Semitone (interval region)

A semitone is an interval that spans exactly one step of a 12-tone chromatic scale. In just intonation, an interval may be classified as a semitone if it is reasonably mapped to 2\24. The use of 24edo's 2\24 as the mapping criteria here rather than 12edo's 1\12 better captures the characteristics of many intervals in the 11- and 13-limit. Semitones come in two functional categories based on their number of steps in the diatonic scale:

• Diatonic semitones, minor seconds (m2) or limmas,
• Chromatic semitones, augmented unisons (A1) or chromas.

As a concrete interval region, it is typically near 100 cents in size, distinct from commas and dieses (less than 60 cents), and from neutral seconds (about 150 cents). A rough tuning range for the semitone is about 60 cents to 125 cents according to Margo Schulter's theory of interval regions.

In just intonation

By prime limit

In the low prime limits, up to the 5-limit, in which the West has developed a formal system of diatonic harmony, the distinction between diatonic and chromatic semitones is the clearest, so a pair of 2 semitones will be provided for each. However, higher than the 5-limit, function as diatonic vs. chromatic tends to become less clear, and larger intervals can be seen as belonging to neither category.

• In the 3-limit:
  • The limma, or Pythagorean diatonic semitone, is a ratio of 256/243, and is about 90 cents.
  • The apotome, or Pythagorean chromatic semitone, is a ratio of 2187/2048, and is about 114 cents.
• In the 5-limit:
  • The classical diatonic semitone is a ratio of 16/15, and is about 112 cents.
  • The classical chromatic semitone is a ratio of 25/24, and is about 71 cents.
    • There is also a ptolemaic chromatic semitone, which is a ratio of 135/128, and is about 92 cents.
• In higher limits:
  • The 7-limit third-tone is a ratio of 28/27, and is about 63 cents.
  • The 7-limit minor semitone is a ratio of 21/20, and is about 84 cents.
  • The 7-limit major semitone is a ratio of 15/14, and is about 119 cents.
  • The 11-limit minor semitone is a ratio of 22/21, and is about 81 cents.
  • The 13-limit sinaic is a ratio of 14/13, and is about 128 cents.
  • The 13-limit greater 2/3-tone is a ratio of 13/12, and is about 139 cents.
  • The 17-limit large semitone is a ratio of 17/16, and is about 104 cents.
  • The 17-limit small semitone is a ratio of 18/17, and is about 99 cents.

By delta

This table lists just semitones by delta:

Delta 1 (Superparticular)	Cents
13/12	139c
14/13	128c
15/14	119c
16/15	112c
17/16	104c
18/17	99c
19/18	94c
20/19	89c
21/20	85c
22/21	81c
23/22	77c
24/23	74c
25/24	71c
26/25	68c
27/26	65c
28/27	63c
29/28	61c
30/29	59c
31/30	57c
32/31	55c
33/32	53c
34/33	52c
35/34	50c

In edos

The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant edos.

Edo	16/15	25/24	Other semitones
12	100c
15	80c
16	75c
17	141c	71c
19	126c	63c
22	109c	55c
24	100c	50c
25	96c	*
26	92c
27	133c	89c
29	124c	83c
31	116c	77c
34	106c	71c
41	117c	59c	88c ≈ 256/243
53	113c	68c	91c ≈ 256/243

In regular temperaments

Two important, simple semitone ratios are 16/15 and 25/24. The following notable temperaments are generated by them:

Temperaments that use 25/24 as a generator

• Vishnu, which stacks seven 25/24s to make a just perfect fourth of 4/3
• Chlorine, equivalent to 17edo, stacking seventeen 25/24s to make an octave

Temperaments that use 16/15 as a generator

• Miracle, which splits 3/2 into six semitones, each representing both 15/14 and 16/15.
• Negri, which splits 4/3 into four semitones, such that three of them represent 5/4.
• Diaschismic, which is usually described as having a fifth as its second generator, but can alternatively be generated by a half-octave and a semitone.

Compton has one step of 12edo as its first generator, representing 256/243.

When 25/24 is tempered out, it leads to dicot temperament.

When 16/15 is tempered out, it leads to father temperament.