Tritone

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Tritone

A tritone is an interval that spans six steps of a 12-tone chromatic scale. In just intonation, an interval may be classified as a tritone if it is reasonably mapped to 12\24. The use of 24edo's 12\24 as the mapping criteria here rather than 12edo's 6\12 better captures the characteristics of many intervals in the 11- and 13-limit. Tritones come in octave-complementary pairs, called augmented fourth (A4) and diminished fifth (d5) based on their number of steps in the diatonic scale.

As a concrete interval region, it is typically near 600 cents in size. A rough tuning range for the tritone is about 560 to 640 cents according to Margo Schulter's theory of interval regions. Tritone in this sense can also refer to the semi-octave, a tritone of exactly 600 cents found in every even edo, due to the fact that it is 1\2edo.

For the sake of fully covering the range of intervals within the octave, this page also covers semiaugmented fourths of about 550 cents, and semidiminished fifths of about 650 cents. Note that these are not conventionally considered tritones, and are included here for simplicity (and frankly for having more than one pair of simple ratios to use in the EDO section). More info may be found at semiaugmented fourth and semidiminished fifth.

As such, this article covers intervals from 560 to 640 cents, but intervals between 540-560 and 640-660 cents have been "grandfathered in" due to the fact that superfourths and subfifths were not originally given their own articles.

In just intonation

Due to being close to 600 ¢, tritones come in octave-complementary pairs. For low-limit harmony, these pairs are often referred to as "augmented fourth" (A4) and "diminished fifth" (d5) based on their function in diatonic harmony, but in higher limits, the tritones are usually just distinguished by size.

Historically, the term "tritone" referred to the Pythagorean augmented fourth, the ratio of 729/512 reached by stacking three Pythagorean whole tones (hence "tri-tone"), or equivalently, six 3/2s, which is an interval of about 612 ¢. There is also the octave complement, the Pythagorean diminished fifth of 1024/729, which is about 588 ¢ in size.

Much simpler tritones exist in higher limits, however, for example:

• The 5-limit ptolemaic augmented fourth and ptolemaic diminished fifth are ratios of 45/32 and 64/45 respectively, and are about 590 ¢ and 610 ¢ respectively.
  • There are also the classical augmented fourth and classical diminished fifth, which are ratios of 25/18 and 36/25 respectively, and are about 569 ¢ and 631 ¢ respectively.
• The 7-limit narrow tritone and wide tritone are ratios of 7/5 and 10/7 respectively, and are about 583 ¢ and 617 ¢ respectively.
• The 11-limit superfourth and subfifth are ratios of 11/8 and 16/11 respectively, and are about 551 ¢ and 649 ¢ respectively; they are listed here because they barely do not make the cutoff (550 ¢ and 650 ¢) to be included in the pages on fourths and fifths.

In EDOs

The following table lists the tunings of 11/8, 7/5, and their octave complements, as well as other tritones if present, in various significant EDOs. Note that many EDOs map 7/5 and 10/7 to the semioctave.

EDO	11/8	7/5	16/11	10/7	Other tritones
12	600 ¢
15	560 ¢		640 ¢
16	525 ¢	600 ¢	675 ¢	600 ¢
17	565 ¢		635 ¢
19	568 ¢		632 ¢
22	545 ¢	600 ¢	655 ¢	600 ¢
24	550 ¢	600 ¢	650 ¢	600 ¢
25	*	576 ¢	*	624 ¢
26	554 ¢	600 ¢	646 ¢	600 ¢
27	*	578 ¢	*	622 ¢
29	*	579 ¢	*	621 ¢
31	542 ¢	581 ¢	658 ¢	619 ¢
34	565 ¢	600 ¢	635 ¢	600 ¢
41	556 ¢	585 ¢	644 ¢	615 ¢
53	543 ¢	589 ¢	657 ¢	611 ¢	634 ¢ ≈ 36/25, 566 ¢ ≈ 25/18

In regular temperaments

Temperaments involving tritones often involve tempering a pair of tritones together. As such, each pair of tritones has a corresponding temperament, which equates both tritones to the semioctave:

Note that in these temperaments, the tritone is usually considered as the period, even though it is technically also a generator.

Pair of tritones	Temperament
45/32, 64/45	Diaschismic
25/18, 36/25	Diminished
7/5, 10/7	Jubilismic
11/8, 16/11	Temperament of 128/121

Note that sometimes, tritones are used as generators, utilizing the small commas between the tritone pairs to approximate some other interval. The two simplest tritones, 11/8 and 7/5, also happen to be rather far from the semioctave, and as such are rather useful for this purpose:

Tritones as approximations of the semioctave

In some tuning systems having an even number of divisions of the octave (equal or well-tempered), the tritone (defined as three whole tones) is the same as the half-octave; if the divisions of the octave are equal and the octave is tuned pure, the tritone will therefore be exactly the square root of 2. The following table compares selected JI tritone pairs that approximate the half-octave and the commas separating them:

Ratios	Prime limit	Distance from 600 ¢	Comma
729/512, 1024/729	3	11.730	531441/524288
45/32, 64/45	5	9.776	2048/2025
7/5, 10/7	7	17.488	50/49
99/70, 140/99	11	0.088	9801/9800
13/9, 18/13	13	36.618	169/162
24/17, 17/12	17	3.000	289/288
27/19, 38/27	19	8.352	729/722
23/16, 32/23	23	28.274	544/529
41/29, 58/41	41	0.515	1682/1681

However, it is possible for a tuning system to have an even number of notes and a pair of tritones that is not the same as the semioctave.

An example of a tuning system having an even number of notes and a pair of tritones that is not the same as the semioctave is 26edo. In flattone tuning systems such as 26edo, 11/8 is C–F♯, which is the lesser tritone mapping to 12\26 (not the semioctave 13\26); 16/11 is C–G♭, which is the greater tritone mapping to 14\26. Note that these tritones are not necessarily valid generators for tuning systems analogous to flattone but substituting 11/8 for 4/3; for instance, these are not valid generators for 26edo with a whole-octave period, because they instead produce 13edo.

Temperaments that can use tritones as generators

Temperaments that use 11/8 as a generator

In 13edo, 11/8 and 16/11 map to the tritones 6\13 and 7\13, respectively, and are valid generators for it. Furthermore, although 12\26 is not valid as a generator for 26edo with a whole-octave period, it is valid with a half-octave period, for a temperament analogous to Injera but substituting 11/8 for 4/3.

Todo: complete section

Temperaments that use 7/5 as a generator

Todo: complete section (see List_of_rank_two_temperaments_by_generator_and_period#Generator_.7E7.2F5)

In moment-of-symmetry scales

Intervals between 545 and 654 cents 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
545–654 ¢	1L 1s	2L 1s	2L 3s	2L 5s	2L 7s	2L 9s