Tetracot
Tetracot, in this article, is the rank-2 temperament in the 2.3.5.11.13 subgroup generated by a submajor second of about 174–178 ¢ which represents both 10/9 and 11/10. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates 3/2, which cannot occur in 12edo, resulting in 100/99, 144/143, and 243/242 being tempered out. This is in contrast to meantone, where 10/9 is tuned sharper than or equal to just in order to be equated with 9/8.
| Tetracot |
100/99, 243/242 (2.3.5.11)
100/99, 144/143, 243/242 (2.3.5.11.13)
2.3.5.11.13 15-odd-limit: 10.9 ¢
2.3.5.11.13 15-odd-limit: 20 notes
- This page is about the regular temperament. For the ploidacot signature, see Ploidacot/Tetracot.
Tetracot has many extensions for the 7-, 11-, and 13-limit. See Tetracot extensions. Equal temperaments that support tetracot include 27, 34, and 41.
See Tetracot family for more technical data.
Intervals
Interval chain
In the following table, odd harmonics and subharmonics 1–15 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 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
As a detemperament of 7et
Tetracot is considered as a cluster temperament with 7 clusters of notes in an octave, so it is naturally a detemperament of the 7 equal temperament. The diagram on the right shows a 34-tone detempered scale, with a generator range of −16 to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent 40/39, 45/44, 55/54, 65/64, 66/65, 81/80, and 121/120 all at once.
Scales
- Tetracot7 – 6L 1s scale
- Tetracot13 – improper 7L 6s
- Tetracot20 – improper 7L 13s
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 176.0283 ¢ | CWE: ~10/9 = 176.0965 ¢ | POTE: ~10/9 = 176.1598 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 175.7765 ¢ | CWE: ~10/9 = 175.8847 ¢ | POTE: ~10/9 = 175.9849 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 175.8150 ¢ | CWE: ~10/9 = 176.0854 ¢ | POTE: ~10/9 = 176.1965 ¢ |
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 11/10 | 165.004 | ||
| 243/200 | 168.574 | 1/2-comma | |
| 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 | |
| 27/20 | 173.184 | 1/3-comma | |
| 11/9 | 173.704 | ||
| 81/80 | 174.501 | 2/7-comma | |
| 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 | 1/4-comma | |
| 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, 2/9-comma | |
| 13/9 | 176.338 | ||
| 5\34 | 176.471 | ||
| 15/13 | 176.516 | ||
| 5/3 | 176.872 | 1/5-comma | |
| 13/10 | 176.890 | ||
| 13/12 | 176.905 | ||
| 4\27 | 177.778 | 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | |
| 27/25 | 177.794 | 1/6-comma | |
| 243/125 | 178.452 | 1/7-comma | |
| 15/11 | 178.984 | ||
| 13/8 | 179.736 | ||
| 3\20 | 180.000 | 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone | |
| 9/5 | 182.404 |
* Besides the octave
Music
- "October Dieting Plan" from TOTMC Suite (2023–2025) – in modus, 34edo tuning
- Modal Studies in Tetracot (2021) – in 34edo tuning
- Tetracot Perc-Sitar
- Tetracot Jam
- Tetracot Pump – all in modus, 27edo tuning