Tetracot: Difference between revisions
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See [[Tetracot family]] for more technical data. | See [[Tetracot family]] for more technical data. | ||
== Interval chain == | == Intervals == | ||
=== Interval chain === | |||
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''. | |||
{| class="wikitable right-1 right-2" | {| class="wikitable right-1 right-2" | ||
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| 15/13 | | 15/13 | ||
|} | |} | ||
<nowiki />* In 2.3.5.11.13 subgroup CTE tuning | <nowiki/>* In 2.3.5.11.13 subgroup CTE tuning | ||
=== As a detemperament of 7et === | |||
[[File: Tetracot 7et Detempering.png|thumb|Tetracot as a 34-tone 7et detempering]] | |||
Tetracot is considered as a [[cluster temperament]] with 7 clusters of notes in an octave, so it is naturally a [[detemperament]] of the [[7edo|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 == | == Scales == | ||
| Line 122: | Line 128: | ||
| | | | ||
|- | |- | ||
| | | | ||
|243/200 | | 243/200 | ||
|168.574 | | 168.574 | ||
|1/2-comma | | 1/2-comma | ||
|- | |- | ||
| 1\7 | | 1\7 | ||
| Line 132: | Line 138: | ||
| Lower bound of 2.3.5.11 subgroup 11-odd-limit, <br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | | Lower bound of 2.3.5.11 subgroup 11-odd-limit, <br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
|27/20 | | 27/20 | ||
|173.184 | | 173.184 | ||
|1/3-comma | | 1/3-comma | ||
|- | |- | ||
| | | | ||
| Line 142: | Line 148: | ||
| | | | ||
|- | |- | ||
| | | | ||
|81/80 | | 81/80 | ||
|174.501 | | 174.501 | ||
|2/7-comma | | 2/7-comma | ||
|- | |- | ||
| | | | ||
| Line 160: | Line 166: | ||
| 11/8 | | 11/8 | ||
| 175.132 | | 175.132 | ||
| 2.3.5.11 subgroup 11-odd-limit minimax | | 2.3.5.11-subgroup 11-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| 3/2 | | 3/2 | ||
| 175.489 | | 175.489 | ||
|1/4-comma | | 1/4-comma | ||
|- | |- | ||
| 6\41 | | 6\41 | ||
| Line 175: | Line 181: | ||
| 13/11 | | 13/11 | ||
| 175.899 | | 175.899 | ||
| 2.3.5.11.13 subgroup 13- and 15-odd-limit minimax | | 2.3.5.11.13-subgroup 13- and 15-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| Line 205: | Line 211: | ||
| 5/3 | | 5/3 | ||
| 176.872 | | 176.872 | ||
|1/5-comma | | 1/5-comma | ||
|- | |- | ||
| | | | ||
| Line 222: | Line 228: | ||
| Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | | Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
|27/25 | | 27/25 | ||
|177.794 | | 177.794 | ||
|1/6-comma | | 1/6-comma | ||
|- | |- | ||
| | | | ||
|243/125 | | 243/125 | ||
|178.452 | | 178.452 | ||
|1/7-comma | | 1/7-comma | ||
|- | |- | ||
| | | | ||
| Line 245: | Line 251: | ||
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| 180.000 | | 180.000 | ||
| Upper bound of 2.3.5.11 subgroup 11-odd-limit diamond monotone | | Upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
Revision as of 12:00, 29 March 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.
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
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 | 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 | 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