Comma and diesis

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

← Unison

Interval region

Semitone →

Name	Comma, diesis
Lower bound	0¢
Upper bound	40¢ – 60¢
Example JI intervals
Intervals	128/125 (41.1¢) 81/80 (21.5¢)
Related regions
Complement	(Imperfect) octave

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

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

The range of dieses largely overlaps with the range of quartertones (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 the diatonic scale, the analogous concepts are subchromatic and enharmonic steps. A subchromatic step (a "comma") does not change the interval category (for example, in most just notation systems, if you flatten the major third 81/64 by an 81/80 comma to produce 5/4, the latter is still considered a major third). Diatonically, subchromatic steps are perfect unisons (P1), and there are none that are not a unison in a rank-2 diatonic tuning. An enharmonic step (a "diesis", although this is controversial) changes the interval category to an enharmonic interval (for example, a major third to a diminished fourth, or a chromatic semitone to a diatonic semitone). Similarly, enharmonic steps are ascending or descending diminished seconds (d2).

In just intonation

By prime limit

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, intervals are provided 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 ¢.

By delta

As comma and diesis is the smallest interval class, it may be represented by:

• Any superparticular interval smaller than 29/28
• Any delta-2 interval smaller than 57/55
• Any delta-3 interval smaller than 88/85

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). Note that this does not depend on how each EDO tunes the intervals that 128/125 might be derived from, only on which edostep is closest to 128/125's size.

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 a stack of ten 49/48's equals 5/4.

In MOS scales

Intervals less than 100 ¢ generate the following MOS scales:

These tables start from the last monolarge MOS generated by the interval range.

Scales with more than 12 notes are not included.

Range	MOS
0–100 ¢	1L 11s