Semitone (interval region)

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Semitone (interval region)

This page is about the interval region. For the interval size unit of exactly 100 cents, see Interval size measure#Gross.

A semitone is an interval that is near 100 cents in size, distinct from commas and dieses (less than 60 cents), and from major seconds (about 200 cents). A rough tuning range for the semitone is about 50 cents to 140 cents, though this is extremely wide; some might prefer to restrict it to around 70 cents to 130 cents.

Semitones tend to fall into one of two functional categories, based on the system being used:

• diatonic semitones, minor seconds (m2) or limmas,
• chromatic semitones, augmented unisons (A1) or chromas.

This page covers both categories of intervals, as the distinction between them is largely a matter of the diatonic MOS, and is also not the subject of this article.

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

Todo: complete list

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

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