Semitone (interval region)

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Semitone

A semitone is an interval that makes up part of a tone, often as 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 ¢ in size, distinct from commas and dieses (less than 60 ¢), and from neutral seconds (about 150 ¢). A rough tuning range for the semitone is about 60 ¢ to 125 ¢ according to Margo Schulter's theory of interval regions.

The intervals covered in this article range from 50 ¢ to 140 ¢.

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

By delta

This table lists just semitones by delta; simple semitone ratios tend to be superparticular.

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

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

• Valentine, which divides 3/2 into nine small semitones, five of which make 5/4. See also the related Carlos Alpha.
• Vishnu, which stacks seven 25/24s to make a just perfect fourth of 4/3.
• Chlorine, based on 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.

In moment-of-symmetry scales

Intervals between 100 and 133 ¢ generate the following MOS scales:

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

MOSes with more than 12 notes are not included.

Range	MOS
100–109 ¢	1L 10s	11L 1s
109–120 ¢	1L 9s	10L 1s
120–133 ¢	1L 8s	9L 1s

See also

• Semitone (disambiguation page)