Tetracot: Difference between revisions

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@Flirora: it's better to link temperament names to individual temperaments (it's possible - and likely - that they will transformed into individual pages). The categorization via redirect helps us to build an "Index of Temperaments (IoT)"
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#REDIRECT [[Tetracot family #Tetracot]]
{{interwiki
| de = Tetracot
| en = Tetracot
| es =
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}}
'''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.
== 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'''.
{| class="wikitable right-1 right-2"
|-
! #
! 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
|}
<nowiki />* In 2.3.5.11.13 subgroup CTE tuning
== Scales ==
* [[Tetracot7]] – [[6L&nbsp;1s]] scale
* [[Tetracot13]] – improper [[7L&nbsp;6s]]
* [[Tetracot20]] – improper [[7L&nbsp;13s]]
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br />Generator
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
! Generator (¢)
! Comments
|-
|
| 11/10
| 165.004
|
|-
| 1\7
|
| 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
|-
|
| 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
|
|}
<nowiki />* Besides the octave
== Music ==
; [[Flora Canou]]
* [https://soundcloud.com/floracanou/october-dieting-plan?in=floracanou/sets/totmc-suite-vol-1 "October Dieting Plan"] from [https://soundcloud.com/floracanou/sets/totmc-suite-vol-1 ''TOTMC Suite Vol. 1''] (2023) – [[modus]] in 34edo tuning
; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) – in 34edo tuning
; [[Xotla]]
* "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:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Tetracot family]]
{{IoT}}

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

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

Flora Canou
Zhea Erose
Xotla
  • "Electrostat" from Lesser Groove (2020) – Spotify | Bandcamp | YouTube – ambient electro, tetracot[13] in 34edo tuning
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