Tetracot: Difference between revisions
Jump to navigation
Jump to search
m Oops, wrong tuning |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 7: | Line 7: | ||
'''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]]. | '''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 [[generator|generated]] by a sub-major second of about 176 | It can be seen as implying a rank-2 tuning which is [[generator|generated]] by a sub-major second of about 176{{c}} 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 {{EDOs| 27, 34, and 41 }}. | ||
Tetracot has many [[extension]]s for the 7-, 11- and 13-limit. See [[Tetracot extensions]]. | Tetracot has many [[extension]]s for the 7-, 11- and 13-limit. See [[Tetracot extensions]]. | ||
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 == | ||
| Line 18: | Line 18: | ||
{| class="wikitable right-1 right-2" | {| class="wikitable right-1 right-2" | ||
|- | |- | ||
! # | ! # | ||
! Cents | ! Cents* | ||
! Approximate Ratios | ! Approximate Ratios | ||
|- | |- | ||
| Line 86: | Line 86: | ||
| 15/13 | | 15/13 | ||
|} | |} | ||
<nowiki />* In 2.3.5.11.13 subgroup CTE tuning | |||
== Scales == | == Scales == | ||
* [[Tetracot7]] | * [[Tetracot7]] – [[6L 1s]] scale | ||
* [[Tetracot13]] | * [[Tetracot13]] – improper [[7L 6s]] | ||
* [[Tetracot20]] | * [[Tetracot20]] – improper [[7L 13s]] | ||
== Tunings == | == Tunings == | ||
| Line 97: | Line 97: | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! Edo<br>Generator | ! Edo<br />Generator | ||
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]] | ! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
| Line 110: | Line 110: | ||
| | | | ||
| 171.429 | | 171.429 | ||
| 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 | ||
|- | |- | ||
| | | | ||
| Line 212: | Line 212: | ||
| | | | ||
|} | |} | ||
<nowiki>* | <nowiki />* Besides the octave | ||
== Music == | == Music == | ||
| Line 225: | Line 225: | ||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
[[Category:Tetracot| ]] <!-- | [[Category:Tetracot| ]] <!-- Main article --> | ||
[[Category:Tetracot family]] | [[Category:Tetracot family]] | ||
Revision as of 21:19, 13 November 2024
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 ¢ 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, 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