Comma and diesis

"Comma" and "diesis" are two terms used to refer to intervals that are less than about 60 cents in size.

"Comma" also refers to an interval that is tempered out by any given temperament, but that is not the sense of "comma" that this article covers.

Generally, "comma" refers to an interval flatter than about 30 cents, and "diesis" refers to an interval between about 30 and 60 cents.

In just intonation

In just intonation, commas are often seen as the difference between two similar intervals, so it is hard to find intervals within this range that are treated as steps in their own right. The 3-limit interval in this range is the Pythagorean comma of 531441/524288, which can be considered an augmented seventh (octave-reduced), and is about 23 cents.

For the remainder of this list, I have tried to choose intervals that are not mostly treated as commas (in the temperament sense). Higher-limit intervals in the comma and diesis range are:

• The 5-limit augmented diesis is a ratio of 128/125, and is about 41 cents.
  • There is also the 5-limit magic comma of 3125/3072, which is about 30 cents.
• The 7-limit slendro diesis is a ratio of 49/48, and is about 36 cents.
• The 11-limit quarter tone is a ratio of 33/32, and is about 53 cents.
• The 13-limit minor diesis is a ratio of 40/39, and is about 43 cents.

In EDOs

The following table lists the best tuning of 128/125, and other dieses or commas if present, in various significant EDOs. Not included are EDOs (i.e. those smaller than 15) where the best tuning is the unison, or 0c, or those where the best tuning is sharper than 60 cents (i.e. not a diesis or comma).

EDO	128/125	Other commas and dieses
22	54c
24	50c
25	48c
26	46c
27	44c
29	41c
31	39c
34	35c
41	29c	59c ≈ 33/32
53	45c	22c ≈ 81/80

In regular temperaments

The role of commas and dieses in regular temperaments is often as the intervals that are tempered out (i.e. equated to 0 cents). Discussing that is not within the scope of this article; you may learn more at Temperament.

However, there are, rarely, temperaments generated by commas. One example is slender, where ten 49/48s equal 5/4.