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'''Tetracot''', in this article, is the rank-2 [[regular temperament]] | '''Tetracot''', in this article, is the rank-2 [[regular temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a "sub-major" second of about 174–178{{cent}} 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 [[tempering out|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 [[extension]]s for the 7-, 11-, and 13-limit. See [[Tetracot extensions]]. Equal temperaments that support tetracot include {{EDOs| 27, 34, and 41 }}. | |||
See [[Tetracot family]] or [[No-sevens subgroup temperaments#Tetracot]] for more technical data. | See [[Tetracot family]] or [[No-sevens subgroup temperaments#Tetracot]] for more technical data. | ||
== Interval chain == | == 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 | 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]], and [[121/120]] all at once. 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" | ||
| Line 224: | Line 222: | ||
* "Electrostat" from ''Lesser Groove'' (2020) – [https://open.spotify.com/track/5LIPr8n6uQySeLUfM11U2W Spotify] | [https://xotla.bandcamp.com/track/electrostat-tetracot-13 Bandcamp] | [https://www.youtube.com/watch?v=5SAuoyDwpgc YouTube] – ambient electro, tetracot[13] in 34edo tuning | * "Electrostat" from ''Lesser Groove'' (2020) – [https://open.spotify.com/track/5LIPr8n6uQySeLUfM11U2W Spotify] | [https://xotla.bandcamp.com/track/electrostat-tetracot-13 Bandcamp] | [https://www.youtube.com/watch?v=5SAuoyDwpgc YouTube] – ambient electro, tetracot[13] in 34edo tuning | ||
[[Category:Tetracot| ]] <!-- Main article --> | [[Category:Tetracot| ]] <!-- Main article --> | ||
[[Category:Rank-2 temperaments]] | |||
[[Category:Tetracot family]] | [[Category:Tetracot family]] | ||
Latest revision as of 12:07, 6 August 2025
Tetracot, in this article, is the rank-2 regular temperament in the 2.3.5.11.13 subgroup generated by a "sub-major" 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 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, and 121/120 all at once. In the following table, odd harmonics and subharmonics 1–15 are in bold.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.00 | 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
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 | ||
| 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 | |
| 11/9 | 173.704 | ||
| 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 | ||
| 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 | |
| 13/9 | 176.338 | ||
| 5\34 | 176.471 | ||
| 15/13 | 176.516 | ||
| 5/3 | 176.872 | ||
| 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 | |
| 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 Vol. 1 (2023) – modus in 34edo tuning
- Modal Studies in Tetracot (2021) – in 34edo tuning