Acoustics & Music Theory
why it sounds like that
Concept

Moment of symmetry — where it appears

A cardinality at which a chain of one interval, folded into an octave, has exactly two step sizes. For the fifth those cardinalities are 2, 3, 5, 7, 12, 17 and 29, which are also the convergents of the fifth's own logarithm.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The sizes a chain of a pure fifth will make. Chains of a pure fifth of two to 12 notes, each folded into one octave and drawn on a 1200-cent line. A chain has exactly two step sizes at 2, 3, 5, 7, 12 notes and three or more at every other size in this range. The five-note and seven-note cases are the chain's pentatonic and its diatonic set, and they are neighbours in a series rather than two separate facts.

The only sizes a fifth will make

Stack pure fifths and fold them into an octave, and at almost every number of notes the result has three or more different step sizes. At 2, 3, 5, 7, 12, 17 and 29 it has exactly two — and at no other size below thirty. The pentatonic and the diatonic set are not two discoveries. They are neighbours in one series, and it is the comma's series.

scales · The diatonic set The same census, in every universe from four to thirty. How many sets have all four properties, in a universe of n equal steps. None at all when n is two more than a multiple of four — 6, 10, 14, 18, 22, 26, 30 — exactly one when n is a multiple of four, and exactly two when n is odd. The multiples of four each have their survivor at n/2 + 1 notes generated by n/2 − 1 steps, so twelve's seven notes generated by the fifth is the general answer with n put at twelve rather than a fact about twelve.

Every universe has one, or none

Run the census that found the diatonic set in a universe of nineteen equal steps, or twenty-four, or fifty-three. The answer is completely regular and nobody appears to have written it down — none at all when the universe is two more than a multiple of four, exactly one when it is a multiple of four, and exactly two when it is odd.

tuning · The diatonic set

Named alongside it

The objects these essays reach for when they reach for this one.

Diatonic scaleWell-formednessChain of fifthsContinued fractionEqual divisionInterval contentMaximal evennessMicrotonalityPentatonicStep pattern

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