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← Why Twelve Notes · Lesson 3 of 4

Why twelve

About it

Two notes an octave apart vibrate in the ratio 2 : 1; a fifth is close to 3 : 2. Stack twelve fifths and you almost land seven octaves up — almost. (3/2)¹² is about 129.75, and 2⁷ is 128, a gap of about 1.4% called the Pythagorean comma.

Equal temperament closes the gap by making all twelve steps the same ratio, the twelfth root of two, about 1.0595. Every fifth comes out a hair narrow, and every key on the piano works.

The figure

Twelve fifths against seven octaves

twelve fifthsseven octaves(3/2)¹² ≈ 129.752⁷ = 128octavesfifthstops, 25× largerabout 1.4%
Two towers on one scale of pitch. The left is twelve fifths stacked, each multiplying the pitch by 3/2, which comes to about 129.75. The right is seven octaves, each doubling it, which comes to exactly 128. The left tower ends a little higher, about 1.4%: the Pythagorean comma, shown again magnified at the right.
One question

Stepping round the twelve keys by 5 semitones at a time, how many different keys do you visit before coming back to the start?

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