Page  00000001 PATHS ON STERN-BROCOT TREE AND WINDING NUMBERS OF MODES Domenico Vicinanza Musica Inaudita - DMI University of Salerno - Italy ABSTRACT The aim of this paper is to propose a natural definition of a winding number for a m-note mode, generalizing the concept of well formedness (proposed by Carey and Clampitt) providing and discussing some aspects regarding the Stern-Brocot trees. We start by giving an algebraic definition ofa rm-notes mode in a chromatic set of n elements as a composed map between Zm and Z,. Then we introduce the winding number of a mode, as a sort of topological index related to how a certain mode is placed around the circle of fifths, and it will be discussed some connections with the concept of well formedness. Finally we shall focus on Stern-Brocot trees, discussing an interesting relation existing between winding numbers and some particular paths on the tree. Paths which can be viewed as the result of a dynamical evolution, driven by the winding numbers, from the root of the tree to the leaf corresponding to a well-formed mode. 1. MODES WITHIN A CHROMATIC SET AS A COMPOSED MAP We shall start by proposing a definition of a m-elements mode in a chromatic set of n elements (m < n) as a map from Zm to Z,: P E,m < n Just to fix the idea, one can look at the elements of the usual chromatic set C, C#, D, D#,..., B as represented by the elements of Z12: 0, 1..., 11. In this fashon, the diatonic scale C, D, E, F, G, A, B will be represented by the choice of the seven elements 0, 2, 4, 5, 7, 9, 11 among the twelve ofZ12, i.C. A.dia: 7 -Z 12: 0 -> 0, 1 -> 2, 2 -> 4, 3 -> 5, 4 -> 7, 5 -> 9, 6 11. Then, one can consider a second map between Z, and Z, defined as: g: Z,-> gZ, = Z,, (g, n) = 1I g: x G Z'- gx GZ' Using the diatonic scale as example, and taking g = 7, one has: 0 ->0, 2 ->2, 4 ->4, 5 ->11, 7 ->1, 9 ->3, 11 ->5. Finally it is possible to consider the composed map A = g o p. Taking as an example the diatonic scale, Adia would be: 0 -> 0 -> 0, 1--> 2--> 2, 2 -4 4, 3--> 5 ->11, 4 -> 7 -> 1, 5 -> 8 -> 3, 6 ->11 -5, where /dia go 0diai dia: Z7 )12, 9:X 12 *7x 12. While taking as an example the whole-tone scale (C, D, E, F#, G#, B, i.e. choosing the six elements 0, 2, 4, 6, 8, 10 among the chromatic set), the composed map /whole would be: 0 ->0 -> 0,1 ->2 -> 2,2 ->4 -> 4,3--> 6 -> 6, 4 -> 8 -> 8, 5 -> 10 -> 10. 6 -> 11 >15. where Awhole= go9P, A: Z6 ) 121 g: X G Z12 7xG Z12 We are now ready to define a mode: Definition 1 (Mode) A m-note mode in a chromatic set of n elements is the triple (Zm, In, A)2. WELL FORMEDNESS In 1989 Norman Carey and David Clampitt [5] proposed a definition for the concept of well formedness for musical scales within a certain chromatic set, based on the existence of some particular automorphism of the group Zm. One of the standard way to set up the diatonic scale is take the sequence of seven perfect fifths and order them within an octave. Representing the sequence of the seven fifths as seven points regularly spaced around a circle F, C, G, D, A, E, B and joining them a first time following the fifths order (fig. 1, left) and a second time following the scale order (fig. 1, right). Looking at the two pictures one can notice a regular heptagon on the left, with seven distinct rotations which bring the figure into itself and a star on the right, with the same number of distinct rotations which leave the graph unchanged. In other words the two representations of the diatonic set, show the same kind of rotational symmetry. While any sequence of fifths can be represented as a regular polygon once equally spaced on the circle and connected in the fifth order, the conservation of such a rotational symmetry discussed above is not a general rule. One can observe, for example, that the sequence of six consecutive fifths, F, C, G, D, A, E once turned into F, G, A, C, D, E by the scale ordering, does not show the same symmetry of the hexagon. One can start testing sequences of consecutive fifths, from one to twelve and classify the modes according to whether the scale does fulfill or not the rotational symmetry preservation.

Page  00000002 while, taking the whole tone scale we have: A 0 ax. Figure 1. Well formedness of the diatonic scale, following Carey and Clampitt [5]. Left: diationic set following the fifth order, right: diatonic set following the scale order. Definition 2 (Well-formed scales) Scales generated by consecutive fifths in which symmetry is preserved by scale ordering are called well-formed scales [5]. Among the twelve scales obtained by collecting consecutive fifths, one discovers six well-formed scales (1, 2, 3, 5, 7, 12) and six not well-formed ones (4, 6, 8, 9, 10, 11). Let us consider now the diatonic scale numbering the notes (F, C, G, D, A, E, H) as the elements ofZ7: 0,1, 2, 3, 4, 5, 6, as seen in the previous section. The rearrangement into scale order: 0, 2, 4, 6, 1, 3, 5 (corresponding to F, G, A, B, C, D, E) is represented algebraically by the fact that it exists an element of the group Z7, 2 in this case, such that the multiplication by 2 modulo 7 (the order of the group) arranges the notes into scale order: 0 x 2mod 7 0, 1 x 2mod7 2, 2 x 2mod7 4, 3 x 2mod 7 6, 4 x 2mod 7 1, 5 x 2mod7 3, 6 x 2mod 7 5. In general for each well-formed scale of m notes one is able to find and element b e Z7 such that the multiplication by this element, modulo the order of the group, arranges the notes in the scale order. 3. WINDING NUMBERS 3.1. Definition In this section we shall look at modes and well formedness from a slightly different point of view defining a sort of topological index, the winding number of a mode around the circle of fifths. Definition 3 (Winding number) Given a mode (Z,7, Z, ) we shall call winding number, wnm,nr, or shortly, where it does not generate ambiguities, wn, the integer so defined Wnm,n, - wn6,124, =(2 + 2 + 2 + 2 + 2 +2)/12 = 12/12 = 1 3.2. Winding number and well formedness The definition of a winding number wnm,n,A as it has just proposed, allows us to associate an integer to a mode. The question which naturally arises is what is the meaning of such a number with respect to the other quantities we already know. Looking at its definition, wnm,n,A is a sort of metric in the sense that it contains information about how each point of the circle which represent the / map is far from the subsequent one, i.e. given the k-th point, how far is the k + 1-th. Since the distance is computed on the circle, this distance is the number of points (of Z,) from k to k + 1. According to this definition, the diatonic scale fulfill the Myhill property, while the melodic minor scale (having three sizes of fourth: diminished, perfect, and augmented) does not. In the same way the whole-tone scale does not fulfill the Myhill property, having only one specific size for each generic interval. The winding number so defined is strictly connected to the well-formedness of the scale, thanks to the following result (see [7] for a sketch of the proof): Theorem 1 Given a well-formed mode (Z,, Zn, f), the Carey and Clampitt's multiplication element b is the winding number wnm,n,n. 3.3. Position of the fifth within the mode and winding numbers Let us focus again on a m-element mode in a chromatic set of n elements (i.e. a triple Z~, Zn, p), m < n. It is possible to observe that the position of the fifth, is the inverse mod m of the winding number (see for ex. [5], [7]). In other words one can write the fraction p/m ("position of the fifth" over "number of notes of the mode") for a m-note mode as p position of the fifth m number of notes (WN) 1d mod m Em=1 [ /(k - 1)lmod n + [/(0) I(n - 1)]mod n have: For example, for the octave-fifth scale, WN= 1 and m = 2, p/m (1)od 2/2 = 1/2; for the structural scale, WN= 2 and m = 3, so p/m = (2)d- 3/3 = 2/3; for the pentatonic scale, WN= 2 and m = 5, p/m (2)r d 5/5 = 3/5; for the diatonic scale, WN= 2 and S 7, p/m = (2)rod 7/7 = 4/7; finally, for the chromatic scale, WN= 7 and m = 12, p/m (7)od 1/12 7/12. We can conclude, that it holds the following characterization for the winding number: Definition 4 Given a m-element mode and given the ratio position of thefifth the winding number is the number of notes m' t n n number which multiplied by pmod m returns I To fix the ideas, taking the diatonic scale we {[ /(1) - /(0)lmod 12 + - ' ' + [/(0) - /A(6)mod2 12} /12 (2 + 2 + 7 + 2 + 2 + 2 + 7)/12 = 24/12 = 2

Page  00000003 4. STERN-BROCOT TREES AND WELL-FORMED MODES The first idea about connections between Stern-Brocot trees and well-formed modes was due to Norman Carey (Wellformed Scales and Stern-Brocot Tree [4]) and it has been presented at the at the AMS Sectional Meeting in Phoenix, 2004 (Special Session on Mathematical Techniques in Musical Analysis). What is published in the following pages is an ideal development (the attempt to define a dynamic on the tree controlled by the winding numbers) of what was initially proposed by N. Carey. 4.1. Stern-Brocot trees Let mi/ni and m2/n2 be two given fractions, it is possible to define the mediant of these fractions as (mi + m2)/(nl + n2). Starting from 0/1 and 1/0 as starting fractions (1/0, of course is not a fraction, but it allows to describe Stern-Brocot trees is a simple manner). The mediant of 0/1 and 1/0 is 1/1, going ahead we can compute the mediant of 0/1 and 1/1, which is 1/2 and the one of 1/1 and 1/0 which is 2/1. On the next stage of construction we get the following fractions 1/3 (from 0/1 and 1/2), 2/3 (from 1/1 and 1/2), 3/2 (from 1/1 and 2/1), 3/1 (from 1/1 and 2/1). Proceeding this way we get the infinite tree known as Stern-Brocot tree. That tree contains all possible non-negative fractions expressed in lowest terms and preserve the natural order among the rationals (as a consequence, it follows that a certain fraction can appear only once in the tree). 4.2. Well-formed modes and paths on the Stern-Brocot tree For a given well formed mode with m elements, it remains determined one and only one fraction p/m, where p is the position of the fifth in the m-note mode. We can then focus our attentions to the fractions, 2,,, 1 looking at them on the tree. octave scale (the scale with just one element, the tonic, repeated at all the octaves), to the octave-fifth scale, represented by the fraction, and so forth up to the chromatic scale, whose representative fraction on the tree is 12 Let us define two matrices S and T (M6bius matrices) as [1]: s-( 1 } T=(1 1) 0 1 The action of the two matrices on a generic vector is: These two matrices allow to move from a certain node to one of its descendent moving to the down-left or the down-right node. Starting from {, which we can represent as a vector. 1 ) Coming back to our fractions, representatives of the well-formed modes,,,,, 7 one can look for the right "evolution matrices", obtained by composing the S and T matrices, which allow to move from the root of the tree to each of the fractions. It is easy to verify that those matrices are the following ones: S 1 1 0\ 1 _ 1 ST 1 1 2 1 3) STS ( STSS ( STSST ( ) STSSTT( ) 2 1 5 3 2 3 1 1 4 5 2 1 7 3 4 7 \ 5 7 10 in '-' ^""- - 2 2/ F,/ _ / [1\ 4 5 4 3 3 1 2 3 4 5 5 4 5 7 7 5 7 8 7 7 5 4 5 5 4 4.3. Paths on the Stern-Brocot tree, evolution matrices and winding numbers Let us study with a greater detail the previous sequence of matrices which have been collected again here: '/, 3 3 3 2 1 3321 S- 1 ~ ST1 I I ) (I 1 S)S 1 2 STS (2]1 32) Figure 2. The Stern-Brocot tree, where it has been highlighted the path from 1, 2,, 4 As it is possible to see in fig. 2, there exists a path connecting all the cited fractions, so it exists something like a dynamic process which allows to "evolve" along the tree starting from the root, the fraction 1, corresponding to the STSS ( ) STSST 3 4 S5 7 STSSTT =( 2 S- 5 12) If we consider those matrices as a sequence of strings (S, ST, STS,...), the bold and italic style in the previous

Page  00000004 list help to highlight the columns unchanged when moving from a certain string to the subsequent one. One can observe that if we move from a string xxx to xxxS (i.e. appending a S) the second columns remains unchanged, while passing from yyy to yyyT (i.e. appending a T) leaves unchanged the first column. In addition to that, one can notice that appending a S to a matrix string representing the transition from the starting vector ( to a certain node N, will give the matrix which drive ( to the descendent of N, located at the subsequent row, at the left. In the same manner, appending a T will drive to the descendent of g, located at the subsequent row, at the right. In other words, from the list S, ST, STS, STSS, STSST, STSSTT,..., we can get the path from the first element of the tree, the fraction 1 to the descendent nods'2 3 4 7 1 as nodes 2' 3 4 7 12 s left, left-right, left-rightleft, left-right-left-left, left-right-left-left-right, left-rightleft-left-right-right,... Let us come back on the 2 x 2-matrices S, ST, STS, STSS,.... If we call a1, a12, a21 a22 (al1 a12 the elements it is possible to prove the following result which relates the matrices and the winding number of the m-element mode: Theorem 2 Let us consider a m-element mode (Zm, n, i P). Let p/m be the ratio position of thefifth and let M the number of notes matrix a11 a12 such that M P a21 a22 / \)/ rn then a22=WN((Zm, n, P)) PROOF: We start proving that (a,1 + a12) *-a22 Moda21 a22 1. One can easily observe that the following property holds for all the matrices product of S and T: M S11 a12 a21 a22 As an example we report the matrix STSST (Mode with 7 elements, position of the fifth=4, WN=2), where in bold is emphasized the winding number. STSS (3 1 STSS =5 2 5. CONCLUSIONS The paper started proposing a formalization of the concept of m-notes mode in a chromatic set of n elements as a composed map Zm - Z, and discussing the twofold classification of the modes in the usual chromatic set (C, D, E, F, G, A, B) in well-formed and not-well-formed modes, following Carey and Clampitt. Then it has been introduced a sort of topological index related to a mode, the winding number around the circle of fifths, and it has been discussed its connections with the concept of well formedness. Finally well-formed modes has been studied with the help of the Stern-Brocot tree, discovering an interesting relation existing between winding numbers and some path on the tree, path which can be viewed as the result of a dynamical evolution from the root of the tree to the leaf corresponding to the well-formed mode, driven by the WNs. 6. REFERENCES [1] Cafagna, V. and Noll, T. (2003): Algebraic investigations into enharmonic identification and temperament, proc. of UCM 2003 International Conference, Caserta, December 2003 [2] Cafagna, V. and Vicinanza, D. (2004): Winding modes around the circle offifths (in preparation). [3] Carey, N. (1998): Distribution Modulo 1 and Musical Scales, Dissertation, University of Rochester, MTO Dissertation Listings, Volume 4.6 1988 [4] Carey, N. (2004): Well-formed Scales and the Stern Brocot Tree, Handout, distributed at the Special Session on Mathematical Techniques in Musical Analysis at the AMS Sectional Meeting in Phoenix, 2004, January 10. [5] Carey, N. and Clampitt, D. (1989): Aspects of WellFormed Scales, Music Theory Spectrum, vol. 11, no. 2, pp. 187-206. in Musical Analysis at the AMS Sectional Meeting in Baton Rouge, 2003, March 14-16. [6] Clough, J. and Myerson, G. (1986): Music scales and the generalized circle offjifths, Math. Monthly, November 1986 [7] Vicinanza, D. (2004): Some remarks about well formedness, Slides of the presentation held at Special Session on Mathematical Techniques in Musical Analysis at the AMS Meeting in Evanston, October 2004. http://www.musicainaudita. it/ papers/evanston04.pdf aa22- a21a12 =1 - (1) (al1 + a12) *-a22 Moda21a22 1I iff k eGN: (al1 + a12) *-a22 - (a21 + a22) - k 1 so it is sufficient to prove that it exists such a k eN (al1 + a12) *-a22 - (a21 + a22) - k = 1 + a21a12 + a12a22 - (a21 + a22) - k 1 where we have used the equation (1). k a=a12 eGN fulfills the condition. Now it remains to notice that: p _all+ a12 Mr a21+a22 so a22 is such that (position of the fifth) - a22Modm 1= and then from the previous characterization of the winding number a22 mWN((m, Z,p)). D