Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Towards a generative theoryof melody

Lindblom, B. and Sundberg, J.

journal: STL-QPSRvolume: 10number: 4year: 1969pages: 053-086

http://www.speech.kth.se/qpsr

STL-QPSR 4/1969

1. Some appropriate goals for musical theory

Much scientific endeavor i s directed towards the construction of

theories. This is s o because in empir ical sciences i t is in the f o r m of

a theory that scient is ts express the i r understanding of the phenomena

that they have chosen to investigate. Informally the t e r m theory can be

used to r e fe r to a sys tem of laws by means of which observed facts can

be derived a s consequences and can be explained and by means of which

s t i l l unobserved phenomena may be cor rec t ly predicted.

This usage of the word theory is not hard to i l lustrate with examples

f r o m the history of physics. F o r instance, the geocentric sys tem of

Ptolemy (70- 147 A. 9. ) was an accomplishment that made i t possible to I I explain'' many features of celestial motion and to predict in advance the

positions of planets with reasonable accuracy. It is t rue that i n the 16th

century the Ptalemaic view was shown to be "wrong". After much contro-

ve r sy it was replaced by the heliocentric sys t em advocated by Copernicus

(1473-1543), l a t e r defended by Galileo (1564-1642) and refined and de-

scr ibed mathematically by Kepler (1571-1630). A theory had emerged

whose predictive and explanatory power was g rea te r than that of the

Ptolemaic sys tem and therefore gained acceptance. F o r fur ther i l lustra-

tions s e e ( ' ) and (2) . To some of us i t may appear doubtful whether the notion of theory

a s sketched above and a s commonly found in many branches of physics

and natural science c ~ u l d eve r be applied with success to subjects in the

humanities, in par t icular to "aesthetic" disciplines such a s the study of

music. We might, f o r instance, feel tempted to a s k if there is any as-

pect of music that can be "understood" in the s a m e manner a s physicists

say that they understand plasmas, mechanics, o r wave motion etc; what

is there to be explained and what is there to be predicted, anyway? In

the i r everyday use of music many people in Western societies might a s -

sociate i t p r imar i ly with the sensuous pleasure they der ive f rom listening

to i t o r i ts ability to affect the i r s ta tes of mind and emotions. Stressing

these propert ies of music some people might feel that i t is too "sacred"

to be analyzed scientifically and that once you begin to understand, the

above qualities of music will disappear. This reaction, however, may

be assoc ia ted with beliefs not likely to be entertained by the ser ious stu-

dent of music. Rather, if he had asked the above questions he might have

done s o because in the study of psychological processes investigators a r e

too close to the phenomena that they a r e dealing with. This familiarity

often makes i t difficult t o s e e how ' these phenomena can pose ser ious

problems o r call f o r intricate explanatory theories. One is incl ined to

take them for granted a s necessary o r somehow ' natural ' . This

point is made by Chomsky about the study of language. As f a r a s the

present wr i t e r s know i t i s extremely r a r e to find the need f o r explanatory

I theories of music a s behavior (composing, performing and listening) dis- I cussed in the literntwre. Consequently i t might be suggested that Chomsky' s

point about language is relevant a l so in the c a s e of describing the "mental

facts" of music. Hindemith (4) r emarks in 1937, and his point may s t i l l

be valid, that "It is an astounding fact that instruction in composition has

never developed a theory of melody".

Let us now go back to an ea r l i e r paragraph where we wrote that a

theory r e f e r s to ' a sys tem of laws by means of which observed facts can

be derived a s consequences' . A few moments' reflection will convince

us that i t i s perfectly possible to interpret this definition i n musical t e rms ,

Let us f i r s t consider composing and musical s t ructure. The facts ob-

served f o r musical compositions can be found in the composer ' s written

record, i.e. the notation. They compr ise symbols for such things as the

duration of notes, the i r pitches, etc. It i s c l ea r that a l l facts in the

written representat isn of music can be regarded a s consequences of an

underlying sys t em of rules that the composer consciously o r not, a p ~ l i e s

in putting the piece together. There can be no doubt that in psychological

t e r m s composing is highly structured, rule-governed behavior, a c i r -

cumstance that musicologists have always been aware of a s the la rge lit-

e r a tu re on counterpoint, harmony, "Forrnlehre" etc. bea r s witness of,

Consequently to s ta te that the scientific study of music has produced a

theory of Bach' s music ( o r ra ther of some well-defined homogenous par t

thereof) would imply that scient is ts have ar r ived a t a basic se t of prin-

ciples o r "laws" by means of which they a r e able to der ive a l l the written

music under analysis. This theory would then provide a scientific explan-

ation of Bach' s works until these r e sea rche r s o r the i r colleagues pro-

duced a rival theory which could be shown to be be t te r according to gen-

e r a l scientific c r i t e r i a of theory evaluation. Moreover, this theory

would also constitute a hypothesis about the organization of the psychol-

ogical processes that constituted Bach' s knowledge of his craft . It would

not descr ibe how thoughts and ideas whirled round in his mind while he - wrote the pieces in question1 Nor would it s ay anything about the o r d e r

in which Bach wrote down such and such a sequence of bars , khe the r he

used o r did not use, an instrument a s an aid in inventing a par t icular

tonal sequence etc. Rather, the ' sys tem of laws' should be interpreted

psychologically a s general conditions n well-formedness f o r Bach' s

music, a s boundary conditions that serve a s constraints i n delimiting

the c l a s s of permissible musical structures.* Fur thermore , if in the

future, a theory can be devised that Corllectly descr ibes and explains the

facts observable in ~ a c h ' s music i t wil l not be limited to historically

documented matepial. It will pel'mit extensions. It will permit der iva-

tion$ of music that no one has seen nor heard but which i s produced in

accordance with the laws that have been found to charac ter ize Bach' s

music!, C3hseqtlently the$e cases constitute predictions about mite;ic

that Bach might have wriiten had he lived longer, had he had time, had

he been p=irticuiarly inte reg ted in the part icular fokm of dampositioll

under examination etc. , etcr

Summarizing what has been said s o f a r i t should, in our opinion be

possible to apply the notion of theory a s traditionally conceived of in the

philosophy of science a l so t a the study of music and musical behavior.

A successful theory of a given style of music would provide an under-

standing of the psychological processes that s e rve a s generative prin-

ciples in composing. In the scientific sense of the word i t would ' explain'

these processes . Such a theory if constructed would indeed constitute a

significant contribution to the study of the human mind and human nature.

So far , however, we have refer red to theories which deal only with

the more abs t rac t s t ruc tura l propert ies of music. But s imi l a r considera-

tions can be made also f o r other aspects of musical communication e.g.,

fo r performed music. The observed facts a r e in this c a s e sound waves

that a r e derived f rom information on the relative positions of notes in

pitch and t ime and rules that t ranslate this representation into an acous-

tic waveform. It is to be expected that such rules will reflect perceptual

constraints, principles of sound-generation character is t ic of instruments

o r , in the c a s e of singing, of the human vocal apparatur, a s well a s

various style-dependent conventions. Clear ly a scheme for computing

acoustic waveforms based on principles of this type could be said to X

Cf. the linguistic t e r m s competence and performance that r e fe r to a speake r ' s and listener 's knowledge of the rules of his nat've lan- guage and his actual use of this knowledge respectively (lzaf.

STL-QPSR 4/1969 57.

constitute a theory of musical performance. Let us quote some contri-

butions in this area.

The musical instrument and the human voice represent cent ra l fac-

t ~ r s in the generation of musical sound waves. Thei r propert ies may

be described and explained i n t e r m s of a rule sys tem that permits the

derivation of appropriate sound waves,

E a r l i e r i t was thought that the formant-like peaks in the sound spec-

t r a of double reed instruments were consequences of the acoustic pro-

per t ies of the tube resonator. Fransson discovered (5) that the formant

shape remained even when the resonator of the instrument was replaced

by a simple cylindrical tube. The interpretation proposed was that the

formants der ive f r o m the mechanical propert ies of the double reed. He

found suppart f ~ r this theory by exciting the instrument with an ionophone

sound source fi lkredthrough simple formant circuits. The acoustic out-

put f rom the instrument excited in that manner matched the sound spec-

t r a of the blown instrument quite well.

Another example may be quoted f r o m the field of the acoustics of the

singing voice. The "singing formant", an abnormally high spec t ra l level

in the frequency region around 2.8 kHz, was e a r l i e r explained a s due to

'head resonance". Sundberg (6 ) assumed that i t was associated with a de-

c r e a s e of the frequency distance between the third and fifth formants.

Simulation experiments showed that i f this frequency distance was de-

c reased to the values observed in the spec t ra of sung vowels, the output

spec t rum displayed a c l ea r "singing formant". Per fec t agreement be-

tween sung and simulated vowel spec t ra presupposed a glottal source

spec t rum abnormally r ich in overtones. Investigations of the source

spec t rum of sung vowels confirmed this assumption. Here, the genera-

t i ~ n technique provided a confirmation of the theory of the origin of the

"singing formant".

A complete understanding of the sound wave requires knowledge of the

rule sys tem used by the player in decoding the notes. Bengtsson (7) has

initiated a project intended t a shed light on the question of rhythm i n

musical performance. The question that he r a i se s is what a r e the rules

that t ransform the nominal durations indicated in writ ten representations

of music into the durations of acoustic segments characterizing the per-

formance? What a r e the generative rules that make a Vienna waltz

sound genuinely Viennese?

STL-QPSR 4/196 9 58.

We might assume, that the rule sys tem 3f the musical instrument

has a n isolated interest without implications fo r musical campcsition,

This would however be an erroneous belief, Knowledge of the rule sys-

t e m of an instrument is frequently required for a full understanding of

musical structure. We may just recal l the t e r m "idiomatic music" re-

ferr ing to music composed in o r d e r to suit the t imbre charac ter i s t ics

and playing technique of a cer ta in instrument especially well, Let u s

take a n example to i l lustrate this more clear ly (8). Two folkloristic

musicians playing the ' spilspipa' performed the s a m e melodies quite

differently, one using a major third, the other a minor third. An ex-

amination of the two ' spilgpipas' showed that this difference was due to

the resonance propert ies of the instruments. Moreover, the player us-

ing the minor third adopted a major third in playing the same melody

on his violin. Evidently, the melody existed in a major tonality in the

mind of this player, but the rule sys tem of his instrument forced a

minor tonality upon the melody.

Thus, to summarize, the technique used in working out a generative

theory of music that explains the s t ruc ture of music a s i t appears in the

wri t ten record, may a lso be used for a theory of musical performance.

The rule sys tems hereby developed may also help us to explain the

written record of music.

2 . Analysis and synthesis of nursery songs

2.1 Mate rial

The following observations concerning melodic s t ruc ture were made

in connection with studies of the music of Alice Tegnhr (1864-1943);

(henceforth AT). Many songs fo r which she wrote the words and/or the

music have enjcyed grea t popularity among children and adults of the

las t few generations. Her melodies a r e knawn in practically a l l Swedish

homes. They a r e the Swedish equivalents of ' Jack and J i l l ' , ' The Muffin

Man', ' Humpty Dumpty' , ' London Bridge' , ' Three Blind Mice' and

~ t h e r s that a r e well-known in Anglo-Saxon countries. We have selected

her music f o r its popularity assuming that the nu r se ry song, although

not always a product of remarkable compositional c raftmanship and a r t -

i s te ry , would offer a convenient starting point owing to i t s relative

simplicity. However, recalling the r e sea rch program outlined initially

STL-QPSR 4/1969 5 9.

in this paper and i t s emphasis on psychologically relevant explanatians

3f melodic s t ructure, we real ize that simple does not necessar i ly mean

easy and the difficulties associated with the t a sk of analyzing even the

' simplest ' m e l ~ d i e s must not be underestimated.

It should be remembered that a sma l l number of the songs that AT

published were written by other composers. Nevertheless the melodies

exhibit a reasonable degree of stylistic homogenity. The mate r i a l inves - tigated consis ts of the following melodies :

8 b a r melodies ------A-

in 4 /4 t ime: 1.7 J a n s a min docka in 2/4-time: 1.15 Ro r o barnet 1.8 Ekorrn sat t i granen 3.2 Kissekatt

1.10 Brollopomarsch 4.14 Majas visa 3.8 Mors lilla Olle in 3/4-time: 1.1 Julafton

3.10 Bldsippor 1.4 Solen skin' pb vd r a 3.11 9 e n doda gdsungen kyrketak 4. 1 Lasse l i ten 2.7 Krgketosernas visa 4.3 Turnmeliten 2.1 1 B revduvan 4.7 F roken kxlla 3.3 Sockerbagarn 5.3 Jumbommar 4.2 Dansvisa 7.4 Mdns klumpedump 4.13 Vad f l ickorna 7.7 Gunghasten vilja bli

7.13 Solstrdlen 16 b a r melodies ---------

in 2/4-time: 1.2 Nar jag va r liten i n 3/4-time: 2.5 Katten och svansen 1 .5 Ba, ba, vita lamm 2.6 Tuppen och honan 3.5 Hanseman 2.13 Marl XI1 3.6 Fodelsedagsvisa 3.13 I Ulkbacken 4.6 "Lille ju l a f t~n" 4.5 Smdjantorna

4.11 Arst iderna 4.15 Frit jof och Ingebarg 6.1 Julboc ken 5.1 Elring julg ranen 6.2 Fisken i badkaret 6.3 3 a n s i det grona 7.9 Ann-Sofi och Gret-Mari 6.6 Lillgubben och

7.14 "En glad och munter hnns flickor vandring sman" 7.8 Jans lb t

Fiigcln s i t t e r i parontrad Borgmastar Munthe Var t ska' du gd? Gossen h a r e n l i ten gullvagn Videvisan Vad gossarna vilja bli Vsrhalsning I skogen (vandrings13t) T r e s jomansflickor Hemdt i regnvader

2.2 Observations

Since the melodies studied cannot a l l be presented in print, for com- I

plete information on the original data, we r e fe r the reader to "Sjung med

oss mamma", vol, 1-7, AB Seelig & Co,, St3ckholm. Instead we shal l

use a single i l lustrative example (Fig. V-A-1). Concerning this m e l ~ d y

the following observations can be made. W e find half-notes, quar te r -

notes, sometimes dotted, and eight-notes. The longest notes a r e found

in the second, fourth, sixth, and eight b a r s and within b a r s on the f i r s t

and third beats. Metrically the f i r s t half period, o r opening phrase, is

equal to the second half (except for minor discrepancies). A s imi l a r

identity between the f i r s t and second halves we note within each half per -

iod if we d is regard the interspersed notes on the second beat of b a r two

and six.

Fig, V-A-2 shows the met r ica l organization of the tune just described

and eleven other songs. It appears that, by and la rge , the above observa-

tions a r e c ~ r r e c t a l so for the r e s t of the mater ia l with a few exceptions.

F o r instance, sixteenths occur in a few cases a f te r dotted eighths o r ow-

ing to the demands of vcrsification.

Traditionally harmanic events a r e analyzed with respect to the func-

tional role that each chord plays in a chord progression. The present

mater ia l has been examined in t e r m s of a parameter which may be said

to descr ibe such functional relations: harmonic distance f rom tonic.

This distance can be measured in t e r m s 3f the number of harmonic s teps

that must be taken f rom any given chord ( o r chord inversion) before

reaching the tonic. Thus to take a simple c a s e the dominant would be

one s tep f r o m the tonic whereas the double dominant would be two. Fig.

V - J L - ~ i l lustrates the distances that we have tentatively assigned to the

chords on the bas is of observations in the present material . Clear ly

there a r e cer ta in problems associated with the interpretation of harmonic

distances, The present assigments were made pr imar i ly f o r purposes

of g r o s s analysis. Obviously the tonic may be reached along many paths

f rom a given chord and the progressions possible f rom this chord may

be of varying lengths. Some paths may be typical of a cer ta in composer

o r s ty le and s o on. The numbers in Fig. V-A-3 re fe r to minimal ob-

served distances. Fig. V-A-4 shows a plot of harmonic distance a s a

function of t ime o r position along melody. Similar plots were prepared

h-lodcrato.

- Las - se, Las - se li - ten, Las - se, Las - se li - ten, Las - se, Las - se li - ten,

t--t------t------IC-l- r~ I rl. I - -- PA- ----- TI

stor - re &n ;lu ngn - sin tror, Las - se, i a s - be li - ten. men Gud r& - der o - ver allt, Las - se, Las - se li - ten. bor - ta bra, men hem -ma bast. Las - se, Las - se li - ten.

Fig. V-A-1 . Notation of the Swedish nursery song "Laere liten".

Fig . V - A - 2 . The metric structure of the 12 melodies in 4/4-time analyzed. The melodies are those used in the seven volume edition of Alice Tegnbr' s nursery songs (Stockholm 1892).

Mel. . n r.

1 :7 1 : 8

1 :I0 3 :8 3:lO 3 :II 4 : l

4 : 7 5 :3 7:4

.7:7 -

B A R N U M B E R

JJ.

1

J J J I J I J J J3nJ J find f i ~

J A J J J'J JJJ J. J J J J J J

f i f i n f i f i J J J n~ n~

JJ. JJ

2

A J J J f i ~

J J 2 J J J J J J 7 J A J J J J l J l J IIJ ad

J J an^ J nn~

J J

3

J J J J J J J 3 n J J

J J n~

J nJ J I J J J J

J. J J J d ~ : ~ J J J J J ~ J I J ~ J J J J J I J I J ' ~ J J J J J I J J J ? J J

f i f i f i f i J 3 J JJIJ J

yrnnnnm~ $J. JJ

4

L i A J J J 1

J J t n J

J J JJIJ J

J r.

5

J y J l J n J ~ A J A J J J I J J J J I J 7 J J2J f i J

JJJ n J J I J J J .

J3 f i f i JJJXJ J ~ A J J

yznn~ J ad

6

J J I J J J J J I J J J ~ J J J J J ? UJnJXJ-3JJJ

A J JJJ

J'IJ 2

J n~ J ' J J J

J J n n ~ J

n m ~ . J

7

J J AJ J J J J J J A J z J nJ J l J

J J J J. J J J J J J a L l J l

J3fiJ3JU3J . .

annn~ ~rf inm~.u.u J J JA

8

A N f i J J J J J t

J I J ?

J J ?

J J 2

1

7

d

T = major tonic D=major dominant

S=major subdominant

r=minor relative b= double dominant

Fig . V - A - 3 . Values asc r ibed to different chords giving t h e i r harmonic dis tance f r o m the tonic.

TONIC 0

2 3 4 5 6 7 8

BAR NUMBER

Fig . V-A-4. The harmonic d i s tance f r o m the tonic of the chords appear ing in the song shown in F ig . V - A - 1 .

STL-QPSR 4/1969 61,

f o r a l l melodies analyzed. As already pointed out, our interpretation of

the notion of harmonic distance should be fur ther refined before a l l the

details of s ta i rcase curves such a s that ~f Fig. V-A-4 can be evaluated.

Presumably the following observations would not be affected by fur ther

refinements. As a rule, the f i r s t b a r s t a r t s out f r o m the tonic. A new

chord may be introduced half-way through this ba r , in the next b a r o r

in the third. In the fourth b a r e i ther the tonic o r the dominant appears,

A s imi l a r pattern i s found in the second half of the period. Here, how-

ever , only the tonic ~ c c u r s a s the las t chord of the eighth bar. The shape

that the harmonic distance curve assumes for the l a s t four b a r s in Fig.

V-A-4 is encountered over and over again in our material: f i r s t an ex-

cursion f rom the tonic and then the typical "slower" s t a i r case approach

to the target chord. Each half of an eight-bar period may display one o r

two o r four such "depart-return" patterns. Harmonic "departuresff a r e

character is t ical ly disc ontinous and may often involve fair ly wide l c a ~ s

"Return" paths, on the other hand, a r e continuous in the sense that chord

progressions develop in a more o r l e s s stepwise fashion the harmonic

distance f rom the target ch3rd gradually diminishing. PA chord change

a l ~ n g a re turn path is thus a movement in the direction of the tonic f rom

one p i n t to another adjacent to i t in the network of harmonic relations.

Sometimes a s in Fig. V-A-4 departures a r e more extensive in the second

half of the period than in the f i rs t .

With respect t o tonal movement in ' Lasse liten' (Fig. V-A-1) the

following facts emerge. Symbolising s teps by S, repetitions by R, and

leaps by L and upward and downward movement by + and - respectively

the sequence ~f tones i n ' Lasse liten' can be represented a s follows :

where the raised symbols indicate tonal relations ac rgss bars . A c e r -

tain economy charac ter izes the melodic processes . The f i r s t four b a r s

have s teps and leaps in the same places a s the las t four. With respect

t o signs, b a r s 3 and 4 = 7 and 8, 2 = 6. Cornparing b a r s 1 and 5 we

find opposite signs. Moreover although not identical the f i r s t and third

b a r s display s imilar i t ies . All these complete and part ia l identities

between the different par t s of the song exemplify, what i s known a s se - - quencing. We a lso note that in the opening phrase tonal movement has

an upward direction. In the second half i t is downward. The song ends

on A4 the key being A major. Chord notes a r e found e. g., on the f i r s t

and third beats in every bar . Non-chord notes may appear in other posi-

tions. Regularities of the above kind a r e prevalent in the ent i re material .

A c lose r look a t the distribution of chord notes reveals that the third

beat of even-numbered b a r s takes only chord notes; the f i r s t beat of a l l

b a r s and the third of odd-numbered b a r s may take non-chord notes that

constitute suspensions; a l l remaining positions including interspersed

eighth-notes take al l types of notes, passing, auxiliary notes etc.

2 . 3 Interpretations

2.3.1 Choice of framework

Many of the metr ic , harmonic, and tonal facts mentioned above

probably s t r ike the professional musicologist a s famil iar o r elementary.

But we must now a s k why the melodies have these properties. The prob-

l e m i s this: Given a cer tain well-defined c l a s s of melodies, what a r e the

principles and laws by means of which the met r ic , harmonic, and tonal

facts of these melodies can be derived?

Previously, attempts have been rnade to descr ibe musical processes

in t e r m s of Markovian o r stochastic models ( 9 ) . In such frameworks

which originated within information theory, regularit ies in tonal organiza-

tion a r e described using conditional probabilities. We believe that the

"grammaticality" of a well-formed melody cannot be adequately captured

in probabilistic t e rms . It s eems c l ea r that a theory of musical s t ruc ture

should not be dependent on chance and our good luck, fo r i f i t were, when-

ever we we r e unlucky an 'lung rammatical" melody would be derived.

But r a the r than enter into a detailed analysis of the applicability of in-

formation theory to the description of music we would like to r e fe r the (10) reader to Slawson's discussion of this question . We quote:

"Suppose, f o r example, that a Markov device has been se t up to generate the pitches of a melody in the style, say, of the ear ly 18th Century. The machine, having just chosen a B natural, i s in a cer tain s ta te - that associated with the B natural. The machine i s faced with a number of possibilities fo r i t s next choice. Since the machine i s ignorant of i t s own history, i t cannot use information about i t s ea r l i e r choices, such as , fo r example, whether the melody began in C o r E major, A choice

must be made on the basis of some assigned probabilities. If the e a r l i e r choices had established C major, the B natural would very likely have to be followed by C. If E major had been established, a much wider choice would be permitted, but a C natural could be stylistically proper only under very unusual circumstances. Since the device cannot "remember" the key, i t cannot make the C natural highly probable i n one case and highly improbable in another. It simply lacks the necessary information. Such remote dependencies a r e beyond even "higher order" Markov processes because melodies vary freely in length. That i s to say, our musical intuition se t s no fixed upper bound to the number of notes in a melody. Thus, in addition to being highly inefficient, the extension of a Markov process t 3 some fixed higher o r d e r is completely arbi t rary. Such psychological factors a s memory and attention span a r e what l imit the lengths ~f melodies, not some principle that is an inherent par t of the musical language",

La te r in his review Slawson uses the 25th variation of Bach' s Goldberg

variations a s an example of musical 2 r ~ n n i z a t i n wh!;st: complexity is

beyond the descriptive power of Markov models:

"Each chord of the basic progression, which controls each of the va.riations, is filled out o r embellished, in the 25th varia- tion, with a miniature progression of i t s own. Each of the sma l l scale progressions is perfectly lawful within itself and in rela- tion to the part icular member of the basic progression being embellished . . . . . . . . . To generate, o r "explain" this var ia- tion, we must r e so r t to rules that can r e fe r to themselves - rules that permit "self-embedding ". . . . . . . . . . The analogy in language to this kind of organization is the possibility of em- bedding relative clauses within relative clauses within relative clauses , etc. , without any fixed limit. It has been shown that a Markovian process , no mat te r how complex, cannot produce such s t ruc tures ( see Chomsky, Syntactic Structures , Mouton, 1962, Chapt. 3) . "

The framework i n t e r m s of which we shall formulate our interpreta-

tions of the present musical data i s that of generative o r transformational

g r a m m a r which was developed during the last few decades i n the study

of language. The c lass ica l p r e s e n t a t i ~ n of generative g r a m m a r is

Chomsky' s Syntactic Structures ( I 1 ) t.i which Slawson re fe r s in his r e -

view. Since the appearance of Chomsky's book generative o r t ransform-

atianal linguistic studies have been intense and a number af cmtr ibut i3ns

have been made that constitute fur ther developments of the ~ r i g i n a l sug-

gestions by C h ~ m s k y and others (12). Let u s take a simple sentence in

English:

John beats his s i s t e r (2)

we may want to express the information contained in (5) by introducing

labeled brackets :

These considerations i l lustrate that our melody has constituent s t ruc ture

and imply that some of i t s basic building blocks can be generated with a

so-called phrase-s t ruc ture g r a m m a r I*). This demonstration hardly

provides any new insight into the s t ruc ture of melody. The .>nly thing that

may s t r ike the reader a s novel might be the terminology of generative

grammar .

But le t us go on to examine some of the consequences that the syntactic

organization of spoken utterances have with respect to pronunciation. It

can be shown that syntactic s t ruc ture is one of the determinants of the

' s t r e s s contour ' , a character is t ic feature of every utterance (12b). B~

s t r e s s contour a linguist would normally mean the degrees of relative

emphasis, o r prominence, that speakers and l i s teners would assign to the

successive syllables of an utterance. Let us give a few examples that lend

credence to our claim. Again take sentence (2).

We can represent the distribution of s t r e s s e s that might be observed

under normal, idealized conditions, a s

2 3 1 John beats his s i s t e r

where the heaviest, main o r pr imary s t e s s s = 1, secondary = 2, and

t e r t i a ry = 3. In a g r a m m a r of English it must be possible to der ive this

par t icular way of s t ress ing the words. The s t r e s s contour 2 3 1 associated

with this sentence is a grammatical fact of English. Individual and dialectal

variations between different speakers of English may sometimes complicate

the task of finding the rules that cor rec t ly predict s t r e s s e s but this problem

is usually solved by looking a t one variable a t the t ime ignoring o r keeping

constant such things a s individual, dialectal o r other factors , which a r e not

a t the focus of the linguist' s immediate interests . Within the framework

of generative phonology i t has been suggested that a sequence of s t r e s s e s

such a s that proposed above for (2), be generated in the following manner.

Essentially two different types of rules a r e used:

a ) rules that assign s t r e s s e s within words and

b) rules that assign s t r e s ses within compound nouns and phrases. Some-

t imes the application of a rule is cyclic, that is, the rule is applied sev-

e r a l t imes before a derivation is terminated. We can take a s our starting

point (4) which we repeat for clarity:

((( John)N)Np ((beats IV ((his )Fron ( s i s t e r ) N ) NP)VP)S (4)

In short , the computation of the s t r e s s contour begins inside the inner-

most pa i r of brackets. We s t a r t out by applying rules that assign s t r e s s e s

to the individual words. The deepest constituents a r e - his and s i s te r .

The la t te r receives s t r e s s on the f i r s t syllable. The f o r m e r is a pronoun

and does not receive s t r e s s . Next we e r a s e the innermost pa i r of brackets

and again look fo r the deepest constituents. This t ime we find ( ~ o h n ) N beat^)^ and (his sister)Np. At this level a lso ' John' and 'beats ' receive

s t r e s s according to a rule of type a). E r a s e innermost parentheses once 1 1

more and we get ( ~ o h n ) and (beats his At this stage the NP

conditions fo r applying a rule of type b) a r e met. The V P contains two

main s t r e s ses . Since i t is not a compound noun but a phrase, a ' right-

priority ' rule ( ~ u c l e a r S t r e s s ~ u l e ) i s applied that i s , the rightmost

s t r e s s remains and the other one is lowered by onedegree . The Com-

pound rule i s a left-priority rule. 2b) After this operation brackets a r e

e rased and we now have the sequence (1 2 The right pr ior i ty rule

gives 2 3 1 and that is exactly what we intended to get.

Let u s take one more linguistic example that may contribute to fur ther

clarifying the fact that syntactic s t ruc ture determines s t r e s s .

- . I

smal l I I

boy 's school I

s mal l I

boys' I

school

The string ' smal l boys' school' i s ambiguous in conventional orthography.

But in the pronunciation of an English ta lker there would be no ambiguity

a t all. A native l i s tener would be able to decide whether the ta lker i s r e -

fe r r ing to a boys' school that is smal l o r to a school fo r smal l boys,

since the s t r e s s patterns a r e different in the two cases . The Compound

rule and the Nuclear S t ress rule predict this difference, a s i t seems, in

the co r rec t manner. F o r (8) we obtain 2 1 3 by applying the left-priority

STL-QPSR 4/1969

rule to the compound noun 'boys' school' and then the right-priority rule

within the NPA F o r (9) the application of these rules in the opposite o r d e r

gives 3 1 2. Consequently there can be l i t t le doubt that in English syn-

tactic s t ruc ture is one of the determinants of the phonetic shape of ut-

terances.

We would 92w like to peksuade the r eade r to play the following game

with us. Let us t r ea t the melody of F ig r v-A-1 a s i f it were an English

sentence. At the top of Fig' v-A-5 we recognize the melody of ' Lasse

liten'. Immediately below the labeled bracketing corresponding to ( 5 )

and ( 6 ) a r e shown. Suppose that we apply the right-priority rule of Eng-

l i sh prosody cyclically to this bracketing& The successive derivatiohs

ape shown below, In the f i r s t cycle a i l b a r s receive one' s, After the

innermost parentheses have been arasedj the pa i r s of ones a r e turned

into pa i rs of 2 1 , After t w more cycles the final result becomes

4 3 4 2 4 3 4 1. What do these numbers mean? In spoken languages

and in particular, in Germanic languages like English, German, and

Swedish, s t r e s s is realized in t e r m s of ar t iculatory timing: i n a f i r s t

approximation the more prominent o r s t r e s sed a syllable is the longer i t s

vowel and to a l e s s e r extent its consonant segments tend to be (I5). But

what is the musical interpretation of ' prominence' a s defined by these

numbers and the underlying procedure?

2.3.2 The met r ica l o3aniza t ion of the 8-bar ~ e r i o d ------------ ------------ --- Let us assume that the above numbers do reflect something that

psychologically has to do with how prominent, "heavy" o r s t r e s sed a

given b a r seems in relatian to al l other bars . Let it be fur ther assulxed

that like in speech such relative prominence is signaled a lso in the t ime

dimension, 3bviously on the written record (but not in actual performance)

the duration of each b a r i s invariant. Suppose instead that the g rea te r

"weight" of a b a r i s manifested in t e r m s of note lengths: the more prom-

inent a given b a r the longer notes, o r equivalently, the fewer notes it

should contain. Let us examine this hypothesis and s e e whether i t is a t a l l

reasonable. There i s of course no a pr ior i reason why rules postulated

to account for observations about speech should be applicable a l so to mu-

s ica l phenomena.

To t e s t the conjecture we have gone through our ent i re mater ia l with

respect to the number sf notes pe r b a r found in 8 -ba r melodies wri t ten

in 2/!-, 3/s4 - and 4/4-time, The contents of the final eighth b a r has been

se t to unity by subtracting an appropriate number. Thus when the

eighth b a r contains three notes we subtract by two$ when i t contains two

notes by one etc. This subtraction i s a iso ca r r i ed out for the remaining

bars . After this normalization the average number of notes pe r b a r is

computed. In Fig. V-A-6 the value of this normalized and averaged number

i s indicated fo r each b a r individually, By definition the eighth b a r now has

one note. It can be seen that on the average a l l the other b a r s have more

than one, PLs noted ea r l i e r , minima occur in t h e fourth b a r and in the

seCond and sixth b a r s alth9ugh the la t te r a r e not a s pronounced, d o m e -

quently b a r s with add numbers appear to contain m o r e notes than the others.

Now fo r our hypothesis to survive the degree of "s t ress" camputed fo r

each b a r on the bas is of the period constituent s t ruc ture should be c o r r e -

lated with the measurements just described. In Fig. V-A-4 the solid l ine

indicates the relative "weights" assigned to each bar. It can be seen that

not only do the data points and the s ta i rcase curve follow each other quali-

tatively, there is a l so almost complete quantitative agreement for a l l

measures although the ordinates a r e not quite conlpatible. In fact if the

decimals of the measurement means a r e truncated and rounded off there is

perfect agreement between the predicted and the observed numbers. Con-

I sequently i t is tempting to conclude that applying the rules of linguistic

prosody to the type of music under analysis is not meaningless hocus -p~cus .

I

The resul ts a r e not likely t-, be due to chance and deserve fur ther invest-

igation.

2 . 3 . 3 AT'S musical use of the ve r se mete+e ........................ Since AT 'S music i s pr imari ly meant to be sung the met r ic s t ruc ture

of the melody should be intimately related to that of the meter. We must

consequently investigate a l so the relation between ve r se and melody i n

o rde r to attain an understanding of how h e r melodies a r e built. A few

examples will be given below that i l lustrate some main points. To the

left the prominent words have been marked, To the right the correspanding

.ry We have profited great ly f rom discussions with doc. S. Malmstrom, Stockholm, about these matters .

COMPUTED RELATIVE PROMINENCE

COMPUTED RELATIVE PROMINENCE

Fig. V-A-6. Avera ed and normalized number of notes per bar in a) [.I 12 melodie. in 4/4-time and X ) 8 melodies in 3/4-time (upper graph) and, b) 10 melodies in 4/4-time, ( X 10 melodies in 3/4-time, and (0) 10 melodier

in 2/4-time (lower graph). I

STL-QPSR 4/1969 70.

of the underlying segment of the poems. This phenomenon is well known

in the study of prosody and i s denoted by the t e r m catalexis. However,

we cannot s tay content with such an explanation for why should the ve r se

have this property in the f i r s t place? On the other hand if we regard

songs written i n the style of AT a s a simultaneous verbal and musical

realization of an underlying abs t rac t s t ruc ture and sys tem of prosodic

rules that a r e largely common to both versification and musical composi-

tion we might have an explanation of the asymmetr ica l contents of v e r s e

segments within reach. The explanation is that given above to account

for the met r ica l organization of the AT melodies. Accordingly it might

be proposed that the t r e e s postulated and the rules of prosody permit us

to predict the relative weights not only of b a r s but a l so of the correspand-

ing ve r se segments prominence o r weight in these segments being signaled

in t e r m s of the number of syllables they contain. Halle and Keyser (16)

interpret the notion of me te r a s a highly abs t rac t pattern which like the

invisible par t of an iceberg is hidden under the surface and whose prin-

ciples may a lso be hidden deep in the poet' s mind beyond the reach of

his consciousness. If we look upon me te r in that manner the explanation

proposed above should appear acceptable.

2.3.4 The met r ica l o s a n i z a t i o n of the 16-bar period ------------ ---------------- The lower half of Fig. V-A-6 shows the average notes/bars for 16-bar

melodies. T 3 compute a s ta i rcase curve that provides a good fit to the

data observed the constituent s t ruc ture i n the lower par t of Fig. V-A-7

must be postulated. Comparing this with the t r e e of the 8-bar melodies

which is shown in the upper par t of Fig. V-A-7 we s e e that the f i r s t

eight' b a r s of the 16-bar period has roughly the same s t ruc ture a s the

8-bar periods themselves whereas the second half does not contain the

two four-bar constituents but fo rm four 2-bar units direct ly dominated by

the 8-bar constituent. The average 16-bar period i s thus a s t ruc ture

divided into 4 + 4 + 8, o r ( 2 + 2)(2 + 2)(2 + 2 + 2 + 2), according to this

analysis. It may be objected that this s t ruc ture is an ad hoc ~ b j e c t that

we have constructed just in o rde r to save the rules and the analogy with

linguistic prosody. Until there i s independent justificatian for i t i t is not

unreasonable to remain sceptical about i t s significance.

Fig . V - A - 7 . T r e e d i a g r a m s (consti tuent s t r u c t u r e ) postulated t o predic t re la t ive prominence of number of notes wer b a r (solid l ines in Fig . V - A - 6 ) . Consti tuents a r e labeled according t o the number of b a r s that they contain.

STL-QPSR 4/1969 71.

2 . 3 . 5 Chord prosressions --------- ----- However, there does s e e m to be such independent evidence in the

patterning of the harmonic data. We suggested in our discussion of har -

monic facts that chord progressions a r e typically organized in t e r m s of

"departure- return" patterns. Recall that each phrase of an 8 -bar mel-

ody would exhibit one, two, o r sometimes four such patterns. Fig. I V-A-8 shows the average harmonic distance f rom the tonic plotted a s

a function of t ime for twelve 8-bar and ten 16-bar melodies. Note that

the upper diagram (pertaining to 8 -ba r tunes) indicates two major humps I whereas the lower d iagram shows two major humps for the initial eight

b a r s of the 16-bar songs but four "depart-return" patterns for the final

eight bars . Clearly in the 8-bar tunes chord changes in the direction

of the tonic a r e organized in t e r m s of two blocks of four b a r s each o r

in t e r m s of 2 + 2 + 4. Similarly, harmonic events within 1 h-bar mel-

odies appear to be arranged in two 4-bar blocks followed by four 2-bar

blocks. The paral le l between 8 -ba r and 16-bar harmonic s t ruc tures

and the corresponding constituent s t ruc tures should be obvious: 8-bar

songs a r e on the average built according to a * t 4 s c h e m e b o t h (2 + 2)

metrical ly and harmonically. Similarly there is not mere ly the met r ic

and durational data to support the 4 + 4 t 2 + 2 + 2 + 2 s t ruc ture fo r

the average 16-bar period.

2 . 3 . 6 Melodic l ines ----------- An analysis of the melodic mater ial in t e r m s of patterns such a s

(1) shows that a very common tonal relation between two consecutive

notes is the interval of a second o r step, - S. Another frequent relation

can a lso be regarded a s a s tep not along the diatonic sca le but along

the chord. We can symbolize this relation by A. Leaps, L, a lso occur

but not without restriction, In ' L a s s e liten' and other songs they occur

once o r twice during the whole melody and appear to have the effect of

creating constrast and increasing melodic ' tension' raising the melodic

line to i t s peak. In the second half of these songs leaps may reappear

but then with the opposite sign. The fourth type of tonal relation i s re -

petition, R. AT ' s music i s constructed pr imari ly on the bas is of S, A

and R relations, It i s tempting to interpret this c i rcumstance a s pointing

to the operation of what we might call an adjacency principle. Why such

a principle should operate i s not too hard to suggest a reason for , AT's

1 melodies a r e vocal not instrumental, When singing, an increase in pitch, I

-I 0 Z

HARMONIC DISTANCE -I 0 z HARMONIC DISTANCE z

everything e lse being equal, i s normally associated with a correspon-

ding increase in vocal chord tension subglottal p res su re and general

physiological effort. Thus for energy expenditure to proceed smoothly

minimal changes of pitch such a s s teps along a scale o r he tween ad-

jacent chord notes would be appropriate.

When we sing o r play a melody note by note without introducing

chords we can sti l l in many cases 'hear' those chords. The tonal

sequence 'implies' the underlying chords. Also the AT melody possesses

this property of harmonic implication. This phenomenon has to do with

the rules that govern the distribution of chord-notes and non-chord

notes. F o r the AT melodies examined we obtain a profile for two con-

secutive b a r s that can be drawn a s

BAR I I1

BEAT: 1 2 3 4 1 2 3 4 Type of non chord note: (pas sing, auxiliary, Suspension suspen- sion etc. )

None (=chard- note)

It i s interesting to note that the most pronounced minimum occurs on

II:3 and other minima occur on odd-numbered beats. This distribution

is correlated with the ' s t ress ' contour that can be derived for the

events within pa i rs of bars . Such a derivation might proceed a s follows:

((0 0) (0 0) ( 0 0) ('0 0)) - .

P r i m a r y s t r e s s on all beats 1 1 1 1 1 1 1 1

left-priority rule 1 1: 1 2 1 2 1 2 (14)

right -priority rule 2 3 2 3 2 3 1 3

The final resul t is thus compatible with the profile shown in (13) but p re -

supposes that there a r e no brackets for bars . This analysis is in agree-

ment with the observation that a melodic line i s normally not segmented

into ba r s but often moves continuously within longer t ime spans e. g.,

pai rs of bars . Since b a r s do appear a s constituents in the met r ic and

harmonic analyses the question a r i s e s whether tone assignment rules

pressupose a slightly different interpretation of s t r e s s rules. This ques-

tion cal ls for investigation. Fur the r correlat ions between the relative

weight of a given note and the tonal interpretation of that note can be

found in the maximally heavy (vI I I :~) position where the fundamental of

the tonic i s the only premissible choice and i n IV where likewise only

a limited choice is available owing to the restr ic ted inventory of chords,

We have made extensive use of prominence in interpreting met r ic

harmonic and tonal facts, We have argued that prominence i s related

to the hierarchical relations within syntactic s t ructures . If these inter-

pretations appear somewhat far-fetched we would like to remind the

reader of Riemann' s theories (I7' and the analyses of the met r ic patterns (18) of music into hierarchical levels performed by Cooper and Meyer ,

In fact Riemann attributes to the 8 -ba r period of quantitative 'p romi-

nence profile' that he gives a s 1 2 1 3 1 2 1 4. Each number i n

this s e r i e s represents a cer ta in ' Gewichtsstufe' . Note that this con-

tour is identical to the one derived above except that in our notation

numbers re fer to prominence rank whereas Riemann' s stand fo r

weight.

2.4 Sketch of the rule sys tem

In the following section we shall present a tentative sys tem of rules

with the aid of which simple melodies can be generated. The major com-

ponents of this sys tem a r e shown i n Fig. V-A-10. These rules have been

designed in such a way that many, but a t present certainly not all , of

the important facts about AT' s songs can be derived. Fig. V-A-9 shows

some ear ly results. Tunes labeled op. 1 differ f rom that marked op. 2

and (40) below mainly i n t e r m s of l e s s seve re restr ic t ions on leaps.

Thus partly for expository reasons partly for a lack of understanding

of a l l details our proposal is presented in a deliberately simplified form.

The consequences of formulating the sys tem a s i t now stands must b e

fur ther explored and extensive checking against original AT data will

be necessary before a more pe rmanent theory of the AT melody can b e

established. The major principles and rules that we hypothesize can be

informally described a s follows.

* We a r e indebted to Prof. Ingmar Bengtsson f o r drawing our attention to Riemann' s ideas on rhythm and metr ics .

SYNTAX w CONSTITUENT STRUCTURE

T IM ING RULES

DURATION

-

HARMONIC RULES

CHORD PROGRESSIONS

I

> TONAL RULES

I HARMONIZED MELODY

Fig . V - A - 9 . Tentative s y s t e m of ru les proposed fo r the generat ion of s imple melodies .

2.4.1 Syntax ----------- Basic to all derivations a r c tic syntactical processes that underlie 1

a given mclody. Tie suggest that tlicsc processes can be adequately

expresscd in t e r m s of the machinery provided by phrase s t ructure gram-

mar , In the following wc shall r e s t r i c t our discussion to C-bar nlelo-

dies in 4/4 tirile. Vye construct a g r s m m a r of this type in ouch a way

that it generates the following s tructure

As in Fig. V-A-7 constituents a r e for simplicity labeled according to

the number of b a r s they contain. It is easy to extend this g r a m m a r in

such a way that other s t ruc tures can be produced e . g.,

Such t r e e s a r e generated b i r-&e mkx (11) such a s

I ) u r ; ~ t i o n n l pattern of eighth b a r

( 1 ) Tllc sccond half of b a r VIII i s a half-note, or(note -t rest)of equi-

valent duration.

(2) Choose number of notes (n) in the f i r s t half of b a r VIII. n=O, 1 ,2 , 3

o r 4. Illustrations: I

(3) Number of notes per b a r

Compute the number of notes i n each b a r (N). This number is

obtained by adding the number of notes (n) that b a r VIII contains

in excess of the obligatory half-note, and the number representing

the prominence rank of the b a r in question. Accordingly,

N = n + p (26)

(4) Durational pattern of f i r s t b a r

Begin the assignment of durational values in b a r I. This assign-

ment is made by choosigg any permissible permutation of the N 1

notes obtained for b a r I in s tep 3. 'Permiss ib le ' is defined by

the following met r ic output constraint. -

Bar: I I1 111 IV V

Beat: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 f l 2 3 4 1 2

I no

We infer f r o m (27) that two eighth-notes must never occue on the

third beat of even-numbered bars . A half-note is the only pos-

sible realization of VIII:34 (the latter half of b a r eight) etc.

STL-QPSR 4/1969 7 7.

( 5) Durational pattern of period

Step 4 thus resul ts in a durational pattern for the f i r s t bar. The

second b a r is either a me t r i c copy of the f i r s t b a r o r an entirely differ- I ent configuration. In the la t te r case the procedure i s a s for the f i r s t

bar. A choice is made of a permutation of N notes compatible with the 2 met r i c output constraint (27). In the fo rmer case a simple copying i s

performed again bearing the output constraint in mind. Consequently

the rules should allow for both AA and AB a s me t r i c patterns. In the

following two b a r s I11 and IV no new met r i c mater ia l must be introduced.

The third and fourth b a r s make use only of A o r B patterns as f a r a s

the output constraints permit. The closing phrase i s invariably a me t r i c

copy of the opening phrase. The total inventory of patterns is thus ra ther

restricted. So i s that observed in the AT melodies provided that we

disregard minor differences between patterns. Melody 3:8 (Mors l i l la

Olle) has three durational patterns: A B A C A C A C. Since the only

difference between the B and C patterns i s an interspersed eighth-note

it appears justified to descr ibe this tune metr ical ly a s A B A B' A B' A I?! I

the B and B f patterns being related by an interspersion transformation.

Such differences a r e likely to be associated with minor i r regular i t ies

in verse structure. If regarded a s essential they can easily be accounted

for in the system of rules by allowing a random perturbation of N by

some smal l number.

This concludes the description of the timing rules. To summarize

the derivation of the me t r i c pattern of an 8 -bar melody proceeds a s

follows:

1. Compute durational pattern of eighth bar.

2. I ' number of notes per bar.

3. " durational pattern of f i r s t bar .

4. " durational pattern of whole period.

2.4.4 Chord rules ------------ Summarizing our o t servations of the harmonic character is t ics of

8-bar melodies we should draw attention to the following facts which

seem to constitute some of the obligatory restr ic t ions and important

degrees of freedom available to the composer.

STL-QPSR 4/1969

Obligatory chord assignments

(1) In VIII:34 only the tonic is allowed,

(2) The harmonic interpretation of IV:34 is restr ic ted to e i ther

dominant o r tonic.

(3 ) Harmonic goals, departure and return patterns

Target chords a r e approached along a pattern of chord progres-

sions in which the harmonic distance f rom the tonic i s successively

decreased fo r each step. Given the harmonic point of departure

and the harmonic goal only a finite number of re turn paths a r e

available. These conside rations suggest that a composer may

choose the ' amplitude' of the harmonic departure , the harmonic

goal and one of the available re turn paths.

(4) Number of depart-return paths pe r phrase

Some of the be t te r known melodies (Lasse liten, Sockerbagarn,

Dansa rnin docka, ~ l d s i p p o r ) exhibit a single depar t - re turn pat tern

pe r phrase. Others a l so fairly well-known ( ~ o r s l i l le Olle,

Majas visa, ~ u m m e l i t e n ) have two. A few r a r e cases have four

(Ekorrn sat t i granen, Jungfru ~ a l l a ) .

(5) Harmonic contents of f i r s t b a r

In general the f i r s t b a r contains the tonic although there a r e a

couple of exceptions (Jungfru Kalla, Majas visa) which have the

dominant D. In these songs however the tonic i s implied i n the

anacrusis.

(6) Size of harmonic unit

Chord changes take place between bars, sometimes within bars .

In the la t te r case they generally occur between the second and

third beats thus dividing b a r s harmonically into halves. F o r

some melodies (Jungfru Kalla, ~ u m b o m m a r ) such int ra-bar

changes a r e over-all charac ter i s t ics every b a r exhibiting a chord

change. F o r others (BlHsippo r , Mors lilla Olle, Klumpedump,

~ u m m e l i t e n ) they appear particularly frequen$ly i n b a r s 5 and 6. Disregarding the melodies for which two chords pe r b a r is an

o v e ~ l l property we find the following distribution of 2-chord

b a r s fo r the tunes listecl in Fig. V-A-2.

Bar: I I1 111 IV V VI V I1 VIII

Number of 2-chord

(28)

b a r s out 0 2 3 2 5 6 2 1 of 9:

I

The available choices concerning the s ize of the constituent which is to

be harmonically interpreted a r e the b a r o r the half-bar. Note that we

a r e s o f a r s t i l l talking only about melodies in 4/4-time. The computa-

tion of chord progressions must be based on a determination of the

chord change pattern, a choice among alternative target chords in IV

and initial chord in I, a determination of the number of 'depart-return '

patterns p e r phrase and a choice of one of the premissible n-step re turn

paths - n being given by the chord change pattern, The following ' ques-

tionaries' i l lustrate the available options,

Determination of chord change pattern

(1) Is the b a r the harmonic unit?

(1.1) If yes, a r e b a r s 5-6 harmonically divided?

(1.11) If yes, a r e b a r s 3-4 and/or 7-8 harmonically divided?

(1, a If the answer i s no i n s tep 1 al l b a r s a r e harmonically divided.

Illustrations of possible output patterns:

Bar: I I1 I11 IV V VI V I1 V I11

Ix ( # X I x / x I xx I = I x 1 X I (29)

l x l x / x x 1 - 1 xx I x x 1 x I xl etc.

(2) Is the chord in IV:34 the tonic o r dominant?

(Is the chord in I:12 the tonic o r dominant?)

(3) How many ' depart-return ' patterns p e r phrase?

(4) At this point the chord change pattern, the chords in I:12, IV:34

and VIII:34 and the number of depart-return patterns a r e known.

The following pattern i s one of the possible inputs to s tep 4.

Bar: I I1 111 IV V VI V I1 VIII

In s tep 3 we determine that I-IV shal l have one 'depart-return'

pattern, V-VIII two,

The f i r s t departure f rom the initial tonic can occur either a t the

1-11 o r VI-VII boundary. Let us choose the 1-11 boundary, The T of IV

is the goal chord and should be reached in two steps. To determine the

chord of VI we choose among the possible 2-step progressions taking

u s back to T (Fig. V-A-3). One possible interpretation of 11 i s Sr.

(1) Is tonal movement directed upward o r downward i n the opening

phrase?

Suppose we explore the 'downward' alternative. As the result

of an obligatory rule the closing phrase has downward tonal

movement. i , T tunes with downward tonal movement in the

opening phrase generally repeat these f i r s t four b a r s with only

slight modifications in the closing phrase ( ~ a n s a min docka,

Brdllopsmarsch) provided that the harmonic events a r e the

s a m e within the phrases. Since this condition is met i n (33)

our problem is to generate a tonal sequence fo r the f i r s t four

b a r s and then simply repeat it.

(2) Select key.

Since the melodies should be easy to sing they must not en-

compass a wider range than say, B3-E5. As a consequence of

this fixed range, highest possible chord-note and other features

on which rules operate have key-dependent definitions. Assume

that E flat major has been selected.

(3) Harmonic implication.

This s tep should lead to a decision concerning the notes that

a r e c a r r i e r s of t h e harmonic implication associated with a given

melodic segment. This implication is expressed essentially in

t e r m s of the tonal relations A and R, that is, chord s teps and

repetitions. Within a t ime segment that is harmonically homoge-

neous, S- relations occur only a s ' fill-in' material , not to sug-

ges t an underlying chord. That would of course be impossible

by definition. Consequently in the f i r s t b a r A' s and R' s must

be selected f rom the notes of the E flat major chord that fall

within the range. Thus we obtain E4 flat G4 B4 flat and E5 flat.

Tone assignments proceed in a n o r d e r which is determined by

the relative prominence of notes. F o r (32) we compute s t r e s s

contours fo r each b a r a s follows:

left-priority rule

- I 1 - (34) - I t -

STL-QPSR 4/1969 8 2.

B a r 11: (((I ) ( n )) ( d )) 1 1 2 1 left pr ior i ty rule

where 1 and 2 fall on 'heavy' beats and 3 and 4 a r e light notes. In

b a r I we s t a r t the assignments by asking: What is the relation between

the notes on the f i r s t and the third bea ts? A o r R ? Next we repeat

this question fo r the notes with s t r e s s 2 and 3, Suppose that we end

up choosing R a s the relation i n both cases , Which note a r e we sup-

posed to repeat? Before making a choice we must recal l what hap-

pened in s tep 1. Here we selected a .l-.wnward direction for the tone

contour of the initial phrase. Since s tar t ing out on E4 flat o r G4

would leave no leeway for such development we must +k B4 flat o r

E5 flat. Tossing a coin about these alternatives gives us B4 flat but

e i ther should work. We now have B4 flat, x l , B4 flat, x2, B4 flat, x3

where x represents a note of a s yet unknown pitch. The f i r s t two x ' s

should be interpreted in accordance with the adjacency principle.

This means that these x ' s should be related to adjacent notes i n t e r m s

of S' s o r A' s. Alternative sequences a r e given below:

Tonal relation *1 - X2 - A + E5 flat E5 flat

This table indicates 16 possible realizations of the interspersed notes.

Are they a l l grammatical in the given metr ical context? In our opinion,

yes. One of them is: B4 flat - A4-B4 flat - E5 flat - B4 flat. Now to

determine the pitch of x we must look ahead. The second b a r contains 3 the subdominant, that i s , some inversion of an A flat major chord. As

before, we a s k what is the relation between the heaviest two notes, A

o r R? This t ime our coin te l ls u s to select the A relation. To make

a choice of tones that b e a r the A relation to each other we must look

back. x j is preceded by B4 flat and should approach the following note

smoothly that is, by a step. Another requirement is that i n going f r o m

B4 flat to x a smooth (non-L) transit ion must a l so be selected. 3 !

Choosing C4 a s the f i r s t note of ba r I1 i s not compatible with these con-

ditions for x will bear an L relation t o either of i t s neighbors what- 3

ever we do. E4 flat i s a l so impossible. Although, f rom the point of

view of the f i r s t bar , it i s perfectly possiblc to interpret x a s G4 and 3

then have E4 flat, a s we a r e moving into A flat major , this note plays

the role of leading-note preparing the ground for A4 flat. The A rela-

tion previously established now offers a choice between E4 flat and C5.

Our coin te l ls us to explore the C5 alternative. In view of the adja-

cency principle the task i s now to choose the fill-in notes between A4

flat and C5. A l l relations between A4 flat and the following note ex-

cept L fulfill this ' fill-in' requirement. Thus we ask whether we

should select A t , R or S t , We obtain R. The final note i s selected

f r o m the s a k e alternatives, for example S t . The sequence of notes

has now been determined for the f i r s t two bars :

(4) Sequencing

The further development of the melody may be based on the introduc-

tion of new tonal mater ia l o r on sequencing. If the sequencing a l te r -

native i s chosen two facts should be considered in interpreting b a r 111.

The over-all tonal movement i s downward and the tonal relation pat-

t e rn for b a r I1 and thus if possible, a l so for b a r 111, i s R, S t , s+. The chord i n b a r 111 i s D7 (33). The sequence of notes compatible

with these requirements and the adjacency rule i s :

At the end of this b a r the melody note i s A4 flat and the underlying

chord i s D7. In the following b a r the tonic appears again. F o r the

same reason a s the final G4 of b a r I must be followed by A4 flat, the

A4 flat i s in this case obligatorily followed by G1. Sequencing requires

R, R, S t and we obtain

Thc iirlal phrase i s to be a copy of the opening phrase except fo r the

obligatory occurrence of E4 flat on the final note of V.111 and the

slightly different met r ic appearance of this b a r which i s

the value of n being se t to two. The sequencing procedure forces '

us to interpret the f i r s t note a s G4. The second note constitutes a

fill-in note and must be selected a s having an S or R relation to the

surrounding notes owing to the adjacency principle. Three possible

alternatives exist:

G4 G4 E4 flat

G4 E4 flat E4 flat

G4 F4 E4 flat

We have now derived the following tune:

2.5 Summary of rule sys tem

The components of the rule system a r e the syntactical rules, the

timing rules, the chord rules, and the tone assignment rules.

An important principle is that timing, harmonic, and tone assignment

rules a r e ' s t r e s s ' -dependent ' s t r e s s ' being determined by syntactic

s t ructure.

Metric sequencing appears in various forms but i s obligatory under

a l l conditions. Essent ial features of chord progressions a r e the concepts

of harmonic goal, depart-return pattern which a r e among the degrees of.

freedom available to the composer. In the generation of melodic lines

the notion of harmonic implication, the adjacency principle and melodic I

sequencing define the major constraints.

refs . on next page

STL-QPSR 4/1969 85..

References:

(1) Hempel, C. : Philosophy of Natural Science (Englewood Cliffs, N. J. 1966).

(2) Kuhn, T . S. : The Structure of Scientific Revolutions (Chicago 1962).

(3) Chomsky, N. : Language and Mind (New York 1968).

(4) Hindemith, P. : The Craf t of Musical Composition o on don 1948).

(5) Fransson, F. : "The Source S Cctrum of Double-Reed Wood-Wind Instruments", STL-QPSR 4) 966, pp. 35-37 and STL-QPSR 1/1967, pp. 25-27.

(6) Sundberg, J. : "Artioulatory Differences Between Spoken and Sung Vowels in Singers", STL-QPSR 1/1969, pp. 33-46..

(7) Bengtsson, I. : "Empirisk rytmforskning", Swedish Journal of Musicology - 51 (1 969), pp. 49-1 18.

(8) Fransson, F., Sundberg, J , , and Tjernlund, P.: "Bestimmung und Berechnung d e r Grundfrequenzen von mit d e r ' Spilgpipa' gespielten Melodien", paper submitted fo r publication to Deutsche Akademie d e r Wis senschaften zu Berlin, DDR.

(9) Hiller, L. and Isaacson, L. : Experimental Music (New York 1959). cf. a l so Fucks, W. and Lauter , J. : Exaktwissenschaftliche Musikanalyse (Koln und Opladen 1965).

(1 0) Slaw son, W. : "Review of G. Lefkoff, ed. , 'Computer Applications in Music' ", Journal of Music Theory - 12:l (1 968), pp., 105-1 11,.

(1 3 ) Chomsky, N. : Syntactic Structures (' s-Gravenhage 1962).

(1 2a) Chomsky, N. : Aspects of the Theory of Syntax (Cambridge, Mass. 1965).

(1 2b) Chomsky, N. and Halle, M. : Sound Pat te rn of English ( ~ e w York 1968).

(1 2 c ) ~ a k o f f , G. : Generative Semantics (to be published).

(13) Slawson, W. : "Review of Meta + Hodos, ' A Phenomenology of Twentieth Century Musical Materials and an Approach to the Study of Form' ", Journal of Music Theory - l O i l (1966), pp, 156-163.

For te , A. : "A Program fo r Analytic Reading of Scores", Journal of Musical Theory i2:2 (1968), pp. 331 -365.

Brender , M. and render, R. : "Computer Transcript ion and Analysis of Mid-Thirteenth Century Musical Notation", Journal of Music Theory 11:2 (1967), pp. 199-221.

Winograd, T. : " ~ i n g u i s t i c s and the Computer Analysis of Tonal Harmony", Journal of Music Theory - 12:l (1968), pp. 3-49.