Acoustics & Music Theory
why it sounds like that
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Small whole numbers

Consonance is frequency ratios of small integers, and it has been known since Pythagoras. Almost everything interesting follows from those integers refusing to line up.
CC♯DE♭EFF♯GA♭AB♭Bstartseven octavestwelve fifthsthey miss by 23.46 centsone turn is one octave · angle is pitch class · radius is how far it has climbed Pitch and tuning

Twelve fifths and seven octaves, which are not the same thing

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.

 024681012-20-1001020semitones above the tonicCC♯DE♭EFF♯GA♭AB♭BCjust intonationPythagoreanquarter-comma meantonesharp of equalflat of equal Pitch and tuning

Where to hide the comma, which is the only real question

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.

 E♭B♭FCGDAEBF♯C♯G♯D♯-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4+35.7each fifth's departure from a pure 3:2, in centsone fifth carries the whole error Pitch and tuning

The wolf at the end of the chain

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.

 one frame = one repeat of the combined wavelower — 2 per frameupper — 3 per frametheir sumthe pattern repeats after 2 cycles of the lower tone Intervals and chords

Two notes and a ratio, which is the whole of consonance

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.

 00.511.52seconds3 beats per second — the difference, exactlythe carrier is drawn slower than it sounds, or it would be a solid band Intervals and chords

Beats are arithmetic that anybody can hear

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.

 semitones above the lower tone6/55/44/33/25/32/1CC♯DE♭EFF♯GA♭AB♭BCroughest at about a semitonesmooth at the simple ratios Intervals and chords

Roughness can be computed, and the answer looks like a scale

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.

 CC♯DE♭EFF♯GA♭AB♭B2 · 2 · 1 · 2 · 2 · 2 · 1steps, in semitonesthe short chords are the semitones — two of them, unevenly spaced Scales and modes

Seven of the twelve, chosen unevenly

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.

 C1/1E5/4G3/243major0–4–7C1/1E♭6/5G3/234minor0–3–7C1/1E♭6/5F♯33diminished0–3–6C1/1E5/4A♭8/544augmented0–4–8semitones above the root Harmony and voice leading

Three notes at once, and why these three

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.

 2C majorA minortotal 2 semitones12C majorF majortotal 3 semitones231C majorF♯ majortotal 6 semitonesthe shortest change is the one nobody has to sing far Harmony and voice leading

The shortest move, which is what a chord change is

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.

 E(2,4)the simplest divisionE(3,8)tresillo — Cuba, and half the world's pop musicE(5,8)cinquilloE(4,9)Turkish aksakE(5,12)West African bell patternE(7,16)Brazilian necklaceone cell per step · filled cells are struck Rhythm and metre

As evenly as possible, which turns out to be a famous rhythm

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.

 32they coincide every 6 subdivisionscommon grid: 6 equal steps Rhythm and metre

Two clocks at once, and where they agree

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.

 1130.8 HzC2261.6 HzC3392.4 HzG4523.3 HzC5654.1 HzE -14¢6784.9 HzG7915.7 HzB♭ -31¢81046.5 HzCpartialthe dots are the nodes — the places that do not move Timbre and acoustics

A string does everything at once

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.

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