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Music · the physics of intervals

Twenty-three hundredths

The piano in your living room wasn't tuned wrong. It was tuned to a three-hundred-year-old compromise, so that all twelve musical keys could share a single keyboard — and of everything it plays, only the octave comes out clean.

Pluck a string and stop it exactly at the middle: the pitch jumps to double what it was, whatever the string, the instrument, or the century. It's the one musical interval that math settles perfectly, with nothing left over — an octave is always the ratio 2 to 1. The other intervals we hear as "in tune" — the fifth, the major third — are tied to simple fractions too: 3 to 2, and 5 to 4. Build a keyboard instrument from nothing but those fractions, though, and the math stops coming out perfectly a second time.

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The comma

Climb from a note, fifth by fifth, twelve times in a row — C, G, D, A, and on around the entire circle of keys — and you land back on a C, seven octaves higher. If every fifth stayed perfectly pure, at the ratio 3 to 2, the twelve stacked fifths should land exactly on those seven octaves. They don't. The stack overshoots seven octaves by a fixed gap, known for more than five centuries as the Pythagorean comma: 23.46 hundredths of a semitone. Musicians divide each semitone, the smallest step on a piano, into a hundred equal parts called cents; here, simply, hundredths.

The comma has to be hidden somewhere, and tuners of the past had to choose where. Meantone temperament, widely used from the sixteenth through the eighteenth century, kept major thirds close to pure and dumped the entire comma into a single fifth, usually the one between G-sharp and E-flat — a fifth so far off that tuners called it the wolf, and composers simply avoided the keys that passed through it. Equal temperament, which gained ground gradually across the nineteenth century and has since become the standard for pianos and guitars alike, chose the other path: it sliced the comma into twelve identical pieces and shaved exactly one twelfth of it — 1.955 hundredths — off every single fifth. The wolf disappeared. In exchange, every fifth on a piano is left slightly, but systematically, narrower than the fifth your ear actually asks for.

Pick an interval and press play. The first tone is tuned pure, the second is tuned exactly as it would be on a piano — the gap between them produces the beating you hear.

pure piano
pure (3:2 / 5:4) piano (equal temperament)

Difference: 1.96 hundredths of a semitone

0.74 beats per second · once every 1.34 s

23.46hundredths of a semitone — the Pythagorean comma: how far twelve pure fifths overshoot the seven octaves they should exactly fill
1.955hundredths — how much equal temperament shaves off every piano fifth, to split the comma evenly and erase the wolf
13.69hundredths — how sharp the piano's major third ends up instead, measured against the pure third the ear actually asks for

The ear feels the difference even when it can't name it. On a piano, a fifth beats just once every 1.34 seconds — rarely enough that it registers only as a faint restlessness in the sound, not a flaw. A major third beats 4.37 times a second, a flutter most listeners recognize without knowing why it sounds slightly "off," like a radio caught between two stations.

Voices and fretless, fingered strings — the violin, viola, cello — can choose their pitch note by note, free of keys fixed once and for all. A vocal quartet whose chords ring clearer than the same chords on a piano isn't singing more accurately by accident: the singers are instinctively pulling their thirds toward the pure 5-to-4 ratio, something no piano, locked to a fixed tuning for over a century, is able to do.

The figures above are computed directly from the standard twelve-tone equal-temperament formula — 2 to the power of 1 over 12, for each semitone — and from the pure interval ratios, 3 to 2 and 5 to 4. The tones in the instrument are generated live, at those same frequencies, from an A of 440 hertz, today's standard orchestral pitch.