Semitone (interval region)

A semitone, as a concrete interval region, 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.

← Comma and diesis

Semitone, minor second, augmented unison

Neutral second →

Name	Semitone, minor second, augmented unison
Lower bound	60¢ – 75¢
Upper bound	125¢ – 140¢
Generated MOSes	1L 8s, 9L 1s, 1L 9s
Example JI intervals
Intervals	16/15 (111.7¢) 25/24 (70.7¢)
Related regions
Complement	Major seventh

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Semitone

Functionally, a semitone is an interval that makes up part of a tone, often as one step of a 12-tone chromatic scale, which is a possible criterion for the classification of an interval as a semitone in just intonation.

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.

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)