Acoustics & Music Theory
why it sounds like that
Concept

Wolf interval — where it appears

The interval left over where a chain of tempered fifths fails to close, far out of tune because it takes up all the error the others have shed. It is what a meantone keyboard pays for its pure thirds and what a split key moves to a place where it is less often needed.

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 A meantone keyboard is a stretch of the chain of fifths, and every split key lengthens it by one place at an end. 14 keys hold the positions from A♭ to D♯. That is 10 major and 10 minor triads; major keys E♭, B♭, F, C, G, D, A, E; minor keys with a raised leading note C, G, D, A, E.

A split key buys one of everything, and two buy C and E

An organ builder with money for two extra keys could have put them anywhere, and the arithmetic of triads cannot say where. On a meantone keyboard every added key buys exactly one major triad, one minor triad, one major key and one minor key, wherever it goes. So the count is no guide, and the choice was about which keys. The two accidentals most often divided, G♯/A♭ and D♯/E♭, are one on each side of the chain, and they are exactly what completes the minor keys on C and E. What they buy by the roughness sum is seven per cent.

tuning · Split keys

Named alongside it

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

EnharmonicSplit keysBeatingKey colourMeantoneTemperament

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