Tetracot: Difference between revisions
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[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}. | [[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}. | ||
Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which | Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1|11th]] and [[13/1|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1|7th harmonic]]. | ||
* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>; | * [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>; | ||
* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>; | * [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>; | ||
Revision as of 11:24, 18 May 2026
| 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, 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.
Equal temperaments that support tetracot include 27, 34, and 41.
Tetracot has four strong extensions for the 7-, 11-, and 13-limit, which use the same methods of obtaining the 11th and 13th harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the 7th harmonic.
- Monkey (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out 875/864 and thereby equating 7/4 with (6/5)3;
- Bunya (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out 225/224 and thereby equating 7/2 with (15/8)2;
- Modus (27e & 34d) obtains the 7th harmonic at 8 generators down, tempering out 64/63 and thereby equating 7/4 with 16/9;
- Wollemia (27e & 34) obtains the 7th harmonic at 19 generators up, tempering out 126/125 and thereby equating 7/1 with (5/3)3(3/2).
See Tetracot family for 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