graph topological representation of melody scores

8/6/2019 Graph Topological Representation of Melody Scores

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A G r a p h Topological Representation

o f M e l o d y S c o r e s

Leonardo Peusner

I introduce here a graphical method for mappingthe sequential aspects of a melody into a static, global pre-sentation of the music [1]. In the process, the mapping inci-

dentally makes explicit some of the aesthetic informationburied in the score of the melody: Pleasant melodies can be

represented by pleasant drawings. (The converse statement isnot necessarily true.) Although the visual patterns loosely cor-

relate to the normal way in which a listener segments themelody, the graph introduces new grouping elements that arenot part of the listener's perception [2].

Of course, there have been many previous applications of

topological and geometrical principles to music. The suc-cessful applications of geometry and topology to modern mu-

sicology-e.g. the pioneering work of Roeder [3] on class pitchseries, Kuster's analyses of pitch rows in Stravinsky's TheFlood[4] and Chambouropoulos's feature surfaces [5], to name

only a few-have demonstrated the benefits of applyingstructured-object representations to music theory [6]. In par-ticular, graph theory [7] is implicitly incorporated in manyformal aspects of modern music: artificial intelligence and mu-

sicology [8], cognition [9], modeling [10] and algorithmiccomposition [11,12]. In addition, graphs potentially relateboth to computer models and to grammatical forms and pro-vide the backbone to all connectionist neural models [13-16].

I should point out in passing that most of our fundamental

understanding of human physical reality is recorded in

graphs-e.g. molecular skeletons in chemistry, fundamental

particle interactions in physics (in Feynman graphs) and ther-

modynamic networks [17].It would be misleading, however, to claim that I purposely

set out use graphs in order to discover a universal musical prin-ciple, theory of analysis, new notational form or cognitive the-

ory of music.

My motivation when I began this project had little to do withmusic. I was writing a book on the genesis of irreversibility inBrownian motion-the haphazard, random-walk pattern ofparticles and molecules suspended in the bulk of a liquid. Dur-

ing a break, I picked up my Steinberger electric guitar and

began to play Albeniz's Asturias (1886), while my mind kepton thinking about the direction of time flow. For some rea-son, something in the hesitant though unyielding nature ofthe Spanish music reminded me of the physical problem ofthe vibrating molecules. Might it be possible to find a globalpattern to the melody using techniques similar to those I hadused in the problem of molecular motion? Given that I was

utilizing fractals in my thermodynamic work, my suspicion at

the time was that some pattern of "deterministic chaos" mightsurface-e.g. the familiar Lorenz butterfly, which is a signa-ture of many complex processes [18]. The result was strikinglydifferent: I always ended up with spiderwebs and other hope-

lessly messy designs. Several months A B S T R A C T

(and many frustrating attempts at T rstructuring melodies) later, I ruth- Ihis article s an nformal

lessly applied William of Ockham's presentation fa rather rivialobservation: fa melody core srazor of simplicity: I threw away my translated nto graph, hework and decided to let the music resultant rawing asaestheticitself tell the story. This time I qualities hatparallel n he

marked the distinct notes that ap- visual omain hepleasure fexperiencing hemusic n hepeared in a given melody as dots, auory rea Meoe thiauditory ealm. Moreover, hisand did what a child might have beauty asseveral nterestingdone: I joined the dots whenever a elements fprecision hat he

particular transition appeared. As I author xplores sing hesimple

repeated the process for many though igorous oolsof graphr ,. ,~~~~. r,theory.melodies, distinct patterns began toemerge.

Graph theory is just that, the artand science of joining dots by lines or curves! It is useful be-cause it allows us to represent structural, logical or sequentialinformation without the need to use "real" math. Graphs work

particularly well in situations in which one is simply tran-

scribing connectivity characteristics without the benefit of ageneral guiding theory or model.

Given that graph theory is used here only as a language totranslate a linear melody score into a drawing, it does not in-troduce extraneous or irrelevant elements to the originalmelody, or add anything new from outside the score. Ofcourse, by its very primitive nature, the graph technique haslimitations. A graph is like a bare Christmas tree, if you will:its decorations and bright lights must be provided by ancil-

lary disciplines-e.g. geometry or topology. For this reason,we will first review the fundamental ideas of graph theory and

topology, and only then proceed with the presentation of mu-sical examples and conclusions. Although my approach here

is heuristic-and I make no pretense at axiomatic deduc-tion-there is no reason to shun some elementary formal-ization.

What can we expect to learn from this exercise beyond anew aesthetic point of view? In my opinion, the prime lessonis the fact that a melody, as well as a given composer, has global,time-independent signatures-i.e. the patterns that we displayas graphs. These patterns allow melodies to be classified intoa finite number of equivalence classes. Moreover, the modelindicates how to partition the graph in unique ways, to obtainthe elementary phonemes that comprise the melody, so thatthe whole can be recovered from the constituent parts.

Leonardo Peusner (scientist, musician, writer), Boulevard Maritimo 2865, Mar del Plata,Argentina. E-mail: .

LEONARDO MUSICJOURNAL, Vol. 12, pp. 33-40, 2002 332002 ISAST


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Among the feasible practical and edu-cational applications, we can mention:the teaching of composition from a

global point of view; the use of graph"light organs" as listening aids for the

hearing impaired; a notational aid to theperformer; and the generation ofmelodies by computer (both as a com-position-interaction device and as a com-

puter-generated design) [19].

GRAPHS

Let us recall that graphs consist of

points-also referred to as nodes or ver-tices-connected by edges [20]. Morerigorously, one can define a graph as a

mapping that presents pairs of nodes assets of edges. It is important to stress that,although we are accustomed to seeinggraphs as explicit drawings, they can beexpressed independent of any drawingby means of incidence matrices. Theseare 2D arrays that specify which point isconnected to which. In the particularcase of the melody graphs used in the ex-

amples below, the simplest constructionconsists of pairing the notes with thenodes of the graph and connecting themwith arrows, which specify transitions be-tween notes.

Graphs in which edges specify a direc-tion (arrows from one point to another)are called directed graphs. If their edgesare associated with, for example, the

transfer of information or money, or withphysical flows, the graphs are referred toas networks. Graphs are planar if they canbe drawn without having an edge crossanother; maximally connected planargraphs connect as many nodes to othernodes as possible without crossing.

There are many technically interestingproperties of graphs, some of which have

caught popular fancy-the four-colortheorem, for example. From our point ofriew, t suffices to state that graphs pro-

vide a direct, simple, natural way to view

topological relations between neighbor-ing points. Topological relations are a

pre-metric notion-i.e. they are inde-pendent of the distance separating twogiven points (of course, distance is notdefined in topology) and of the shape ofthe curve connecting two points: a curveof any shape and a straight line are equiv-alent in the realm of topology; the onlything we care about is whether two pointsare connected or not. For that reason, wemay say that topology is geometry doneon a rubber sheet.

sen because classical, romantic and popmusic are the easiest melodies to intro-duce in the present context without un-

necessary complication.

EXAMPLESOF MELODY GRAPHS

Consider the standard notation to theopening bars of Mozart's G minor Sym-phony K550 (1788) (Fig. 1, top) [21]. Ihave assigned three levels of grey to threegroups of notes, corresponding to how alistener organizes the notes of a melodyinto segmented phrases [22]. In line 1underneath the standard notation, themelody is denoted as a linear trajectoryconsisting of a note plus a duration [23],while line 2 abandons all reference to du-ration (notes are replaced by points iden-tified by the name of the note). Takingadvantage of the flexibility allowed by to-

pology, I created a loop from one Re to

1.

2.

;JjMti Re Re M;b Re Re

a neighboring Re, without altering theneighboring relations. In line 3, I"folded" the melody path onto itself sothat different appearances of the samenotes are placed near each other; finally,in line 4 notes are "glued" together-allMib's to Mib's and, independently, allRe's to Re's-while corresponding pathsare collapsed into a single path. Arrowsleaving a note are numbered to denotethe order in which transitions take placefrom that note to a neighboring note.

The resultant graph displays roundarcs and circles, but these are completelyarbitrary: the topological circle denotesonly the closure of the themes and sub-themes in the melody, which are solelyspecified as a return to a node of originalong some arbitrary path. The graphconsists of four vertices-the set [Mib,Re, Sib, rest]; five edges-Mib-Re, Re-Re(a self-loop), Re-Mib, Re-Sib, and Sib-rest;and two faces, or loops.

Mib Re Re Sib Rest

M.- Re Re Mi R f Mb Re Re S>Mib Re. Re Mt.1. Re R? M b Re Re S.b Res'

3. ** -"0-X ~A-0.

Re Re Sit, Rest

In the following examples, I analyzethe opening measures of several familiarmelodies using a graph-topological rep-resentation. These examples were cho-

34 Peusner, A Graph Topology of Melody Scores

Fig. 1. The standard notation to the well-known opening bars of Mozart's G Minor SymphonyK550 (1788) shows three groups of notes corresponding to how a listener organizes the notesin a melody into segmented phrases [42]. Line 1, underneath the standard notation, denotesthe melody as a linear trajectory, while in line 2 notes are replaced by points, and loops arecreated from Re to neighboring Re, without altering the neighboring relations. In line 3, themelody path is "folded" on itself so that different occurrences of the same note are placednear each other and, finally, in line 4, all Mib's are "glued" to all other Mib's and, indepen-dently, all Re's to Re's, while corresponding paths are collapsed into a single path. Arrowsleaving the note Re are numbered to denote the order in which transitions take place fromthat note to neighboring notes. Transitions from other notes have no ambiguity. (? KarinaZerillo-Cazzaro)


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The most notable and noticeable

change from the traditional score is the

appearance of two faces. Thus, from aone-dimensional program, the melodic

trajectory specified by the score, we havedefined a 2D object, a graph that con-tains faces in addition to nodes and

edges. Faces indicate the closure ofthemes and subthemes within the

melody line, as registered in the foldedscore. In addition to the listener's natu-ral grouping of themes, new visual group-ings have appeared, which are not all partof the listener's tool kit-e.g. the returnarc Re-Mib, which completes the eyeglassshape of the graph.

As a second example, consider the first

eight measures of Henry Mancini's"Moon River" (1961) melody, which hasbeen folded in Fig. 2 so that different oc-currences of the same note are near eachother. Anyone attempting this procedurewould soon find that it is very difficult todo without "getting the cables tangled."(I realized Fig. 2 only after much trial anderror.) For that reason, I have adopted a

simplified approach, drawing single ar-rows between connected notes. Figure 3

depicts the resultant graph, obtained di-

rectly from the written melody by first as-

signing a distinct node or vertex to eachof the notes and then joining any binarysequence of notes with an arrow.

Clearly, the resultant drawing not onlyrepresents "Moon River" but can also rep-resent other melodies as well, providedthat new notes are assigned to the nodes.We may say that they are in the same

equivalence class.The simplified graph of "Moon River"

is planar, as are those of many of themelodies analyzed here; it has 9 vertices,6 faces and 13 edges.

For comparison, I should point outthat if I had drawn dots at random on a

piece of paper and connected them in all

possible ways without crossing lines, therewould be 21 edges and 14 faces. Note thatthe faces do not all have the same num-ber of sides.

Even more variability occurs in Bach's

"Gavotte" from the Fifth French Suite(1772), shown in Fig. 4, in which thereare seven three-sided faces, two five-sidedfaces, one loop, two six-sided faces andtwo nonplanar faces-a triangular and a

hexagonal face. Note that topology doesnot distinguish between straight lines

Fig. 2. The first eight measures of "Moon River" 1961), by Henry Mancini, plotted as atrajectory that starts in Sol and ends in La-the start and end points are marked as circleswith a dot and a cross inside, respectively. Duration is indicated by the use of standard notes.With the exception of the crossover in the links running from Sol to low Do, all "cables" are

untangled. (? Karina Zerillo-Cazzaro)

Mi

Dot

Do,~.r~ La

and curves, so that when I loosely referto a triangle or a hexagon, I mean that a

portion of the plane is bounded by threeor six curves, or whatever. Keep in mind,however, that we are not allowed to mea-sure distances in this game!

As a further example, Fig. 5 depicts thefirst few measures of the Autumn Concerto

by C. Bargoni (1956). The pattern of a

few closing loops or "circles" is ubiqui-tous in pleasant pop melodies, and it isshared in its basic elements with D. Mod-

ugno's "Nel Blue di Pinto di Blue"(1958)-the pop song known as "Volare"

(Fig. 6).On a completely different note, Fig. 7

depicts the first eight measures ofWeber's opera sonata Mouvement Perpetuel(1812). The graph appears more com-

plex than it is, because of the presenceof a larger number of notes (16). It con-tains 2 loops, 9 triangular faces, 3 four-

sided faces and one 14-sided face (theexternal boundary, if we count loops).(Again, note that the four-sided faces arenot squares, as the sides are unequal and

may be curved. The same thing is true ofthe triangular faces.) This compares with42 possible edges and 28 "triangular"faces in the maximally connected graph.It is clear that both this fragment andBach's "Gavotte" ntroduce many more

two-way transitions than do popular"tunes." Curiously, in the Rolling Stones'

"LadyJane" (1966), shown in Fig. 8, arest, not a note, serves as a pivot for threecircles. There are 9 notes, 11 faces and18 transitions joining the notes in this

melody.In a final example, Fig. 9 depicts Alexan-

der Guilmant's Tarantelle (Op. 48). Alltransitions are explicitly marked with ar-

Fig. 3. "Moon River" plotted to indicate onlythe basic connectivity (the topology), withouttracking individual transitions. There are 13edges, 9 notes, and 6 closing loops or faces.(The maximally connected graph would have9 vertices, 21 edges and 14 faces.) (? KarinaZerillo-Cazzaro)

Peusner A Graph Topology of Melody Scores 35


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Sol

1,5,7,8

Fig. 4. "Gavotte" rom the Fifth French uiteNo. 5 in G (BWV 16) byJ.S. Bach (1722). This isa nonplanar graph, because of the links that join Re with Mi and the low Si. If one eliminatesthe nonplanar faces, the graph has 11 notes (vertices, or nodes), 20 edges and 11 planarfaces. (The corresponding maximally connected graph of 11 notes would have 32 edges and22 faces.) (? Karina Zerillo-Cazzaro)

rows, to show the number of times a giveninterval connects one note to another.

A feature of this graph that deserves

particular attention is the fact that thenumber of arrows entering any one node

equals the number of arrows exiting thesame node. This is a kind of "conserva-tion of flow law" analogous to Kirchoff'scurrent law. This is not surprising, as itsimply means that the melody, once itstarts, continues until it finishes.

In principle, a note in Tarantelle ouldhave a number of transitions equal toeach of its allowable neighbors, or, at theother extreme, there could be a singletransition to a single neighbor-this isthe problem of ambiguity [24]. In thefirst case, the melody would be com-

pletely uninteresting; in the second, itwould be too predictable.

Three general types of nodes appear inthis graph: (1) unambiguous node tran-sitions-i.e. passing notes from whichthere is only one possible connection toa neighbor; (2) nodes that connect equalnumbers of times to a cluster of nodes-e.g. the high Mi; and, finally, (3) am-biguous nodes with unequally distributedtransitions to neighboring nodes. Pre-sumably, it is the latter set that injects theelement of surprise into the melody.

For reference and amusement pur-poses I include, in Fig. 10, the graph thatstarted all this, Albeniz's Asturias. Clearly,this simple flower is quite different from

Fig. 5. The first few measures of Concerto d~Autunno by C. Bargoni(1956) have been drawn as a graph. Numbers indicate the order inwhich transitions take place. Black dots represent the notes. Thispattern of epicycles appears very often in pleasant pop melodies.(? Karina Zerillo-Cazzaro)

1 Fig. 6. "Nel Blue di Pinto di Blue" ("Volare") by D. Modugno (1958)consists of 6 notes tied by 9 edges and closing in 5 faces. The num-bers denote simple transition order. (? Karina Zerillo-Cazzaro)



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Fig. 7. The first eight measures of Weber's Mouvement erpetuel(Rondo from the Sonata op. 24) (1812). (? Karina Zerillo-Cazzaro)

Fig. 8. The Rolling Stones' "LadyJane" 1966) (first 8 measures).This piece by M. Jagger and K. Richards s quite ntricate n its sim-plicity. There are 9 vertices, 11 faces and 18 edges. Three circlespivot around a rest instead of a note. (? Karina Zerillo-Cazzaro)

the multivariable "phase space" withwhich I started.

MELODY FRAGMENTSAND THE MUSICAL PHONEME

Suppose we fragment the graph intosmaller "phoneme" subgraphs consistingof each note plus the arrows leaving thenote together with the notes at the headsof the arrows. What do we end up with?

Obviously, there will be as many separategraphs as notes that were in the initial

melody. I will describe each new sub-

graph, or melody fragment, by placingall the notes that appear in the new graphinside brackets, beginning with the noteat the tail of the arrows-the "source," f

you will. For example, in the case of

"Volare" he melody graph would parti-tion into the following phoneme graphs:[Do, Do, Si, Sol], [Si, Fa, End], [Fa, Fa,La], [La, Do], [Re, Do], [Sol, Re],[Begin, Re], in which Begin and Endhave been treated as notes. As I statedabove, the first note listed connects toeach and every other note inside thebrackets. However, nothing else can besaid about whether the remaining notesconnect among themselves-that factmust be established by looking at the(two) phoneme graphs that begin with

the notes under scrutiny. Continuingwith the "Volare" example, consider the[Do, Do, Si, Sol] subgraph. Do connectsto itself, to Si and to Sol. However, we do

not know simply by looking at this sub-

graph whether Si and Sol are connectedto each other. That requires looking forthe phoneme graphs that start with Siand with Sol. (The answer is no.)

Up to this point, I have been looselyreferring to the fragment graphs, or sub-graphs, as phoneme graphs. I will now

attempt to justify the label I have

given them. If these acted, indeed, as

phonemes, they should be able to re-trieve the constitutive notes; they shouldbe able to yield melodic patterns of anylength; and they should be able to re-cover the complete melodic pattern-i.e.the complete graph. In addition, wewould expect them to have the exem-

plary behavior of yielding all of the above

by means of elementary logical opera-tions, not by some contrived procedure.

At this point, we will have to enlargeour minimal "technical" vocabulary a bit.We shall define the AND (synonyms:intersection, logical multiplication) op-eration between two phoneme graphs asthe element that is common to bothphoneme graphs. For example, [Si, Fa,End] and [Fa, Fa, La] have the note Fain common. In fact, we can easily see thatthe initial collection, or set, of notes inthe melody can be recovered, one pointor note at a time, by the pair-wise appli-cation of the AND operation. We now de-fine the OR operation (synonyms: union,addition) between two phoneme graphs,which is a new graph (bigger than either

initial phoneme graph) that contains allthe elements that are in either or bothinitial phoneme graphs, without repeti-tion. For example, the [La, Do] OR [Re,Do] is a new graph that contains an edgegoing from La to Do and an edge goingfrom Re to Do. Clearly, he OR operationis neither more nor less than picking two

phoneme graphs and "gluing" them to-

gether.As a further example, consider the

Mozart fragment of Figure 1. The

phoneme subgraphs in this melody frag-ment are [Mib, Re], [Re, Re, Mib, Sib],[Sib, rest]. (Note that, in the incompletefragment considered, Re moves to bothMib and Sib; however, Mib does not moveto Sib-this is established by the phonemegraph that begins with Mib.) The pair-wiseapplication of the AND operation to thesethree sets-i.e. what is common to a pairof them-yields the individual notes Mib,Re, Mib, Sib. The exception is the rest,which would be made explicit as further

phoneme graphs were included. Theunions of the sets- (the pair-wise ogi-cal additions without repetition)-yields[Mib, Re, Re, Sib], [Mib, Re, Re, Sib, rest]and [Re, Re, Mib, Sib, rest]. These are the

larger patterns that can be extracted fromthe melody fragment.

Of course, some phoneme graphs will

not have anything in common; they willbe disconnected upon "gluing." This sit-uation would correspond to being ableto listen to two disjoint parts of the

Peusner A Graph Topology of Melody Scores 37


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Start

Lab 'a4 FaJ

Fa

End

Do1 Re,Fig. 9. Alexander Guilmant's Tarantelle Op. 48) (1878). The actualnumber of transitions etween wo nodes is denoted by repeatedarrows. t is clear that there are different evels of ambiguity or

different nodes. (? Karina Zerillo-Cazzaro)

Fig. 10. Basic heme of Isaac Albeniz's Asturias, rom the SpanishSuite 1886). Although written or piano, it is usually performed onthe classical guitar; he first guitar ranscription ppears o be that ofDomingo Prat (1920). (? Karina Zerillo-Cazzaro)

melody at once. I now return to themillion-dollar question: Can we recoverthe whole melody graph? Yes. If we applythe OR operation to two, three or more

subgraphs we can obtain arbitrary frag-ments of the melody until we recover the

complete melody graph as desired. Itmakes sense, then, to suggest that thesefragments, not the individual notes, arethe musical phonemes. (The notes, orvertices, cannot recover the completegraph by themselves using some primitivelogical operation and nothing more.)

The question arises whether this is a

unique partition, or whether there are

optional ways to fragment the graph. Theanswer is that there is one more possibleway to fragment the melody graph so thatwe retain the initial information: take anote and all the incomingarrows, ogetherwith the notes at the tail of the arrow. The

steps to be followed are completely anal-ogous to the above.

These two types of fragments (basedon outgoing and incoming arrows) cor-

respond to what topologists call openand closed sets, respectively [25].

SUMMARY CONCLUSIONS

In addition to the qualitative observationthat the melody graphs have a pleasant

visual design, the graph also gives a qual-itative idea of the musical vocabulary uti-lized by the composer. This includes thedegree of ambiguity and closure ofthemes and subthemes intended-de-fined as faces or loops in the graph. As abonus track, the model suggests a graph-topological alternative to the standard

concept of musical phoneme, the openmelody set. In the present context, restshave been treated as part of the

phoneme structure-"the sounds of si-lence." They are not an indication of tim-

ing, duration, rhythm or temporal flow.The empty set is always present in any to-

pology, but, as the topologist G. McCartypoints out, it is easily overlooked or,worse, forgotten altogether [26].

A word should be added about the "cir-cular" designs of each graph. Althoughthey are superficially reminiscent of the

early perceptual designs of the Gestalttheory of perception [271, there is littleconnection between one and the other.It should be stressed again that the pres-ent model is not a theory of perception,in sharp contrast to the images that canbe extracted from context-dependentpitch perception [28] or spectrogramsand other measures of auditory percep-tion [29]. All those processes work withsound signals rather than with the score.

On the other hand, the possibility of ob-

taining the complete graph from modu-lar components-hooking up openmelody sets in specific ways-makes thisa decomposable representation [30].This fact loosely accords with the Gestaltidea that the properties of the whole de-

pend not only on the individual proper-ties of the parts, but also on their

relationships.Clearly, the limited and introductory

characteristics of the present work do notallow much room for generalization. It isconceivable, however, that with consid-erable taxonomic efforts, the "topogra-phy" of these graphs might be classifiableinto distinct composer graph signatures,or groupings within a graph-as done in

cognitive psychology for simultaneous,sequential and segment grouping [31 ]. Ialso suggest that the incorporation ofcurrent cognitive models [32-35] mighthelp to determine elements of themelody-making process, especially if it be-comes possible to subsume the graph ina formal theory of grammar that cen-trally incorporates connectionist com-putation, in the sense advocated bySmolensky [36].

As a final speculation, a question of nominor significance arises as to the originof the graph. Whether such patterns are


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spontaneous or are innate in the case of

experienced, natural composers, or

whether, as Leman argues [37], the

emergence of musical perception is a re-sult of the interaction with physical fac-tors in the environment are openquestions that require further study.

APPENDIXGraph ambiguity can be formalized byconverting the number of transitions toa probability [38], while maintaining thesame incidence relations between nodesand edges as in the original graph. Theresult would be a (stochastic) Markovtransition graph.

For example, in the case of Tarantelle,there are six transitions from high Re to

high Mi, and two transitions from highRe to high Do. If these transitions wereto be converted to frequencies of event

occurrence, the corresponding proba-bilities would be given by p (Re2) = (6/8,2/8)-two unequal probabilities. The re-

maining (out) transition probabilitieswould be P (Re2) = (1/3, 1/3, 1/3),P(Do2) = (4/6, 1/6, 1/6), P (rest) =

(1/5,4/5), P (Sol) = (1/11, 1/11,2/11,2/11, 5/11), P (Fa) = P (Fa sharp) = P

(Mil) = P (Rel) = P (Dol) = 1.In general, the unevenness of the dis-

tributions of various probabilities for ar-rows exiting a given node would reflectthe degree of surprise or predictability in

the melody. Obviously, information the-

ory is best suited to handle the formal-ization of such networks of messages,because it has a built-in measure of am-

biguity that varies between 0 and 1, giv-ing a measure of messages that are,

respectively, too predictable or too ran-dom. Using information theory, wewould simply look at the connections as

messages with certain probabilities of oc-currence and would calculate a degreeof order or disorder, information or en-

tropy, respectively. Although these con-

siderations have been applied to manyfields, including biology [39] and aes-thetics [40], they are outside the scopeof this paper.

AcknowledgmentI wish to thank Karina Zerillo-Cazzaro for her experthelp and patience in redoing my scribbled drawings.All designs are original; they are based on at most

eight measures of published scores. I also want tothank Nicolas Collins for his encouragement whenI began working on this paper.

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17.J.W. Kennedy and L.V. Quintas, eds., Applicationsof Graphs n Chemistry nd Physics Amsterdam, NewYork, Oxford: North-Holland, 1988); S. Weinberg,The Quantum Theory fFields Cambridge, U.K: Cam-bridge Univ. Press, 1995); L. Peusner, Studies n Net-work Thermodynamics Amsterdam: Elsevier, 1986).

18. M. Schroeder, Fractals, Chaos, and Power Laws(New York: W.H. Freeman and Company, 1990); L.Peusner, Las mdquinas del tiempo Buenos Aires: Li-breria y Editorial Alsina, 1998).

19. Some of these applications are in the process ofbeing patented, registered and/or copyrighted.

20. Harary [7].

21. Alphabetic terminology (notes C, D, E, etc.) isnot used here in order to avoid confusion with graphterminology, in which G is used for "graph" and allthe capital letters are used to signify points. "Si" sequivalent to "Ti," which is used in the United Statesfor the same note.

22. Lerdahl andJackendoff [2].

23. P. Dessain and H. Honing, "LOCO: A Composi-tion Microworld in Logo," Computer MusicJournal 12,No. 3, 30-42 (1988).

24. Longuet-Higgins [9].

25. Considered rigorously, this is not exactly true.The reason is that the present musical situation, con-taining self-loops, is foreign to topology, as a pointcannot be a neighbor to itself.

26. G. McCarty, Topology: n Introduction with Appli-cation to Topological Groups New York: Dover, 1988).

27. M. Wertheimer, "Untersuchungen zur Lehre vonder Gestalt," Psychologische Forschung 4 (1923)pp. 301-350.

28. Todd and Loy [15]; M. Leman, "Relevance ofNeuromusicology for Music Research,"Journal fNewMusic Research 8, No. 3, 186-199 (1999); M. Leman,ed., Music, Gestalt and Computing: tudies n Cognitiveand Systematic Musicology (Berlin, Heidelberg:Springer-Verlag, 1997); M. Leman, "Naturalistic Ap-proaches to Musical Semiotics and the Study ofCausal Musical Signification," Semiotic nd CognitiveStudies n Music (Bratislava, Slovakia: ASCO Art andScience, 1968) pp. 11-38.

29. Leman [28].

30. Honing [1]; H.A. Simon, The Sciences of the Artifi-cial (Cambridge, MA: MIT Press, 1969); T. Winograd,"Linguistics and the Computer Analysis of Tonal Har-mony,"Journal ofMusic Theory 2, No. 1, 2-49 (1968).

31. Honing [1].

32. W.S. Dowling and D.L. Harwood, Music Cogni-tion (New York: Academic Press, 1985).

33. S. McAdams and I. Deliege, eds., La Musique etles Science Cognitives Brussels: Mardaga, 1989).

34. J.G. Roeder, Introduction to the Physics and Psy-chophysics fMusic (New York: Springer-Verlag, 1975).

35. J. Sloboda, The Musical Mind: The Cognitive Psy-chology of Music (Oxford, U.K.: Clarendon Press,1985).

36. Smolensky [14].

37. Leman [28].

38. R. Bod, "What Is the Minimal Set of FragmentsThat Achieves Maximal Parse Accuracy?" ProceedingsACL 2001 (Toulouse, France, 2001).

39. Leonardo Peusner, Concepts n Bioenergetics En-glewood Cliffs, NJ: Prentice-Hall, 1974).

40. R. Posner and H.P. Reinecke, eds., Zeichenprozesse:Semiotischeforschung n der einzelwissenschaften Weis-baden, Germany: Akademische VerlagsgesellschaftAthenaion, 1977).

41. Honing [1].

42. Lerdahl andJackendoff [2].

Glossary

connectionism-in artificial intelligence (AI), thecommitment to representing cognition as distributedacross multiple neural networks; in this view, themind does not have a central processing unit.

entropy-a thermodynamic function of a state thatmeasures the likelihood that a given transition willtake place.

Euler-Poincar6 relation-specifies that, in planargraphs, the sum of vertices plus faces equals the num-ber of edges plus two. In the case of "Moon River,"for example, there are 9 vertices and 6 faces, whichequals the number of edges (13) plus 2.

graph theory-a branch of mathematics that dealswith primitive combinatorial properties of discretetopological elements (points) and their connections.

information theory-one of the founding disciplinesof cybernetics, information theory deals with the ef-ficiency of transmitting messages in a background ofnoise, without considering the meaning of the trans-mitted signal.

Peusne, A Graph Topology of Melody Scores 39


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music representation-the structuring of musical in-formation, from either a technical notation, com-

puter music, publishing-and-retrieval or cognitivepoint of view (including both human and machinecognition). According to Honing [41], a represen-tational system should be formal, declarative, explicitand multiple, and should be associated with time in-tervals.

neighborhood-in topology, the characteristics ofthe points that can be reached from another. To de-scribe neighborhoods we introduce both the openset-i.e. the

point plusthe

neighboring points,ex-

cluding the boundary-and the closed set, which in-cludes the boundary as well.

topology-the branch of mathematics that deals onlywith primitive logical relations among neighboringpoints, without considering distances. In the presentcontext, a topology for a given melody can be de-rived from individual set elements of the directedgraph. It consists of all the open sets in the melody,plus their intersections (the elements that are iden-tical in two sets), plus their unions (the sum of allthe elements, without repetition), plus the empty set(rest), plus the universal set-in the present case, thecomplete melody.

Manuscript received 2January 2002.

Leonardo eusner eceived is BS in ElectricalEngineeringifrom IT (1965)--where he com-posed music for Tech Show-and a Ph.D. inBiophysics rom Harvard University 1971).He plays classical and electric guitar and com-poses "tunes. Peusner was aFulbright cholarin 1990. Credited with the irst professionaluse of the Fender Stratocaster uitar in Ar-gentina n 1959, he is the author of three ub-lished novels and several scientific books ndarticles. His big claim ofame is having heardthe Beatles ive-in Baltimore, 964.

II N rno

The Never-Ending Role of Artistsand Scientists in Times of War

Threeyears ago Leonardo Editorial Advisor Michele Emmer asked: What can artists and scientists do whenthere is a war? How can we be useful? How can we help to find solutions? How can we avoid the use of the

military while at the same time protecting the lives of innocent civilians? What educational work can we doto avoid violence and war?

These are questions that do not have a deadline, unfortunately.Leonardo and Guest Editor Michele Emmer continue to seek papers discussing these and other topics that

address the role of artists and scientists in times of war.Please send manuscripts or manuscript proposals to Michele Emmer or to

the Leonardo ditorial office: LEONARDO, 425 Market Street, 2nd Floor, San Francisco, CA 94105, U.S.A.E-mail: .

Texts that are being published as part of this project include the following:

Published in Vol. 34, No. 1 (2001):* MICHELE EMMER: Artists and War: Answers?* BULAT GALEYEV: Open Letter to Ray Bradbury

*JOSEPH NECHVATAL: La beaut6 tragique: Mapping the Militarization of Spatial Cultural Consciousness

Published in Vol. 34, No. 4 (2001):* UBIRATAN D'AMBROSIO: Mathematics and Peace: Our Responsibilities* ALEJANDRO DUQUE: New Media as Resistance: Colombia* SHEILA PINKEL: Thermonuclear ardens: nformation Art Works about the U.S. Military-Industrial Complex

Published in Vol. 35, No. 2 (2002):* MATJUSKA KRASEK: The Role of Artists and Scientists in Times of War

Published in Vol. 35, No. 3 (2002)* ANDREAS BROECKMANN: Small Channels for Deep Europe (Almost a Sermon)

40 Peusne, A Graph Topology of Melody Scores

graph topological representation of melody scores

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