The cycle on twelve points is a well-known representation of the twelve pitch classes of the traditional scale. We treat a more general situation where the number of pitch classes can be different from twelve and where, moreover, other measures of closeness are taken into account. We determine all situations for which the Generalized Hexachord Theorem continues to hold. \u

Althuis, T.A.

Göbel, F.

English

The cycle on twelve points is a well-known representation of the twelve pitch classes of the traditional scale. We treat a more general situation where the number of pitch classes can be different from twelve and where, moreover, other measures of closeness are taken into account. We determine all situations for which the Generalized Hexachord Theorem continues to hold

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Faculty of Mathematical SciencesUniversity of TwenteUniversity for Technical and Social SciencesP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlMemorandum No. 1573Graph theoretic aspects of music theoryT.A. Althuis and F. GöbelFebruary 2001ISSN 0169-2690Graph theoretic aspects of music theoryAbstractThe cycle on twelve points is a well-known representation of the twelvepitch classes of the traditional scale. We treat a more general situationwhere the number of pitch classes can be different from twelve andwhere, moreover, other measures of closeness are taken into account.We determine all situations for which the Generalized Hexachord The-orem continues to hold.11 IntroductionWe consider microtonal systems in which the number of tones or pitchesin one octave is n. The letter n will be used throughout this paper inthat sense. A pitch class is a set of pitches where octave equivalent andenharmonically equivalent pitches are identified. The elements of a pitchclass are all represented by an integer between 0 and n−1 inclusive; c = 0, csharp = d flat = 1, etc. So a pitch class set, abbreviated PC set can be seenas a set of integers reduced modulo n. By the order of a PC set we meanthe number of pitch classes in the PC set. The complement of a PC set Sis the set of all pitch classes that are not in the PC set S. The complementof S is denoted by S.Two PC sets A and B are said to be equivalent if there is a number t suchthat for each p ∈ A there is a q ∈ B such that either p + t = q(mod n) orp + q = t(mod n). In the first case B is a transposition of A, denoted byB = TtA, in the second case it is an inversion, denoted by B = ItA.The interval between two pitch classes p and q with p < q is the minimumof q−p and n+p−q. The interval vector V (S) of a PC set S is a vector thei-th entry of which is the number of intervals of length i, where i runs from1 to bn/2c. Let S be a PC set of order at most n/2. Then the differencevector 4V (S) is defined as the vector V (S) − V (S). If the order of S isgreater than n/2, 4V (S) is defined as V (S)− V (S).Example 1 Let n = 12 and take S={0, 1, 2, 5, 8}. Then V (S)=[212221].The complement S={3, 4, 6, 7, 9, 10, 11} and V (S)=[434442],so V (S)− V (S)=[222221].The special ‘almost constant’ form of the difference vector is not accidental.In fact, the Generalized Hexachord Theorem (GHT) says that the differencevector only depends on the number of pitches in S. For details, see Regener(1974).As is well-known, the twelve pitch classes of the traditional scale can berepresented on a cycle, consisting of twelve points and a line between twopoints when the difference between the corresponding pitches is a semitone.More generally, a microtonal system with n pitch classes can be representedby a cycle with n points. In these cases, lines represent ’closeness’ of tonesin terms of pitch. Other measures of closeness are conceivable, e.g. two C’son different manuals of a church organ. A different example is given by acycle in which the diagonals that correspond to a fifth have been added.In the general case, we have a graph, that is: a set of points (representingtones or pitch classes) and lines between certain pairs of points (indicatingsome type of closeness). The GHT refers to the graph Cn: the cycle on npoints.2In the next sections we solve the following problem: determine the graphsfor which the difference vector 4V (S) depends on S only through the orderof S. For convenience we call such graphs robust.2 Prerequisites from graph theoryWhen we replace the well-known cycle by a different graph (e.g. the graphof Figure 1), the concept of ‘interval’ from music theory is generalized to theconcept of ‘distance’. The distance between points P and Q is the minimumnumber of lines in a ‘walk’ between P and Q. The length of the ic-vector isthe diameter of the graph, i.e. the largest distance occurring in the graph.(The graph of Figure 1 has diameter 3). The degree of a point P is thenumber of lines incident with P . If all points of a graph G have the samedegree, G is called regular. (The graph of Figure 1 is 3-regular, i.e. regularof degree 3). Since the points of the graph correspond to pitch classes, theletter n will likewise be used for the number of points of the graph.Example 2 Now consider a graph G. Take a PC set S of one element P ,i.e. S = {P}. All components of the interval vector V (S) are 0. Sincein the graph G, the element P is a point, the first component of V (S) isequal to the number of lines between points different from P . Hence thefirst component equals the total number of lines in G minus the degree ofP . Therefore 4V (S) is independent on P only if the first element of V (S)is the same for all points P , i.e. if the graph G is regular.The regularity-condition is not sufficient, as is shown by the following ex-ample.Example 3 The graph G in Figure 1 is regular. However, for the PCset S = {A,B} we have V (S)=[100], V (S)=[744], 4V (S) =[644], whereasS = {B,C} gives V (S)=[100], V (S)=[753], 4V (S) =[653].BACFigure 1: graph GIt turns out that in graph theory a concept exists that is equivalent torobustness.Definition 1 A graph G is distance degree regular (DDR) if for each pointP of G, the number of points at distance j from P is independent of P .3Definition 2 The Cartesian product A× B of the sets A and B is the setof all ordered pairs (x, y) with x ∈ A and y ∈ B.Example 4 If A is the set of suits from a deck of cards (hence A = {spades,hearts, clubs, diamonds}), and B is the set of values (hence B = {2, 3, ..., 10,jack, queen, king, ace}), then A×B is just the deck of cards.Definition 3 Consider two graphs G1 and G2 with point sets P1 and P2respectively. The product graph G1 ×G2 is the graph whose point set is theproduct of sets P1×P2. For any two points (u1, u2) and (v1, v2) in P1×P2,there is a line between these points in G1 ×G2 whenever [u1 = v1 and u2v2is a line in G2] or [u2 = v2 and u1v1 is a line in G1].Example 5 The product K2 × C12 of the graphs K2 and C12 is shown inFigure 2 (The graph K2 has two points and one line). The two cycles canbe seen as the pitch classes of two manuals on a church organ. The linesbetween the cycles indicate the ’closeness’ of the endpoints: they differ intimbre only.Figure 2: the product graph K2 × C123 The main resultTheorem 1 If G is DDR, then G is robust, and conversely.Proof Let G be a DDR graph. Split its point set into two classes: S and S.In the following definitions, d(u, v) denotes the distance between the pointsu and v.xj = the number of pairs u, v with u, v ∈ S and d(u, v) = j,xj = the number of pairs u, v with u, v ∈ S and d(u, v) = j,yj = the number of pairs u, v with u ∈ S, v ∈ S and d(u, v) = j.Then xj + xj + yj = number of pairs in G at distance j. This numberdepends on G only. Choose a path of length j for each pair of points atdistance j. The points at distance j from a fixed point in S lie either inS or in S. There are xj S-S-pairs and yj S-S-pairs. The total number ofend-points in S of such paths is 2xj + yj = kj |S|, where kj is a constant (G4is DDR). Since also xj + xj + yj is a constant, it follows that xj − xj is aconstant. Hence G is robust.Conversely, let G be robust. Take S = P , hence |S| = 1. Obviously, xj = 0for all j. Since G is robust, xj − xj depends on |S| only. In this case weconclude that xj is independent of the choice of P . From the fact thatxj + xj + yj is constant, we know that yj is independent of P . Hence G isDDR. 2If a graph is point-transitive (roughly speaking: if the points are mutu-ally indistinguishable as in Cn and K2 × C12), then it is obviously a DDRgraph. Hence according to Theorem 1 we have the following result.Corollary 1 All point-transitive graphs are robust.Remark Point-transitivity refers to equivalence of points, whereas robust-ness is a property of sets of points!Theorem 2 If the graphs G and H are robust, then the graph G × H isrobust.Proof Suppose G and H are robust graphs. Let v be a point in G, and w apoint in H, hence (v,w) a point in G×H. Let kG(d, p) denote the numberof points in G at distance d from p. ThenkG×H(d, (v,w)) =d∑i=1kG(i, v) × kH(d− i, w).Since kG(i, v) and kH(d − i, w) are independent of v and w, respectively, itfollows that G×H is robust. 24 ExamplesThe graph K2 × C12, see Figure 2, is point transitive, hence robust. So foreach set of m points the difference vector has the same form. For example,if m = 3, we have 4V (S) = [27, 36, 36, 36, 36, 27, 9].Several types of robust graphs exist that are not point-transitive. For ex-ample, each regular graph of diameter 2 is obviously robust. Examples withdiameter larger than 2 can be obtained by multiplying such a graph by anarbitrary robust graph (e.g. a point-transitive graph). Still other examplesexist. In Figure 3 we present the three 3-regular robust graphs on 12 pointsthat are not point-transitive (this we determined by inspecting the list of allconnected 3-regular graphs on 12 vertices in Bussemaker c.s. (1976)).5Figure 3: the three 3-regular robust graphs G1, G2 and G3 on 12 points5 TableIn Table 1 we present difference vectors for some robust graphs. n denotesthe number of pitch classes, and m the order of the PC set.Graph Difference vector 4V (S)Cycle Cn n odd [(n− 2m)...(n− 2m)]n even [(n− 2m)...(n− 2m)(n/2 −m)]Prism K2 × Ct t odd [(3n− 3m)(4n − 4m)...(4n − 4m)(2n − 2m)](n = 2t) t even [(3n− 3m)(4n − 4m)...(4n − 4m)(3n − 3m)(n−m)]G1 [(18− 3n)(30 − 5n)(18 − 3n)]G2 [(18− 3n)(36 − 6n)(12 − 2n)]G3 [(18− 3n)(36 − 6n)(12 − 2n)]Table 1: difference vectors of some robust graphsReferencesBussemaker, F.C., Cobeljic, S., Cvetkovic, D.M. and Seidel, J.J., Computerinvestigation of cubic graphs, T.H.-report 76-WSK-01, Technological Uni-versity Eindhoven, 1976.Regener, E., On Allen Forte’s theory of chords, Perspectives of New Music13.1 (1974), p.191-212.6

Graph theoretic aspects of music theory

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Graph theoretic aspects of music theory

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