Concept

Circulating temperament — where it appears

A tuning whose chain of fifths closes, so that every one of the twelve keys can be played even where some are much worse than others. The unevenness is the point: it gives each key its own interval sizes and therefore its own character.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as well temperament — the same set of essays touches all of them, so they are one junction rather than several.

Key character in Werckmeister III. The twelve major keys in circle-of-fifths order, each with a bar as long as its major third is sharp of a pure 5:4. The bars run from 3.9 to 21.5 cents, so the keys genuinely differ.

Keys that had characters, and could be measured

Eighteenth-century writers described E flat major as devotional and F sharp major as harsh, and modern readers file it under synaesthesia. On the instruments those writers used, the difference between the two keys was seventeen cents of third, and that is a quantity.

tuning · The comma

The same total, distributed differently. The departure of each of the twelve major thirds from a pure 5:4, one row per temperament, keys ordered round the circle of fifths. The totals are equal — every closing temperament's twelve thirds add to 4800 cents — so what a temperament chooses is not how much error there is but which keys carry it.

What a temperament cannot do

Add up the twelve major thirds of any keyboard tuning whose chain of fifths closes and the answer is 4,800 cents. Every time, in every system, before a single fifth has been chosen. So no temperament has ever made the average third less wrong than any other one, and four hundred years of argument were about a distribution rather than about accuracy.

tuning · The comma

Named alongside it

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

Cents Well temperament Chain of fifths Equal temperament Key character Key colour Pythagorean comma Temperament

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