Tetracot

Tetracot, in this article, is the rank-2 regular temperament for the 2.3.5.11.13 subgroup defined by tempering out 100/99, 144/143, and 243/242.

It can be seen as implying a rank-2 tuning which is generated by a sub-major second of about 176 cents which represents both 10/9 and 11/10. It is so named because the generator is a quarter of fifth: four generators make a fifth which approximates 3/2, which cannot occur in 12edo. Equal temperaments that support tetracot include 27, 34, and 41.

Tetracot has many extensions for the 7-, 11- and 13-limit. See Tetracot extensions.

See Tetracot family or No-sevens subgroup temperaments #Tetracot for more technical data.

Interval chain

Tetracot is considered as a cluster temperament with seven clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents 40/39 ~ 45/44 ~ 55/54 ~ 65/64 ~ 66/65 ~ 81/80 ~ 121/120 all tempered together. In the following table, odd harmonics and subharmonics 1–15 are in bold.

#	Cents*	Approximate Ratios
0	0.00	1/1
1	175.8	11/10, 10/9
2	350.6	11/9, 16/13
3	527.4	15/11
4	703.3	3/2
5	879.1	5/3
6	1054.9	11/6, 24/13
7	30.7	55/54, 45/44, 40/39
8	206.5	9/8
9	382.3	5/4
10	558.2	11/8, 18/13
11	734.0	20/13
12	909.8	22/13
13	1085.6	15/8
14	61.4	33/32, 27/26, 25/24
15	237.2	15/13

* in 2.3.5.11.13 subgroup CTE tuning

Scales

Tetracot7 – 6L 1s scale
Tetracot13 – improper 7L 6s
Tetracot20 – improper 7L 13s

Tunings

Tuning spectrum

Edo Generator	Eigenmonzo (Unchanged-interval)*	Generator (¢)	Comments
	11/10	165.004
1\7		171.429	Lower bound of 2.3.5.11 subgroup 11-odd-limit, 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	11/9	173.704
	11/6	174.894
7\48		175.000
	11/8	175.132	2.3.5.11 subgroup 11-odd-limit minimax
	3/2	175.489
6\41		175.610
	13/11	175.899	2.3.5.11.13 subgroup 13- and 15-odd-limit minimax
	15/8	176.021
	5/4	176.257	5-odd-limit and 5-limit 9-odd-limit minimax
	13/9	176.338
5\34		176.471
	15/13	176.516
	5/3	176.872
	13/10	176.890
	13/12	176.905
4\27		177.778	Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	15/11	178.984
	13/8	179.736
3\20		180.000	Upper bound of 2.3.5.11 subgroup 11-odd-limit diamond monotone
	9/5	182.404