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, distinct from the semiaugmented fourth of roughly 550 cents and the semidiminished fifth of roughly 650 cents. 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.

In just intonation

Historically, the term "tritone" referred specifically to the Pythagorean augmented fourth, the ratio of 729/512 reached by stacking three Pythagorean whole tones (hence "tri-tone"), or equivalently, six 3/2's, 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.

In edos

The following table lists the tunings of 7/5 and its octave complement, 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	7/5	10/7	Other tritones
12	600 ¢	600 ¢
15	560 ¢	640 ¢
16	600 ¢	600 ¢
17	565 ¢	635 ¢
19	568 ¢	632 ¢
22	600 ¢	600 ¢
24	600 ¢	600 ¢
25	576 ¢	624 ¢
26	600 ¢	600 ¢
27	578 ¢	622 ¢
29	579 ¢	621 ¢
31	581 ¢	619 ¢
34	600 ¢	600 ¢
41	585 ¢	615 ¢
53	589 ¢	611 ¢	566 ¢ ≈ 25/18, 634 ¢ ≈ 36/25

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

Tritones as approximations of the semioctave

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