a declarative model of atonal analysis

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8/18/2019 A Declarative Model of Atonal Analysis http://slidepdf.com/reader/full/a-declarative-model-of-atonal-analysis 1/15 A Declarative Model of Atonal Analysis Author(s): John Roeder Source: Music Perception: An Interdisciplinary Journal, Vol. 6, No. 1 (Fall, 1988), pp. 21-34 Published by: University of California Press Stable URL: http://www.jstor.org/stable/40285414 Accessed: 30/07/2010 02:26 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ucal . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. University of California Press is collaborating with JSTOR to digitize, preserve and extend access to Music Perception: An Interdisciplinary Journal. http://www.jstor.org

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Page 1: A Declarative Model of Atonal Analysis

8/18/2019 A Declarative Model of Atonal Analysis

http://slidepdf.com/reader/full/a-declarative-model-of-atonal-analysis 1/15

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

Accessed: 30/07/2010 02:26

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=ucal.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of 

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

University of California Press is collaborating with JSTOR to digitize, preserve and extend access to Music

Perception: An Interdisciplinary Journal.

http://www.jstor.org

Page 2: 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

Roeder

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

of

Atonal

Analysis

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

John

Roeder

References

Alphonce,

B.H.Music

analysisby computer-

A field or

theory

ormation.

Computer

Music

Journal,1980, 4(2), 26-35.

Arbib,

M. A. Local

organizing

rocesses

nd motionschémas

n

visual

perception.

Machine

Intelligence,

979, 9,

287-298.

Babbitt,

M.

Remarks

n

the

recent

Stravinsky.

n

Boretz

&

E. T. Cone

(Eds.),

Perspectives

on

Schoenberg

nd

Stravinsky.

New York:

Norton, 1972, pp.

165-185.

Beach,

D. Pitchstructure nd the

analytic

process

n

atonal

music:

An

interpretation

f the

theory

of

sets.

Music

TheorySpectrum, 979, 1,

7-22.

Berry,

W. Structural

unctions

n

music.

EnglewoodCliffs,

NT:

Prentice-Hall,

976.

Boretz,

B. Meta-Variations

I). Perspectives f

New

Music, 1969, 8(8),

1-74.

Boretz,

B. Conversation

with ElliottCarter.

Perspectives f

New

Music,

1970a, 8(2),

1-22.

Boretz,

B.

Sketch

f

a musical

ystem

Meta-Variations,

art

I).

Perspectives

f

New

Music,

1970b, 8(2),

49-111.

Clocksin,

W.

F.,

&

Mellish,

C. S.

Programming

n

Prolog

(2d éd.).

New

York:Springer-

Verlag,1984.

Forte,

A.

A

program

or the

analyticreading

of scores.

Journal

of

Music

Theory,

1966,

10(2),

330-364.

Forte,

A. Set and nonsets

n

Schoenberg's

tonal

music.

Perspectives

f

New

Music,1972,

11(1),

43-64.

Forte,

A.

The structure

f

atonal

music.

New

Haven

and London:Yale

University

Press,

1973.

Forte,

A.

AnalysisSymposium

Webern,

Orchestral

Pieces

(1913),

Movement

(Bewegt).

Journal

of

Music

Theory,

197

4, 18,

13-43.

Forte,

A.

Pitch-class et

analysis oday.

Music

Analysis,1985,

4(1

&

2),

29-58.

Hasty,

C.

Segmentation

nd

process

n

post-tonal

music.

Music

TheorySpectrum,

981, 3,

54-73.

Laske,O. Keith:A rule-systemormakingmusic-analyticaliscoveries.n M. Baroni& L.

Callegari

Eds.),

Musical

grammars

nd

computer

analysis,

Florence:

Leo S.

Olschki,

1984,

165-199.

Lenat,

D.

B.,

&

Harris,

G.

Designing

a

rule

system

hat searches

or scientific

discoveries.

In

D. A. Waterman

&

F.

Hayes-Roth,

Eds.),

Pattern-Directed

nference ystems.

New

York:

Academic

Press, 1978,

25-52.

Rahn,

J.

Logic,

set

theory,

music

heory.

College

Music

Symposium,

979,

19(1),

114-127.

Rahn,J.

Basic atonal

theory.

New York:

Longman,

1980.

Regener,

E. On Allen Forte's

heory

of

chords.

Perspectives

f

New

Music,

197

4, 13(1),

191-212.

Roads,

C. An overview

f music

representations.

n

M.

Baroni

&

L.

Callegar

Eds.),

Musical

grammars

nd

computer

analysis.

Florence:

Leo S.

Olschki, 1984,

7-37.

Smoliar,

S.

W. Reviewof

Musical

rammars

nd

computer

nalysis.

ournal

of

MusicThe-

ory, 1986, 30(1), 130-141.

Williams,

E.

W.

Jr.

On Mod

12

complementary

nterval ets.

In

Theory

Only,

1983, 7(2),

34-43.

Wintle,

C. An

early

version

of derivation:

Webern's

Op.

ll/III.

Perspectives

f

New

Music,

1975,

13(12),

165-177.