26070754 analytic reading of scores allen forte

7/30/2019 26070754 Analytic Reading of Scores Allen Forte

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Yale University Department of Music

A Program for the Analytic Reading of ScoresAuthor(s): Allen ForteSource: Journal of Music Theory, Vol. 10, No. 2 (Winter, 1966), pp. 330-364Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843247

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3 3 0

A Program

f o r

Introduction

As is well known to most musicians, musical composition under-

went a remarkable and revolutionary transformation during theyears 1908-1922. This radical change was due almost entirelyto the innovations of Arnold Schoenberg and his students, AntonWebern and Alban Berg. The term "atonal", commonly usedto describe the music of that time, is construed as "not ex-

hibiting any of the structural characteristics of the music of

traditional, triadic tonality". However inadequate the term

may be, it does indicate that the structural determinants ofthis complex music are not well understood. The score- readingprogram described in this paper is part of a larger research

project concerned with the development of a general structural



3 3 1

t h e Analytic Reading

o fScores

ALLEN FORTEdescription of atonal music. *1

Certain basic decisions that led to the development of the score-

reader are intimately connected with the goal of the larger pro-

ject indicated above. The most fundamental of these decisions,the choice of atonal music as the object of research, bringswith it the need for an extensive revision of outlook. For ex-

ample, conventional descriptive language ('melody', 'harmony',

'counterpoint')is not

very useful,and

mayeven be a

signifi-cant hindrance to new formulations, since it is oriented toward

older music. When, in addition, problems of structural analy-sis are stated in terms accessible to computer programming,

many conventions are set aside and many familiar concepts are

rejected after serious scrutiny. These considerations and



3 3 2

others mentioned below, led ultimately to the design of the basic

tool described in this paper.

APPROACH TO A SYNTAX-BASED METHOD

The Score as a System of Graphic Signs

One can obtain information about a particular compositionin a

number of ways: for example, by reading statements made bythe composer which purport to describe his composing tech-

niques or his aesthetic views, or by asking a trained listener

to supply a description of some kind. Such information is al-

ways incomplete, however, and does not provide an adequatebase for an analytic system. The musical score, on the other

hand, constitutes a complete system of graphic signs and, pro-

perly represented for computer input, may be analyzed by a

program as a logical image of the unfolding musical events that

make up the composition. This, then, was the point of depar-ture for the score reading program: the analysis of formal

characteristics of sequences of graphic signs in the music in-

put language discussed below.

Advantages of Automatic Data-Structuring

When the trained analyst examines a musical score he asso-

ciates certain signs with others to form units and makes a

series of basic decisions about the temporal spans of such unitsand their internal structuring. In a metaphorical sense he

places a template over the score to frame patterns and to show

how they are interwoven to form local contexts.

Although the decisions which the analyst makes may rest upon

years of practical experience they are often unsystematic and

are subject to many influences that are not easily identified.

Nevertheless, it is apparent that at least part of the time he

makes decisionsaccording

to rules of some kind. Thissug-

gests the possibility that rules could be stated. Such rules,

interpreted as algorithms and stated in a programming language,would yield a complete and precise structuring of the data pre-

sented by the score. A tool of this kind should produce results

at least as good as those produced by the human analyst. It

shouldnot give provisional readings that require further hand-

editing or interpretation of some kind. Indeed, it would be

appropriate to aim for a reader which would produce results

superior to those obtained by the human analyst. The use of

the expression "human analyst" in this context is not intended



3 3 3

to imply that the programmed score-reading process is basedupon methods commonly used by human beings. The logicalbase of the program and its empirical relevance to a fundamen-tal musical parameter, time, will become evident as we pro-ceed.

The Music Representation

Before processing, scores are encoded by hand (keypunched)in a language designed by Stefan Bauer-Mengelberg*2 in con-nection with a computer-implemented project to print scores

automatically. This representation has unique attributes:

(1) It is isomorphic to standard music notation.

(2) It is highly mnemonic, hence easy to learn.

(3) The responsibilities of the encoder are minimal. The

encoding rules are unambiguous and do not requirethe encoder to make arbitrary decisions.

(4) It is economical. The amount of code required for a

complete representation is remarkably small. (SeeExample 1.)

The representation uses a currently non-standard characterset of 64 graphics, but a set of equivalent codes is availablewhen the expanded set cannot be obtained. Example 1 shows a

listing made from punched cards that contain the completecode, in Bauer-Mengelberg's music representation, corres-

ponding to a score of one page in length.

The Programming Language

The high level programming language SNOBOL3, designed byFarber, Griswold, and Polonsky*3, is used for the program.This language handles free-form strings with facility, and its

generality renders it most appropriate for the kind of experi-mentation that necessarily accompanies a research project ofthis kind. The score-reader program itself consists mainlyof a number of SNOBOL-encoded functions. A small amountof basic information is stored in lists, but the power of theprogram resides in the functions. These cope with any even-tuality in the input string, including encoding errors.

Output Forms

The current output forms are provisional. An appropriate syn-



3 3 4

EXAMPLE

1

WEBERN,FOUR PIECES FOR VIOLIN AND PIANO,OP.7 (1910)2. I01XVIOLIN,/102XPIANO,/101 XXGDXXT=$SEHR LANGSAM (E=CA 50)$ =XM2X.4 VPP::Ll,21-HJ,24-N4J,XAMIT$DAEMPFER=A / VXLlXG2,21HJ,24N4J XG2 / ;XMR,.4 21Q RQ xX3EIX.2 REI,"XVESPRESS.$ VPP,30xQI1JL+l / uXM2X.4 102 XXG 3XM2.I4 RW / VPPP,214XH,26XXH,30s=Hx- / XXMR,.4 RQ RQ (VPPP,16X"XE,18XE,22:=EEL+l 17 XEJ, 19=EJ,23x=EJ)/ XXM2x,4+ =XF =XM2:,4 RW / 79:=H,84XSXH:- / :XM3::.4 RE VPPPXLI,69-EL+1

77XXE V:L1IG2,76::Q.L,+ VxG2 / uXM2u.4 101 XXG :XM2,.4 V'Ll,30 'E 29cxQVXLI 27:=E / XXM3x.4 V=G1,23-E 29==Q.L41 V=G1 RE,XXVWEICH GEZOGEN$,-XVC

OL LEGNO$ VPPPSEMPRE,26X=EL+lx- / ((28-Sx- 26:=S:- 28S:- 26SL+41-)) ((28SL+lx- 26S:- 28x5- 26S=-)) ((28S:- 26S:- 28Sx- 26S:-)) / XXM2:.4 102 XXG

XXM2X.4 XX3Q2=,2 VXL2, 17:H2, 19-=H2,23.=H2 VXL2,19=:Q2J,21IXQ2J,25SXQ2J/ XXM3=,4 19Q,1Q,25Q 18x=MHJ,20XHJ,25-HJ / VG2,18Q,20Q,25Q VXG2,18-QJ

,21-QJ,24#XQJ 18Q1,21Q1,24Q1L+l VPP,33==E1L+10,XVAEUSSERST ZART$ / XXM2

X,4 X+ XXF XXM2X.4 VUL2,69-QL+1 VxL2,77xxQJ / XXM3,.4 77Q. VXG2 72-E 74XXQJ VuG2 / 74L+1 RE VPPP,63X=Q.L+1 / XXM2,.4 101 XXGCXM2X.4 ((28-SL+1X-

26x=SX- 28SX- 26Sx-)) ((28Sx-,xXTRIT.$ 26Sx- 28S:- 26SL+1:-)) / ((VCGl,28-SL+1:- 263=S::- 27:=SL4+1-)) RS RE ((28SO VxG1,27SO)) / RW // 102 XXGXXM2.,4 (V::G2,30-xE10 24-E10 21:XE10) VUG2,17=:XQJL+1- / 17Q RQ RE VPPP,

18-Q,21-Q,23=:XQ RE // ::+ XF XXM2o.4 63u=H / RQ2 VPP'G2,76-Q2L+l VXG2,19CxQ2JL+l / 19Q. RE //



3 3 5

tactic structure for output will be developed as the reading pro-cess is tested and refined. In all likelihood the output formswill ultimately employ a subset of the input language, since theresults of an analysis could then be printed automatically inconventional notation.

In early phases of the processing the forms of the output (in-termediate) strings are constrained by the syntax of the input

language because of the relation between input and output strings.The nature of that relation will be made clear in the next sec-tion, which begins with a summary of terms, symbols, abbre-viations, and conventions used to describe the parsing processcarried out by the score-reader.

DESCRIPTION OF THE PARSING SYSTEM

Definitions of Terms

Character

One of the 48 standard characters in six-bit mode. (Theextra character ':' is used in the intermediate strings.)

Space-Code

A sequence of characters in theinput language

which con-tributes to the specification of pitch.

Accidental-Code

A sequence of characters in the input language which,like a space-code, contributes to the specification ofpitch.

Pitch-Code

A space-code followed by an accidental-code.

Time-Code

A sequence of characters in the input language which con-tains information needed to specify the duration of anevent.



3 3 6

Element

A sequence of characters in the input language repre-

senting a distinct symbol in music notation. For exam-

ple, '.' represents the dot of music notation.

String

A one-dimensional sequence of characters which hasmarkers to delimit beginning and end.

Data String

The original data before processing: a string in the in-

put language.

Input String

A string of characters that is scanned to form a new

string. The data string is the first input string.

Intermediate String

A string, constructed by the program from an input string,which represents an intermediate stage in a parsing op-eration.

Output String

A processed string.

Position-Value

An integer in an intermediate or output string which spec-ifies the temporal location of an element or sequence of

elements.

Segment

A distinct unit within an output string, delimited by '(' on

the left and ')' on the right.

Block

A distinct unit within a segment, delimited in the output

string by 'P' on the left and a single blank on the right.

The corresponding unit in the input language is delimited



3 3 7

by blanks on left and right.

Parsing

The extraction, from an input string, of strings of seg-ments representing classes of event-configurations in

the score.

Terminal String

The final output form for a class of segments.

Definitions of Symbols

In the definitions that follow, 'A' and 'B' name arbitrary input,

output, or intermediate strings, and the defined affixes denote

relations between, or operations upon, pairs of such strings.

A--B A is replaced by B.

A B Concatenation of A and B.

A n B The set of segments shared by A and B.

A B A has the same unordered contents as B.

A c B A is asubstring (proper

orimproper)

of B.

A DB A is a superstring (proper or improper) of

B.

Abbreviations

1 Special Classes

SG

PSGI

PSGT

SSGI

SSGT

RSSGT

of Strings:

Segment

The string of primary segments for input.

The terminal string of primary segments.

The input string of secondary segments.

The terminal string of secondary segments.

A replica of SSGT.



3 3 8

RPSGT A replica of PSGT.

2 Programmed operations on strings:

BCP Block-concatenation parsing operation.

SCP Segment-concatenation parsing operation.

3 Grammatical categories for input, intermediate, or termi-nal strings. (The abbreviations that follow will be used for

describing the form of strings according to conventions givenbelow.):

AC Accidental-Code

TC Time-Code

SCSpace-Code

PC Pitch-class, represented by an integer in

the set ( 0, 1, 2, 1.,11. The expression

'pitch-class' or the abbreviation PC (after

Babbitt *4) is usually used in the sense

'pitch-class representative'. The meaningis clear from the context.

PV Position-value.

RC Rest-Code, indicating a silence.

]6 A (literal) blank.

Conventions

Names of strings, segments, blocks, functions, function argu-ments and indices are in upper case alphabetic characters.

Usage conforms to the conventions of the programming language:

Indices are separated from generic names by '.'

Literals are enclosed within single quotation marks.

Function arguments are enclosed within parentheses, and

multiple arguments are separated by commas.

Examples:

String name A.

String name with (variable) index I.

A

A. I



3 3 9

A. '1' String name A with index 1.

B(X, Y) Function name B with arguments X and Y.

Variables used as indices, such as I and J, do not imply any-thing about sequence, but merely indicate distinctness.

Grammatical categories listed under 3 of Abbreviations areused in this paper to specify the form of a string as follows:

If X and Y are grammatical categories, then the expressionX Y designates the result of writing, in the input language, a

representative of the grammatical category X followed imme-

diately by a representative of Y, with no intervening space.Occasionally literals are used where the string-form describedcontains invariant items; for example, the expression

PV '$'

displays the form of a string consisting of some PV followedimmediately by the dollar sign.

Preliminary Operations

For the remainder of the paper a single composition will beused to illustrate the work done by the program. This is shownin Example 2, together with the corresponding data string en-coded in the input language.

Two basic operations are performed on the data string beforethe main parsing begins. (1) Since the encoding is basically a

one-pass procedure, the encoder progressing from left to rightacross each system on the page for each instrumental part, thecharacters which belong to a particular instrument are distrib-uted throughout the data string. The program therefore beginsby scanning the data string to assemble the complete code-

sequence for each instrument. It must deal with such situationsas the following: In an orchestral score of 200 measures the

bass clarinet enters on the last beat of bar 75, plays a staccatothirty-second note, and is not heard from again during the re-mainder of the piece. In this and all such cases where the partis not present either at the beginning or the end of the score,the program finds the complete instrumental part. (2) Thesecond of the two preliminary operations establishes the basisfor the derivation of complex event-configurations by the ap-plication of the parsing rules. This consists of the assignmentof a position-value PV to each code which implies a temporal

position. Two passes through the input string are required.On the first pass, information relevant to the duration-value



3 4 0

EXAMPLE

2

- 48"

64 80 212C0V1 28

0 6 96 168 176

If,9 0~ ,,h.--- -... ~114460

104 128

*4Q92 352 480 548 5681

^92 432 5j

8 j. bJtr ^ n iiY |

7 7K264 312 464 488 512

1

~CbR A 15G A

34u384

Copyright 1924 by Universal Edition

Renewed copyright 1952 by Anton Webern's ErbenUsed by permission of Theodore Presser Company

2- 6



3 4 1

of time-codes is accumulated. In particular, a normative orlist value is assumed for each time-code. This value is ad-

justed if a groupette occurs in the same bar. For example,if the meter signature is 2/4 an eighth-note triplet is a group-ette and indicates three eighth notes in the (normative) time oftwo. For each bar a number is computed which represents the

product of all unique groupette divisors. The number is 1 ifthere are no groupettes in any part in a particular bar. Onthe

second pass a duration-value is assigned to each time-code byconsulting the list of normative values and the number which

represents the product of the groupette divisors. The dura-

tion-value DV for groupette members is then computed by theSNOBOL statement

DV = (VL * GP2) * (Z / GP1)

where VLis the list value of the time-code, GP2 is the norma-tive

division of that value,Z

is the groupette product, and GP1is the groupette divisor. For all other time-codes in a barwhich contains a groupette or groupettes,

DV = VL Z

For bars in which groupettes do not occur, the list value is theduration-value assigned.

Once the position-values have been assigned - and this is

straightforward after the duration values have been computed- position value becomes the main parameter in the parsingsince it indicates temporal position. The string can then beviewed as a succession of blocks within segments, each block

containing the total of codes associated with the point in time

represented by the position value. (To deal with codes that arerelevant to a block but exterior to it - for example, the codefor the current clef sign - the program maintains an associa-tion-list. ) In this way it is possible to refer to any moment in

a composition and to relate any event to any other event withrespect to a time-continuum. In this regard, as well as inmany others, the analytic technique differs fundamentallyfromthose employed by researchers in other fields where the stringis the basic data structure. The structural linguists, for ex-ample, are not concerned with time-placement in this extendedsense, whereas for music research the temporal dimension is

always primary. *5

The association of each event and eachevent-configuration

witha set of position-values has the additional advantage that order



3 4 2

characteristics are preserved. Thus, complexities of rhythmicinteraction can be deduced from the set of numbers attached to

particular configurations. For example, collation functionscan be written to search for categories that might be of specialinterest, such as all segments that have the same first position-value.

Parsing Rules: General Comments

For the description of the parsings, information falls conven-

iently into these categories:

1 Names of the strings being parsed.

2 An informal description of the parsing technique and acharacterization of the event-configurations that it ex-tracts from the input string.

3 A symbolic expression representing the form of the

segment in the output string.

4 A formal statement of the parsing rules.

5 A description of the deletion conditions for segmentsin the output string.

6 Illustrations of program-generated output, music no-

tation, and scanning strategy.

Parsing Rules: Primary Segments

Underlying any segmentation or parsing process such as theone executed by the score-reader, are certain assumptionsregarding the discreteness of events. What conditions deter-mine a unit? Which classes of signs serve as disconnectives?How can one speak with precision of components of larger

event-configurations? Questions such as these are in the fore-ground of the description of parsing rules, and an effort ismade to state the answers clearly.

For the first parsing of the input string, each substring rep-

resenting a complete instrumental part is scanned to extractall primary segments. A primary segment is a continuousunit delimited by the graphic sign denoting silence: a rest inmusical notation, RC in the input language. It begins with aPC which has been preceded by a rest (or by an understood

(logical) rest if the pitch is the first in a part), contains no in-



3 4 3

ternal rests, and ends with a rest. Thus, it is set off from

other substrings and is regarded provisionally as a disjunct

sequence in a one-dimensional environment. The scan for pri-

mary segments in an instrumental part terminates when the

input language code '//' (double bar) is encountered.

In the input language a primary segment has the form

(1) RC ] SC AC TC SG d RC ]

With respect to sequence in the string the block SC AC TC is

always next. Since there may be more than one RC in a sub-

string of the input string without an intervening SC AC TC -

for example,

RC RC RC SC AC TC SG RC RC RC

the first RC in the formgiven

above at(1)

isalways

theright-most with respect to SC, while the second RC shown in the form

is always the leftmost with respect to SG.

After parsing, the form of the segment (now in an intermediate

string) is

(2) '*' PC 'KA' PV '$'... 'KZ' PV '$'

where 'KA' and 'KZ' designate attack and release, respectively,and refer to the position-value which follows. In the case of'KA' the position-value always refers to the first (attack) PC.In the case of 'KZ' the position-value refers either to the ter-minal delimiter (some RC) or, if the next code is '//,', to somecode which is an understood (logical) RC. An example of thelatter can be seen in the viola part in Example 2.

The form given in (2) above is the input form for a primarysegment. The terminal form is:

(3) '(' 'P' PV '*' PC '*' PC ... '*' PC ')'

The transformation from input form to terminal form involvestwo kinds of change: deletions and deletions with replacements.Characters deleted (without replacement) are 'A', 'KZ', ], PV

(when preceded by 'KZ'), groupette definers, and meter signa-tures. Each pitch-code is replaced by an integer representinga pitch-class PC. The appropriate mapping is executed by afunction PCM, which utilizes modulo 7 arithmetic. A numberof contextual details require attention here. First, if the in-



3 4 4

EXAMPLE

3

48̂ 9 64 80, bj l ,

80- J I J2dI'"2

0 t6,I! 96, 68,876,

,~,~, 4j j "",F

e i 160

'I" I I 104 128

, 52 - 480- t 548568I 32, 52, 432 0 1J 548 56

1336 I t S It

-,.

-b

,p,yr 4lr

-,t

X7 ; 312 464 4881 512 A~~~~~~~~~Dli~~~~~~-

I 2 ID r34 o1 384

iJ.

PSGT(PO::1 P16:3 )(PO::O )(PO::4 )(P48:2 P64::5 P80:3 )(P64::4 P96::4 )(P64::11 P96

:11 )(P104::5 P128::5 )(P120::6::10 P128::6::10 )(P120::11 P128::ll )(P144::0 )(P

160:8 P192::8 P264::10 )(P160::7 )(P168::6 P176::6 )(P192:9 )(P224::7 P240::7 )

(P276::8 )(P312::4 )(P336::3 )(P352::5 )(P384::2 P408::1 )(P432::0 P480::0 )(P46

4::11 P480:11 P488::11 )(P480::6 )(P512::8 )(P512::9 )(P544::7 )(P548::8 )(P568::3 )(P568::11 )(P568:1 P576:2 )(P568:4 )



3 4 5

EXAMPLE

4

Al RISG.I S __

EAR tSG.J *_

A2 S2 R2

Al R1SG.I S1

AER t

SG.JdA2 S2 R2

Al R1SG,I S1 53EAGR +

SG,J 4

A2 S2 R2

Al RlSG,I S1 S2

LAER t t

SG.JA2 53 R2

Al R1SG,I S1 52

GAR t t

SG,J 4,

A2 S3 S4 R2

SG,ILAGR

SG,J

Al51 52

A2 S R

A2 s4 R2

R153

SG.1 = S1SG.2 = S2SG.3 = S3

SG.4 = 52 S4

SG.1 = S1 52

SG,1 = S1 S2

SG.1 = S1SG.2 = S2SG.3 = S2 53

SG.1 = S1SG.2 = S2SG,3 = 52 53

SG.1 = S1SG.2 = 52SG.3 = 53SG4 = S4SG.5 = S1 S4



3 4 7

strument is a transposing instrument, the appropriate modulo12 arithmetic must be performed. Second, a scan is made for

space-codes that have implicit accidentals, under two familiarconventions of notation, and those accidentals are inserted.

Third, the current clef sign must be made available for each

pitch-class mapping. Example 3 shows the complete stringPSGT, together with the music notation, where primary seg-ments (31 in all) are indicated by brackets.

Parsing Rules: Secondary Segments

The string of primary segments is made up of substrings, onefor each instrumental part. Every (unordered) pair of these

substrings is scanned for secondary segments. This is thefirst and most important step in the progression from the one-dimensional primary segments to a representation of the ana-

lyzed two-dimensional score. At this stage the program is

concerned with the ways in which events represented by pairsof primary segments interact in local contexts. Indeed, oneof the important consequences of the parsing for secondarysegments is to show how the primary segments are affected bymotion in other parts of the context. The following combinatorysituation provides the basis for a formal description of the

parsing.

Given three temporal relations: less than (before), equal to

(coinciding),and

greaterthan

(after), togetherwith the

position-values of the attack (A) and release (R) for each of two primarysegments SG. I and SG.J (where SG. I-PSGT. I and SG. J C

PSGT. J), there are (32 + 4) - 1 ways in which the two seg-ments interact, excluding the non-contigous case. Example 4shows the 11 possible interactions reduced to 6 classes by vir-tue of the symmetry of the defined relations. Each class is

assigned a mnemonic name in a quasi-Polish notation; for ex-

ample, EAR means that the relation equal to obtains betweenboth A's and both R's, while LAER means that the first A is

less than (before) the second, while the two R's coincide. Thesame name is applied to the second (symmetrical) relation ineach class and a subscript is used to distinguish the two.

As indicated above, a given pair of primary segments is scanned(left to right) for Al, R1, A2, and R2. Example 5 is a graphof the decision structure that is traversed to determine theparticular relation between the pair.

When the relation has been determined, one of four functionsis called and secondary segments are formed as shown by the



3 4 8

EXAMPLE

6

SSGT(PO::O::1 P16::3 )(PO::1::4 P16::3 )(PO::O::4 )(P48::2 )(P64::5 P80::3 )(P64::4 )(P6

4::4::5 P80::3 )(P64::11 )(P64::5::11 P80::3 )(P64::4::11 P96::4::1l )(P64::4 P96::4

P104::5 )(P64::11 P96::ll P104::5 )(P96::4 )(P96::11 )(P104::5 )(P104::5 P128::5

P144::0 )(P120::6::10::11- P128::6::10::11 )(P120::6::10 P128::6::10 P144::0 )(P120::6

::10 P128::6::10 P160::8 P192::8 P264::10 )(P120::6::10 P128:5::6::10 )(P120::6::10P128::6::10 P160::7 )(P120::11 P128::11 P144::0 )(P120::11 P128::11 P160::8 P192::8 P264::10 )(P120::11 P128::5::11 )(P120::11 P128::11 P160::7 )(P128::5 )(P160::8

)(P160::7 P192::9 )(P160::8 P168::6 P176::6 )(P160::8 P192:8 )(P160::7 P168::6P176::6 )(P160::7::8 )(P160::8 P192::8 P264::10 P276::8 )(P168::6 P176::6 P192::9

)(P192::9 P224::7 P240::7 )(P192::8 P264::10 )(P192::8::9 P264::10 )(P192::9 P276

::8 )(P224::7 P240::7 P264::10 )(P224::7 P240::7 P276::8 )(P264::10 )(P312::4 P35

2::5 )(P312::4 P336::3 )(P312::4 P384::2 P480::l )(P336::3 P352::5 )(P336::3 P384

::2 P408::1 )(P384::2 P408::1 P432::0 )(P432::0 )(P464:;11 )(P464::11 P480::0::ll

P488::ll )(P480::0 )(P480::0:;6 )(P480::11 P488::11 )(P480::6::11 P488::11 )(P512

::8::9 )(P544::7 P548::8 )(P568::3::11 )(P568::1 )(P568::1::3 )(P568::3::4 )(P568::1

::11 )(P568;:4::11 )(P568::1::4 )(P576::2 )

7

vv

vo

V1V2V3V4

SNM ::Nl/'1:: N2/'1:: ' = /F(WRTE)HNM = SNM

HNM N1 :N3/'1' ',' = /S(V1)HNM N2 :N3/'1: ',' = /S(V2)HNM ::N3/t1t:: N2 ,t = /F(V4)S(V3)$('SG N1 N3) = DP2($C'SG' N1 N2),$('SG' N1 N3))

$('SG' N2 N3) = DP2($('SG' N N2),$('SG' N2 N3))$('SG' N3 N2) = DP2($('SG' N1 N2),$('SG' N3 N2))

$S(SG' N1 N2) = DP2(PSGT,$('SG' NI N2))SSGT SSGT $('SG' NI N2) /(VV)

/(VO)/(VO)/(VO)



3 4 9

arrows in Example 4. Observe that the basis of segmentationis attack and release operating reciprocally on the constituent

primary segments. Since the disconnectives (attack and re-

lease markers) of the original primary segments now them-

selves point to other disconnectives, it is evident that the pars-

ing is recursive in this limited respect.

The scanning strategy is straightforward: if k is the number

of substrings in the string of primary segments, each substringrepresenting an instrumental part, then the number of scans

required to find all secondary segments is the same as the

number of distinct pairs in the unordered cross-product of a

set of k elements, (k(k+l)/2)-k.

Each scan of a pair of strings of primary segments yields a

string (possibly null) of secondary segments. The name of this

string is SSGI. J K, where the variable index J is the index of

one member of the pair of primary segments, and K is the in-dex of the other. Example 6 shows the complete output stringSSGT.

The scanning strategy for the deletion of duplicate secondary

segments utilizes the fact that for any pair of strings of sec-

ondary segments SSGI. J and SSGI. K, where J and K representcatenated indices of primary segments as described above,

duplicate segments can occur only if one element of the com-

pound index J is the same as an element of the compound indexK. This strategy greatly reduces the number of scans requiredand permits the operation to be controlled by a non-arithmetic

program segment in a fashion idiomatic to SNOBOL. The pro-

gram segment that performs this operation is shown in Exam-

ple 7.

Parsing Rules: BCP(SSGT)

A secondary segment results from the interaction in time of

the attacks and releases of two primary segments. Possibly aparticular secondary segment is a constituent of a still larger

event-configuration; that is, it may be associated with otherevents in the local context. The parsing function BCP (forblock-concatenation parsing) finds such situations and derivesa new class of segments.

BCP operates on SSGT and its replica RSSGT as follows. Eachblock in each segment of SSGT and RSSGT has the form

'P' PV '*' PC.'1' '*' PC. '2' '*' . . . PC.N t



^ $I ^$0>y ~ 8~ :'

2 0>- 352 480 548 568312 432 4 88

0 wr 336

l 694 312 a 464 488 512

7<08/,fQs^Ta

7 6

r 408

384

544

BCP(SSGT)(P64::4:5::11 P80:3 )(P64::4::5::11 )(P64::4::5::11 P96:::11 P104::5 )(P96::4::11 )

(P104::5 P128:5::6::10:11 P144:0 )(P120::6::10::11 P128::5::6::10 :11 P144::0 )(P120:6::10:1 P128::5::6::10::11 P160::7::8 P192::8::9 P264::10 )(P120::6:10::11 P128::5

:::10::11 P160::7::8 )(P128::5::6::10::11 )(P16::7:8 P192::8::9 )(P160::7::8 P168:6

P176::6 )(P16O:7::8 P192::8::9 P264::10 P276::8 )(P1686 P176:: P192::8::9 )(P192::8::9 P224::7 P240::7 )(P192::8::9 P276::8 )(P480:0:6:: 11 P488::11 )

3 5 0

EXAMPLE

8

(6



3 5 1

For every block B.I in a segment SG.I (SG.I c SSGT), with PV.I,and every block B.J in SG.J (SG.J cRSSGT), with PV.J,

if PV. I = PV. Jthen B.I--B. I B.J

In this way a secondary segment SG.I is expanded by the infixingof all blocks in all other segments which intersect intime with

some block in SG. I. Two additional aspects should be noted.First, concatenation is followed by the deletion of duplicatePC's. Second, the process is not recursive, in the sense that

only the original blocks in a segment are operated upon; con-catenation does not extend to blocks infixed during the parsing.

Possibly some secondary segment does not contain any PVblock which intersects with the PV of some other block in the

way defined. In this case the segment is complete in itself and

does not belong to a larger context. Such residual segmentsare deleted when the execution of the function has been com-

pleted. It is also possible that a segment in the parsed stringwill have one or more duplicates. These internal duplicatesare deleted after BCP has been executed.

Because the concatenation does not take into consideration theorder of position-values or the order of pitch-class represent-atives, it is necessary to call collation functions within BCP.Pitch-classes are placed in ascending order at present. How-

ever, the set of indices representing original ordering couldbe retained, wherever that ordering is significant. Example 8shows the output string BCP(SSGT) and the corresponding nota-tion, with segments delimited by frames.

Parsing Rules: BCP(PSGT)

When BCP operates on PSGT the result is a string each com-

ponent of which is anchored to a position-value that occurs in

two or more instruments. It should be remarked here that the'instrumental part' is merely a device. Once the position-values have been assigned, the location of an element within a

particular part is of no fundamental concern. The reading thusis independent of the notions of 'part', 'voice', 'voice-leading'and other interpretive formulations which belong to a higheranalytic level.

Deletion conditions for BCP(PSGT) are as follows. Residual

primary segments are deleted. Internal duplicates are deleted.Since the rules for BCP when applied to PSGT may form dupli-



3 5 2

EXAMPLE

9

3252 480 548 568T,3^ 432-- --

336 r

26B?r t180

s57 ,4264 1 8 525i2276 408 M 544 l

384 k

13

BCP(PSGT)(P0::0::1:4 P16::3 )(POO::0:1::4 )(P48::2 P64:4::5::11 P80:3 )(P64::4::5:11 P96::4::11 )(P14::5 P128::5:6::10::11 )(P120:6::10:1 1 P128::5::6::10::11 )(P160::7::8 P192::

8::9 P264:10 )(P192::8::9 )(P432:: 0480::0::6::11 )(P464::11 P480::0::6::11 P488:11 )(P480O::0::6:11 )(P568::1::3::4::11 )(P568::1::3::4::11 P576::2 )



3 5 3

EXAMPLE

10

64 80 b21

t i i '^ I/ IOPrt0 16 96 168 176

160P j

0-

Oa ( }I4

,J

^ A 14H rT ,4 17

SCP(SSGT)

(P64::4:5*:11 P80::3 P96"4:'ll P104:5 P120::6::10::11 P128'5::6::10::11 P144:O P16

0:7::8 P168::6 P176:6 P192::8::9 P224::7 P240::7 P264::10 P276:8 )(P96::4::11 P10

4::5 P120::6:10::11 P128::5::6::10::11 P144::0 P160::7':8 P168::6 P176::6 P192::8::9 P

224':7 P240::7 P264:'10 P276::8 )(P104::5 P120::6::10::11 P128::5::6::10':1 1 P144::0

P160::7::8 P168:6 P176:: P192::8:9 P22P26 P240 P26410 P276:8 )(P120:6:10

P128::5::6:;10::11 P144::0 P1607:8 P1686 P176:6 P19288:9 P224:7 P240:7: P26

4:10 P276: 8 )(P128::5::6::10:11 P160::7: P168::7 P176::6 P192::8::9 P224::7 P240

`7 P264:10 P276%8 )(P160::7::8 P168::6 P176::6 P192::8::9 P224::7 P240:: 7 P264::10 P276::8 )(P168::6 P176:6 P192::8::9 P224::7 P240:7 P264::10 P276:8 )(P192:8::

9 P224::7 P240::7 P264:10 P276::8 )(P224::7 P240:7 P264::10276::8 )(P312::4 P336::3 P352:5 P384::2 P408::1 P432:0 )(P336::3 P352::5 P384::2 P408:l P432::0 )

IV-+ ILo



3 5 4

cates of secondary segments, a scan must be made to deletesuch duplicates. Example 9 shows the complete output stringBCP(PSGT) and corresponding notation.

Parsing Rules: SCP(SSGT)

The content of segments derived by BCP is limited with respectto span since the concatenation is restricted to units of block

form. The function SCP (for segment-concatenation parsing)may extend to event-configurations of greater span since itdefines the segment as the concatenation unit. For every PV.Iin SG. I (SG. I _ SSGT) and every PV. J in SG. J (SG. J RSSGT),

if PV.I = PV.Jthen SG. I --SG. I SG. J

As remarked above, SCP finds event-configurations which are

connected by (at least)one common PV. The SCP-derived

segments thus represent continuity over longer spans, even

though disjunctions may occur in some subparts of the musicalcontext. As a consequence SCP yields very intricate and sub-tle readings.

The scanning procedure for SCP differs significantly from thescan for BCP. In the case of BCP the scan for a new segmentalways begins at the head of the string. This is not true forSCP. There the first PV in each segment is a potential primegenerator of a segment in SCP. If a particular first PV hasnot previously been a prime generator it is qualified to becomeone. Let us assume that some first PV, PV. '1' is a primegenerator. The scan to find segments for concatenation beginswith the first segment to the right of SG. '1', SG. '2' in RSSGT.If the scan finds a match for PV. '1' in SG.'2' the entire seg-ment is concatenated with SG. '1' after duplicate blocks havebeen removed. The PV. '2' in SG. '1' then becomes a (second-

ary) generator; the scan begins again with the first segment to

the right of SG. '2' in RSSGT, and so on. Possibly the segmentconcatenated contains position-values that are not in the orig-inal SG.'1'. Such position-values become secondary generators.In this way links are formed to other segments and the new

segment in SCP is expanded until there are no position-valuesfor which a complete scan has not been made. Thus, unlike

BCP, the scan for SCP is ordered, in the sense that there is

no retracing and in the sense that a prime generator heads onlyone segment. The characteristic nesting of segments generated

bySCP is evident in Example 10, which gives the complete

output string and a partial representation of the corresponding



3 5 5

notation for SCP(SSGT). This output string contains 11 seg-ments in all. Because of density, however, it is not feasible

to give all in notation. Example 11 gives SCP(PSGT) in its

entirety.

SUMMARY: STRUCTURE OF THE PARSING SYSTEM

From the preceding descriptions of the parsings that make upthe system it should be evident that the system is hierarchic,in the initial sense that primary segments are formed before

secondary segments. Further, from the definitions of block

and segment and the rules for BCP and SCP it follows that there

is also a hierarchic relation between the function BCP and the

function SCP. Specifically, if we regard the function calls as

string names: (1) Every SG in BCP(PSGT) is a subsegment

(possibly improper) of some segment in SCP(PSGT); (2) EverySG in

BCP(SSGT)is a

subsegment (possibly improper)of some

SG in SCP(SSGT). Moreover, the nesting of BCP and SCP does

not produce additional distinct strings of segments. For ex-

ample:

SCP(PSGT) 2 SCP(BCP(PSGT))

SCP(SSGT)2 SCP(BCP(SSGT))

BCP(PSGT) 2 BCP(SCP(PSGT))

BCP(SSGT) 2 BCP(SCP(SSGT))

Also,

BCP(SSGT, PSGT)_ BCP(PSGT) n BCP(SSGT)

and

SCP(SSGT, PSGT)2 BCP(SSGT)n SCP(SSGT)

Finally, every SG in BCP(SSGT, PSGT) is a segment in BCP

(PSGT) U BCP(SSGT), and every SG in SCP(SSGT, PSGT) is asegment in BCP(SSGT) USCP(SSGT). Thus, the most efficient(in the sense of shortest scan) and the complete set of distinct

parsings possible within the system is represented by the fol-

lowing 6 names:

PSGT SCP(PSGT)SSGT BCP(SSGT)BCP(PSGT) SCP(SSGT)

The output strings are generated in the above order. Accord-



3 5 6

EXAMPLE

11

8 64 80 O28

0 16 96 168 176

Pt ~p I.1ri ,'T^ . 1P JB|

iif _

'A

("C1 4 160

>

[104

128

cJ ^ - -?J j J, v92 352 480 548 568

336

4264^ 2 -f 1264 312 44 48 512

-b~~ I16 t,V '1U"4VU384

SCP(PSGT)

(P48::2 P64::4::5::11 P80'3 P96::-411 )(P104::5 P120::6::10::11 P128::5::6::10::11 )(P432:: P464::11 P480::0::6::11 P488:11 )

e [ 544



3 5 7

ingly, the schema of string relations for the deletion of all du-

plicate segments is as follows:

IN: DELETE DUPLICATE SEGMENTS OF:

PSGT PSGT

SSGT SSGT PSGT

BCP(PSGT) BCP(PSGT) SSGT PSGT

SCP(PSGT) SCP(PSGT) BCP(PSGT) SSGT PSGTBCP(SSGT BCP(SSGT) SCP(PSGT) BCP(PSGT SSGT PSGT

SCP(SSGT) SCP(SSGT) BCP(SSGT) SCP(PSGT) BCP(PSGT)

SSGT(PSGT)

FROM SCORE READING TO HIGHER ANALYTIC LEVELS

After a reading has been obtained the question arises: How is

the output to be interpreted at a higher level? A comprehensiveanswer to this question would exceed the scope of the present

paper. Nevertheless, some indication of the nature of further

analytic operations can be given. A sequence of linked pro-

grams in the MAD programming language has been written to

process the output from the score reading program in order to

yield increasingly higher representations of structural rela-tions. If the collection of pitch-class representatives in each

segment of the output strings from the reading program isscanned to remove duplicates the resulting collection may be

called a compositional set. Then, from a listing of all the

compositional sets the analysis programs (1) determine theclass to which each set belongs; (2) list and count all occur-rences of each set-class represented; (3) compute, for each

pair of set-class representatives, an index of order-similarity;(4) determine the transposition- inversion relation for each pairof set-class representatives; (5) list, for each set-class repre-sented, those classes which are in one of three defined simil-

arity relations to it and which occur in the work being examined;

(6) summarize in matrix format the set-complex structure ofclasses represented in the work; (7) accumulate and retrievehistorical and other informal comments in natural language.

These programs thus permit environments to be examined for

similarities and differences and for distributional character-

istics which might suggest generalizations about the structure

of the particular work or about the structural determinants of

the entire class of compositions to which the work belongs. To

illustrate this, let us consider certain aspects of the pitch-



3 5 8

EXAMPLE

12

160 168 176 192 224 240 264 276

87

10

6 6

8

78 687 6

7,88

6

6

9

CLASS

2-2

7 7

9

8

86 6 9

98

8,99

7

77

7,8 8,98,9

7,8 8,96 6 8,9

8,9 7

8,9

7,8 6 6 8,9 76 6 8,9 7

8,9 77

7

77

7

77

77

8

2-22-2

2-12-1

10 8 2-22-32-2

10 2-210 3-1

8 2-110 2-3

8 2-110

10 4-12-1

10 8 4-13-23-1

8 2-1

10 8 5-110 8 4-1

10 8 4-110 8 3-2

PV

PSGT1234

56SSGT789101112131415161718192021BCP(PSGT)2223BCP(SSGT)24252627SCP(SSGT)2829

3031



3 5 9

EXAMPLE

13

PV 312 336 352 384 408 432 464 480 488 CLASS

43

2 1

4 54 34

3 53

4 3 53 5

2 12 1

2 12 1

01 1 1 1 1 1

6

00

2- 1

2-12- 13- 2

2- 23- 13- 1

11

0

0,6 2-611 1 16 ,1 1 1 1 2- 5

0 0, 6, 11 3- 5

0,6 ,1 1I1 3- 51 1 0,6 ,1I1 3- 5

0 11 0,6,11 11 3-5

0,6, 11 11 3- 5

0 6-10 5- 1

PSGT

23

567SSGT891 0

1112131415161718a1920

BCP(PSGT)212 223S CPCPSGT)24BCP(SSGT)

25SCPCSSGT)262 7



3 6 0

structure of two contiguous contexts in the Webern work thatwas parsed by the score-reading program.

The first context, extending from PV = 160 to PV = 276, is

represented in Example 12. The position-values are arrangedfrom left to right at the top of the illustration. Each row belowthen gives the pitch-class representatives for one segment.The name of the string from which the segment derives is in-

dicated by the headings at the left margin. The name of theset-class to which the corresponding compositional set belongsis given in the rightmost column. The first integer in the name

gives the cardinality of the set, the second gives the ordinal

position of the class on a stored list. Criteria of class-mem-

bership are based upon interval content and will not be explainedhere. *1 (Since a set containing one element does not form an

interval it is not distinctive with respect to interval content. )

Thelargest

set inExample

12 is the set numbered 28: class

5-1. It is evident by inspection that all the other sets are sub-

sets of this one and therefore that all components of the con-

text belong to a single set-complex. It should be said that in

other cases the determination of set-complex structure is not

as easy as this. In all, 7 of the 15 set-classes in the complexare represented:5-1, 4-1, 3-1, 3-2, 2-1, 2-2, and 2-3.

The second context is displayed in Example 13. There the

largest set is numbered 26: set class 6-1. The number of

set-classes represented is 9: 6-1, 5-1, 3-1, 3-2, 3-5, 2-1,2-2, 2-5, and 2-6. All but two of these belong to the set-com-

plex about class 6-1. The two referred to are: 3-5 and 2-6.

The set 2-6 is a subset of 3-5, which first occurs in the output

string BCP(PSGT) and can be seen in the score as the simul-

taneity at bar 10. Thus, within the context there is a distinct

division into two parts on the basis of set-complex structure,a division which is signalled by an entrance of pitch-name C at

PV = 432.

A comparison of the two environments shown in Examples 12

and 13 reveals that all the sets in the first are transformed

subsets of set number 26 in the second, set class 6-1. This

provides a basic and audible measure of change since the event-

configuration which begins at PV = 312 is a continuation of the

intervallic complex of the preceding context. The next distinc-

tive event, from the standpoint of pitch-relations, is the for-

mation discussed above: the pitch-class collection 0, 6, 11,

which is a member of class 3-5.



EXAM PLE

14

B

A

15

(A)

PRIMARY SEGMENT



3 6 2

Many detailed contextual associations are revealed by inter-

pretation beyond the score reading stage. To illustrate, Ex-

ample 14a shows a secondary segment distributed between viola

and cello. When this is compared with the primary segmentat B the two are seen to be related: They are representativesof the same set-class. The first (X) is a transposition of the

second (Y) at the interval of 11 half-steps. The relation is

non-trivial, for the two notes F and Eb that sound with the E

in X are elements of Y. The association of the two contexts isfurther strengthened by the simultaneities V and W. Again,these are members of the same class: V is a transposition of

W at the interval of 7 half-steps.

It is hoped that this necessarily incomplete treatment demon-

strates the way in which a basis for a still higher level of anal-

ysis can be constructed in terms of a system of pitch and in-

terval relations interpreted with the aid of theoretical concepts.

EXTENSIONS AND IMPLICATIONS

A simplified input string was used in order to facilitate devel-

opment of the parsing program. This limitation in no way af-

fects the generality of the system. Any syntactic class of in-

terest to the analyst can be associated with a position-value,

and an appropriate output format can be designed to accomodate

as many properties of event-configurations as desired.

Since the parsing rules are based on the occurrence of syntactic

structures in the input language (which is capable of repre-

senting every syntactic class), it is possible to extend the sys-

tem in many ways. It could be extended, for example, with

respect to depth of detail, as explained below.

In Example 15 the primary segment, delimited by the bracket

below the staff, contains two subsegments of interest: (A) and

(B). In particular, (B) is delimited by the slur, with end pointsmarked by '*'. In the corresponding input language statement

given below, these end points are represented by 'L+1'. It

would not be difficult to write a parsing function to isolate all

subsegments marked by slurs within primary segments. Any

syntactic class or combination of classes could be defined as

delimiters in a similar way, thus creating new analytic strata.

It might be pointed out in this connection that aspects of an

individual composer's "style" can be investigated to any depth,

provided, of course, that the researcher can specify the syn-

tactic conditions with sufficient precision.



3 6 3

The second possibility for extending the program is more so-

phisticated and correspondingly more difficult: that of providingan inductive capability such that scans would be modified undercertain conditions to consider other possible parsings or toscan for certain structures conditionally. This heuristic facilitywould greatly increase the power of the program, and wouldmake it possible to investigate context-sensitivity in greaterdepth.

In conclusion it should be recorded that the score-reading

program suggests a feasible long-range project: the develop-ment of a precise descriptive language for event-configurations.Such a language would be valuable for the study of musicalstatements in general and would be especially accessible to

computer-implemented studies. Much work remains to bedone.

ACKNOWLEDGMENTS

Work reported herein was supported in part by Project MAC, and M. I. T. researchprogram sponsored by the Advanced Research Projects Agency, Department of De-

fense, under Office of Naval Research Contract Number Nonr-4102(01).

The author is indebted to Stefan Bauer-Mengelberg, President of the Mannes Collegeof Music, for instruction in the use of his music representation and for several ofthe notions that underlie the work described in this article. The author is alsograteful to Gretchen Stein, who encoded the music for computer input and did thekeypunching for all data-sets.

For his careful reading of the manuscript and for many valuable suggestions JohnRothgeb deserves special thanks.

The score-reading program was developed during the academic year 1965-66 whilethe author held a fellowship in the American Council of Learned Societies. The

fellowship was sponsored by the International Business Machines Corporation.Grateful acknowledgment is made to both organizations for their essential assistance.



3 b 4

R E F E R EN C E S

1 Forte, Allen. 'A Theory of Set Complexes for Music", Journalof Music Theory,

8/2(1964).

2 An abstract of the project is published in Bowles, Edmund, Comp., 'Computer-ized Research in the Humanities", ACLS Newsletter, in Special Supplement

(June, 1966), p. 38.

3 Farber, D. J., R. E. Griswold, and I. P. Polonsky. "The SNOBOL3 Program-

ming Language", Bell System Technical Journal (July-August, 1966), pp. 875-

944.

4 Babbitt, Milton. 'Set Structure as a Compositional Determinant", Journal of

Music Theory, 5/1(1961).

26070754 analytic reading of scores allen forte

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