diesis

A diesis is a comma type of musical interval, usually meaning the difference between an octave (in the ratio 2:1) and three justly tuned major thirds (tuned in the frequency ratio 5:4), equal to 128:125 or about 41.06 cents. In 12-tone equal temperament (on a piano for example) three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short.

In quarter-comma meantone tuning, the diesis retains its value of 128:125. In other meantone tunings the defect from an octave, whatever it is, from three major thirds is the diminished second, sometimes also called a diesis. It is then the interval involved between pairs of notes which are enharmonic in equal temperament; for instance the interval between E and F♭, however large that is, is a diesis.

The word diesis has also been used to describe a large number of intervals, of varying sizes, but typically around 50 cents. Philolaus used it to describe the interval now usually called a limma, that of a justly tuned perfect fourth (4:3) minus two whole tones (9:8), equal to 256:243, or around 90 cents. Other theorists have used it for various other intervals.

Some acoustics texts use the term greater diesis for the difference between four 6:5 minor thirds and an octave, which is equal to three syntonic commas minus a schisma, equal to 648:625 or about 62.57 cents (virtually one 63.16-cent division in 19 equal temperament). Being larger, this comma was termed "greater" while the 128:125 comma was termed "lesser". ^[1]

	Diesis (128:125) demonstration First an octave, C to C, is played (2:1). Then three major thirds (5:4): C, E, G♯, B♯. The 2:1 octave C and the 125:64 B♯ are then compared. The difference, 128:125, is the diesis comma.
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See also

• Septimal diesis
• Ditonus

References