Acoustics & Music Theory
why it sounds like that
Concept

Lesser diesis — where it appears

The gap of about 41 cents left when three pure major thirds fail to reach an octave, the ratio 128 to 125. It is the difference between G sharp and A flat in quarter-comma meantone, which is why keyboards of that period were built with split keys.

Named by 2 essays across 2 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 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 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.

Chain of fifthsEnharmonicMeantoneNotationCategorical perceptionIntervalPythagorean commaSplit keysSyntonic comma

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