Tritone

A tritone is an interval that is near 600 cents in size, distinct from the perfect fifth of roughly 700 cents and the perfect fourth of roughly 500 cents. A rough tuning range for the tritone is about 540 to 660 cents, however people tend to narrow that range to around 570 to 630 cents in order to treat superfourths and subfifths as distinct categories. In this case, for the sake of conciseness, however, they are treated as tritones.

The term "tritone" can also refer to the semi-octave, a tritone of exactly 600 cents found in every even EDO. This is not the main subject of this page, but the semi-octave is significant to the nature of tritones so it will be referenced further.

In just intonation

Due to being close to 600 cents, tritones come in octave-complementary pairs. For low-limit harmony, these pairs are often referred to as "augmented fourth" and "diminished fifth" 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 cents. There is also the octave complement, the Pythagorean diminished fifth of 1024/729, which is about 588 cents 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 cents 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 cents respectively.
• The 7-limit narrow tritone and wide tritone are ratios of 7/5 and 10/7 respectively, and are about 583 and 617 cents respectively.
• The 11-limit superfourth and subfifth are ratios of 11/8 and 16/11 respectively, and are about 551 and 649 cents respectively; they are listed here because they barely do not make the cutoff (550 and 650 cents) 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	600c
15	560c		640c
16	525c	600c	675c	600c
17	565c		635c
19	568c		632c
22	545c	600c	655c	600c
24	550c	600c	650c	600c
25	*	576c	*	624c
26	554c	600c	646c	600c
27	*	578c	*	622c
29	*	579c	*	621c
31	542c	581c	658c	619c
34	565c	600c	635c	600c
41	556c	585c	644c	615c
53	543c	589c	657c	611c	634c ≈ 36/25, 566c ≈ 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 these technically do not have the semioctave as a generator, since making it a fraction of an octave causes it to become a period.

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:

Temperaments that use 11/8 as a generator

• TBD

Temperaments that use 7/5 as a generator

• TBD