Interfacing Music and Mathematics: A Case for More Engagement

Introduction

Music is as old as humankind and, since the time of ancient civilization, there have been knowledge seekers interested in exploring various elements of music. Mathematics, in both elementary and advanced forms, has been used in the analysis of lower-level elements of music such as tempo, pitch, timing, chord formation, and meter. Music-related physical phenomena, such as acoustics, and their implication on the development of musical instruments have readily employed mathematics for analysis. Essential information-age appliances, such as computers and cell phones, that have played a substantial role in the dissemination and preservation of music rely heavily on mathematics for their implementation and continual improvement. These are but a few areas of interface between mathematics and music, yet the subject of “mathematics of music,” which I will henceforth refer to as musical mathematics, has been relegated to the fringe and seems to only be touchable to bold algebraists and number theorists who do not mind being labeled hobbyists. The assertion of this article is that from the classroom through private and government organizations interested in STEM, there is a need to create more room for the exploration of the interface between music and mathematics.

There is Historical Precedent for Engagement

The Pythagoreans in ancient Greece were the first researchers to link musical scales and the principles of numbers Pla74, although there are records showing that the ancient Chinese, Indians, and Egyptians studied the mathematical principles of sound Bri87. Pythagoras’s apocryphal experiment with vibrating strings showed him that if he plucked two strings of equal tension, in which the length of one is of a certain proportion to the length of the other, he would get a certain type of harmonious sound, referred to today as consonance. In particular, if the ratio of the lengths of the two strings is 2:1, one gets an octave, that is, two of the same note with the pitch frequency of the shorter string being double that of the longer string; if their ratio is 3:2, the pitch frequency of shorter one is $1\frac{1}{2}$ times that of the longer one, and their combined sound—simultaneously or temporally apart—is referred to as a perfect fifth; and if their ratio is 4:3, their combined sound is referred to as a perfect fourth. Pythagoras’s vibrating string experiment ushered the incorporation of musical sound into the philosophical framework of the Pythagoreans, ultimately helping to shape their central doctrine that “all nature consists of harmony arising out of numbers” Jea68. Although some may argue that this music-driven doctrine of the Pythagoreans had no quantifiable significant positive influence on the development of science and mathematics, it inspired later scientists like Johannes Kepler and Galileo Galilei; Kepler attempted to find consonant music intervals in the orbits of the planets Kep97.

There is Foundational Work in Modern Mathematics to Facilitate Engagement

Even though earlier researchers, such as the Pythagoreans, devoted efforts to studying sound and harmony in relation to numbers, their efforts did not materialize into documented axiomatic underpinnings for music in modern mathematics. Nevertheless, the mathematical exploration of musical sound and structure has found a home in many areas of modern mathematics.

While discussing the geometric series in a calculus class a few summers ago, a student of mine pointed out a subtle use of this series in determining the duration of a musical note or rest with dots. The majority of the class, including two students who are musicians, expressed surprise that a concept used in calculus could be easily applied to reading musical notes. In conventional Western music, it is common for a whole note to get four beats; a half note gets two beats; a quarter note gets one beat; an eighth note gets a half beat, etc. (see Fig 1).

Figure 1.

Basic musical notes and their temporal durations.

If a note has a dot behind it, its duration is extended by one-half its original length, that is, the dot multiplies the original duration by $\frac{3}{2}$. For instance the duration of a dotted eighth note is given by

$$\begin{equation*} \frac{1}{2} + \frac{1}{2} \left(\frac{1}{2} \right) =\frac{3}{4}. \end{equation*}$$

Sometimes, however, composers use multiple dots to effectively convey their duration on a note.

Figure 2.

Modification of a half-note by dots.

Digital automated notations also readily use multiple dots to efficiently notate the precise duration that sometimes only a computer can capture. A second dot beside a note implies an additional duration of one-fourth the original duration, and so on. Fig 2 shows how different numbers of multiple dots modify the half-note. This means an eighth-note with two dots would have a duration of

$$\begin{equation*} \frac{1}{2} + \frac{1}{2}\left(\frac{1}{2}\right) + \frac{1}{4} \left(\frac{1}{2} \right)= \frac{7}{8}. \end{equation*}$$

Thus $D_N$, the length of a note of duration $D$ followed by $N$ dots is given by

$$\begin{align} D_N&= D\left( 1+ \frac{1}{2} + (\frac{1}{2})^2 + \ldots + (\frac{1}{2})^N \right) \\ &= D \sum _{k=0}^N \left( \frac{1}{2} \right)^k \\ &= D \left( \frac{1- (1/2)^{N+1}}{1-(1/2)} \right) \\ &= D \left( 2 - (1/2)^N \right). \tag{1} \end{align}$$

One way in which most music strives to interest and please the listener is through the presentation of temporal patterns with acoustic cohesion. Mathematicians have used set theory and abstract algebra to explore this compositional attribute of music. Starting at a C, the set of pitches in the chromatic scale is

$$\begin{equation*} \{ C, D\flat , D, E\flat , E, F, G\flat ,G,A\flat , A, B\flat , B \}. \end{equation*}$$

The chromatic scale has a free and transitive action of the cyclic group ${\mathbb{Z} /12\mathbb{Z} }$, with the action being defined via transposition of notes. So the chromatic scale can be thought of as a torsor for the group. If we label these notes with integers, i.e., {0,1,2,3,4,5,6,7,8,9,10,11}, we can the define an X major chord, in it simplest form, as the collection {X, X+4, X+7} modulo $12\mathbb{Z}$. For instance, the C major chord {C,E,G} can be expressed as {0,4,7}. A transposition in music is analogous to a translation in mathematics, and in fact, the latter can be used to model the former. Transpositionally related chords are the same up to translation. Hence, the C major chord is transpositionally related to the G major chord, {G,B,D} or {7,11,2}, because $\{7,11,2\} \equiv \{ 0+7,4+ 7,7 +7\}$ modulo $12\mathbb{Z}$. An inversion in music is analogous to a reflection in mathematics and corresponds to subtraction from a constant value. Chords related by an inversion are the same up to reflection; hence, the C major chord is related to the C minor chord, {C, E$\flat$,G} or {0,3,7}, by inversion because {0,3,7} $\equiv${7-7,7-4,7-0} modulo $12\mathbb{Z}$. Just like translation and reflection preserve the characters of a map in mathematics, musical transposition and inversion are important because they preserve the character of a chord.

Starting with elementary foundational formulations involving mathematical transformations (such as the ones described in the preceding paragraph) for organizing motives and chords, one can analyze a piece of music to discover deep structures in it. One example is Tym06 in which Dmitri Tymoczko’s goal is to model composers’ assessment of the relative sizes of voice leadings. Even though one cannot assume that these assessments will be consistent with any mathematical norm or metric, the fact that transposition and inversion can be treated as preserving musical distance allowed him to require that voice-leading comparisons be invariant under transposition and inversion of any individual musical voice.

Another example is the representation of regular temperaments. In music, a regular temperament is any tempered system of tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. In twelve-tone equal temperament (12-TET), the tuning system used in most Western music, the octave is divided into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale. The theory of regular temperaments has been extensively developed with a wide range of sophisticated mathematics, for example by associating each regular temperament with a rational point on a Grassmannian Maz18 (The Grassmannian $G_r(k, V)$ is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V). In essence, this formulation allows one to replace sound events in a musical piece with operators that act on all possible musical parameterizations.

These examples, though far from exhaustive, illustrate that there is enough foundational work and a fertile ground in modern mathematics to facilitate research work towards a rich and sustainable interface between mathematics and music. A comprehensive compilation on the use of mathematics in music can be found in Guerino Mazzola et al’s four volume book, “The Topos of Music” (Maz17a, Maz17b, MGMD0509, Maz02), and some recent advances in the area can be found in the Journal of Mathematics and Music published by Taylor and Francis Online.

The Field of Music is Vast and Every Part of it Needs Mathematical Engagement

There is a wide array of acoustic endeavors that can be classified as music. From the perspective of the listener, the classification of these endeavors may be synonymous with the mainstream music genres (i.e., classical, jazz, Afrobeat, R&B, techno, etc). A utilitarian, however, may choose to classify musical endeavors as dance, elevator music, children’s music, film music etc. The practitioners in each of these fields of music have needs that open avenues for more interactions with the mathematical world, just as mathematical interactions have emerged for each field of say, biology. The use of number theory and abstract algebra in music is currently restricted to classical music—jazz, with its complex use of dissonance, for instance, has not fully benefited from these activities—and even at that, most of the work done in this regard is hardly given the exposure (say, at conferences and in textbooks) that it deserves. One common reason that is usually given for this is that the language of mathematics is already difficult and that adding the extra task of learning the language of music theory makes research work in musical mathematics inaccessible. While there is some truth to this, other fields that were once deemed very far from mathematics in terms of function and language, have successfully created niches inside mathematics. A good example is biology: while biology as a scientific field has existed for almost as long as mathematics, the subject of mathematical biology experienced a surge in interest only in the 1900s, and mathematicians have managed to identify the different ways to use mathematics in ways that serve the particular need of each subfield. Today we have niches such as theoretical neuroscience, computational immunology, mathematical epidemiology, etc. In sum, this same phenomenon can also happen in musical mathematics, though it would require a comprehensive identification, assessment, and inclusion of all the possible niches that make up musical mathematics.

Mathematics Has a Role in Interfacing Science and Music

Music clearly provides great examples of many interesting phenomena in hearing, and as such is a constant source of inspiration for basic hearing cognition research (see GMTB18 and AHV$^{+}$21 for examples). There are also active clinical endeavors that employ musical sounds to directly improve lives. Clinical music therapy, for instance, has been linked to stress reduction, mood improvement, and improvement in self-expression; and is regarded by many in the health field as an evidence-based therapy (see MST$^{+}$20 and GLW$^{+}$06 for examples). Hence, there are long-standing interests in the scientific use of music within several scientific fields (such as neuroscience, psychology, and cognitive science). Broad research questions in this line of work have included: what makes music pleasurable? Why do some music pieces sound good and others do not, and why is this judgment subjective? And why do we have music to begin with? There also have been very targeted questions like how neurons in a certain area of the auditory cortex (the part of the brain responsible for processing auditory signals) become selective to music over time, even without musical training BNHMK21. These are big questions that cannot be answered scientifically alone. It requires the efforts of the interested scientist to guide the right experiments as well as those of a music mathematician to perform the right theoretical analysis. Furthermore, being able to tap into these questions, I believe, will help unravel some emergent musical phenomena. For instance, the intricate interaction between speed, harmony, and melody that jazz improvisation presents yearns for explorations from every tangential field – psychology, neuroscience, and anatomy. More engagement from mathematicians will help to develop a better and more complete framework of exploration.

The Music Community is Ready and Looks Up to Mathematicians

My conversations with musicians inform me that they are very aware of the connection between mathematics and music. Most of them express the desire to explore this connection and truly believe that they stand to gain much if they can only take a step in the right direction. Even though many of them do not have the mathematical expertise to meaningfully contribute to the mathematics of music, a number of them have contributed to the discussion of the mathematics-music interface in artistically substantial ways. For instance, in music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths (see Fig 3). A well-known story in the jazz community is that the famous saxophonist John Coltrane had a long-lasting fascination with the connection between mathematics and music. It is said that he constructed “Coltrane’s Circle of Tones” (see https://www.openculture.com/2017/04/the-tone-circle-john-coltrane-drew-to-illustrate-the-theory-behind-his-most-famous-compositions-1967.html) in an attempt to formulate an axiomatic connection between geometry and music. Many musicians believe that there is still a lot to uncover mathematically with the circle of fifths and other examples of musicians making efforts to interface mathematics and music abound.

Figure 3.

The circle of fifths.

Martin Scherzinger, an ethnomusicologist, has devoted a good amount of effort towards mathematically analyzing southern African dance music through the lens of Zimbabwean Matepe and Mbira music Sch17. In 2016, the Herbie Hancock Institute of Jazz began the first phase of Math, Science, and Music, an initiative that uses music as a tool to teach math and science to young people in public schools across the United States. Initiatives of this type, though non-technical, could make good use of mathematicians’ insight for success.

The NSF Needs to Get Involved

The National Science Foundation (NSF) could do a lot to amplify the mathematics of music. This can start with a long-term workshop that brings together a diverse group of experts in the areas of STEM that intersect with music. The overarching goal should be to address mathematical (and scientific) issues that are relevant to this research community and to expose possible areas of opportunity for multidisciplinary music-related research deeply rooted in mathematics and science. A workshop of this nature would be instrumental in identifying the most substantive research questions that can be addressed by researchers in the field of musical mathematics. It will also help recognize the community needs, knowledge gaps, and barriers to research progress in this area. Furthermore, it will help in identifying future directions that cut across disciplinary boundaries that are likely to lead to transformative multidisciplinary research in musical mathematics.

Conclusion

We do not need a survey or study to prove that exploring music is a meaningful and gratifying experience for anyone engaging in it. Even so, studies continue to show that music education enhances learning skills, creativity, teamwork, discipline, cultural awareness, and self-esteem DW94KWS20. Mathematics has a big void to fill in the quest to continue exploiting all the benefits of music to humankind.