Maths and music, together in harmony

4 January 2018

There’s a deep and well-established link between music and mathematics. For example, both deal with frequencies, waves, and ratios – even if they use different terminology. Yet, when these subjects are taught at school, the connection is usually completely severed. However, reuniting the two can only increase the enjoyment students get from learning mathematics, while also enriching and deepening all aspects of their music-making, from casual singing to composing their own pieces. Here, we take a look at some of the connections to motivate the teaching of the two subjects together, in perfect harmony.

Thinking rationally

It takes mathematical reasoning to make even the simplest instrument – be it considering surface shapes and areas in the design, the spacing between finger holes, or the various modes of string vibration. A classic example of this is the monochord – a one-stringed instrument whose invention is attributed to Pythagoras, but certainly goes back to when humans first started experimenting with music:

In classical times, musical scales were determined by simple integer ratios of string lengths.

The difference in pitch between plucking an open string and plucking a string stopped at the halfway point is exactly one octave (low “do” to high “do”). Stopping the string at 1/3 yields “sol”, introducing the musical interval of a fifth. These values are relative, and so can be compounded: to get the “sol” an octave higher, one multiplies: $1/3 \times 1/2 = 1/6$ and plucks the string in that ratio. Equally, multiplying by the inverse yields “sol” an octave below: $1/3 \times 2/1 = 2/3$. These operations are the basis of what is known as the Pythagorean tuning, and are at the core of Pythagoreanism — an ancient Greek school of thought where musical and mathematical principles had mystical importance. Curiously, it has been shown that the form of Plato’s dialogues, when counted line by line, is organised around such musical/mathematical ratios, making the case for Plato secretly being a Pythagorean.

And, talking of organising text in a mathematical way, Fibonacci numbers were employed to this end long before Fibonacci himself famously used it to count rabbit-population growth. In Sanskrit prosody, there are long ($L$) and short ($S$) syllables, and a single long one is worth two short ones. Therefore, enumerating possible combinations based on total length, we have:

Length 1: $S$ (1 possibility)
Length 2: $SS$, $L$ (2 possibilities)
Length 3: $SSS$, $SL$, $LS$ (3 possibilities)
Length 4: $SSSS$, $SSL$, $SLS$, $LSS$, $LL$ (5 possibilities)
Length 5: $SSSSS$, $SSSL$, $SSLS$, $SLSS$, $LSSS$, $SLL$, $LSL$, $LLS$ (8 possibilities)

And so on. Such enumeration was mentioned in Indian writings from at least as early as the 12th century AD.

Got rhythm

This brings us to another extremely prominent link between music and mathematics – that is, the mathematical properties of rhythm. Godfried Toussaint, a professor at McGill University, Canada, has shown how it is possible to obtain the rhythms of traditional music from around the world by employing the famous Euclidean division algorithm. For example, the bembé, a common West African bell pattern, can be created by distributing 7 beats over 12 slots (that is, by “dividing” 12 by 7). We will represent the beats with 1, and the empty slots (“silences”) with 0:
$$
[1 1 1 1 1 1 1 0 0 0 0 0]
$$ This is equivalent to saying that $12 = (7 \times 1) + 5$. We now distribute the empty slots over the beats, obtaining:
$$
[1 0] [1 0] [1 0] [1 0] [1 0] [1] [1]
$$ The shorter groups (the two singe beats at the end) becomes the new remainder, thus: $7 = (5 \times 1) + 2$. This remainder is now distributed among the larger groups, and repeat…
$$
[1 0 1] [1 0 1] [1 0] [1 0] [1 0] \\
[1 0 1 1 0] [1 0 1 1 0] [1 0] \\
[1 0 1 1 0 1 0] [1 0 1 1 0] \\
[1 0 1 1 0 1 0 1 0 1 1 0] \\
$$ This pattern is then repeated, and so it makes sense to visualise it in a circle:

The bembé beat; ring the bell every time you reach a black circle (or a 1 in the patterns above).

Here, the black dots represent the 1s, and the white dots represent the 0s. The pattern has several interesting mathematical properties. To begin with, the 7 beats are distributed as uniformly as possible over the 12 slots. At the same time, the pattern has no rotational symmetry. This means that every rotation is unique and can be instantly identified by ear, maximising the richness of musical choices and ensuring no repetition. Therefore, it is likely no coincidence that the very same pattern represents exactly the “tone-tone-semitone” spacing of the Western major scale, in which black dots represent the white keys on the piano, and white dots represent the black keys. Even more curiously, this is also exactly the pattern of the distribution of long and short months – 31 and 30-28 days – in the calendar year.

Let us stick with the 12-slots-in-a-circle representation for a while. If we colour every fourth dot and connect them, the result is an equilateral triangle (green below). If we execute this pattern as beats, we get a steady 3-beat cycle. If we map the same pattern onto the piano keyboard and play the notes, we hear the characteristic sound of an augmented triad – listen to the opening chord of “Oh! Darling” by The Beatles, for example. If we colour every third dot, we get a square (blue):

If we were to listen to both patterns simultaneously, as a pattern of beats and silences, we would hear a “3 over 4” polyrhythm. Knowing that pitch is frequency (for example, the note A is tuned at 440Hz, so vibrates at 440 cycles per second), and thinking back to the aforementioned monochord, we can deduce that this very same representation also describes musical pitch relationships (the relationship between the vibration of an open string and a string stopped at 3/4 of its length).

The beat goes on

So far, we have been focusing mainly on fractions, sequences, and –- to an extent -– geometry, but there is much more of an overlap between music and mathematics. There are connections to be found in group theory, probability, and even between tuning systems and the Riemann zeta function. Indeed, so many are the links that it would be impossible to make a comprehensive list. Every single one of these parallels is an opportunity to present to pupils something as seemingly abstract and inaccessible as mathematics in musical terms, making it applied, audible and emotionally charged.

By making the study of mathematics more musical, education in both fields becomes more inspiring, creative and grounded. Drawing on our experience in music, mathematics and education, we founded Harmonite – a platform for teaching music through maths, via a series of online games. Harmonite is currently in its fundraising stage. The funding for the minimum viable product is being sought via a Kickstarter campaign ending on 11 January 2018. After the campaign has run its course, we will reach out to seed investors to secure further funds. The final platform will consist of several modules that will help students learn about, for instance, fractions, graph manipulation or trigonometry by exploring how they overlap with music theory. As a result, they will not only deepen their understanding of mathematics, but also gain practical musical skills, such as sight-reading rhythms, or recognising chords by ear. Harmonite will be marketed both to secondary school heads of music and maths, as well as to individual clients.

With funding for the arts being cut, and a shortage of creative approaches to the maths curriculum resulting in students rote-learning for tests and not enjoying the process, making maths more musical kills two birds with one stone: it makes room for more music in school, and offers a fresh perspective on maths. And that’s what we call added value.

https://www.kickstarter.com/projects/1186413098/harmonite

Ally Dowson & Matt Burdynowski

Ally and Matt met each other while playing in the same street percussion band. Matt has been working as a choir leader, arranger, singer and educator for 15 years, and has an interest in mathematics – especially in how it applies to music. Ally has 10 years of experience teaching mathematics and science at a secondary level, and also pursues music as a hobby. For a period of time, they both worked at the same secondary school. With Matt having studied in Poland, and Ally in Australia, they are very excited about bringing their enthusiasm for learning to the UK, and eventually worldwide.