The math behind music

A note is a number: how many times per second the air moves back and forth. Intervals, consonance, keys, why parallel fifths "fuse": every one of them is a relationship between those numbers. Most tools gloss over this. Stated plainly, you are hearing arithmetic, and once you follow the arithmetic far enough it does something wonderful: it breaks, and music has to compromise to keep going.

Watch two notes add up

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

• 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, which is why parallel octaves collapse two melodies into one. Mathematically, they nearly are one.
• Perfect fifth (3:2): three cycles against two. 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 is beating: the difference between the two frequencies, made audible. Hold onto beating. It comes back as the villain of this whole story.

Consonance is not a matter of taste that theory later rationalized. It is small-whole-number arithmetic your ears do for free, hundreds of times a second. Dissonance is just harder math.

Why twelve, and why they don't quite fit

Here is the fact that everything downstream depends on, and that almost no beginner is ever shown. Start on any note and go up a perfect fifth, the strong 3:2. Do it again, and again, twelve times. You pass through all twelve notes and land back on a version of where you started, seven octaves higher. That is the circle of fifths, and it is why Western music has exactly twelve notes: twelve fifths bring you home.

Except they don't. Multiply the ratios and check. Twelve fifths is (3/2) multiplied by itself twelve times, which is about 129.75. Seven octaves is 2 multiplied by itself seven times, which is exactly 128. Those are not the same number. Twelve pure fifths overshoot seven octaves by a ratio of about 129.75 to 128, a gap of roughly 23 cents, just under a quarter of a semitone. The circle of fifths, the neat wheel the whole key system rides on, does not actually close.

This gap has a name, the Pythagorean comma, and it is not a rounding error or a flaw in anyone's instrument. It is a permanent, provable fact about the numbers 2 and 3: no stack of pure fifths will ever land exactly on a pure octave, because no power of 3/2 will ever equal a power of 2. The universe simply did not arrange for music's two most basic intervals to agree.

The compromise everyone agreed to

So music cheats, and the cheat is so total that you have never once heard a piano play a pure fifth. The modern fix is equal temperament: forget the pure ratios, and instead chop the octave into twelve identical steps. Each step is the number that, multiplied by itself twelve times, gives exactly 2: the twelfth root of 2, about 1.05946. Every semitone on every piano is that one irrational number.

Doing this smears the comma evenly across all twelve fifths. Each tempered fifth is about 2 cents flat of pure, small enough that almost nobody notices, and twelve of those tiny shortfalls add up to precisely the 23-cent comma, now paid off a little at a time instead of dumped in one ugly lump. The trade is exact and it is the deal underneath all Western music: give up being perfectly in tune in any single key, and in return be equally, only slightly, out of tune in all of them. Before this deal, an instrument tuned pure in C was unusable in F sharp. After it, one tuning plays everything, which is the entire reason a piano can exist.

You can hear the price. The tempered major third pays the most: it lands about 14 cents sharp of the pure 5:4, enough that its overtones grind against the root's and produce exactly the beating from the first demo:

The just chord sits perfectly still. The equal-tempered chord shimmers, about four beats a second, and that shimmer is the comma, spread thin and paid in full on every major chord your favorite records are built from. Once you have heard it you cannot unhear it, and you also understand why it is worth it.

Where this is going

This page is the front door to the math trail, and the trail runs deep. The idea that pitch itself is just rhythm too fast to count is Organizing time. Still ahead: why a single played note is secretly a stack of many (harmonics, and why timbre is a recipe), sine waves as circular motion, and, for the Max and gen~ crowd, what those patches are actually computing when an LFO climbs into audio rate. It is generative calculus wearing a hoodie. I mean, c'mon.