The math behind music

A note is a number: how many times per second the air moves back and forth. Every idea in music theory including intervals, consonance, keys, and why parallel fifths "fuse" is a relationship between those numbers. Tools usually gloss over this. Stated plainly: you are hearing arithmetic.

Watch two notes add up

Below are two sound waves and their sum as well as what actually reaches your ear when both notes play at once. Try each interval and watch the sum (the dark line):

What you just saw

• Octave (2:1): the top wave fits exactly twice inside the bottom one, so the sum repeats in perfect lockstep. Your ear hears the two notes as almost the same thing. That's why parallel octaves collapse two melodies into one. Mathematically, they nearly are one.
• Perfect fifth (3:2): three cycles against two. This is still a tiny whole number ratio, so the sum settles into a short repeating pattern: consonant, stable, "empty" enough to fuse. Hence parallel fifths.
• Slightly detuned: the ratio is almost 1:1 but not quite, so the waves drift in and out of agreement. The slow swelling you see (and would hear) is beating: the difference between the two frequencies, made audible.

Consonance isn't a matter of taste that theory later rationalized, it is small whole number arithmetic your ears do for free. They do that shit hundreds of times a second too. Dissonance is just harder math.

Where this is going

This page is the first in a series of prototypes that if fleshed out will go into detail about sine waves as circular motion, harmonics and why timbre is a recipe, equal temperament as compromise with a 12th root in it, and for me (and the other max msp people), what those patches are actually doing. It's generative fucking calculus. I mean, c'mon.