Soundbops in the Classroom: Music & Numeracy
In ancient Greece, music was revered as the study of patterns. Not only did they understand that numerical patterns occurred in music, but they saw the relationships between these numbers as a framework for understanding the composition of the universe! This is pretty existential, even for the ancient Greeks, who spent most of their time ruining good parties by talking about philosophy (we’ve all met one). So, what could these links between maths and music possibly tell us about ourselves and the world today?
Pythagoras (also famous for his obsession with triangles) experimented with an ancient instrument known as the monocord, which was like a flattened guitar with one string. As he plucked, he saw how sound is caused by vibrations, and that by shortening or lengthening the string - and affecting how quickly it vibrated - he could also affect the pitch.
Today we understand the speed of vibrations per second as hertz (Hz), and the higher the hertz the higher the perceived pitch. Pythagoras realised that certain pitches sounded pleasant together, while others were harsh and dissonant. To understand why, we’re going to need some numbers.
The harmonious pitches were produced when the string was split into simple fractions, like halving or thirding its length. For instance, if you play an open string resonating at around 220Hz - a low A -, holding it at the halfway point will double the speed of vibrations (440Hz) and would sound as the octave above the original note; with a ratio of 2:1. Meanwhile dissonances were produced when more complicated divisions of the string’s length were made, or by holding down lengths randomly without measuring.
This means if these notes were played together the vibrations would sync up every second wave (as above).
Combinations of notes that conform to other simple ratios - 3:2, 4:3, 5:4, etc. - are how the other consonant intervals found commonly in Western music are measured. Everything, from Beethoven to Bee Gees, is underpinned by these numerical ratios and our ear’s preference for simple patterns over complex ones.
One of the most dissonant intervals on the piano is that which is found between any two adjacent notes (a minor second), with a ratio of 11:10, meaning the vibrations only synchronise every 11 waves! Although this note combination is still used in lots of music to add tension, and is in no way a ‘wrong note’, it does show that there is a relatively low threshold on what patterns the ear can musically make sense of. This is also why the piano doesn’t need a key tuned to every Hertz, just frequencies that are simple enough for us to understand as patterns.
Let Music do the Maths
Now, if you’re reading this thinking: ‘What robot listens to music in this way?!,’ don’t worry this is obviously not meant to be descriptive of how we think when we hear these sounds! Our preference for patterns is a subconscious process, the purpose of which is still stumping neurologists today. However, although we might not understand why, we can explore how, and this is where it gets interesting!
Average human pitch perception ranges from 20Hz-20,000Hz. Anything higher than this limit is inaudible to us, and anything lower becomes too slow for us to hear it as continuous. Imagine an electric fan starting up and note how, as it gets faster, we begin to hear the rotation of the blades as a low hum. This is because the cycle is repeating more than 20 times a second - too fast for our brain to process the rotations individually - and enters our perception, instead, as a consistent pitch.
The range of pitch perception varies across different animal species, note that creatures that communicate with sonar have a much higher threshold
This means that rhythm and pitch are the same thing, just occurring at different speeds. Any consistent pulse can be transformed into a pitch if it occurs more than 20 times a second. In a way, what we experience as pitch is just the brain’s shorthand way of communicating that there are a lot of vibrations in the air - too many to count - thus we must rely on this innate, subconscious ability to distinguish between and feel numerical patterns!
Usually when dealing with notes the number of Hertz is in the hundreds or thousands, so it is a fascinating feat that the brain simplifies and identifies patterns according to these ratios, without any conscious effort. But that means this goes both ways, and that these musical patterns are also rhythmic.
A Number’s Game
A rhythm with two or more consistent and synchronous patterns is known as a polyrhythm. The easiest way to demonstrate this is with the octave (the 2:1 ratio from before). Try playing the following rhythm with your hands:
If it was possible to play this fast enough that there were 40 (right hand) and 20 (left hand) beats per second, we would hear them as pitches an octave apart. Let’s try a more complex one; start slow and try to keep the time of each beat consistent - maybe try a breath on the blank spaces to help you keep time:
This is a 3:2 ratio, which would become a perfect fifth when sped up (the distance between the first two notes of Twinkle Twinkle Little Star). Perfect fifths are the next most consonant interval after the octave, not just because they sound pleasant but because they are literally the next most simple ratio.
This last one is 4:3. Although at first it may seem complex (with so many blank spaces to keep track of) the beats are still synchronising every 12 pulses - the lowest number that they both have in common. As it becomes more comfortable to play, you may recognise this rhythm as a common component of modern EDM and dance music, and once it is fully sped up (into the realm of pitch perception) it sounds as a perfect fourth: the first two notes of Auld Lang Syne.
The numbers we choose to play with drastically affect the feeling of a rhythm and how we experience that moment in time. For instance, rhythms with 5s or 7s feel very alien compared to the familiar march of 2s and 4s, whereas 3s and 6s are bouncy and provide a lot of motion. Some are ludicrously difficult to feel a comfortable groove with, whereas others like 2:1, 3:2, and 4:3 are used all of the time in popular music today.
The threshold between pitch and rhythm is rarely crossed as it’s impossible to move your hands so quickly (although I don’t doubt kids will have a fun time trying), and instead this blog should prompt you to connect with a side of numbers we don’t often think of: that they can be felt, and that music and rhythm are the kinetic experience of numeracy.
Although the ancient Greeks may have had some questionable ideas about these ratios being a divine blueprint for the creation of the universe, they weren’t wrong about our ability to understand and perceive patterns far greater than what is immediately apparent to us. A few thousand years on and we still have so much to learn about ourselves, and our inherent connection to numbers.