Acoustics & Music Theory
why it sounds like that
Concept

Enharmonic — where it appears

The relation between two note names that fall on one key of a twelve-note keyboard, such as a major third and a diminished fourth. They are one pitch only where the chain of fifths closes; in meantone they are two pitches forty-one cents apart.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The chain of fifths in quarter-comma meantone. The fifths laid end to end as the chain they are. The bar under each shows how far that fifth departs from a pure three-to-two, and one of them — G♯ to D♯, the 12th link, where the chain is forced to close — is the wolf, at 35.7 cents.

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.

tuning · The comma The chain of fifths in quarter-comma meantone. The fifths laid end to end as the chain they are. The bar under each shows how far that fifth departs from a pure three-to-two, and one of them — G♯ to D♯, the 12th link, where the chain is forced to close — is the wolf, at 35.7 cents.

The same distance, under two names

Four hundred cents is a major third or a diminished fourth, and on a keyboard nothing in the sound distinguishes them. An earlier essay was about the boundary between two categories; this is about two categories at one acoustic value, and the surprise is where the ambiguity comes from. In quarter-comma meantone a major third is 386 cents and a diminished fourth is 427 — two names, two pitches, forty-one cents apart. Equal temperament collapsed them, and what a listener now supplies from context used to be in the sound.

intervals · Categorical-hearing The chain the page counts on. Every way of writing a pitch with up to one accidental, laid out along the chain of fifths and grouped by the pitch class each one sounds as. The vertical axis is what a keyboard has and the horizontal axis is what the page has: the spellings of one pitch class are seven steps of a fifth apart, which is exactly the distance a comma is measured over. Notation did not choose an arbitrary redundancy; it kept the coordinate a tuning system is built in, and equal temperament is the projection that loses it.

The stave is not a ruler

A hundred and eighty essays here draw pitch against an axis somebody computed. The one axis every reader already owns is the five lines, and it is not a pitch axis at all: it counts letters. Seven positions carry twelve pitches, so the same vertical distance is two intervals before an accidental is allowed and six after — and the accidental is not an extra symbol on a complete scale but the repair for a scale with five values missing.

scales · Notation Where the two spellings meet, and where they cross. G♯ minus A♭ against the fraction of a comma each fifth is narrowed by — twelve fifths against seven octaves, and nothing else in the calculation. In Pythagorean tuning G♯ is 23.46 cents ABOVE A♭; in quarter-comma meantone it is 41.06 cents BELOW it; the two spellings coincide at 0.09090 of a comma, which is what equal temperament is. A page that distinguishes the two names is exact in every tuning on this line except one point on it, and at that point it is wrong by 0.0014 cents rather than by nothing, because one eleventh is not quite the crossing.

Two names for one key

A keyboard has one key between G and A and the page has two names for it. That looks like redundancy and it is not: the two names are twelve fifths apart on a chain, and in every tuning anybody played before the nineteenth century they are two different pitches. The size of the difference is twelve fifths against seven octaves and nothing else — twenty-three cents one way in Pythagorean, forty-one the other way in meantone, and zero at exactly one point in between.

tuning · Notation What eleven positions cover, at seven to the octave and at twelve. A clef's 11 positions, read as a range, in three systems. At seven positions to the octave they cover 1.57 octaves — an octave and a fourth, which is the seventh rung's own number. At twelve they cover 0.92. Writing a 44-semitone range then takes 3 staves and about 15 ledger positions on the staff, and 4 staves and 33 on a chromatic one. The accidentals a chromatic staff removes are paid for in vertical space, at a rate the two integers fix.

The notations invented for the overflow

Every proposal to replace the staff since the seventeenth century is a response to a specific overflow, and the commonest one — a chromatic staff with twelve positions per octave instead of seven — makes a trade computable from the same two integers the clef essay counted. It buys the accidentals outright and pays 40 per cent of the range, three ledger positions for one, and twenty-three enharmonic distinctions that are not merely absent from the page but unrecoverable.

scales · Notation

Named alongside it

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

NotationChain of fifthsIntervalKey signatureLesser diesisMeantoneSplit keysBeatingCategorical perceptionClefDiatonic scaleKey colour

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