formalizes of Move 1 — The ear calls small ratios 'sweet.'
The Ratio and the Ear
Harmony begins as arithmetic and ends as compromise. A dialogue on why a perfectly tuned instrument cannot play in every key.
illustrates of Move 2 — One string already contains the chord; the ratios are physical, not chosen.
Sheet music: Overtone Series on C, key C, 3/4 time
challenges of Move 2 — Stack twelve pure fifths against seven octaves and they miss. The Pythagorean comma: the circle will not close.
interprets of Move 4 — You cannot have every interval pure at once.
So a keyboard tuned in pure fifths drifts steadily sharp, and somewhere one interval must absorb the whole error: the 'wolf,' howling and unplayable. Perfection in one key spoils another.
echoes of Move 4 — The soul counts; the comma is the remainder it cannot quite hide.
Music is the pleasure the human soul experiences from counting without being aware that it is counting.
Gottfried Wilhelm Leibniz (1712), Letter to Goldbach
formalizes of Move 5 — Equal temperament: smear the error evenly across all twelve semitones. None pure, none unbearable.
interprets of Move 7
Now every fifth is a hair flat and every major third noticeably sharp, and in exchange you may play in all twelve keys and modulate freely between them. We trade purity for the freedom to move.
applies of Move 8 — The free wandering through keys that the compromise buys.
Sheet music: Modulation, key C, 4/4 time
grounds of Move 7
complicates of Move 8 — Living players lean toward just intonation when they can.
Yet free a violinist of the frets and she drifts back toward the pure third; the ear still wants what the keyboard surrendered. Unaccompanied singers do the same.
formalizes of Move 11 — The sweeter third, and the small gap a trained ear hears and leans away from.
interprets of Move 11
So the played note is rarely the computed note. Vibrato, lean, and drift are the ear correcting the compromise in real time, key by key, phrase by phrase.
returns of Move 4 — Back to the comma: the remainder that cannot be spent, only moved.
The comma never disappears. It is only ever distributed: into a temperament, into a wolf, into the player's thousand tiny adjustments. The flaw is conserved; we only choose where to keep it.
illustrates of Move 14 — The arrival is sweeter for the tension the compromise leaves humming inside it.
Sheet music: Cadence Home, key C, 3/4 time
concludes of Move 1 — From the monochord's clean ratio to the living, slightly out-of-tune chord.
Harmony, then, is not the absence of dissonance but its skillful placement: arithmetic that agreed to bend a little, so that the music could move.
The dialogue is concluded
A perfectly tuned instrument can play one key flawlessly; we chose, instead, to play them all.