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	176.3	11/10, 10/9
2	352.7	11/9, 16/13
3	529.0	15/11
4	705.3	3/2
5	881.7	5/3
6	1058.0	11/6, 24/13
7	34.4	55/54, 45/44, 40/39
8	210.7	9/8
9	387.0	5/4
10	563.4	11/8, 18/13
11	739.7	20/13
12	916.0	22/13
13	1092.4	15/8
14	68.7	33/32, 27/26, 25/24
15	245.0	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

Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
11/10	165.004
11/9	173.704
12/11	174.894
11/8	175.132
4/3	175.489
13/11	175.899
16/15	176.021
5/4	176.257	5-odd-limit minimax
18/13	176.338
15/13	176.516
6/5	176.872
13/10	176.890
13/12	176.905
15/11	178.984
16/13	179.736
10/9	182.404