Acoustics & Music Theory
why it sounds like that
Generator

Every keyboard temperament before 1800, on one line

The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number.
Every keyboard temperament before 1800, on one line. The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 13 of its 13 placements carry sound, built from the same numbers as the drawing, offering these buttons: A major triad, Pythagorean, A major triad, equal, A major triad, quarter-comma meantone, A tempered fifth at a quarter comma, Both, in equal temperament, G♯ and A♭ in Pythagorean, G♯ and A♭ in quarter-comma meantone, Major third, Pythagorean and 4 more.

Called by 2 essays

the blast radius of changing it

Every keyboard temperament before 1800, on one line. The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number.

A fraction of a comma

The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.

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

All figures · What can be heard