Generator

Which traditions make the requirements that produce a comma

Four requirements and five tuning traditions. A comma is what is left over when a scale is generated by fifths, the octave is kept pure, and the chain has to close so that any degree can be a tonic — so a tradition that declines any one of them has no comma to hide. The best fifth each system actually contains is printed beside it, because declining a requirement has its own price.

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. 6 of its 6 placements carry sound, built from the same numbers as the drawing, offering these buttons: A major triad with both intervals pure: the third five to four, the fifth three to two, The same triad in equal temperament, where the third is 14 cents sharp and beats.

Called by 1 essay

the blast radius of changing it

Where the chain was never closed

Eleven earlier essays treat the comma as a fact about music. It is a fact about three requirements — that the scale is built from fifths, that the octave stays pure, and that the chain closes so any degree can be a tonic — and most of the world's tuning traditions decline at least one of them. What they pay instead is computable, and it is paid somewhere else entirely.

scales · The comma

All figures · What can be heard