,
and
its
duration.
Each event
is
a
set
of
specific
values
for
each
of
those
four
attributes.
event(Pitch,
Instrument, Attack,
Duration).
Any
collection
of such
events forms
a
context
in
which the events
may
have meaning,
and
all event
relations
and
structures
are
associated with
musical
contexts.
The
system
only recognizes
structures
and
relations of
events
if the events
belong
to the musical
context under
consideration.
A
context
may
be a
segment,
a
section,
an
entire
piece,
or even a
collection
of
pieces,
and each
segment
possesses
its
own
local
structures and
relations.
Since
an event
only
has
meaning
in a
context
of which it is
part,
the
sys-
tem
must
recognize
the
membership
of an
event
in
a context.
This
is accom-
plished
by
the
following
declarations:
element(E,[E|T],T).
An
event
E
is
an
element of
a
context that
begins
with
E
and ends with the
context
T.
element(E,[Y|T],[Y|Tl]):-
An
event
E
is
an
element of a context that
begins
element(E,T,Tl).
with
an
event
Y
and ends with the context
T
if E
is
an
element
of
the context
T.
These
and
subsequent
relational declarations
are
expressed
in the Edin-
burgh
syntax
of
the
programming
language
Prolog
(Clocksin
&
Mellish,
1984),
and
correspond
to
Horn clauses
in
first-order
predicate logic.
The
2.
Along
with these
formal
specifications,
the
design
of the
system
was also constrained
to avoid metalogical constructs (such as cuts and asserts in Prolog), in order to be as de-
clarative
as
possible
within the limitations of
a
procedural
machine architecture.
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A
Declarative
Model
f
Atonal
Analysis
27
piece
webern,op
I_no3,
Webern's
Opus
11,
No. 3 is a
collectionof events
[event(27,cello,0,12),
including
an Et
3 and a
Ft
3
played
by
the cello
event(28,cello,0,12),
at the start
of
the
piece
for
the duration
of 12
event(48,cello,12,8), triplet ixteenths,a cello C4 12triplet ixteenths
event(38,piano,15,15),
..]).
afterthe firstevent
of
the
piece,
a
D3
playedby
the
piano
15
triplet
ixteenths
after he firstevent
of
the
piece
for the durationof
15
triplet
six-
teenths,
etc.
This
representation
will allow the
system
to use informationabout one
work
to direct ts
analysis
of another.
Every
eventrelation
has
the same
form,
partitioning
he context into re-
lated events
and unrelated
events.
sameJnstrument(event(Pl,I,Al,Dl),
Two eventsarerelated f the same instru-
event(P2,I,A2,D2),C,
R)
- ment
plays
hemboth
n
the
context
C.
The
subset([event(Pl,I,Al,Dl),
remainder
f
the events
n
the context orm
event(P2,I,A2,D2)],C,R).
R.
same_attack
event(Pl,Il,A,Dl),
Two
events
are
related
if
they
have the
event
P2,I2,A,D2),
C, R):-
same
attack ime
in
the
context C. The re-
subset([event(Pl,Il,A,Dl),
mainderof
the events
n
the context form
event(P2,I2,A,D2)],
C,
R).
R.
Similarly,
vents
may
be
temporally_adjacent,
r
sound_together,
f
they
belongto the same contextand have the appropriate emporalrelations.
These
low-level
predicates
express
the basic
relations
a listener
may per-
ceive
among
musicalevents.
Accordingly,
he most fundamental
nalytical
statement
he
system
can
makeabout
a
context s the associationof all
pairs
of events
n all
possible
relations,
uch
that several
different elationsobtain
for
every
event.3
A
primary egment
s
definedas a collection
of
events that are related
in the same
way:
primary([H,X|T],
Context,Rem,Relation):-
A
collection
of
events
containing
Goal= ..[Relation,H,X,Context,R], eventsH and X is aprimary egment
call(Goal),
under the
stipulated
Relation in a
primary([X|T],[X|R],Rem,Relation).
given
Context
f H
is so related
o
X
in
that
Context,
and if all
the events
in
the collection
except
H
are
a
pri-
mary segment
under
the same Rela-
tion
in
the same Context.
primary
[H],Context,Remainder,_):-
A
single
event
H
belonging
o
a
con-
element(H,Context,Remainder).
text is a
primary
egment.
3. Competing interpretationsof unorganized data also characterize the local organizing
processes
in
Arbib's
(1979)
model
of
visual
cognition.
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28
John
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That
is,
the
cognitively
based
relations are the basis
for the
cognitively
most
important types
of
segments.
Most of
Forte's conventional
primary segments
are covered
by
this
definition. An instrumental
part,
for
example,
is a collection of events
associated
by
the relation
sameJnstrument.
A chord is
a collection
of
events
with the same
attack.
In
a melodic
line the events
are
temporally
adjacent,
that
is,
every
event
in the
line is
immediately preceded
or followed
by
another
event
in the line.
Thus the
formal structures
of
seemingly
dif-
ferent
types
of
segments
are
in
fact
identical:
a
segment
of
every type
is
a
collection
of events
associated
by
one
of the
basic,
and
formally
identical,
event relations.4
Some
of Forte's more
complex segments
can
be
expressed
as
primary
seg-
ments of one type contained
within the
context
of
primarysegments
of
an-
other
type.
Consider,
for
example,
a declarative
definition
of the
rest-
delimited melodic
lines
in
the instrumental
parts
of
a
piece:
primary(IP,Context,
,same_instrument),
A
rest-delimited
nstrumental
art
is
primary
RDIP,IP,
,temporally_adjacent).
a
primary egment
RDIP of
tempo-
rally
adjacent
vents
n
the context
of
a
primary
segment,
IP,
of events
played
by
the
same instrument.
Although
a
comprehensive
set
of definitions
of all
types
of
primary
seg-
ments is beyond the scope of this briefdescription of the system, the system
similarly represents
them
all
as collections
of
cognitively
associated
events
in various contexts.
The
segmentation
of contexts
according
to various
de-
fined musical
relations
constitutes
the basic
analytical
capability
of the
sys-
tem.
A
somewhat
more
sophisticated
analysis exposing
the
multiple
function-
alities
of
events
can also
be achieved
using only
the
declarations
cited
above.
Events
belonging
to
more than one
primary
segment
in the
same
context
are
describing by
the
conjunction
of
clauses,
for
example:
primary([E
_],Context,same_attack),
An eventE ispartof a chordalpri-
primary([E
_],Context,temporally_adjacent).
mary segment
and
also
part
of a
melodic
primary
egment.
This collection
of
Prolog
clauses
thus
constitutes
a functional
segmenter
that
can
identify
and
relate
many
sorts
of
primary
segments.
A
query
by
the user
is
expressed
in the form
of a
goal,
which the
system
satisfies
by
ap-
plying
the
cognitively
based relations
it knows
to the
facts
of the
piece.
Con-
4.
Other
types
of
primary segments
recognized
in atonal
analysis
may
also
be
expressed
declaratively For instance, in one kind of primarysegment all events arerelatedin the same
way
to
one event
in the
segment,
but not
necessarily
to
each other.
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A
Declarative
Model
of
Atonal
Analysis
29
siderthis declaration
of an
exhaustive
partition
of
a
context into
primary
segments
of a
single type:
primary_segmentation([S|R],Context,Relation):-
A list of event collections is a
pri-
primary(S,Context,Rem,Relation),
mary
segmentation
of a
Context
primary_segmentation(R,Rem,Relation)
for
a
specified
Relation
if
the first
collection S is
a
primary segment
for the
specified
Context and Re-
lation,
and if
the rest of
the
list R
is
a
primary segmentation
of
the
rest of the
Context under the
same Relation.
primary_segmentation
[],[],_).
An
empty
context has an
empty
primary segmentation.
This
higher
evel
predicate
an be used to
expressanalytical
goals
that
may
be satisfied
n
a
variety
of
ways
consistentwith the
cognitively
based
seg-
mentation
rules. For
example, any
chord
in
Webern's
Opus
11,
No. 3
is
expressed
declaratively
s
a
primary
segment by
the
conjunction
of two
clauses:
piece(webern,opll_no3,Context),
SJist
is
a
list
of
event collec-
primary_segmentation(S_list,Context,same_attack).
ions such that the
events
in
each collection
belong
to
the
context
of
Webern's
Op.
11,
No. 3 and are attacked
at
the
same
time.
The event
collections
atisfying
his relationare istedabove
Figure
1. True
to the declarative
epresentation,
o
procedure
orms
or
compares
struc-
tures,
he
systemsimplyrecognizes
he
presence
of
primary
egments
based
of the network
of
cognitively
based relations
n
the
data,
and it
will do so
identically
or all knownrelations.
n
fact,
by
rewriting
his
conjunction
us-
ing
otherdeclared
elations,
uchas
sameJnstrument,
emporally_adjacent,
same-duration,
nd
same_pc,
we can
represent
nterestingaspects
of the
segmentai
tructure
of
this
piece,
as
shown
below
Figure
1. The
first
two
lines
below the score show the
segments
based
upon
the
relations
sameJnstrument
nd
temporally_adjacent;
hese
correspond
o
what we
conceive o
be
the
individual nstrumental
arts
and
the rest-delimitedme-
lodic
lines,
respectively.
The last two lines show
that
interesting egments
can also
consistof
nonadjacent
vents.
For
nstance,
nearlyevery
event
has
the same
durationas
another vent
n
the
piece
(Berry,
976,
pp.
397-400);
the relation
partitions
he
piece
into
several
egments
containing
one,
two,
or threeevents.Also, nearly everyeventhas the samepitch class as an-
other
event:the bottom line
under he
score,
which
indicates he
segments
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30
John
Roeder
containing
events with the same
pitch
class,
reveals that the second half of
the
piece
recapitulates
the
pcs
of
the first half
(Wintle, 1975).
The
general
definition
of
a
segmentation
for an
arbitrary
ist of
relations
is:
segmentation([X|Y],Context,[H|T]):-
A
list
of
primary
segmentations
is
a
primary_segmentation(X,Context,H), segmentation
of a
Context
for a
spec-
segmentation(Y,Context,T).
ified list
of
relations
if
the first
pri-
segmentation([],
_,[]).
mary segmentation
on
the
list, X,
is
a
primary segmentation
of
the
spec-
ified Context under the relation
H,
and
if the
rest
of
the list
Y
is
a
seg-
mentation
of the Context under the
other relations.
The crucial
analytical
statements
in a
pc-set analysis
assert that the
pitch-
class contents
of
two
or more
segments
are identical under
transposition
or
inversion,
so that the
segments
belong
to the same
Tn/TnI-equivalence
class
(Rahn,
1980; Forte, 1973).
The
analyst normally
determines the
class,
or
type,
of a
pc
collection
by
using
a
procedure
to reduce the collection
to
a
standard form that can be
found
in a
table
of set
types.
However,
this set-
classification
procedure
can be
very
simply
declared as the relation
of
the
collection
to
the standard
form
of
the abstract
set-type
in a
particular
con-
text:
set_type(Set,Type,Context):-
A
Set
belongs
to
a certain
Type
in a
intervaLnormal_form(Type,Int_Series),
Context,
if
an Interval-Series
associ-
subset(Ordering,Set,[]),
ated
with
that
Type spans
some
Or-
pcJntervaLseries(Ordering,
dering
of
the Set
(Regener,
1974).
Int_Series,Context)
The
clause
pc
JntervaLseries
expresses
the relation of
an
ordered
collection
of pitched events to the ordered series of pitch-class intervals that spanthem
in
a
particular
context. Consistent
with
this
relation,
the standard table
of
set classes is declared
as
a
collection of relations
among
interval series
and
set-class labels:
interval_normal_form('3-l',[l,l]).
The intervalnormal
form of a
set
be-
intervaLnormal
orm('3-2',[1,2])...
longing
o class3-1 is the
series
of
pc
in-
tervals
[1,1],
etc.
With these added relations the
system
can
express
the relation of
segments
to set-class
labels,
so
that the
following
conjunction
of
clauses:
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A
DeclarativeModel
of
Atonal
Analysis
3
1
piece(webern,opll_no3,Context),
P
is
a
primary
egment
belonging
o
primary(P,Context,,temporally
adjacent),
set
type
Type,
and
consisting
of
tem-
set_type(P,Type,Context). porally
adjacent
events that
belong
to the context of Webern'sOp. 11,
No. 3 and
are
attackedat
the same
time.
is satisfied
by every
rest-delimited melodic line
P
that
belongs
to the
set-class
Type
in
the Context of the Webern
piece.
Note
that the declarative definition
of
set-type
is not
restricted to
primary
segments;
so the declaration
piece(webern,opl
_no3,Context),
set_type(P, ype,
Context).
is satisfiedby any collection P of events having an intervalnormal formrec-
ognized
by
the
system,
whether or not
P
is a
primary
segment.
Analysts
of-
ten
consider such
a
complex
segment
as
significant
if
it
belongs
to
the
same
type
as
a
primary segment.
Forte's
composite
segments
are a
case
in
point.
The collection
of events
in
a
composite
segment
are
not
uniformly
related
as
they
would be
in a
primary segment,
but Forte
stipulates
that each
event
is
contiguous
with another event
in
the collection.
Although
Forte
does
not
formally
define
continguity,
his
analyses
suggest
the
following
rule:
contiguous(X,Y,Context,Remainder):-
Events
X
and
Y
are
contiguous
n
sound_together(X,Y,Context,Rem); a Contextif theysoundtogether,
temporally
adjacent(X,
,Context,Rem)
or if
they
are
temporally djacent,
not
between(X,_,Y,Context,Rem).
or if
there is no
event
between
them
temporally
n
the
context.
The relation
of two
contiguous primary
segments
in a
composite
segment
can then be declared as:
composite(C,Context,Relations):-
A
composite
segment
C in a
Con-
segmentation(S,Context,Relations),
text
is the union of
two
primary
element(Ll,S,_),
lement(L2,5,_),
segments
n
the context such
that
element(Pl,Ll,_),element(P2,L2,_),
an
element of
one
primary
seg-
notPl=P2, ment is contiguouswith an ele-
element(El,Pl,_),element(E2,P2,_),
ment
of
the other
primary seg-
contiguous(El,E2,Context,_),
ment.
union(Pl,P2,C).
But a more
general type
of
contiguous
segment
can
also be
declared:
contiguous_segment(Seg,Context)
A
collectionof events
Seg
s
a
con-
all_contiguous(Seg,Seg,Context).
tiguous
egment
n
a
Contextifev-
all_contiguous([H|T],S,Context):-
ery
event
n
it is
contiguous
o an-
element(X,S,_),not
=
H,
other event
in
it.
contiguous(X,H,Context,_),
all_contiguous(T,S,Context)
alLcon
iguous
[] _,_)
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32
John
Roeder
Accordingly,
the
system
can
express
the relation of
any primary
segment
to all
more
complexly contiguous segments
of the same
type:
primary
_segmentation(Segmentation,Context,Relation),
element(Primary_Segment,Segmentation,_),
In a
given
Context,
Segment
is
any
set_type(Primary_Segment,Type,Context),
contiguous segment
that
belongs
set_type(Segment,Type,Context),
to the
same set
Type
as a
contiguous_segment(Segment,Context).
Primary Segment.
This
conjunction
of relations
can be used
in
the
Prolog system
to
list
all
composite
segments
that
belong
to the same set
type
as the
primary
seg-
ments. As
an
illustration,
Figure
2
shows the
composite segments
the
pro-
gram
finds
that
belong
to the same
set
type (Forte
number
3-3)
as the
first
piano
verticality,
in which the events
are related
by
same_attack
(Williams,
1983).
Each of
these
segments
is
perceptually
coherent,
in
the sense that ev-
ery
event
in
the
segment
is
contiguous,
in
one
of
the three
ways
we
have
defined,
to another
event
in
the same
segment. Interestingly,
this
composite
segmentation
provides
a
meaningful
context for
every
event of the
piece.
In
all,
this
system
exhibits some
basic structural characteristics of atonal
pc-set
analysis.
It
represents
abstract
pc-set-analytical understanding
in
stages:
more
complex
structures
and relations are
logical
conjunctions
of
simpler
ones,
and an entire
network
of
segmentai
relations are
demonstra-
bly founded upon a few cognitively based event relations. Analytical
knowledge
is
distributed
throughout
the
system
in
the
form
of
clauses that
Fig.
2.
Segments
consisting
of
contiguous
events
and
belonging
to the same
set
type
(tran-
scribed
from
program
output).
{x,y,z}
identifies
a
pitch-class
collection of
type
3-3
(inter-
valnormal
form
<l,3>or<3,l>).
[c]
indicates
which
type
of
contiguity
obtains
between
the
corresponding
events
(see
the
Appendix:
t means
the events are
temporally
adjacent,
s
means the events sound together, and n means that there is no intervening event between
the two events.
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A Declarative Model
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33
specify
he structure
f each musicalevent as a
collectionof
audible
prop-
erties.
The clauses attribute
he musical
meaning
of an
event
to the mul-
tiplicity
of relations t
bears o other eventsand to
the
multiplicity
f struc-
turesto which it
belongs.
They
relate musical events
according
o their
audible
properties,
define he
form of
segmentsaccording
o those
musical
relations,
classify
the
segments
by
type,
and relate different
segments
be-
longing
to the
same
type.
Since
the
system
structure
orresponds
o
theo-
rists'
conceptions
of the
structure
f
atonal
analytical
knowledge,
and
since
the
system
supports
he
same
kind
of
pc-set
analytical
tatements hat
hu-
mans
make,
it would
appear
o be
a
good
model
of
human
analytical
un-
derstanding
f
atonal
music,
and it
suggests
hat
a
declarative
ystem
might
be
a useful
model
of more
general
music-analytical
nowledge
as well.
Appendix
subset([],X,X).
subset([H|T],X,R):-
element(H,X,D),
subset(T,D,R).
sound_together(event(Pl,
I, Al, Dl), event(P2, 12, A2, D2), Context,
Remainder)-
subset(
[event(Pl,Il,Al,Dl),
event(P2,12,A2,D2)], Context,
Remainder),
AKA2
+
D2,A2<A1+D1.
successive(event(Pl,Il,Al,Dl),event(P2,I2,A2,D2),Context,
emainder):-
subset([event(Pl,Il,Al,Dl),event(P2,I2,A2,D2)],
Context,
Remainder),
A2is
Al
+
Dl.
temporally_adjacent
X,Y,Context,Remainder)
-
successive
X,
Y,Context,Remainder)
successive
Y,X,Context,Remainder)
order(event(Pl,Il,Al,Dl),event(P2,I2,A2,D2),Context,Remainder):-
subset([event(Pl,Il,Al,Dl),event(P2,I2,A2,D2)],Context,Remainder),
AKA2.
order
First,Middle,Last,Context,Remainder)
order(First,Last,Context,Remainder),
order
First,Middle,Context,_),
order
Middle,Last,Context,_)
between(First,Middle,Last,Context,Remainder):-
order(First,Middle,Last,Context,Remainder);
order(Last,Middle,First,Context,Remainder).
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34
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Webern's
Op.
ll/III.
Perspectives
f
New
Music,
1975,
13(12),
165-177.