Acoustics & Music Theory — why it sounds like that An illustrated collection of essays about why music sounds the way it does — tuning, intervals, harmony, rhythm and timbre — with every figure computed and every sound synthesised from the same parameters. https://www.acoustics-music-theory.com/ 2026-08-03T18:48:22.626Z Twelve fifths and seven octaves, which are not the same thing https://www.acoustics-music-theory.com/essays/twelve-fifths-and-seven-octaves/ 2026-08-03T18:48:22.626Z Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that. Where to hide the comma, which is the only real question https://www.acoustics-music-theory.com/essays/where-to-hide-the-comma/ 2026-08-03T18:48:22.626Z Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written. The wolf at the end of the chain https://www.acoustics-music-theory.com/essays/the-wolf-at-the-end-of-the-chain/ 2026-08-03T18:48:22.626Z Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose. Two notes and a ratio, which is the whole of consonance https://www.acoustics-music-theory.com/essays/two-notes-and-a-ratio/ 2026-08-03T18:48:22.626Z Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation. Beats are arithmetic that anybody can hear https://www.acoustics-music-theory.com/essays/beats-are-arithmetic/ 2026-08-03T18:48:22.626Z Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned. Roughness can be computed, and the answer looks like a scale https://www.acoustics-music-theory.com/essays/roughness-can-be-computed/ 2026-08-03T18:48:22.626Z Feed two spectra into a model of how nearby frequencies interfere, sweep one past the other, and the curve that comes out has dips exactly where the consonant intervals are. Nobody put them there. Seven of the twelve, chosen unevenly https://www.acoustics-music-theory.com/essays/seven-of-the-twelve/ 2026-08-03T18:48:22.626Z A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable. The same seven, started later https://www.acoustics-music-theory.com/essays/the-same-seven-started-later/ 2026-08-03T18:48:22.626Z A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to. Keys are neighbours, and the map is computed https://www.acoustics-music-theory.com/essays/keys-are-neighbours/ 2026-08-03T18:48:22.626Z Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away. Three notes at once, and why these three https://www.acoustics-music-theory.com/essays/three-notes-at-once/ 2026-08-03T18:48:22.626Z A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple. The shortest move, which is what a chord change is https://www.acoustics-music-theory.com/essays/the-shortest-move/ 2026-08-03T18:48:22.626Z Between any two chords there is an assignment of voices that moves the least. Compute it and the changes that composers actually use turn out to be the short ones. A progression is a path, and the map can be drawn https://www.acoustics-music-theory.com/essays/a-progression-is-a-path/ 2026-08-03T18:48:22.626Z Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering. Rhythm is a circle, and the bar line is a choice https://www.acoustics-music-theory.com/essays/rhythm-is-a-circle/ 2026-08-03T18:48:22.626Z Draw a rhythm as a line and it looks like a sequence of decisions. Draw it as a cycle and the same pattern turns out to be shared across continents, differing only in where somebody decided to start counting. As evenly as possible, which turns out to be a famous rhythm https://www.acoustics-music-theory.com/essays/as-evenly-as-possible/ 2026-08-03T18:48:22.626Z Ask an algorithm to space five strikes over eight beats as evenly as it can. It produces the cinquillo. Ask for three over eight and it produces the tresillo. Nobody told it about Cuba. Two clocks at once, and where they agree https://www.acoustics-music-theory.com/essays/two-clocks-at-once/ 2026-08-03T18:48:22.626Z Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number. A string does everything at once https://www.acoustics-music-theory.com/essays/a-string-does-everything-at-once/ 2026-08-03T18:48:22.626Z A plucked string does not vibrate at one frequency. It vibrates at all the whole-number multiples of one frequency simultaneously, and nearly everything in music theory is downstream of that fact. The ear hears the list, not the shape https://www.acoustics-music-theory.com/essays/the-ear-hears-the-list/ 2026-08-03T18:48:22.626Z Two sounds with the same partials and different phases have completely different waveforms and sound identical. What the ear extracts is a list of frequencies and strengths, and everything else is discarded. The shape of a note, which is most of what an instrument is https://www.acoustics-music-theory.com/essays/the-shape-of-a-note/ 2026-08-03T18:48:22.626Z Cut the first fifty milliseconds off a recorded piano and listeners stop calling it a piano. The attack carries more identity than the steady tone it leads into, and it is the part every spectrum plot leaves out. The room is part of the instrument https://www.acoustics-music-theory.com/essays/the-room-is-part-of-the-instrument/ 2026-08-03T18:48:22.626Z A room has frequencies it supports and frequencies it will not. In a small one those frequencies are far apart, so some bass notes are loud in one corner and absent in another — and no equipment fixes it.