Rhythm

The most simple connection between music and math is rhythm. To first read music, you have to first understand how to read rhythm. Any piece of music has a beat, which is the basic unit of time, the pulse, of the mensural level. Its units are in beats per minute. 60 beats per minute is approximately the stepping rate of a walking human. On the bottom, you can see 5 measures of notes, each with different lengths. The key is in a 4, 4 time signature, meaning that every measure gets 4 beats and the quarter note gets the beat. The whole note to the way left gets 4 beats, the 2 half notes get 2 beats each, the 4 quarter notes get 1 beat each, the 8 eighth each get 1/2 of a beat, and the 16 sixteenth notes each get 1/4 of the beat and so on. These rhythms are essentially all fractions, and with a combination of these notes, you can create endless amounts of unique rhythms. Using a little combinatorics, a 4,4 measure can have 5272 (have no idea if this is actually correct, you literally got it from a reddit post) different rhythms using only whole notes to sixteenth notes.

There are also other rhythms like triplets, which divides a beat into thirds like the rhythm shown on the left. Trickier rhythms exist such as the one on the right. The rhythm on the right signifies that 3 notes take up 2 beats, which means that each note takes 2/3 of a beat.

To further demonstrate that rhythm is all about fractions, to the left there are 3 notes, which all contain smaller notes within them. The top most one is called dotted half note, which essentially means that it is a half note with an added half of a half note which is a quarter note, meaning that a dotted half note is worth 2+1 or 3 beats. The middle note is called a double-dotted half note, which is just adding half of a quarter note to the dotted half note, or an eighth note more than a just a dotted half note, meaning that a double-dotted half note is worth 3 and 1/2 beats. Following this logic, the last note which is a triple-dotted half note should be worth 2+1+1/2+1/4 beats or 3 and 3/4 beats.

Polyrhythms

Polyrhythms are even trickier rhythms. They are essentially the simultaneous use of two or more rhythms that are not necessarily derived from one another. Polyrhythms are frequently used in jazz music. The simplest example is a playing triplets and eighth notes at the same time. A visualization of polyrhythms is on the left. Geometrical representations of polyrhythms can look very complicated, as you can see by the image second image. You can read more about polyrhythms in the paper below.

ExploringPerfectlyBalancedRhythmsAndScalesPrePrint.pdf