Comma and diesis

This article is about "comma" and "diesis" as interval regions. For other senses of these two words, see comma and diesis.

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

"Comma" also refers to an interval that is tempered out by any given temperament.

In terms of interval regions, "comma" refers to an interval flatter than about 30 ¢, and "diesis" refers to an interval between about 30 and 60 ¢.

The range of dieses largely overlaps with the range of quarter tones (between 40 and 60 ¢, reasonably mapped to 1/24edo), which, according to systems that determine consonance in terms of proximity to simple just ratios, is one of the most dissonant interval regions. This also corresponds to an interseptimal interval range. However, quarter tones are still covered here to provide a resource for them in the same format as the other interval region pages.

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 ¢.

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

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 0 ¢, or those where the best tuning is sharper than 60 ¢ (i.e. not a diesis or comma).

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

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