Abstract. CHAZAL, YVONNE D ANDREA. Mathematical Structures in Musical Spaces. (Under the direction of Dr. Radmila Sazdanović.)

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1 Abstract CHAZAL, YVONNE D ANDREA. Mathematical Structures in Musical Spaces. (Under the direction of Dr. Radmila Sazdanović.) It has been known for centuries that there is an inherent connection between mathematics and music. In his book A Geometry of Music, D. Tymoczko described and utilized the representation of n-note chords by n-dimensional orbifolds, opening up many possibilities for musical analysis. In their paper Musical Actions of Dihedral Groups, A. Crans, T. Fiore, and R. Satyendra formulate dihedral group actions from musical concepts whose commutativity is present in examples that span centuries. In this paper, we give a survey of the topological structures arising in Tymoczko s model and the group actions described by Crans et al., explore possible extensions of the methods in Musical Actions of Dihedral Groups to higher dimensions, as well as illustrate the interactions between the orbifold and the dihedral group models.

2 Mathematical Structures in Musical Spaces by Yvonne D Andrea Chazal A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Master of Science Mathematics Raleigh, North Carolina 2017 APPROVED BY: Dr. Molly Fenn Dr. Nathan Reading Dr. Radmila Sazdanović Chair of Advisory Committee

3 Biography Yvonne Chazal was raised in Morehead City, NC, where she was introduced very early to piano and dance at Crystal Coast School of the Arts under the instruction of Ms. Emily Nelson and Ms. Wendy Powers. Throughout her childhood, she developed a simultaneous fascination with both music and mathematics. She relocated to Durham, NC, for her junior and senior year of high school at the North Carolina School of Science and Mathematics, where she continued to foster these parallel passions. Her senior year, she was in charge of running sound in the school s auditorium and for other events on campus, a co-captain of the school s Dance Ensemble, and a student ambassador, in addition to taking a multitude of the available math courses. She attended North Carolina State University following her graduation from high school, where she is in the process of completing the Accelerated Bachelor s/master s Program in Mathematics. From day three at NC State, she was involved at WKNC 88.1 FM, the student-run radio station on campus, where she would eventually hold the roles of Operations Manager, Promotions Director, and Program Director. Throughout her four years at NC State, she also became a Caldwell Fellow, danced with Just Cuz Dance Crew, played bass with the band Beverly Tender, and tutored with the Academic Support Program for Student Athletes. While Chazal is not sure exactly what will follow the completion of her master s, she is certain that she will continue to pursue both music and mathematics. She hopes to one day become an instructor at the community college level. ii

4 Acknowledgements First and foremost, I would like to thank Dr. Radmila Sazdanović for all of her guidance and support. Her enthusiasm and patience throughout the course of the project were instrumental to making it what it is, and I am incredibly grateful for the opportunity to have worked with someone so passionate, energetic, and incredibly intelligent. There are several other professors that deserve thanks for helping me grow to the mathematician that I am today. These include but are not limited to: Dr. Nathan Reading, Dr. Min Kang, Dr. Alina Duca, Dr. Dan Teague, Maria Hernandez, Dr. Sandra Paur, Dr. Ernest Stitzinger, Ashley Walls, Dr. John Griggs, Dr. Chris Judge, and Dr. Robert Martin. I would also like to thank Dr. Molly Fenn both for sparking my interest in algebra and for encouraging my pursuit of a career in education when few others did. The enthusiasm for mathematics shown by each and every person above is truly inspirational, and I feel so fortunate to have received such an incredible education at The NC School of Science and Mathematics and NC State University. Thank you to The Caldwell Fellows program for providing world class leadership development, for facilitating reflection throughout the past three years, and for providing the support that allowed me to make the most out of my college experience. I am so proud to be a member of this community, without which I would not be where I am today. Any reflection upon my experience at NC State would be incomplete without mention of WKNC 88.1 FM. I would not have made it through the past four years had it not been for my WKNC family, and I know that the skills, memories, and relationships formed among those checkerboard walls will last a lifetime. Lastly, I cannot thank enough Jamie Lynn Gilbert, Dr. Janice Odom, Shivani Chudasama, my parents, and my loving friends for all of their support, advice, and willingness to listen. Without you, this would not have been possible. Thank you. iii

5 Table of Contents LIST OF TABLES... LIST OF FIGURES... v vi Chapter 1 Introduction... 1 Chapter 2 Background Mathematics Background Musical Prelude Pitch and Pitch Class Chord and Chord Classes... 6 Chapter 3 Musical Modeling An Orbifold of Chords Seeing Chords of Length Permutation of Pitches The Musical Orbifold Seeing Chords of Length 3 and Beyond Commutative Chords Classification of Chords Musical Actions of Dihedral Groups Commutativity in Higher Dimensions Commutativity Among Cubes Conclusion Bibliography iv

6 LIST OF TABLES Table 2.1 Standard approximations of integer pitch numbers... 6 Table 3.1 Elements in bijection by b v

7 LIST OF FIGURES Figure 2.1 Circular representation of C... 5 Figure 3.1 Change of basis of R Figure 3.2 R n with identifications by ' Figure 3.3 Left: common polygonal representation of a torus, right: alternative polygonal representation of a torus Figure 3.4 C 2 with identifications by the action of S 2 on our alternative representation of a torus Figure 3.5 Möbius band representation of P Figure 3.6 Visualization of regions identified by S 3 with colors and associated permutations Figure 3.7 Construction of the prism representation of P Figure 3.8 Left: base of P 3 relative to the colored prism representation in Fig. 3.7c; right: general cross section of the prism representation parallel to the base 19 Figure 3.9 Prism representation of P 3 containing nearly even c-orbits (black) and perfectly even c-orbits (red) Figure gon representation of P 3 with c-orbit [3, 8, 11] Figure 3.11 Notion of commutativity in the first four chords of Pachelbel s Canon in D Figure 3.12 Notion of commutativity in four chords of Pile s Yellow Room Figure 3.13 One of three distinct 4-dimensional polytopes sharing a similar structure; half-diminished seventh (purple), French sixth (blue), minor seventh (red), dominant seventh (green), perfectly even (black) Figure gon representation of nearly even c-orbits of length 4 (dashed black) containing nearly even c-orbits of length 3 (dashed blue); left: [0,2,6,9]* and [2,6,9]*; right: [3,5,8,11]* and [3,8,11]* Figure 3.15 Subspace of P 3 consisting of perfectly and nearly even c-orbits Figure 3.16 Illustrations of various properties of M Figure 3.17 PLR-group represented on our cubic structure: P (red), L (purple), R (blue) Figure 3.18 Our examples from Section 3.3 realized in our cubic structure vi

8 Chapter 1 Introduction It has been known for centuries that mathematics and music have an inherent connection; however, it is my belief that we will never cease to discover new ways in which to interpret their profound relationship to one another. As we continue to develop the language of mathematics, we find new sentences with which we can describe musical structures and sequences, and we gain new lenses with which to view the beauty that arises from our favorite songs, tunes, and chords. In this paper, we give a survey of a few of the mathematical models for musical analysis. It is by no means comprehensive, and it contains a mere two of the methods that I found particularly intriguing as well as how we can combine the two approaches. In the book A Geometry of Music [8], Dmitri Tymoczko articulates the idea of using orbifolds to express a space of chords in order to analyze musical works. The beauty of his model lies in its robustness, as it encompasses many of the models that are typically considered to be separate and can accommodate many different musical paradigms. In the paper Musical Actions of Dihedral Groups [3] by Alissa Crans, Thomas Fiore, and Ramon Satyendra, it is proven that two sets of common musical ideas give rise to group actions that have dihedral group structures that have some notion of commutativity. This paper expands slightly on their work, looking at a few higher dimensional cases. Additionally, these group actions can be realized within a subset of Tymoczko s model, giving rise to a few fascinating visualizations. 1

9 Chapter 2 Background 2.1 Mathematics Background We will first introduce the necessary mathematics concepts before proceeding with our musical discussion. The following facts and definitions are consistent with the those given in Artin s Algebra [2], Munkreses Topology [5], and Hatcher s Algebraic Topology [4]. Further discussion on the necessary background material can be found in these works. Let us first establish what we mean by standard topology on R and R n, spaces we will utilize throughout this paper. We will employ the standard topology on R denoted T R, consisting of unions of open intervals (a, b) such that a, b 2 R and a < b. We will use the product topology on R n, which we will denote T R n. Definition 1. Let X be a topological space. X is called Hausdorff if for any two points x 1, x 2 2 X, there exist two disjoint, open sets U x1 3 x 1 and U x2 3 x 2. Definition 2. Given topological spaces X and Y, a map f : X! Y is called an embedding if f is injective and continuous. Thus, an embedding f is a homeomorphism from X to its image f (X) given the subspace topology. Much of Tymoczko s work involves taking quotients by given equivalences, so we give a few definitions of various kinds of quotients. The first of which is one may come across in an algebra course, while the second is topological in nature, giving us a space with a topology that preserves openness from its parent space. Definition 3. Let S be a set with equivalence relation. The set of equivalence classes of S under is called the quotient set and is denoted S/. Definition 4. Given topological spaces X and Y, a map f : X! Y is called a quotient map if it is surjective and U2T Y () f 1 (U) 2T X. Given a surjective map q : X! Y, the topology induced 2

10 on Y such that q is quotient is called the quotient topology. Let X be a partition of the space X and give X the quotient topology induced by q : X! X such that q(x) =A implies x 2 A. Then X is the quotient space of X. Our third flavor of quotients will be both algebraic and topological, so we will build up all of the necessary definitions. Recall that the set of homeomorphisms H(X) from a topological space X to itself form a group under composition [4]. Definition 5. Let G be a group and let X be a topological space. An action of G on X is a homomorphism F : G! H(X). We say G acts on X. For brevity, we will denote F(g) by g. Thus, for all g 2 G, g : X! X is a homeomorphism. F is called simply transitive if for every x 1, x 2 g(x 1 )=x 2. 2 X, there is a unique g 2 G such that The orbit O x of an element x 2 X is the set of images of x under g for all g 2 G. That is, O x = {x 0 2 X : x 0 = g(x), g 2 G} The quotient X/G of a topological space X by the action of a group G is called the orbit space and is the set of orbits of X under G. That is, X/G = {O x : x 2 X}. In some cases, group actions may preserve the properties of a topological space. For instance, in the case that we have a free and properly discontinuous group acting on a manifold, the resulting orbit space is a manifold [7]. Otherwise, we obtain an orbifold. Orbifolds can be informally understood as manifolds containing singularities and have many applications across many various mathematical disciplines. We want to make use of these structures to understand musical concepts. The following definitions are consistent with those given in [7]. Definition 6. An n-dimensional orbifold B is a Hausdorff topological space X with an open cover {U i } i2i where I is some indexing set, closed under finite intersections. For all i, there is a finite group G i acting on Ũ i R n and a homeomorphism f i : U i! Ũ i /G i. For U i U j, there exists injective group homomorphism f ij : G i! G j and embedding f ij : Ũ i! Ũ j such that for all g 2 G i and x 2 Ũ i, we have f ij (g(x)) = f ij (g)( f ij (x)). Additionally, given r i : Ũ i! Ũ i /G i sending x 2 Ũ i to its orbit O x, similar r j for j, and inclusion map î : U i!u j the following diagram commutes: 3

11 Ũ i f ij Ũ j r i r j Ũ i /G i f ij /G i Ũ i /G j f ij f i f j /G j U i î U j f j 2.2 Musical Prelude Now we move to our discussion of music. The music theoretical background necessary for this paper will be light, and this section will mostly serve both to introduce a few basic musical concepts as well as to construct the musical spaces discussed later on Pitch and Pitch Class The fundamental musical object we will discuss is a pitch. For the purposes of this paper, we will define pitch as follows. Definition 7. A pitch is the tone occurring at a given frequency, identified by its frequency. For musically inclined readers, the frequency that we will use to identify a pitch will be the fundamental frequency. For instance, A4 or pitch standard as it is called in music is the pitch 440, the pitch that is 9 notes above middle C on a standard keyboard. Unlike the notes we may be familiar with on the five line staff, pitches are continuous. Thus, even though we may associate integers like 60 with middle C and 61 with middle C], is also a valid pitch. For this reason, we can identify the space of pitches with the real line, noting that only a small range of these pitches can be heard by human ears. Let F denote the set of pitches. We will map F into the real numbers via F : F! R defined as follows: F( f )= log 2 f 440 This embedding is standard in music theory. Definition 8. We will refer to F( f ) as the pitch number of pitch f. We will refer to F(F) =R as the space of pitch numbers. In the space of pitch numbers, a unit is called a semitone. An octave is an interval of 12 semitones. 4

12 Pitch numbers a and b = a + 12 are separated by an octave. a and c = a + 36 are three octaves apart. We can define the notion of octave equivalence via the following relation: Definition 9. Define the relation ' as follows: a ' b if and only if a = b + 12z for some z 2 Z. In other words, a and b are equivalent under ' if and only if they are separated by an integral number of octaves. We will say a and b are octave equivalent. Relation ' is an equivalence relation and therefore partitions the space of pitch numbers into equivalence classes. Definition 10. The equivalence classes in R under ' are called pitch classes and are represented by [x] where x 2 [0, 12) R. Let C denote the set of pitch classes {[x] : x 2 [0, 12)}. For example, ( 1) ' (0) ' (1) ' (2), so [3] ={..., 9, 3, 15, 27,...}. This is the equivalence class encompassing all of the pitch numbers associated with D]. Note that C is cyclic in nature and is homeomorphic to S 1 via the map [x] 7! e 2pi 12 x, so we can represent the space C on a circle as shown in Fig This is a common practice in music theory. [11] [0] [1] [10] [2] [9] [3] [8] [4] [7] [6] [5] Figure 2.1: Circular representation of C The standard nomenclature for pitches gives the pitch class followed by the octave number. For example, C4 represents what is commonly referred to as middle C, the C key that is positioned near the center of a standard keyboard. 4 corresponds to the octave in which it is positioned on a standard keyboard. It is typical in music theory for octaves to cycle beginning with the pitch class C, even though C is no more musically distinguished than any other pitch class. Table 2.1 gives a range of the standard approximations for integer values of this identification. This nomenclature will allow us to more easily discuss common musical objects and movements, such as chords. 5

13 Table 2.1: Standard approximations of integer pitch numbers... G3 G]3 A3 A]3 B3 C4 C]4 D4 D]4 E4 F F]4 G4 G]4 A4 A]4 B4 C5 C]5 D5 D]5 E Chord and Chord Classes While a musical melody may consist of only one pitch played at once, most music consists of multiple pitches played simultaneously. For this reason, we want to be able to consider higher dimensional musical objects that represent such simultaneous pitches, otherwise known as chords. Definition 11. A chord c 2 R R... R = R n of length n is an n-tuple of pitch numbers x =(x 1, x 2,...x n ). A chord class g 2C C... C = C n of length n is an n-tuple of pitch classes g =([g 1 ], [g 2 ],..., [g n ]). For brevity, we will delimit chord classes with brackets as such g =[g 1, g 2,..., g n ], omitting brackets on pitch classes. We will use parenthesis to delimit chords. For example, the major C4 chord of length 3 is (60,64,67)2 R 3. The C-major chord class of length 3 is [0,4,7]2 C 3. Let us consider the product of pitch class spaces C C= C 2. Each tuple is a chord class. We can use component-wise modular arithmetic to determine the chord class of a chord. For example, the chord (58, 70) belongs to the chord class (58 mod 12, 74 mod 12) =[10, 2]. Definition 12. Define r n : R n!c n such that for x =(x 1, x 2,..., x n ) 2 R n, r n (x) =[x 1 mod 12, x 2 mod 12,..., x n mod 12]. For every [g] =[g 1, g 2,..., g n ] 2C n, (g 1, g 2,..., g n ) 2 R n is an element of [g], so r n is surjective. Next, we define the following topology on the set of chord classes. Definition 13. Let T C n be the collection {V C n : rn 1 (V) 2T R n}. Proposition 1. T C n is a topology on C n. Proof. r n ( ) =, so 2T C n. r n (R n )=C n, so C n 2T C n. Let {V a } a2j be an arbitrary collection of open sets. Then r 1 n (V a )=U a for some U a 2 R n. rn 1 ( [ V a )= [ rn 1 (V a )= [ U a ) a2j a2j a2j 6

14 But this set is open because T R n is closed under arbitrary union. Thus S a2j V a 2T C n. Let {V i } i=0 m be a finite collection of open sets. Then r 1 n (V i )=U i for some U i 2 R n. m\ rn 1 ( V i )= i=0 m\ i=0 r 1 n (V i )= m\ U i i=0 But this set is open because T R n is closed under finite intersection. Thus T n i=0 V i 2T C n. Note that a set U = rn 1 (V) is open in C n if and only if V is open in R n, so r n is a quotient map. The set of pitch classes C is a partition of R under the equivalence relation '. Additionally, x 2 r 1 (x) =[x mod 12], so C is a quotient space of R, i.e. C = R/ '. Proposition 2. C n is the quotient space of R n under '. Proof. The map r n : R n!c n is quotient as it is surjective and open. Thus, T C n is the quotient topology on C n. Additionally, C n is a partition of R n, and x =(x 1, x 2,..., x n ) 2 r n ((x 1, x 2,..., x n )) = [x 1 mod 12, x 2 mod 12,..., x n mod 12] So C n is a quotient space of R n. Note that we can view chords of length n as points in R n, which we will hereon refer to as the space of chords of length n. We can view chord classes of length n as points in C n, called the space of chord classes of length n. Consider the two bijective canonical maps sending chord (x 1, x 2,..., x n ) to point (x 1, x 2,..., x n ) in R n and mapping chord class [g 1, g 2,..., g n ] to point ([g 1 ], [g 2 ],..., [g n ]) in C n, which we will also denote [g 1, g 2,..., g n ] for brevity. Thus, they are easily embeddings into n-space, and we will represent them as such momentarily. 7

15 Chapter 3 Musical Modeling Now that we have established the spaces within which we will explore different models for musical analysis, we will introduce two models from the [8] and [3] in Sections 3.1 and 3.2 respectively. In Section 3.3 we will see how these two models interact, and use them in conjunction to view a section of Johann Pachelbel s Canon in D and a modern rock song by Boston band Pile. 3.1 An Orbifold of Chords We begin with some of the ideas from A Geometry of Music [8]. In his work, Tymoczko constructs numerous models with which one is able to analyze musical objects and movements, and he gives extensive analysis as well as historical and theoretical background. The information given in this section certainly does not exhaust the complete model, and further discussion of the material in this section can be found [8]. Tymoczko s orbifold model is comprised of every possible chord one could imagine. In reality, the chords that are used in music are only a relatively small subset of every possible chord. While we may not utilize the entire space of chords, there are many models that arise within Tymoczko s model which are typically considered to be separate. It encompasses models that describe standard Western music theory as well as models that illustrate non- Western musical ideas. For example, while there are many discrete models that explore the chromatic scale common to Western music theory, Tymoczko s orbifold space can also accommodate the Carnatic scale of classical Indian music that contains twenty-two unique tones. All of these tones cannot be realized in a model containing a discrete set of twelve pitch classes, but it is certainly possible to realize them in a continuous space. The method of using a continuous orbit space to model chords is presented in a more musical flavor in A Geometry of Music [8]. This section will provide proofs and definitions for these concepts of a more mathematical flavor. 8

16 3.1.1 Seeing Chords of Length 2 Let us first consider the space of chords of length 2 embedded in 2-space. This will allow us to introduce a few more ideas and explore a simple space to gain intuition for what is happening in higher dimensions. Let the x-axis record the first pitch number in the two-note chord, and the y-axis record the second. We can reparameterize this space using the following rotation matrix: " # " p cos( p = 4 ) sin( p 4 ) #" # x c sin( p 4 ) cos( p 4 ) y This change of basis corresponds to a rotation of 45 clockwise as shown in Fig (0,12) (3,12) (6,12) (9,12) (12,12) (-9,9) (-6,12) (-3,15) (0,18) (-6,9) (-3,12) (0,15) (0,9) (3,9) (6,9) (9,9) (12,9) (-6,6) (-3,9) (0,12) (3,15) (0,6) (3,6) (6,6) (9,6) (12,6) (-3,6) (0,9) (3,12) (-3,3) (0,6) (3,9) (6,12) (0,3) (3,3) (6,3) (9,3) (12,3) (0,3) (3,6) (6,9) (0,0) (3,0) (6,0) (9,0) (12,0) (0,0) (3,3) (6,6) (9,9) (a) R 2 with standard basis (b) R 2 with rotated basis Figure 3.1: Change of basis of R 2 The line c = 0 is the image of the line y = x, so the p-axis consists of chords of the form (x 1, x 1 ). The line c = a (a 2 R) is the line y = x + a, so chords along this line are of the form (x 1, x 1 + a). Definition 14. Let x =(x 1, x 2,..., x n ) be a chord in R n and let g =[g 1, g 2,..., g n ] be a chord class in C n. 1. A translation of x by c 2 R is a group action t : R! H(R n ), and t(c) =t c where t c : R n! R n is the homeomorphism defined by t c (x) =(x 1 + c, x 2 + c,..., x n + c). 9

17 2. A translation of a chord class g by c is a group action T : R! H(C n ) such that T(c) =T c where T c : C n!c n is defined by T c (g) =[(g 1 + c) mod 12, (g 2 + c) mod 12,..., (g n + c) mod 12]. Consider two points on the line c = a, a =(a 1, a 1 + a) and b =(b 1, b 1 + a). Movement from a to b along c = a can be described by translation T c such that c = b 1 a 1. Then we have: t c (a) =(a 1 + c, a 1 + a + c) =(a 1 +(b 1 a 1 ), a 1 + a +(b 1 a 1 )) = (b 1, b 1 + a) =b We are interested in reducing the redundancy in Fig. 3.1 by taking the quotient by octave equivalence to obtain our chord class space C 2, pictured in Fig Here, we identify chords that are related by octaves component-wise. For example, [10, 2] and [10, 24] will be identified, as will [5, 1] and [29, 13]. [0,0] [3,3] [6,6] [9,9] [0,0] [3,0] [6,3] [9,6] [0,9] [3,9] [6,0] [9,3] [0,6] [3,9] [6,9] [9,0] [0,3] [3,6] [6,6] [9,9] [0,0] [3,3] [6,6] [9,6] [0,9] [3,0] [6,3] [9,3] [0,6] [3,9] [6,0] [9,3] [0,3] [3,6] [6,9] [9,0] [0,0] [3,3] [6,6] [9,9] [0,0] Figure 3.2: R n with identifications by ' Note that if we pick any arbitrary chord class g, the horizontal line containing g consists of its translations {T c (g)} c2[0,12). Because we know that C is homeomorphic to S 1, then C 2 = C C = S 1 S 1 = T 2 is a torus. While one might be accustomed to considering the square polygonal representation of a torus, because of our change of basis, we want to consider a slightly different representation. Both are represented in Fig. 3.3 with color-coded regions for clarity. Note that every possible chord class is represented exactly once in each figure except for the chord classes at the boundary which are identified. 10

18 [0,12] [3,12] [6,12] [9,12] [12,12] [6,6] [9,9] [0,0] [3,3] [6,6] [0,9] [3,9] [6,9] [9,9] [12,9] [9,6] [0,9] [3,0] [6,3] [9,3] [0,6] [3,9] [6,0] [9,3] [0,6] [3,6] [6,6] [9,6] [12,6] [0,3] [3,6] [6,9] [9,0] [0,3] [3,3] [6,3] [9,3] [12,3] [0,0] [3,3] [6,6] [9,9] [0,0] [0,0] [3,0] [6,0] [9,0] [12,0] Figure 3.3: Left: common polygonal representation of a torus, right: alternative polygonal representation of a torus Permutation of Pitches In music the ordering of pitch classes within a chord class is irrelevant, as they are played simultaneously. For instance, there is no difference between chord classes [10, 2] and [2, 10] as the pitches with pitch numbers c and d (for c, d 2 Z) are played in conjunction. Thus we want to identify chords that are permutations of one another. We will do this by taking the quotient by the action of the symmetric group S 2 on two objects, namely [g 1 ] and [g 2 ] in [g 1, g 2 ]. Let the group S 2 act on C 2 via f (2) defined in the following way: is homeomor- Definition 15. Define the group action f (n) : S n! H(C n ) such that f (n) (s) =f (n) s phism f (n) s : C n!c n defined by f (n) s ([g 1, g 2,..., g n ]) = [g s(1), g s(2),...g s(n) ] For the remainder of this paper, the superscript (n) will be omitted for brevity. The orbit g of a chord class g of length n will consist of each chord class corresponding to a permutation of the elements [g 1 ], [g 2 ],..., [g n ]. We will denote the orbit g of a chord class g by [g a, g b,..., g z ] such that g a apple g b apple... apple g z. Definition 16. Define P n to be the space of chord class orbits (or c-orbit space for short) such that P n = C n /S n, the chord class space quotient by the action of S n. Here we quotient our chord class space C 2 by the group action f, giving us the orbit space C 2 /S 2 of f. This is shown in Fig. 3.4 imposed on the color-coded torus given in Fig

19 Figure 3.4: C 2 with identifications by the action of S 2 on our alternative representation of a torus However, there is again some redundancy in Fig. 3.4 that can be removed. Consider again the square representation of the torus in Fig We can informally think of identifying points [g 1, g 2 ] and [g 2, g 1 ] as folding over the line y = x. This will identify the red region with the green region as well as the blue and yellow. Thus including both our red and green regions as well as both our blue and yellow regions is superfluous, so we represent P 2 using just the red and yellow regions as shown in Fig [6,6]* [7,7]* [8,8]* [9,9]* [10,10]* [11,11]* [0,0]* [6,7]* [7,8]* [8,9]* [9,10]* [10,11]* [11,12]* [5,7]* [6,8]* [7,9]* [8,10]* [9,11]* [0,10]* [1,11]* [5,8]* [6,9]* [7,10]* [8,11]* [0,9]* [1,10]* [4,8]* [5,9]* [6,10]* [7,11]* [0,8]* [1,9]* [2,10]* [4,9]* [5,10]* [6,11]* [0,7]* [1,8]* [2,9]* [3,9]* [4,10]* [5,11]* [0,6]* [1,7]* [2,8]* [3,9]* [3,10]* [4,11]* [0,5]* [1,6]* [2,7]* [3,8]* [2,10]* [3,11]* [0,4]* [1,5]* [2,6]* [3,7]* [4,8]* [2,11]* [0,3]* [1,4]* [2,5]* [3,6]* [4,7]* [1,11]* [0,2]* [1,3]* [2,4]* [3,5]* [4,6]* [5,7]* [0,1]* [1,2]* [2,3]* [3,4]* [4,5]* [5,6]* [0,0]* [1,1]* [2,2]* [3,3]* [4,4]* [5,5]* [6,6]* Figure 3.5: Möbius band representation of P 2 12

20 Note that in Fig. 3.5, the top and bottom boundaries are continuations of the same line and that each corner is identified with the corner diagonal from it. The left and right edges are identified in a reflected manner. Thus, this space is a Möbius band. It is important to note that every c-orbit will not necessarily contain the same number of elements. Consider the individual c-orbits of length 2. c-orbits of the form [g 1, g 2 ] such that g 1 6= g 2 contain two distinct chord classes: [g 1, g 2 ] and [g 2, g 1 ]. While c-orbits of the form [g 1, g 1 ] contain only [g 1, g 1 ]. Thus, the group action of the permutation (12) on C 2 fixes chord classes of the form [g 1, g 1 ]. Additionally, note that this representation of P 2 consists of a 1-simplex with points [ , ] and [ , ], and this simplex is translated along the line x = y until it is translated to the reflection of itself. We will aim for a similar structure in higher dimensions The Musical Orbifold The innovation behind Tymoczko s work lies in his connection of these musical spaces to orbifolds. Let us prove that our c-orbit space of length n is an orbifold. First there are a few short lemmas that will be used in the proof. Lemma 1. C n is locally Euclidean and Hausdorff. Proof. Let [g] 2C. If [g] 6= [0], there exists some 0 < e < g. Consider the interval ([g e], [g + e]). Then if we consider the restriction of our map r : R!Cto (g e, g + e) R, we have that p (g e,g+e) ((g e, g + e)) = ([g e], [g + e]). Thus p (g e,g+e) is a homeomorphism from an open subset of R to an open subset containing [g]. If [g] = [0], we can consider interval ([11], [1]) as a neighborhood of [g]. Then r ( 1,1) (( 1, 1)) = ([11], [1]) is a homeomorphism from an open subset of R to an open subset containing [g]. Therefore C is locally Euclidean. Let us next show C is Hausdorff. Let [g], [d] 2C, [g] 6= [d]. g d d g Let e = min{ 2, 2 }. Consider intervals V g =([g e], [g + e]) and V d =([d e], [d + e]). Then V g \V d =. Because r 1 (V g )= S z2z(g e + 12z, g + e + 12z) is open as the union of open sets, V g is open. V d is open by the same argument. These intervals are disjoint, so C is Hausdorff. Finite products of locally Euclidean spaces are locally Euclidean and the same is true for Hausdorff spaces. Thus C n is locally Euclidean and Hausdorff. Lemma 2. s 2 S n is an element of the stabilizer of g, denoted S g, if and only if for all i such that s(i) =j 6= i, g i = g j. Proof. If s 2 S g, then f s (g) =g, then g i = g s(i) for all i, so the forward direction is true. If for i such that s(i) =j 6= i, g i = g j and otherwise s(i) =i, then for all i we have that g i = g s(i), so s(g) =g and s 2 S g. 13

21 We will use these lemmas to prove the following: Proposition 3. P n has the structure of an orbifold. Proof. Let g =[g 1, g 2,..., g n ] be a c-orbit in P n = C n /S n. Recall that our c-orbits will be represented such that g 1 apple g 2 apple... apple g n. Consider g =[g 1, g 2,..., g n ]. Note that f id (g) =g and r(g) =g. We first show that there is an open neighborhood Ũ g of g such that for all s 2 S g, f s (Ũ g )=Ũ g and for all s 0 2 S n \ S g, f s 0(Ũ g ) \ Ũ g =. Suppose g =[g 1, g 1,..., g 1 ]. Then g =[g 1, g 1,..., g 1 ] is the only chord class contained in g. S g in this case is S n, as every permutation of g fixes g. Then Ũ g = C n is such an open neighborhood. Now suppose that there exists pair i 0, j 0 2 [n] such that g i0 6= g j0. Consider the open neighborhood Ũ g of g such that Ũ g = i=1 n ([g i e], [g i + e]) of g where e = 1 2 min{ g` g m : g` 6= g m } over `, m 2 [n]. Note that this is an interval of pitch classes, rather than a chord of length 2. Given s 2 S g, we want to show s(ũ g )=Ũ g. Let g 0 2 Ũ g. Then for all i, gi 0 2 ([g i e], [g i + e]). If s(i) = i, then gs(i) 0 = g0 i 2 ([g i e], [g i + e]). If s(i) =j 6= i, then by Lemma 2, s 2 S g gives us g i = g j, so gs(i) 0 = g0 j 2 ([g j e], [g j + e]) = ([g i e], [g i + e]). Thus f s (g 0 ) 2 Ũ g, so f s (Ũ g ) Ũ g. We know that S g is a subgroup of S n, and therefore contains an inverse for each of its elements. Thus s 1 2 S g. Then we have f s 1(Ũ g )=fs 1 (Ũ g ) Ũ g. Therefore, f s (Ũ g )=Ũ g. Now suppose s 2 S n \ S g. Then there exists i 0 such that s(i 0 )=j 6= i 0, g i0 6= g j. Consider g 0 2 Ũ g. g i0 g 0 j + g0 j g j g i0 g 0 j + g0 j g j = g i0 g j min{ g` g m : g` 6= g m } g i0 g 0 j min{ g` g m : g` 6= g m } g 0 j g j min{ g` g m : g` 6= g m } e = 2e e = e Thus g 0 j /2 ([g i0 e], [g i0 + e]), so f s (Ũ g ) \ Ũ g =. Therefore, we have a suitable Ũ g for all g 2C n. We can restrict f to S g for all g to obtain a group action of S g on Ũ g. Because C n is locally Euclidean, Ũ g is homeomorphic to some subset of R n. Then U g = Ũ g /S g is an open neighborhood of g. Let f : U g! Ũ g /S g be the identity. While {U g } g 2P n is an open cover of P n, it may not necessarily be closed under finite intersections, so let us augment this cover. If U = \ k a=1 U ga 6=, if Ũ ga is the open neighborhood of g a associated with U g a for all a, then Ũ = \ k a=1ũg a 6=. Furthermore, if S = \ k a=1 S g a, for every s 2 S, f s (Ũ) =f s (\ k a=1ũg a )=\ k a=1 f s(ũ ga )=\ k a=1ũg a = Ũ If s /2 S, then there exists some b such that s /2 S gb, and f s (Ũ gb ) \ Ũ gb =, so f s (Ũ) \ Ũ is also empty. So our set S constitutes a group action on Ũ. Take f : U!Ũ /S taken to be the identity map. If we augment {U g } g 2P n with all such open sets \ k a=1 U ga, we have an 14

22 open cover of P n that is closed under finite intersections. We now show that we have a proper injective homomorphism and embedding for all U g 1 U g 2. Suppose we have U g 1 U g 2 for U g 1 = Ũ g1 /S g1 and U g 2 = Ũ g2 /S g2. Then Ũ g1 Ũ g2, so if f 1,2 : Ũ g1! Ũ g2 is the inclusion map, we have a embedding. For all s 2 S g1, by construction f s (Ũ g1 )=Ũ g1 Ũ g2. Thus, because the image of Ũ g2 under elements outside of S g2 is disjoint from Ũ g2, S g1 S g2. Then the canonical map f 1,2 : S g1! S g2 sending s 2 S g1 to itself is an injective group homomorphism. For all s 2 S g1 and for all g 2 Ũ g1, f 1,2 (f s (g)) = f s (g) =f f1,2 (s)( f 1,2 (g)) Therefore, the conditions are satisfied and P n is an orbifold. P n has many interesting properties that can be realized both mathematically and musically. Let us further develop our intuition of P n and in particular, P 3 so that we can visualize some properties of this space Seeing Chords of Length 3 and Beyond Before generalizing our quotient process to chords of any length, we will investigate the space of chords of length 3. Again, we want to quotient by our octave equivalence to obtain C 3, our space of chord classes of length 3. This will leave us with a cube. Let us consider the individual orbits obtained by the action of S 3 on C 3 to first gain some intuition. Let [g 1, g 2, g 3 ] be a c-orbit. If g 1, g 2, g 3 are distinct, this c-orbit contains six elements, one chord class for each permutation in S 3. Suppose that g 1 = g 2 6= g 3. Now there are only three distinct images of [g 1, g 2, g 3 ] under the six permutations: [g 1, g 2, g 3 ]=[g 2, g 1, g 3 ] [g 1, g 3, g 2 ]=[g 2, g 3, g 1 ] [g 3, g 1, g 2 ]=[g 3, g 2, g 1 ] In general when the representative chord class of a c-orbit of length n contains non-distinct pitch classes, to count the number of elements it contains, we count the permutations of the multiset of pitch classes. Thus, is g has k distinct pitch classes with a i copies of the i th pitch class, the number of chord classes it contains is given by n = a 1, a 2,..., a k n a 1!a 2!...a k! Consider a c-orbit space of length 3. A c-orbit consisting of three distinct pitch classes will have six elements in its orbit, one for each of the permutations of those pitch classes. These will each occur in one of six regions in C 3. These regions are given in 3.6a, and they are slightly separated in 3.6b for the reader to better discern their shape and relations. 15

23 (a) Regions identified by the action of S 3 (b) Separation of tetrahedra green: x apple y apple z, id 2 S 3 pink: y apple x apple z, (12) 2 S 3 orange: z apple x apple y, (132) 2 S 3 yellow: x apple z apple y, (23) 2 S 3 blue: y apple z apple x, (123) 2 S 3 red: z apple y apple x, (13) 2 S 3 Figure 3.6: Visualization of regions identified by S 3 with colors and associated permutations Each region has a tetrahedral shape. The boundaries of these tetrahedral shapes are formed by two types of planes: x = y, y = z, and x = z, and the boundary of the cube, C 3. The first type of boundary planes consist of the c-orbits of lesser cardinality, i.e. c-orbits with non-distinct pitch classes. For example, given a c-orbit g =[g 1, g 2, g 3 ] that lies on the plane x = y for x, y < z, then g 1 = g 2, so as shown before, there are only three chord classes in its orbit under S 3. Note that each of the tetrahedrons is adjacent to the line x = y = z. Recall that in P 2, lines parallel to x = y corresponded to translation. This will also be the case here along the line x = y = z or hx, y, zi = ht, t, ti, but for translation defined on c-orbits. Definition 17. A translation of a c-orbit g =[g 1, g 2,..., g n ] is a group action t : R! H(P n ) such that t(c) =t c : P n!p n such that t c (g )=[(g i + c) mod 12, (g i+1 + c) mod 12,..., (g n + c) mod 12] such that g i + c apple g i+1 + c apple... apple g n + c. Both g and any given translation t c (g )=[g 1 + c, g 2 + c, g 3 + c] will lie on the line hx, y, zi = hg 1 + t, g 2 + t, g 3 + ti which is parallel to the line given by x = y = z. The green region (x apple y apple z) contains all of the representatives of our c-orbits. Thus, each of the colored regions will be identified with the green region when we quotient by the action of S 3. As we did in P 2, we want to give a more musically relevant representation of this region. Rather than using the subregion g of the green tetrahedron bounded bounded by x = y, y = z, 16

24 x + y + z = 12, and the boundary of the cube y = 0, we want to use another equivalent region. Applying the permutation (132) or x 7! z, y 7! x, and z 7! y to g gives us the new boundaries z = x, x = y, z + x + y = 12, and x = 0, a subregion of the orange tetrahedron. Similarly, applying the permutation (123) to the subregion h bounded by x = y, y = z, x + y + z = 24, and z = 12, we obtain new boundaries y = z, z = x, y + z + x = 24, and x = 12. The image of g under (132) and the image of h under (123) are shown in Fig. 3.7a in the colors associated with the regions they belong to. These are merely alternative representations of g and h. Additionally, because the planes z = 0 and z = 12 are identified as well as the plane y = 0 and y = 12, we can consider alternative representations of g and h as shown in Fig. 3.7b, forming the triangular prism in Fig. 3.7c. The bases of this triangular prism are x + y + z = 0 mod 12, while its walls are formed by the planes x = y, y = z, and x = z. Note that the intersections of these planes are x = y = z. Therefore, movement orthogonal to the base of the prism corresponds to translation; however, when we translate by t 4 to the base on top, we have rotated clockwise by 120. Thus, while the bases are identified, they are identified up to this rotation. If we translate by t 8, we will have rotated by 240. This is analogous to the edges of P 2 having opposite orientations: identified at a rotation of 180. Fig. 3.8 gives the base of the prism with the integral c-orbits given. We can see that the sum of each of the pitch classes is 0 mod 12. On the right is a general cross section of the prism parallel to the base at distance g from the base. Because every cross section is a translation of every other cross section, every possible shape of chord is realized in any given cross section. Furthermore, the cross section at distance g from the base lies on the plane x + y + z = 3g. We can generalize this for n-dimensional chords by taking the quotient C n, a hypercube with edges of length 12, by S n to obtain a n-simplex with vertices [g 1, g 2,..., g n ] where g 1, g 2,..., g i = 0 and g i+1, g i+2,..., g n = 12 for i = 1, 2,.., n 1 along with [0, 0,..., 0]. We then cut along each instance of the plane x 1 + x x n = 0 mod 12, so we obtain a generalization of our Möbius band and triangular prism such that we have an (n 1)-simplex that is the intersection of our n-simplex and x 1 + x x n = 0 mod 12 which we will call the base. The points for the base will be ( 0 12 n, 0 12 n ,..., n ), ( n, 1 12 n 1 12 (n 1) 12 (n 1) 12 (n 1) 12,..., n ),..., ( n, n,..., n ) Additionally this base will contain every possible structure of chord of length n. The remaining portion of the prism is a translation of the base up to t 12/n until we hit the second instance of x 1 + x x n = 0 mod 12. This second instance will be rotated cyclically by 2p n, requiring us to twist before identifying. 17

25 (a) Image of g under (132) (blue), image of h under (123) (orange), remaining green subregion (b) Re-identification of orange and blue regions (c) Labeled prism from another perspective Figure 3.7: Construction of the prism representation of P Commutative Chords One of the many models that we can realize within our continuous orbifold model is the one described in [3]. In Section 3.3 we will explore how Crans work can be seen within the orbifold model, but first we explore Crans model itself. For this we need to introduce some notions by which we can classify our chords and chord classes. 18

26 Figure 3.8: Left: base of P 3 relative to the colored prism representation in Fig. 3.7c; right: general cross section of the prism representation parallel to the base Classification of Chords In the c-orbit space, every possible chord class orbit is realized, even if the chord classes comprising it are incredibly dissonant. For this reason, a majority of the chord classes represented are not used often in music. In this section we focus on the so-called major and minor chords that are omnipresent in music. Definition 18. A chord class is called major if there are three distinct pitch classes represented, say a, b, and g, and a =(b 4) mod 12 =(g 7) mod 12. Such a major chord is called a-major, where a is the root pitch class, b is called the third pitch class, and g is the fifth pitch class. A chord class is called minor if a =(b 3) mod 12 =(g 7) mod 12, specifically called a-minor, where a is again the root, with third b and fifth g. A chord is major if it belongs to a major chord class. A chord is minor if it belongs to a minor chord class. [0, 4, 7] is a straightforward C-major chord class, but [0, 4, 4, 0, 7, 4, 7, 7, 0] is also a C-major chord class. The chords (31, 36, 64) =(12(2)+7, 12(3)+0, 12(5)+4) and (36, 54, 24, 52) = (12(3) +0, 12(4) +7, 12(2) +0, 12(4) +4) are both C-major chords. Additionally, chords can be classified by how evenly they divide the octave. Definition 19. A chord class g =[g 1, g 2,..., g n ] is called perfectly even if there exists some permutation s 2 S n such that given g s(1), g s(2),..., g s(n), for all i, g s(i)+1 = g s(i) + 12 n. In other words, if a chord class can be reordered such that each pitch class is 12 n larger than the preceding pitch class, it is perfectly even. For example, [0, 3, 6, 9] is a perfectly even chord of length 3, as is [2.3, 6.3, 10.3]. Definition 20. A chord class g =[g 1, g 2,..., g n ] is nearly even if it differs from a perfectly even chord class of length n in i components for i = 1, 2,...b n 2 c where every differing component is shifted uniformly up or down. 19

27 We can construct a nearly even chord class of length 4 by taking a perfectly even chord class, say [2,5,8,11], and shifting up to b 4 2c = 2 components both up or down. So [2,6,8,11] is a nearly even chord class as well as [2,6,8,0]. Because chords classes within a given c-orbit maintain the same structure, if one chord class in a c-orbit is major, all of the chord classes in that c-orbit will also be major. Thus we can consider a c-orbit to be major if it contains a major chord class. This is also the case with minor, perfectly even, and nearly even. Fig. 3.9 gives a visualization of the structure of these c-orbits in our P 3 space. The perfectly even c-orbits are shown in red, while the nearly even c-orbits are shown in black. The nearly even c-orbits in P 3 correspond exactly to the set of major and minor c-orbits. The thin lines pictured showcase the interesting structure these c-orbits comprise, but they will have no significance to us until Section 3.3. Figure 3.9: Prism representation of P 3 containing nearly even c-orbits (black) and perfectly even c-orbits (red) Much is still unknown about why pitches sound consonant or dissonant to human ears; however, the most widely-accepted theory is that consonant chords are comprised of pitches whose frequencies are related by simple ratios, like 2:1, 3:2, 4:3, 5:4, and 6:5. Because of the logarithmic relationship between frequency and our pitch space, these highly consonant chords correspond to nearly even chords [8]. For example, consider the F-major chord class [0,5,9] and consider the chord (60,65,69) residing in this class. The frequencies related to this chord are f 1 (60) f 1 (65) f 1 (69) =440 20

28 The ratios here are and Thus we have a highly consonant chord. The perfectly even chord class containing F (i.e. [5]) is [1,5,9]. The first pitch class [0] of F-major differs from the first pitch class [1] of the perfectly even chord class, so F-major is nearly even. The same can be shown for any major or minor chord class. Still considering chords of length 3, we can easily construct every nearly even chord that appears in this manner. Major chord classes occur when exactly one pitch class in a perfectly even chord class is shifted up by one semitone, and minor chord classes occur when exactly one pitch class is shifted down by one semitone Musical Actions of Dihedral Groups We now give a brief summary of [3]. For a deeper understanding of the material, the reader is encouraged to read the article by Crans et al. in full. Here we will consider the subspace of P n consisting of only nearly even c-orbits with integral pitch classes. These chords are used very often in Western music as many Western musical instruments are designed to play these integer pitches. Definition 21. Let subspace M n P n contain elements of P n that are nearly even c-orbits consisting of integral pitch classes. Endow these spaces with the subspace topology. Note that the subspace topology on this integral subspace will be the discrete topology, as Z is discrete. Consider M 3, the set of nearly even c-orbits of length 3: major and minor. [3] utilizes M 3 to represent sequences of major and minor integral c-orbits of length 3 or as musicians might call them: chord progressions such that the movement of between c-orbits can be modeled using the group structure of D 12, the dihedral group of order 24. To do this, we can develop two different group actions, the first generated by translation and inversion of c-orbits, while the second is generated by three involutions. While we previously embedded c-orbits of length 3 in a quotient space of a subset of R 3, now we will embed them on a 12-gon, with each vertex representing an integral pitch class. Each pitch class is adjacent to those that are related by exactly one semitone. c-orbits of length n will be represented as n-gons sharing vertices with the 12-gon as shown in in Fig Translation will be defined as in Section Definition 22. An inversion of a c-orbit g =[g 1, g 2,..., g n ] is a group action I : Z/12Z! H(P n ) such that I(c) =I c where I c : P n!p n is a homeomorphism defined by I c (g )=[c g i, c g i 1,..., c g i n ] such that c g i apple c g i 1 apple... apple c g i n. While both t and I are defined on P n, we want to restrict them to give homeomorphisms from M 3 to itself. Every integral minor c-orbit is an inversion of an integral major c-orbit and vice versa, every integral major chord is an integral translation of an integral major c-orbit, and similarly for minor, so these functions are closed on M 3. 21

29 [11] [0] [1] [10] [2] [9] [3] [8] [4] [7] [6] [5] Figure 3.10: 12-gon representation of P 3 with c-orbit [3, 8, 11] t 1 and I 0 are generators of what Crans calls the ti-group with relations t1 12 = id, I0 2 = id, and I 0 t 1 I 0 = t 11 = t 1 1. Thus, t 1 and I 0 define the action of the dihedral group D 12 on our 12-gonal representation, where t 1 corresponds to a rotation of p 12 and I 0 corresponds to a reflection about the line between [0] and [6]. The ti-group is a subgroup of H(M 3 ). In [3], the homeomorphism group is discussed as the symmetric group of the set of major and minor chords, which is the group of bijections from M 3 to itself under composition. Because we have the discrete topology on M 3, every bijection will be a homeomorphism, so the symmetric group is exactly H(M 3 ). The second action of the dihedral group can be realized through three functions: P, L, and R. P maps a major c-orbit to its parallel minor and vice versa. L maps a major c-orbit to the minor obtained by shifting the root down by one semitone and vice versa. R maps a major c-orbit to its relative minor and vice versa. Definition 23. Let P, L, R : M 3!M 3 be involutions. Given major or minor c-orbit x with root a, third b, and fifth g, define these functions as follows: P(x )=I g+a (x ) L(x )=I b+a (x ) R(x )=I b+g (x ) Crans shows that these three functions form what they call the PLR-group. P = R (L R) 3, so L and R alone generate this group. Then x = L R = t 5 which will generate a cyclic subgroup of order 12, y = L is an involution, and furthermore yxy = L (L R) L = R L = x 1, so this group is dihedral of order 24 [3]. These three functions are homeomorphisms from M 3!M 3, so we can define `3 : PLR! H(M 3 ) to the the group action of the PLR-group on M 3. This sets us up to give the following theorem proven in [3]. Theorem 1. The ti and PLR groups act simply transitively on M 3 and induce distinct embeddings of D 12 into H(M 3 ). Additionally, these embeddings are each other s centralizers in H(M 3 ), so their elements commute. 22

30 This notion of commutativity can be found in musical works that span centuries. For example, the first four chords of Johann Pachelbel s Canon in D exhibits it perfectly [3]. t 7 [2, 6, 9] [1, 4, 9] R R t 7 [2, 6, 11] [1, 6, 9] (a) Music notation for the first four chords (b) Commutativity of functions between chords Figure 3.11: Notion of commutativity in the first four chords of Pachelbel s Canon in D The first chord is a c-orbit [2, 6, 9] followed by an c-orbit [1, 4, 9]. The movement from the first to the second is given by t 7 ([2, 6, 9] )=[2 + 7, 6 + 7, 9 + 7] =[1, 4, 9]. The third chord is [11, 2, 6] followed by [1, 6, 9]. Note that t 7 ([11, 2, 6] )=[11 + 7, 2 + 7, 6 + 7] =[1, 6, 9]. Also, we have that the relationship between [2, 6, 9] and [2, 4, 9] is precisely R([2, 6, 9] )=[2, 4, 9], and similarly, R([1, 4, 9] )=[1, 6, 9]. Thus, because (t 7 R)([2, 6, 9] )=[1, 6, 9] =(R t 7 )([2, 6, 9] ), we have commuted our diagram not only mathematically, but also musically. It is not known exactly when Pachelbel wrote this canon, but it is thought to have been composed around 1680 [3]. But this notion commutativity is far from antiquated. Pile s Yellow Room [6] released in 2015 exhibits this commutativity in a succession of four chords heard first at :19 seconds and then at regular intervals throughout the song. R [0, 4, 9] [0, 4, 7] t 5 t 5 R [2, 5, 9] [0, 5, 9] (a) Music notation for the four chords at 0:19 (b) Commutativity of functions between chords Figure 3.12: Notion of commutativity in four chords of Pile s Yellow Room 23

31 3.2.3 Commutativity in Higher Dimensions We can extend the methodology used in [3] to accommodate chords of lengths other than 3. Let us first examine chords of length 2. The perfectly even chord of length two will divide the octave in half, i.e. a chord of the form [g, g + 6]. To obtain a nearly even chord, we shift the first and second notes up and down by one semitone: [g 1, g + 6], [g + 1, g + 6], [g, g + 5], [g, g + 7]. Because of the modular nature of our chords, we can see that they are all of a similar form, consisting of a pitch class and the pitch class that is five semitones above it. [g 1, g + 6] =[g + 11, g + 6] =[(g + 6)+5, g + 6] [g + 1, g + 6] =[g + 1, (g + 1)+5] [g, g + 5] [g, g + 7] =[g + 12, g + 7] =[(g + 7)+5, g + 7] Because these chords have the same form, an inversion of [g, g + 5] will yield [n g, n (g + 5)] =[n g, n g 5] =[(n g 5)+5, n g 5]. We can translate this chord by g (n g 5) to obtain [g + 5, g], which is the same nearly even c-orbit with which we began. Thus the length 2 c-orbits reveal fairly little. Let us consider some larger lengthed chords. Figure 3.13: One of three distinct 4-dimensional polytopes sharing a similar structure; half-diminished seventh (purple), French sixth (blue), minor seventh (red), dominant seventh (green), perfectly even (black) A perfectly even c-orbit of length 4 is [g, g + 3, g + 6, g + 9]. Thus, our nearly even c-orbits, 24

32 elements of M 4 as we have defined them, lie again on a cube of four dimensions as shown in Fig c-orbits that are translations of each other are color coded. The purple c-orbits correspond to c-orbits of the form [g, g + 3, g + 6, g + 10], where exactly one component is shifted up by one semitone. This c-orbit is what music theorists call a half-diminished seventh chord. The red c-orbits (minor seventh) are of the form [g, g + 3, g + 7, g + 10] where two adjacent pitch classes were shifted up (or down) by one semitone. The blue c-orbits (French sixth) are of the form [g, g + 4, g + 6, g + 10] where two non-adjacent pitch classes were shifted up (or down) by one semitone. Lastly, the green c-orbits (dominant seventh) are of the form [g + 1, g + 4, g + 7, g + 10] where one pitch class was shifted down by one semitone. We could find no simple analogy of the PLR-group with so many different kinds of chords. The inversion of a dominant seventh chord will give a half diminished chord and vice versa, but the inversion of a French sixth chord will give a French sixth chord and the inversion of a minor seventh chord will give a minor seventh chord. Therefore, the structure is a bit too complex to hope for a dihedral group structure; however, we can obtain one by slightly restricting our definition of M 4 to what we will call M4 0 defined as follows. Definition 24. Let M 0 n be the subspace of M n defined as the collection of elements of M n that differ from a perfectly even integral c-orbit of length n in exactly one component. We will call such c-orbits most nearly even. One will note that the most nearly even c-orbits of length 2 and length 3 both correspond exactly to the nearly even c-orbits of lengths 2 and 3 respectively, as b 2 2 c = b 3 2c = 1. But for n larger than 3, M 0 n will be a proper subset of M n, as we will see with M4 0. Our most nearly even c-orbits of length 4 will take on the following forms: [g 1, g + 3, g + 6, g + 9] [g, g + 2, g + 6, g + 9] [g, g + 3, g + 5, g + 9] [g, g + 3, g + 6, g + 8] [g + 1, g + 3, g + 6, g + 9] [g, g + 4, g + 6, g + 9] [g, g + 3, g + 7, g + 9] [g, g + 3, g + 6, g + 10] As in M 3, we find two most nearly even c-orbits that are not translations of one another: one of the form [g, g + 4, g + 7, g + 10] and one of the form [g, g + 3, g + 6, g + 10], of which all the other c-orbits are translations. Again, these are the dominant seventh c-orbit and the half-diminished seventh c-orbit respectively. Note that the dominant seventh c-orbit contains a major c-orbit of length 3 ([g, g + 4, g + 7]), and the half-diminished seventh c-orbit contains a minor c-orbit of length 3, ([g + 3, g + 6, g + 10] =[g + 3, (g + 3)+3, (g + 3)+7]). We can also represent these on our 12-gon as shown in Fig To the left, we have [0,2,6,9]*, the D-dominant seventh chord with D-major [2,6,9]*. To the right, we have [3,5,8,11]*, the F-half-diminished seventh chord containing E[-minor [3,8,11]*. We can establish a bijection between c-orbits of length 3 and c-orbits of length 4 to construct the ti- and PLR-groups as in the length 3 case, considering the P, L, and R functions applied to the underlying major and minor c-orbits. Proposition 4. There exists a bijection between M 3 and M

33 [11] [0] [1] [11] [0] [1] [10] [2] [10] [2] [9] [3] [9] [3] [8] [4] [8] [4] [7] [6] [5] [7] [6] [5] Figure 3.14: 12-gon representation of nearly even c-orbits of length 4 (dashed black) containing nearly even c-orbits of length 3 (dashed blue); left: [0,2,6,9]* and [2,6,9]*; right: [3,5,8,11]* and [3,8,11]* Proof. There is exactly one major c-orbit embedded in every dominant seventh c-orbit, and there are no other major or minor chords contained in this chord. Additionally, every major c-orbit is realized within a distinct dominant seventh c-orbit: each of the twelve integral a-major c-orbits [a, a + 4, a + 7] is contained in [a, a + 4, a + 7, a + 10], and there are exactly twelve distinct dominant seventh c-orbits of this form, one for each integral pitch class a. Similarly, there is exactly one minor c-orbit and no major c-orbits realized within in each half-diminished c-orbit: [a + 3, a + 6, a + 10], or (a + 3)-minor, is contained within [a, a + 3, a + 6, a + 10]. Again, there are twelve distinct half-diminished c-orbits, one for each integral pitch class a. Thus there is a bijection between the set of dominant seventh c-orbits and the major c-orbits, as well as a bijection between the set of half-diminished c-orbits and the set of minor c-orbits. Therefore we can combine these bijections to be one bijection b : M 3!M4 0. Again, because M4 0 has the discrete topology, every bijection is a homeomorphism. Table 3.1 explicitly exhibits the elements in bijection between M 3 and M4 0 via b. t n and I n are defined on chords of length n, so they are still defined on chords of length 4. Their compositions are the same, so they still satisfy the same dihedral relations for chords of any length. Because P, L, and R functions form the PLR-group that acts on H(M 3 ), and because we have bijection b : M 3!M4 0, we can define a new group action `4 : PLR! H(M4 0 ) such that (`4) g = b 1 (`3) g b for g 2 PLR. This is a homeomorphism as a composition of homeomorphisms. Thus the PLR-group acts in the same manner on M 3 as it does on M4 0. Therefore, the action r n on M 3 and M4 0 is dihedral of order 24, and we have the same notion of commutativity in c-orbits of these sizes. Because 5 does not divide 12, we have omitted c-orbits of length 5 from this investigation. We will not have a perfectly even chord of length five with integral pitch classes from which 26

34 Table 3.1: Elements in bijection by b M 3 M4 0 [0,4,7]* [0,4,7,10]* [1,5,8]* [1,5,8,11]* [2,6,9]* [0,2,6,9]* [3,7,10]* [1,3,7,10]* [4,8,11]* [2,4,8,11]* [0,5,9]* [0,3,5,9]* [1,6,10]* [1,4,6,10]* [2,7,11]* [2,5,7,11]* [0,3,8]* [0,3,6,8]* [1,4,9]* [1,4,7,9]* [2,5,10]* [2,5,8,10]* [3,6,11]* [3,6,9,11]* M 3 M 4 [0, 3, 7] [0, 3, 7, 9] [1, 4, 8] [1, 4, 8, 10] [2, 5, 9] [2, 5, 9, 11] [3, 6, 10] [0, 3, 6, 10] [4, 7, 11] [1, 4, 7, 11] [0, 5, 8] [0, 2, 5, 8] [1, 6, 9] [1, 3, 6, 9] [2, 7, 10] [2, 4, 7, 10] [3, 8, 11] [3, 5, 8, 11] [0, 4, 9] [0, 4, 6, 9] [1, 5, 10] [1, 5, 7, 10] [2, 6, 11] [2, 6, 8, 11] to obtain our integral nearly even chords. The same will be true of any number that is not an integral factor of 12. Thus the only factor of 12 left for us to check is 6. M 6 will consist of c-orbits with between one and three components differing from a perfectly even chord. There are 24 unique such chords differing in one component, there are 2( 6 2 ) = 30 chords varying in two components and 2(6 3 ) = 40 chords varying in three components with 13 unique musical theoretical c-orbit types up to translation. The complexity of such a structure is beyond my current music theoretical knowledge and, if musically significant, would require a deeper musical background, so this will be left to future work. One can realize a c-orbit from M 3 in M6 0 M 6 as we did for n = 4, but it is not unique. For example, most nearly even c-orbit [0, 2, 5, 6, 8, 10] contains major c-orbit [2, 5, 10] and minor c-orbit [0, 5, 8]. While it is certainly possible to establish another bijection by making a consistent choice of which major or minor c-orbit to choose from a given length 6 c-orbit, it is less illuminating, and therefore we will not show it here. 27

35 3.3 Commutativity Among Cubes Now that we have explored the structure of the model discussed in [3], let us examine its structure within P 3. As shown Section 3.2.1, the set of perfectly and nearly even integral c-orbits of length 3 form a stack of cubes at the center of our prismatic representation of P 3, and the nearly even c-orbits correspond exactly to the set of major and minor c-orbits comprised of integral pitch classes. This cubic structure is shown in Fig with c-orbits listed at each point. The blue faces and connecting lines are merely shown to demonstrate the 3-dimensionality of the structure. Figure 3.15: Subspace of P 3 consisting of perfectly and nearly even c-orbits 28

36 Note that each of the cubes faces are parallel to either the xy-plane, the yz-plane, or the xz-plane, and each edge on any cube corresponds to a unit step in the x, y, or z direction. This is due to the fact that each of the nearly even c-orbits is one unit step from a perfectly even c-orbit. For example, given perfectly even c-orbit [a, b, g], the nearly even c-orbits associated with it are [a + 1, b, g], [a 1, b, g], [a, b + 1, g], [a, b 1, g], [a, b, g + 1], [a, b, g 1] These alone are shown in Fig. 3.16a. By definition of perfectly even, b = a + 4, g = b + 4, and a = g + 4, so t 4 ([a, b, g] )=[a, b, g]. Therefore, there are four distinct perfectly even integral c-orbits of length 3: [a, b, g] and its translates by t 1, t 2, and t 3. This accounts for our entire four cube structure. (a) General subsection of a cube about a perfectly even c-orbit (b) I 0 is given in orange between the c-orbits shown Figure 3.16: Illustrations of various properties of M 3 Fig. 3.16b shows I 0 represented on our cubic structure. I 0 essentially inverts each cube about the base of the prism, the plane x + y + z = 0 mod 12. Only the two cubes adjacent to [0, 4, 8] are pictured to avoid crowding of arrows, but the inversion extends to the two other cubes. Furthermore, translation maps a given corner point the identical point on the cube above it, but is also omitted to avoid crowding of arrows. Thus, the ti group has a natural representation in our cubic structure. 29

37 A bit more surprising is the structure of the PLR-group. As shown in Fig. 3.17, the PLR inversions trace out the edges of our cubic representation of M 3 and the perfectly even chords. P is shown in red, L is shown in purple, and R is shown in blue. Figure 3.17: PLR-group represented on our cubic structure: P (red), L (purple), R (blue) Let us consider movement from a minor c-orbit. [g + 1, g + 4, g + 8]. The minor c-orbit has such a structure because it is one semitone added to one of the components of a perfectly even c-orbit. 30

38 The only movements we can make while remaining in M 3 assuming we only move one component at a time are: Subtracting two semitones from g + 1 (root) Adding one semitone to g + 4 (third) Adding one semitone to g + 8 (fifth) Applying R would subtract two semitones from g + 1, the root pitch class. Applying P adds one semitone to the g + 4, and applying L adds one semitone to g + 8. Because we are working with c-orbits, none of the components are assigned a specific axis (x, y, or z), but because our cubic structure is geometric, we need to reconcile the geometry with where our c-orbits are on the cubes. Thus without assigning a specific axis, let x 0 be the axis on which R lies and let x 1 be the next successive axis (i.e. y if x 0 = x, z if x 0 = y, or x if x 0 = z), and x 2 be the next successive axis after x 1. Because c-orbits are represented such that g 1 apple g 2 apple g 3, if movement in the g 1 component corresponds to movement along the x 0 axis, then movement in g 2 is along x 1 and movement in g 3 is along x 2. Therefore P lies along x 1 and L lies along x 2. We can repeat this process in reverse for major chords. Let us consider the examples from Section 3.2.2, but now realized within our cubic representation of M 3. These are shown on the following page. In the visualization of the first four chords of Canon in D Fig. 3.18a, we can see that when we translate [2, 6, 9] and [2, 6, 11] by 7, the map R is also shifted. Likewise, if we consider the function R, it is as if we shift the map t 7 along R. The same is true for the visualization of Yellow Room in Fig. 3.18b. Thus we are able to, in a sense, commute about the cubes given our ti and PLR-groups. 31

39 (a) Relations between first four chords of Pachelbel s Canon in D :R (blue), t 7 (orange, from D-major to A-major, purple from B-minor to F]-minor) (b) Relations between four chords of Pile s Yellow Room :R (blue), t 5 (orange, from C-major to F-major, purple from A-minor to D-minor) Figure 3.18: Our examples from Section 3.3 realized in our cubic structure 32