Acoustics & Music Theory
why it sounds like that
Concept

Split keys — where it appears

A keyboard whose black keys are divided front and back so that the two enharmonic spellings sound different pitches. They were built from the sixteenth century onward because in meantone G sharp and A flat are forty-one cents apart and one key cannot supply both.

Named by 2 essays across one field — 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 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

Named alongside it

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

EnharmonicBeatingChain of fifthsKey colourLesser diesisMeantoneNotationPythagorean commaSyntonic commaWolf interval

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