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, prime harmonics are in bold.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.00 | 1/1 |
| 1 | 176.20 | 11/10, 10/9 |
| 2 | 352.39 | 11/9, 16/13 |
| 3 | 528.59 | 15/11 |
| 4 | 704.79 | 3/2 |
| 5 | 880.98 | 5/3 |
| 6 | 1057.18 | 11/6, 24/13 |
| 7 | 33.38 | 55/54, 45/44, 40/39 |
| 8 | 209.57 | 9/8 |
| 9 | 385.77 | 5/4 |
| 10 | 561.96 | 11/8, 18/13 |
| 11 | 738.16 | 20/13 |
| 12 | 914.36 | 22/13 |
| 13 | 1090.55 | 15/8 |
| 14 | 66.75 | 33/32, 27/26, 25/24 |
| 15 | 242.95 | 15/13 |
- * in 2.3.5.11.13 POTE tuning
Scales
- Tetracot7 – 6L 1s scale
- Tetracot13 – improper 7L 6s
- Tetracot20 – improper 7L 13s
Music
- "October Dieting Plan" from TOTMC Suite Vol. 1 (2023) – modus in 34edo tuning
- Modal Studies in Tetracot (2021) – in 34edo