Maths + music: patterns in play

The link between maths and music

Hello, music-loving friend,

Do you have a morning routine? On weekdays mine usually goes something like this:

• pat the cat

• make a cup of tea and piece of toast with butter and Vegemite (the amount of which is a subject of great debate amongst all Australians. I prefer a thin scrape, just saying)

• pat the cat

• write in my journal

• feed the cat

• exercise

• and so on

My routine is an excellent example of a sequence. It might vary a bit depending on the mood of the cat and other factors, but the bones of it remain the same. How you organise your grocery list, a recipe, and the route you take on your evening walk are other examples of sequences. And they become patterns when you repeat them.

In mathematics, a sequence and a pattern are related concepts, but they’re not exactly the same. A sequence is a set of numbers or objects arranged in a specific order according to some rule or pattern. A nice example is counting even numbers: 2, 4, 6, 8, 10 and so on. There is a pattern that determines the next number in the sequence. (For more on sequences go to Maths is Fun.)

A pattern refers to a regular and discernible form or arrangement of elements that repeats or has the potential to repeat. Patterns can exist in sequences, like the even numbers, but we can also see patterns in, among other things, shapes, colours, sounds, and behaviours.

So, while every sequence has a pattern (the rule governing its formation), not every pattern necessarily involves a sequence. However, when a pattern repeats regularly it can also be described as a sequence.

Phew!

The connection between maths and music lies in their shared language of structure and order. In maths numbers and shapes follow logical arrangements, while in music the notes do the same. There’s a symmetry, proportion, and repetition found in mathematical concepts that’s very similar to the rhythmic and melodic patterns found in music. They both rely on the repetition of certain elements to create coherent compositions. In maths this could be numerical sequences, geometric shapes, and algebraic relationships. In music it’s rhythmic motifs, melodic sequences, and harmonic progressions.

How music works: sequences and patterns, oh my!

Sequences are a fundamental ingredient in music. When a sequence repeats (a pattern!) our ‘ears’ take note and we start to recognise it each time we hear it. This plays a huge part in making a song or piece of music memorable. Sequences also contribute to the narrative arc of a piece or song helping to tell the story (for more on storytelling go to Tell me a story).

And it doesn’t matter if it’s a children’s nursery rhyme or something much more complex. Think of a song you love and how you respond to the repeated rhythm that runs through it like a backbone, or how you’re familiar with its sequence of verse, chorus, verse, which means you can anticipate that chorus you know is coming round for a second go.

Twinkle twinkle little star, how I wonder what you are

Let’s take a closer look at three examples of how maths and music overlap in the children’s song Twinkle Twinkle Little Star (in which sequences and patterns abound).

1. The song has a regular beat throughout. This is a regular pattern to which you can count 1, 2, 3, 4, over and over again. You can think of this mathematically as a consistent division of time. And you can try this yourself by singing or humming the song and clapping, tapping, or walking in time.

   

2. We can also count the intervals (distance) between the notes in a numerical form. For example, if we look at two notes with three notes in-between we can say the interval is a fifth: C(DEF)G, with only C and G being played, and C as 1 and G as 5.

3. The melody features a sequence of notes that repeats (a pattern) throughout the song. If you look closely at the two sequences of notes below (blue and pink) you’ll see that the notes are descending. (You don’t need to be able to read the music to see this. Just look at how each note appears a little bit lower than the one before). The two sequences are slightly different but similar enough that they form a definite pattern.

Here’s a recording of Twinkle Twinkle Little Star. I suggest you listen a couple of times, once while looking at the notes and how they move and once with your eyes shut to really hear the sequences and the overall pattern:

The same but different?

There are many other examples of maths and music teaming up. But what I’m really interested in is how our brains are able to recognise a song even when it’s been changed.

When we listen to music our brains process the sequences and patterns and form a mental (and emotional!) representation of a song. This is made up of not only the individual notes but also the overall structure and feel of the music.

Our brains hold on to this representation as a reference point and can use it for comparison when we hear a different version of the same song—for example, different style, speed, rhythm, melody, harmony—and still be able to identify the underlying structure and essence of the song. Cool, right?

And to (hopefully) prove my point, here’s a recording of me singing Twinkle Twinkle Little Star again but this time with an improvised (made up) melody:

So here’s to the great love match of mathematics and music. My very small students might roll their eyes when I tell them how much maths and music have in common, but I think it’s pretty amazing. What do you think?

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Thank you for taking the time to read this post. If there's anything else you're curious about or would like me to explore further, let me know! You can comment on this post or send me a message at katepainediscoveringmusic@substack.com

Thanks, Kate ❤️

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