Pitch & Frequency

The ear hears ratios, not differences — the octave is a doubling, and harmony follows.

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The brief

Pluck a string, then pluck an identical string twice as long: the second sounds exactly one octave lower. Halving the length doubles the frequency, and the ear hears the two as the same note — a relation Pythagoras is said to have found around 530 BCE and the seed of Western music theory. The crucial part is that the ear hears ratios, not differences. 440 Hz and 880 Hz feel like one note an octave apart; 880 and 1320, the same gap in Hertz, feel like a fifth. The ear is logarithmic, and the simple whole-number ratios that fall out of that fact — 2:1, 3:2, 4:3 — sit under nearly every scale and chord ever sung.

Pitch and frequency are not the same thing. Frequency is physical, measured in Hertz; pitch is what the mind makes of it, and because perception is logarithmic it is the ratio between two frequencies, not their arithmetic distance, that fixes the interval we hear. This is why tuning is a genuine dilemma rather than a solved equation. The intervals the ear finds most consonant — the 3:2 fifth, the 5:4 major third — are the small whole-number ratios of just intonation, but a keyboard tuned to them plays beautifully in one key and sourly in the rest. Equal temperament, standard since the eighteenth century, divides the octave into twelve identical steps, each very slightly out of tune with those pure ratios; in exchange, every key sounds equally passable and a player can move freely among them. That is the bargain behind Bach's The Well-Tempered Clavier — a small, universal wrongness traded for the freedom of all twenty-four keys. Underneath the mathematics sits a piece of anatomy that explains why any of it is true. The cochlea, the coiled organ of the inner ear, sorts incoming sound by frequency along a membrane whose layout is itself logarithmic — high tones stimulate one end, low tones the other — so it performs, in flesh, something close to the frequency analysis a mathematician would do on paper. The same structure explains consonance: when two tones fall within a single sensing region their vibrations interfere and the ear registers roughness, while tones far enough apart, or locked in simple ratios that share overtones, ring smooth. The ancient discovery that consonance lives in small integers and its physical cause in the shape of the ear turn out to be one discovery seen from two ends.

Why nowNearly all recorded music now lives in equal temperament by default. The MIDI standard hard-codes its twelve tones into every digital workstation, Auto-Tune snaps stray singing to the nearest one, and AI music generators inherit the same grid because their training data is built on it — quietly making a centuries-old European compromise the tuning of the whole streaming world. The alternatives survive at the edges: microtonal composers, and the living traditions of Indian classical, Arabic maqam, and Indonesian gamelan, which never adopted the twelve-tone equal division and still hear intervals the piano cannot play. The logarithmic ear is physics; equal temperament is a choice.