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Journal of Mathematics and Music:Mathematical and ComputationalApproaches to Music Theory, Analysis,Composition and PerformancePublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmam20

Music in the pedagogy of mathematicsMariana Montiela & Francisco Gómezb

a Department of Mathematics and Statistics, Georgia StateUniversity, Atlanta, USAb Department of Applied Mathematics, Technical University ofMadrid, Madrid, SpainPublished online: 22 Sep 2014.

To cite this article: Mariana Montiel & Francisco Gómez (2014) Music in the pedagogyof mathematics, Journal of Mathematics and Music: Mathematical and ComputationalApproaches to Music Theory, Analysis, Composition and Performance, 8:2, 151-166, DOI:10.1080/17459737.2014.936109

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Journal of Mathematics and Music, 2014Vol. 8, No. 2, 151–166, http://dx.doi.org/10.1080/17459737.2014.936109

Music in the pedagogy of mathematics

Mariana Montiela∗ and Francisco Gómezb

aDepartment of Mathematics and Statistics, Georgia State University, Atlanta, USA;bDepartment of Applied Mathematics, Technical University of Madrid, Madrid, Spain

(Received 15 January 2014; accepted 14 June 2014)

The present article addresses the subject of music in the pedagogy of mathematics from the perspectiveof two researchers in mathematical music theory (MMT) belonging to mathematics departments. Ourfirst main topic is the popularization project concerning music and mathematics of the Royal SpanishMathematical Society, and its extension to an international level is proposed. Secondly, we present someideas and outlines for the creation of didactic material for mathematics courses within the framework ofMMT.

Keywords: mathematical music theory; popularization; pedagogical materials; algebraic combinatoricson words; maximal evenness; Rubato Composer

1. Introduction

We begin by contending that mathematical music theory (MMT) is really an interdisciplinaryendeavor and not only a toolbox for the discipline of music, however sophisticated those toolsmay be. In other words, a possible serious misunderstanding about the pedagogy of MMT stems,precisely, from the fact that it has often been seen exclusively as a supporting discipline to musictheory and analysis, composition, and similar areas. While MMT approaches have engenderedenthusiasm among music theorists, and hence its natural presence in music departments, prob-lems that arise in MMT pose challenges deep enough for new ideas and techniques to flourish inmathematics and computer science. This should be seen as really good news, as that interactionincisively defines and cements the interdisciplinary (multidisciplinary) nature of MMT.

The subject of this article, which has been written by two members of mathematics depart-ments, is music in the pedagogy of mathematics. This said, one of us has experience in thecontext of music pedagogy and the other has worked with mixed groups of mathematics andmusic students in which, of course, the subject was made relevant to the music students’ needs,even when they were not specifically the target population.

The present article consists of two overarching sections. In Section 2 we focus on theactivity of popularization and, in particular, Divulgamat, the digital magazine published bythe Royal Spanish Mathematical Society (RSMS, http://www.rsme.es/) and edited by RaúlIbáñez and Marta Macho with exquisite professionalism. Divulgamat illustrates the presenceof mathematics in many disciplines, especially in the arts, in an informative yet rigorous

∗Corresponding author. Email: mmontiel@gsu.edu

c© 2014 Taylor & Francis

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manner. Among its many columns we find Music and Mathematics (present co-author F.Gómez has been in charge since May 2010), which has published 53 popularization articlesto date.

In Section 3, we discuss the production of didactic material for mathematics and computerscience courses within the framework of MMT. The material is mainly intended for, but notlimited to, mathematics and computer science students. At this point, we should mention thatat the heart of these materials lies a genuine interest in the mathematics department of one ofthe present authors and her university (Georgia State University) to elevate MMT to an evenmore favorable position. Until recently, the situation at her department was similar to that ofmany other mathematics departments. Doing research in mathematical music theory was seenas something complementary but not as a principal research agenda. This has changed radicallywith encouragement to form a research area in MMT, to look for graduate students, and developcourses and collaborate closely with the School of Music.

For this reason we think that it is essential to create more didactic materials, in the spirit ofTimothy Johnson’s text (Johnson 2008), based on the multiple subjects of MMT that have nowbecome classical topics but are sometimes reserved to scholarly journals and meetings of special-ists. Some examples of these subjects are scale theory using algebraic combinatorics on words(Noll 2008a, 2008b; Clampitt and Noll 2011; Noll and Montiel 2013), rhythmic canons (Vuza1991, 1992a, 1992b, 1993; Amiot 2005; Andreatta 1996; Agon and Andreatta 2011), multidi-mensional geometry (Tymoczko 2011), and denotator theory (Mazzola 2002). In particular, welook at these topics from the perspective of music in the pedagogy of mathematics. As a matterof fact, it is also important to organize and present the concrete analyses of musical works carriedout using these techniques in a pedagogical way, although this corresponds to mathematics in thepedagogy of music.

Another proposal is that examples taken from MMT be incorporated into mathematics text-books, just as examples from physics, economics, and other areas are incorporated. For instance,the way in which Markov chains, Bayesian probability, neural networks, and genetic algorithmsare employed in music cognition studies (Mavromatis 2005; Temperley 2007) could be given asexamples of applications in textbooks. Indeed, an antecedent in this spirit is the text An Intro-duction to Group Theory: Applications to Mathematical Music Theory, where musical examplesare employed to illustrate notions in group theory (Aceff-Sánchez et al. 2012).

To close this introduction we want to mention an interesting antecedent related to the subjectthat has also been a driving force for our reflections on the need for more didactic materials. Oneof the authors was entrusted with the task of developing a course on MMT for the East ChineseUniversity of Science and Technology in Shanghai, PR China, with which her department hasties and a study abroad program. Indeed, the Chinese university chose the subject of MMT froma pool of options of novel mathematical areas. The course is cross-listed and open to studentsof mathematics and music, both undergraduate and graduate. In the preparation of the course,it was seen that there is little didactic material for a mathematical audience. The goal of thecourse is, in a reasonable manner given the time, to introduce the language and experience ofthinking musically in mathematical terms such as equivalence classes, Z12, homomorphisms,and transformation groups. Once a common language is developed, students can reformulate thedifferent aspects seen in this initial introductory panorama and think about a particular point intowhich they would like to delve more deeply, for instance, by means of a project.

For mathematics students, the musical motivation can lead to the discovery, or at least thecomprehension, of notions and concepts that are fundamental to their progress in the discipline.In this particular course of such a mixture of students, the areas that they can use for their projectsinclude sophisticated notions used in MMT, such as category theory, homology, algebraic topol-ogy (Mazzola 2002, Mazzola et al. 2008) for the graduate students, as well as questions relatedto group theory (Hook 2002; Fiore and Satyendra 2005), and algebraic combinatorics on words,

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or more applied subjects such as probability and statistics. Material is abundant and has beenwritten by readers, authors, and members of the board of this journal. Indeed, this type of coursegives mathematics students the possibility to explore areas that they might never see otherwise.Simultaneously, music students can apply aspects of diatonic theory, Euclidean rhythms, andsimilar notions to analyze melodic, harmonic, or rhythmic aspects of a concrete piece. This typeof application by the music student can give the mathematics students another dimension totheir own grasp on the subject. This sort of interaction is fundamental for both mathematics andmusic students.

2. Popularization of mathematical music theory

Mathematics is not only characterized by the power of abstraction, infinite intellectual depth,insatiable curiosity, and hunger for structural beauty. It also possesses a valuable social com-ponent. Mathematics needs to be communicated in order to realize its potential and accomplishits mission. There is no such thing as un-communicated mathematics. An important form ofcommunication in mathematics – in science, in general – is popularization. Popularization isvital in helping young people discover and begin to pursue their career sooner. Since MMT issuch a young discipline, its practitioners have mainly devoted themselves to research productionand, there is no point in hiding it, to convincing the more traditional scientific community thatMMT is on its own a legitimate, purposeful, scientific discipline. Therefore, popularizing MMTis a timely project and in this article we would like to bring this issue to the attention of MMTpractitioners. As Amiot (2013) has put it,

[a]s a society and individuals we have a formidable task ahead of us in terms of promotion and popularization. It iswell and good to pour out one esoteric MMT paper after another – and I know I am one of the worst offenders – butit is also necessary to promote the idea that mathematics is consubstantial and essential to music (music theory, ormusic understanding, at least).

A venture that can serve as a representative example is that of Divulgamat (1999–the present)and its column on music and mathematics. The spirit of Divulgamat’s column on music andmathematics is inspired by a few basic principles, which concern both disciplines. Mathematicsis a science inasmuch as it is a body of systematic knowledge; from this standpoint, music is alsoa science. Furthermore, music phenomena lend themselves to mathematical study, as music is sofull of patterns and structure. The column is determined to show this parallelism.

Both mathematics and music are instruments to explore the world, whose realms are by nomeans disjoint. Mathematics takes us through the path of abstraction, logic, and creativity. Musicgives sound a meaningful organization through logic – musical logic – and once again creativity.The column proposes to shed light on that common ground. Mathematics and music are artisticactivities in the sense that they seek to create beauty, both intellectual and emotional beauty.Thus, the Divulgamat column proposes to thrill the reader by drawing carefully chosen examplesof that beauty.

Popularization of mathematics and music does not mean oversimplifying content, or evenworse, treating readers as if they had little intellectual stature. More often than desired, wefind popular science bordering on irrelevance, which is counterproductive. Thus, the columnrelentlessly pursues an intellectual respect for the reader. If a topic is complex, as is often thecase, the article is published in series. The ultimate goal is that musicians enjoy mathematics,mathematicians enjoy music, and the general audience celebrates both.

Topics explored so far include the hexachordal theorem, sets of maximum area and musicalharmony, the mathematics of Xenakis’s music, the concept of beauty in both fields of study,

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rhythmical similarity in flamenco music, mathematical distance in musical similarity, mathe-matical measures of syncopation, minimalism and mathematics, the teaching of music throughmathematics, inquiry-based teaching methods for mathematics and music, mathematical andmusical education for children, rotation of rhythms, binarization and ternarization of rhythms,Aksak rhythms, and maximally even rhythms, among others.

In the following we will provide the reader with translations (the articles are written in Span-ish) of some abridged examples taken from actual Divulgamat columns hoping, this way, togive a general idea of the intention. It is worthwhile to mention, as well, that many of the arti-cles contain sound and/or video examples, one of the big advantages of this type of forum. Asof December 2013 there were 53 popularization articles, and we expect that the column willcontinue presenting a new one each month.

2.1. Rhythm and rotations (Divulgamat, May 2012)

This popularization article by present co-author Francisco Gómez about rhythm and rota-tions consists of the following three sections: Mathematical Rotations, Musical Rotations, andSimilarity between Mathematical and Musical Rotations. We follow with some excerpts.

Rotations are transformations that have traditionally sparked interest and, consequently, have been completely andthoroughly studied. A rotation is defined as a rigid movement around a fixed point. This movement takes place insome space that can be as abstract as we wish, but rotations in the plane are among the ones that have been moststudied. [In Figure 1] we have a rotation of a planar object around the point 0 = (0, 0). The rotation is 180 degreesand several intermediate steps are shown.

[. . . ] In this article we are interested in studying the meaning of rotations applied to music and, in particular,to rhythms. Among all rhythms, we chose the so-called clave rhythms. Clave rhythms are rhythms that repeatthemselves throughout a piece, and whose musical functions include maintaining the rhythmic stability, organizingthe phrasing, or serving as temporal references within a piece (Ortiz 1995; Uribe 1996).

[. . . In Figure 2] we see the circle divided into 12 parts and in the center we see the bembé clave. Surrounding it,and in a counterclockwise direction, we see the rotations of the bembé that give rise to other rhythms. The rotationsof the rhythms are clockwise. Thus, the bemba is a 60 degree clockwise rotation of the bembé; the tambú, a 150degree rotation; the yoruba, a rotation of 210 degrees; the ashanti, 270 degrees; and the bembé-2, 330 degrees.

Find the two examples [Figure 3] at the website in the footnote.1

Figure 1. Rotations of an object in the plane.

1 http://divulgamat2.ehu.es/divulgamat15/index.php?option=com_content&view=article&id=14103&directory=67

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Figure 2. Rotations of the claves associated with the bembé pattern.

Figure 3. Two rotations of the bembé pattern.

2.2. Distance and musical similarity (Divulgamat, May, June, July 2011)

This subject, distance and musical similarity, is presented in a series of three articles written byFrancisco Gómez. In the third installment it is explained that,

[t]his is the last article of a series about the mathematical concept of distance and melodic similarity. In the firstarticle we reviewed the main properties of distance as a mathematical object and we made a list of the many fieldsin which this productive concept is used. In that same article we introduced the concept of melodic similarity andwe illustrated it with the famous Mozart variations K. 265 over the popular theme Ah, vous dirai-je, Maman. Inthe second article we entered into more technical details. In the first place we defined the abstract representationsof the melodies and, in the second place, how certain transformations are applied to these representations to get asimilarity measure. Transformations can be of diverse nature, and in the second article we studied some of the mostrelevant ones: pitch transformations, rhythmic transformations, and symbolic measures. Due to space limitations,we left the transformations based on vector-valued measures and harmonic measures for this article. Finally, in this

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article we will describe the experiments of Müllensiefen and Frieler (2004) for the perceptual validation of thesemeasures.

The whole article can be found at the link in the footnote.2

2.3. Suggestions

However, as we mentioned above, the column is published in Spanish. There is an ongoingproject to translate the articles into English. A good idea would be to start up a column in digitalformat where prominent MMT figures would write articles to popularize the discipline. MMT isin need of a figure like Martin Gardner, who brought mathematics to millions through his prose,his clarity of thought, and his understanding of the playfulness of mathematics.

Although the idea of writing a column on MMT is worthwhile and a necessary step, the situa-tion calls for even more action. The word “popular” itself urges us to take on a broader approachto popularization. Here is a list of suggestions, which by no means is exhaustive, of initiatives toreach out to broad audiences:

• a series of videos where MMT concepts are explained in an entertaining yet rigorousway;

• a journal of recreational MMT (in the style of the Journal of Recreational Mathematics);• participation in events of mathematics/music appreciation. F. Gómez has participated in

Maths Week Ireland (2006–2013), the top festival of its kind in the world, for several yearsand the experience has been invaluable;

• a repository of MMT resources for elementary and high school teachers.

Our mention of teachers in the final point was no coincidence. It is of paramount impor-tance to reach that group. In their hands lie our future mathematicians and musicians. Workingwith teachers to help them understand and appreciate the enormous potential of teaching math-ematics via music and music via mathematics should constitute an essential ingredient of ourpopularization work.

As Schneider (1995) reminds us, “we might expect mathematicians, if only through self-interest, to take some responsibility for the ways in which the general public sees and usesmathematics.” If we substitute “mathematicians” by “MMT practitioners,” that statementstill stands.

3. Didactic material

In this section we discuss the development of materials for specific courses for mathemat-ics and computer science students. These courses can be constructed around one particularmathematical subject, or they can consist of units which treat different mathematical sub-jects, some of which are not seen in traditional courses. To give an idea of the possibilitiesthat we envision, although incomplete (there are more subjects that should be included andperhaps prioritized) and simplified (given that there is richness and interrelation among sub-jects that cannot be captured with a scheme), we present the following outline and mention a fewreferences.

2 http://divulgamat2.ehu.es/divulgamat15/index.php?option=com_content&view=article&id=12763:25-mayo-2011-distancia-y-similitud-musical-i&catid=67:ma-y-matemcas&directory=67

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• Uniform triadic transformations, contextual groups, and generalized commuting groups, whichinvolve aspects of advanced group theory. See Hook (2002), Crans, Fiore, and Satyendra(2009), and Fiore and Satyendra (2005).

• Rhythmic canons, which include areas such as Galois theory, tiling, and Fourier analysis onfinite groups. See Vuza (1991, 1992a, 1992b, 1993), Amiot (2005), Andreatta (1996), Agonand Andreatta (2011), and Hall and Klingsberg (2004).

• Scale theory using algebraic combinatorics on words. See Noll (2008a, 2008b), Clampitt andNoll (2011), Noll and Montiel (2013), and many others.

• Counterpoint and harmonic theory using multidimensional geometry. See Tymoczko (2011).• Music cognition using Bayesian statistics, matrix theory (i.e. hidden Markov models), and

category theory. See Mavromatis (2005), Temperley (2007), and Andreatta et al. (2013).• Music software such as Open Music (Agon, Assayag, and Bresson 2013), and Rubato

Composer�, which use category and functor theory, topos theory, and mathematical gesturetheory. See Mazzola (2002) and Milmeister (2009).

• Maximal evenness, a notion whose formalization arose in the context of mathematical musictheory and intersects with combinatorics, mathematical physics, number theory, and distancegeometry. See Clough and Myerson (1985), Clough and Douthett (1991), and Douthett andKrantz (2007).

In particular, the content of a course based on units could be presented as follows (we shouldmention that this is the actual presentation of a concrete proposal for a course of this nature byone of us and is by no means exhaustive; it is just an example).

Content: You will study specific aspects of group theory, algebraic combinatorics on words, topology, and topostheory that are used in the analysis of general objects of music (scales, chords, rhythmic patterns) as well as inspecific applications (development of software, analysis of pieces from different time periods and genres). Youwill acquire a repertoire of mathematical tools and techniques that are not always covered in the core coursesof the major. For example, in group theory you will need to study the semi-direct and wreath products, while inword theory you will be exposed to Christoffel and Sturmian words and morphisms; and you will learn some ofthe basic constructions of category and topos theory, such as limits, colimits, subobject classifier, functors, naturaltransformations, and Yoneda’s lemma. All of this will be put into the context of applications to musical analysis andof some algorithmic procedures in the software Rubato Composer� .

The following four sections present some partial examples of how we envision the develop-ment of these kinds of didactical materials.

3.1. Example: scale theory using algebraic combinatorics on words

Algebraic combinatorics on words is a relatively recent area of study, although its roots can betraced back to the end of the nineteenth century. Algebraic combinatorics on words studies thegeneral properties of discrete sequences of letters, that is, sequences over alphabets. In otherwords, combinatorial aspects of free monoids are studied and, on occasions, extended to investi-gations of free groups. There are myriad applications, given that many different types of discretechains of events can be represented by words. The area is now recognized as an independentmathematical theory and, as announced in the 2002 reference book written by the top experts inthe field under the pseudonym of “Lothaire” (Lothaire 2002), the areas of application of the sub-ject include core algorithms for text processing, natural language processing, speech processing,bioinformatics, and several areas of applied mathematics such as combinatorial enumeration andfractal analysis.

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In his article, Noll (2007) studied musical intervals with a particular focus on their constitutiverole for well-formed scales. Apropos of this, he commented that

[t]he content of the present paper has several close relations to the subfield of algebraic combinatorics of words thatstudies Christoffel words and Sturmian words. [. . .] While revising this paper it became clear to me that the rightmathematical context for the rewriting rules are Sturmian morphisms and morphisms of Christoffel words. (p. 131)3

What followed is history, given that in the subsequent years there was an avalanche of articles(for example, Noll 2008a, 2008b; Domínguez, Clampitt, and Noll 2009; Clampitt and Noll 2011;Noll and Montiel 2013; and many others), in which certain subtleties of intervallic relationswere seen in a new light through the lens of Sturmian morphisms, Christoffel words, and theirconjugacy classes. One of the key articles of the purely mathematical production on the theme ofSturmian involution (Berthé, de Luca, and Reutenauer 2008) used a conjugacy class of Sturmianmorphisms to illustrate duality; the extraordinary coincidence was that their example, withoutthem knowing it, was perfectly suited for the investigation of the six plagal modes.

We will briefly exemplify what was observed by Noll above. Let A = {a, b} be the alphabetof the two letters a and b, and A∗ the monoid formed by all possible words in this alphabet(for example, aab, babb, etc.). Define mappings f : A∗ −→ A∗ by assigning words f (a) andf (b), which correspond to the letters a and b. If a is substituted by the word f (a) = aaba andb is replaced by f (b) = aab, we obtain f (ab) = f (a)f (b) = aabaaab (f is a homomorphism ofmonoids). We then consider the following endomorphisms, which are generators of the monoidof special Sturmian morphisms acting on the monoid A∗ generated by the alphabet A = {a, b}:

G(a) = a, G(b) = ab, D(a) = ba, D(b) = b,

G̃(a) = a, G̃(b) = ba, D̃(a) = ab, D̃(b) = b.

We consider compositions of these morphisms applied to the initial word w = ab. Interpretingthe words resulting from the application to w of certain morphisms, three at a time, by under-standing a’s as ascending major steps and b’s as ascending minor steps, one derives the sixauthentic modes of Glarean. For example, the result of GGD(a|b) = GG(ba|b) = G(aba|ab) =aaba|aab is the Ionian mode (counting major and minor seconds as explained above) andGG̃D(a|b) = GG̃(ba|b) = G(baa|ba) = abaa|aba is the Dorian mode. Notice that these twomodes are rotations of one another, and in general the seven modern modes are all the rotationsof one another.

This apparently fortuitous occurrence spurred the production of more research around the suit-ability of two different dual constructions in the musical context, the plain and twisted adjoints.One of these, the twisted adjoint, coincides with the formal Sturmian involution as presented inthe literature, while the other, the plain adjoint, also has mathematical and musical coherenceand was the source of Noll’s Ionian theorem (Noll 2008a).

These dualities are related to the way well-formed scales are generated. The dual of a Sturmianmorphism, as it appears in the mathematical literature (Berthé, de Luca, and Reutenauer 2008),can be exemplified by the Dorian mode. Instead of applying GG̃D, which led us to the Dorianmode through major and minor seconds, the dual D̃G̃G, obtained by reversing the order andsubstituting D for D̃, will generate the Dorian mode as well; however, in this case, it is generatedby ascending fourths and descending fifths. That is, if a represents a perfect descending fifth andb represents a perfect ascending fourth, then D̃G̃G(a|b) = D̃G̃(a|ba) = D̃(a|aba) = ab|abbab.If we follow the descending fifths and ascending fourths beginning with B, we see that D, thenote that initiates the Dorian “white-key” mode, is the lowest note. See Figure 4.

3 Noll (2008b, Footnote (4)) emphasizes that “with a few exceptions [. . .], musical scale theory and combinatorialword theory have remained unaware of each other, despite having an intersection in methods and results that by now isconsiderable. The theory of words has a long history, with many developments coming in the last few decades [. . .] Ithank Franck Jedrzejewski for an initial reference in word theory and Valérie Berthé for helpful comments.”

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Figure 4. Descending fifths and ascending fourths.

The reason for starting with B is related to the theory behind the plain and twisted adjoints. Inthis case, that of the twisted adjoint, all the folding patterns will start with B, and the lowest noteof each folding pattern generated by a given Sturmian morphism will always correspond to themodal center (finalis) in the diatonic scale pattern generated by the dual of the given morphism.

In algebraic mode theory, the rotations referred to above have turned out to be quite use-ful. They lead to generalizations of the theory of well-formed scales to well-formed modes,where modal patterns are represented as trajectories described through width/height coordinates.A geometric generalization proposed by Berthé, de Luca, and Reutenauer (2008) allows the inter-pretation of those trajectories within a transformational framework. This extension leads to thetopic of Pisot substitutions and symbolic dynamics (Arnoux and Ito 2001; Noll and Montiel2013).

3.2. Example: the music software Rubato Composer�, based on category (topos) theory

The objective of this unit would be to make abstract structures accessible by representationsthat come from tangible, concrete musical objects, thereby introducing students into fields ofmathematics that are usually reserved for those who have completed multiple semesters ofprerequisites.

Advanced areas of mathematics such as group theory and abstract algebra in general, linearalgebra, set theory, category theory, and so on, are basic components of physics, computer sci-ence, logic, geometry, economics, and music theory. For this reason, there exists a strong needto make them accessible to a broad range of undergraduate students. However, because of theirabstract nature, it is difficult to make these subjects meaningful for mathematics and science stu-dents, and even more difficult to do so for students in the humanities. The search for ways toenhance advanced mathematical thinking has led to, among other methods, the use of computersas a tool in the expression of these abstract concepts.

Category theory is the reference behind the development of the software RubatoComposer� (Mazzola 2002; Milmeister 2009). Although the goal of this software is to bringmathematical and computational tools to a level with which a composer or music theorist canfeel comfortable, at this point a certain amount of mathematical sophistication is required tomake things work. This actually gives the software a pedagogical dimension that was not partof its original intention, and it has been seen to be an excellent supplementary learning toolfor some abstract mathematical concepts. In particular, one of the authors has used RubatoComposer� in this way with undergraduate mathematics majors, and has witnessed its util-ity both as an exemplification of abstract concepts, and as a motivation through the musicalpossibilities.

It has been recognized that there is a rift between the formality in modern mathematicsas taught by mathematicians and the mathematics that are perceived to be relevant by non-mathematicians, as well as a good number of students, even mathematics students. One of thefounders of category theory, Saunders Mac Lane, admitted this when he wrote “Even mathemat-ics students themselves often have difficulty in making meaning out of the formal presentationof their subject” (Mac Lane 2005).

In the Rubato Composer� architecture a basic element is the mathematical structure of a mod-ule (roughly speaking, a vector space structure where the scalars can come from a ring, not

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necessarily a field). The mathematical module plays a role similar to a primitive type in a pro-gramming language. However, as it is necessary to work with set constructions to, say, representmusical embellishments that are seen as sets of sets of notes, the theoretical and implementa-tional designers opted for working in the functor category, which consists of presheaves on thecategory of modules (whose objects are the functors). The development of this theory and itsrelation to Rubato Composer� can be found in Mazzola (2002), Thalmann and Mazzola (2008),Mazzola et al. (2008), and Milmeister (2009).

As an example of a completed student project, the task was to take a 64-note bass line and,following the tutorial in Milmeister (2009), create a denotator, that is, a “point” in a theoreti-cal basscoreform4 (roughly speaking, the “space” of all possible scores for bass). The studentconstructed the theoretical basscoreform and the particular bass line (from John Legend’s Allof Me) was played and heard as a midi file through the ScorePlay rubette. The tasks and pro-cedures required are “low level”, where the student must define everything explicitly, from thebasic module morphisms underlying the denotators to the matrices that “spell out” the transfor-mation. Figure 5 shows the bass score that was played as output after creating the morphisms anddenotators representing pitch, onset, duration, volume, and voice (instrument). The morphismsfor pitch, onset, and duration are compositions of an affine morphism5 and an embedding, whichrepresent the 64 pitches, onsets, and durations of the bass line.

Figure 5. Bass score.

These module morphisms take place in the category of all left modules over all possible asso-ciative rings with identity element, whose morphisms form the set of diaffine transformations.6

There is quite a bit of abstract mathematics from module, ring, and category theory implicit inthis last sentence and in the creation of these module morphisms. Pitch, for example, requiresthe composition mp : Z63 −→ Q63 −→ Q, mp = mp,1 ◦ mp,2. In other words, each canonicalaffine basis vector of Z63 is mapped to one of the 64 notes of the piece (in affine space Z63 thecanonical affine basis consists of the 63 usual canonical unit vectors plus the zero vector, that is,64 canonical affine basis elements). Z63 is embedded into Q63, the 63-dimensional product of therationals (when seen as modules, this is an example of a diaffine transformation, where the ringhomomorphism goes from Z −→ Q). Then, by means of a translation vector with 63 compo-nents that does the inner product with each basis vector, plus a translation of 48, which is the firstnote C2 according to the midi convention, the 64 pitches are generated. A similar process is usedfor onset and duration. The creation of the embedding morphism, the affine transformation, andthe composition can be seen in Figures 6, 7, and 8. However, the process itself, under the moti-vation of hearing the music as an outcome and the availability of the tool bars, led the student,who did not have a sophisticated background, to create the function according to the tutorial.

The development of tutorials and projects, interspersed with the relevant theory and exercises,would be a significant contribution to the proposed resources.

4 Forms and denotators are defined in Mazzola (2002). Forms and denotators provide a means for implementing animportant part of computational category theory. Forms are a generalization of the concept of data types, and denotatorsare mathematically defined “pointers.”

5 An affine morphism is a composition of a module morphism with a translation.6 If ϕ : R −→ S is a ring homomorphism and N is an S-module, then N�ϕ� is the R-module defined by the scalar

restriction s · m = ϕ(s) · m, for s in R and m in N . A dilinear homomorphism from an R-module M to an S-module Nis a pair (ϕ : R −→ S, f : M −→ N�ϕ�) consisting of a scalar restriction ϕ and an R-linear homomorphism. A diaffinetransformation is a composition of a dilinear homomorphism and a translation.

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Figure 6. Creation of the embeddingmorphism using Rubato Composer� . Figure 7. Affine transformation in Rubato Composer� .

Figure 8. Composition of module morphisms in Rubato Composer� .

3.3. Example: maximal evenness

Why are the black and white notes of the piano distributed as they are? To answer this question,formulas were discovered and more abstract problems were generated (Clough and Douthett1991; Johnson 2008). Eventually, a connection with models from physics was made (the one-dimensional antiferromagnetic Ising spin model), and this has resulted in alternative calculationspublished in articles in journals such as the Journal of Mathematical Physics (Krantz, Douthett,and Doty 1998), among others.

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Figure 9. Maximally even distribution.

The classical example of maximal evenness is found in the major scale, as an answer to thefollowing question: How can 7 notes (or pulses) be distributed so that the result is as “evenlyspread out” as possible? If we were to spread out 6 notes of 12 we would, of course, divide 12by 6 and the result would be every two notes (or pulses). However, in this case, 7 and 12 arerelatively prime and a technique other than division must be found to calculate the distribution.Indeed, the technique can be generalized to any d and c, d < c, and applied to the study of scalesin general and the multidivison of the octave. Actually several techniques have been found, andthe mathematical interest in the notion of maximal evenness is related to the search for thesetechniques.

In the case of 7 in 12, we can look at the solution shown in Figure 9 (up to a rotation).Indeed this diagram represents the major scale. Of course, rotations would not change the

essence of the distribution and, on the other hand, this is not the only possible pattern. In fact,the seven white note modes are all maximally even.

The gamut of materials that can be developed around this subject is broad and includes ageneral course agenda of the liberal arts type as well as music theory courses and others witha more defined mathematical orientation. A magnificent chapter in Johnson’s Foundations ofDiatonic Theory (2008) already exists; this chapter also plays a multifaceted role which, in thewords of the author, can serve as “a course in music fundamentals, either for majors or non-majors; the review of fundamentals in any course in the core-theory sequence; a course in musicand mathematics; or an advanced course in diatonic set theory.” (p. viii).

However, from the perspective of music in the pedagogy of mathematics adopted here, there isso much to develop. The work carried out by Douthett and Krantz (2007, 2008) around equivalentmanners of defining a maximally even set is a good exercise in different representations, giventhat a maximally even set can be identified using diverse mathematical criteria. For example, amaximally even set, with parameters c and d (using the prototype, of course, of the chromaticand diatonic scales, 12 and 7) can be identified as the one that has the maximum average chordlength when compared with other possible distributions of the parameters. This implies the repre-sentation as a “necklace,” with discrete points on the circle, although the trigonometric formulafor the length of the chord that connects points of a given distance k, 2 sin(kπ/c), belongs tothe continuous realm. Maximal evenness can also be seen from the perspective of integer spec-tra, which is the most common form of understanding the concept from a scale perspective, asintervals. Moreover, a maximally even set can be generated, or corroborated, by the formula⌊

ck + i

d

⌋mod d,

where k = 0, 1, . . . , d − 1 and i is a fixed integer 0 ≤ i ≤ c − 1.It is worth mentioning that Johnson’s book is focused on scale theory (melody). On

the other hand Toussaint, in a series of discoveries around Euclidean rhythms, devel-oped the notion of maximal evenness in the context of rhythm (Demaine et al. 2009;Gómez, Talaskian, and Toussaint 2009). This work, which culminated in the book The Geom-etry of Rhythm (Toussaint 2013), reveals the algorithmic aspect of maximal evenness and relatesit, among other things, to the Bjorklund algorithm used in spallation neutron source accel-erators. This book, while an excellent text, could be complemented with activities, many ofwhich could be carried out using the software Rhythmos, developed by Toussaint and assis-tants (Toussaint 2007). One of the present authors has developed a series of didactic units

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(Gómez and Farigu 2012) around this subject and its application to rhythm, although the level ismore of a general course. However, the subject definitely lends itself to the adaptation of didacticmaterials for mathematics and computer science students.

In this same direction, one of the present authors ran a workshop for high school mathematicsteachers around mathematical music theory, in which the subject of Euclidean rhythms developedby Toussaint was used, as well as the notion of maximal evenness through rhythm. The idea wasfor the teachers to develop lesson plans based on what they learned in the workshop. A studentfrom the School of Music helped with actual rhythmic illustrations and supervision of the groupwork, when the teachers would reinforce the presentation with hands-on activities. The successthat these sessions had was impressive, as was the creativity of the teachers in the developmentof mathematical lessons for their students with the help of the concepts that arose from the notionof geometry through rhythm. There was direct algebraic use of the Euclidean algorithm, and alsolessons for geometry and trigonometry based on permutations and rotations.

3.4. Example: rhythmic canons

The history of rhythmic canons has been elegantly described in Andreatta (2011). The subjectis a natural candidate for the development of didactic materials, given the quantity and qualityof mathematics involved that is not usually studied in standard courses, such as periodic sets,Hajós and non-Hajós groups, Fourier analysis on finite groups, etc. See Vuza (1991, 1992a,1992b, 1993), Andreatta and Agon (2009), Amiot (2009), Kolountzakis and Matolcsi (2009),and Jedrzejewski (2009). Together with these subjects we find a variety of musical examples thatillustrate the mathematical theory and that can be understood without much need of traditionalmusic theory notation. Finally, the implementation that has been done in the OpenMusic VisualProgramming Language (Agon and Andreatta 2011) provides a unique possibility of understand-ing the interaction of mathematics, computer science, and music, as well as offering a space forexperimentation.

Rhythmic canons study the patterns of accents of several simultaneous lines within a peri-odic succession of beats. The study of rhythmic tiling canons demands that there be no overlapbetween the different lines. This can be visualized as shown in Figure 10 (a diagram taken fromAmiot 2009, 72), where the dark squares denote the accent.

Figure 10. Rhythmic tiling canons (Amiot 2009, 72).

In the first decomposition, the set {0, 2, 7} is called the inner rhythm and the set {0,3,6,9,12}is called the outer rhythm. In the second decomposition, the roles are reversed. In the figure,we see an example of the duality between inner and outer rhythms. There are several differentscenarios for analysis and they all intersect with areas such as the actions of cyclic and affinegroups on rhythmic structures, combinatorics and counting in the context of group theory, andtessellation of the integers, among others. The rhythmic tiling canons of maximum category(Vuza 1991) are those in which neither the inner rhythm nor the outer rhythm is periodic. This iswhere questions about decomposition into periodic sets and “bad” groups, not usually includedin standard curriculums but of great pedagogical interest in mathematics, arise.

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We are aware of how this subject has been pedagogically employed in different institutions(IRCAM and Università di Pisa); however, our proposal is to create material that extracts theessence of the scholarly studies (Andreatta 1996; Amiot 2005; Vuza 1991, 1992a, 1992b, 1993;Hall and Klingsberg 2004) and in which the mathematical breadth and musical examples areinterspersed with exercises that would reinforce mathematical knowledge and stimulate intuitionin the context of mathematical studies.

4. Conclusions

Our concern for a way to formalize the overarching idea of music in the pedagogy of mathe-matics is what we have tried to reflect in the content of this article. By no means is this contentexhaustive; it is not even partial. We have not mentioned, for example, the references to the pio-neering work of David Lewin ([1987] 2007). Similarly, subjects such as the Fourier transform,apart from their appearance in the subject of rhythmic canons, also play an important role inpitch-class set theory (Lewin 1959, 1960). On the other hand, we chose the subjects of the twosections because the circumstances of our professional lives have brought us to participate in theRoyal Spanish Mathematical Society’s concrete and formidable project of popularization, and toassume the responsibility of creating materials for mathematics and computer science students.

This article, above all, speaks to future projects. Accordingly, we are convinced that it isfeasible to implement these projects. The popularization work could begin immediately, giventhat there exists a plethora of articles, some of them translated, and it is only a question ofdeciding where they would be presented, designing the format, and beginning to plan and choosesubjects to develop over the next years.

Acknowledgements

We are indebted to Thomas Fiore and Jason Yust for all their hard work in style correction and helpful comments. Wealso thank the anonymous reviewers for their suggestions and corrections.

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