# Music in the pedagogy of mathematics

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<ul><li><p>This article was downloaded by: [Karolinska Institutet, University Library]On: 17 November 2014, At: 06:57Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK</p><p>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</p><p>Music in the pedagogy of mathematicsMariana Montiela & Francisco Gmezba Department of Mathematics and Statistics, Georgia StateUniversity, Atlanta, USAb Department of Applied Mathematics, Technical University ofMadrid, Madrid, SpainPublished online: 22 Sep 2014.</p><p>To cite this article: Mariana Montiel & Francisco Gmez (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</p><p>To link to this article: http://dx.doi.org/10.1080/17459737.2014.936109</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor & Francis makes every effort to ensure the accuracy of all the information (theContent) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.</p><p>This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &</p><p>http://www.tandfonline.com/loi/tmam20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/17459737.2014.936109http://dx.doi.org/10.1080/17459737.2014.936109</p></li><li><p>Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Kar</p><p>olin</p><p>ska </p><p>Inst</p><p>itute</p><p>t, U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rary</p><p>] at</p><p> 06:</p><p>57 1</p><p>7 N</p><p>ovem</p><p>ber </p><p>2014</p><p>http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>Journal of Mathematics and Music, 2014Vol. 8, No. 2, 151166, http://dx.doi.org/10.1080/17459737.2014.936109</p><p>Music in the pedagogy of mathematics</p><p>Mariana Montiela and Francisco Gmezb</p><p>aDepartment of Mathematics and Statistics, Georgia State University, Atlanta, USA;bDepartment of Applied Mathematics, Technical University of Madrid, Madrid, Spain</p><p>(Received 15 January 2014; accepted 14 June 2014)</p><p>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.</p><p>Keywords: mathematical music theory; popularization; pedagogical materials; algebraic combinatoricson words; maximal evenness; Rubato Composer</p><p>1. Introduction</p><p>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.</p><p>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.</p><p>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 RalIbez and Marta Macho with exquisite professionalism. Divulgamat illustrates the presenceof mathematics in many disciplines, especially in the arts, in an informative yet rigorous</p><p>Corresponding author. Email: mmontiel@gsu.edu</p><p>c 2014 Taylor & Francis</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Kar</p><p>olin</p><p>ska </p><p>Inst</p><p>itute</p><p>t, U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rary</p><p>] at</p><p> 06:</p><p>57 1</p><p>7 N</p><p>ovem</p><p>ber </p><p>2014</p><p>http://www.rsme.es/mailto:mmontiel@gsu.edu</p></li><li><p>152 M. Montiel and F. Gmez</p><p>manner. Among its many columns we find Music and Mathematics (present co-author F.Gmez has been in charge since May 2010), which has published 53 popularization articlesto date.</p><p>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.</p><p>For this reason we think that it is essential to create more didactic materials, in the spirit ofTimothy Johnsons 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.</p><p>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-Snchez et al. 2012).</p><p>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.</p><p>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,</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Kar</p><p>olin</p><p>ska </p><p>Inst</p><p>itute</p><p>t, U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rary</p><p>] at</p><p> 06:</p><p>57 1</p><p>7 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>Journal of Mathematics and Music 153</p><p>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.</p><p>2. Popularization of mathematical music theory</p><p>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,</p><p>[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).</p><p>A venture that can serve as a representative example is that of Divulgamat (1999the present)and its column on music and mathematics. The spirit of Divulgamats 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.</p><p>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.</p><p>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.</p><p>Topics explored so far include the hexachordal theorem, sets of maximum area and musicalharmony, the mathematics of Xenakiss music, the concept of beauty in both fields of study,</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Kar</p><p>olin</p><p>ska </p><p>Inst</p><p>itute</p><p>t, U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rary</p><p>] at</p><p> 06:</p><p>57 1</p><p>7 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>154 M. Montiel and F. Gmez</p><p>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.</p><p>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...</p></li></ul>

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