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Tetracot: Difference between revisions

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'''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]] 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]].


It can be seen as implying a rank-2 tuning which is [[generator|generated]] by a sub-major second of about 176 [[cent]]s 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]]. 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]].
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 ~ 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'''.  
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"
|-
|-
! #
! #
! Cents<nowiki>*</nowiki>
! Cents*
! Approximate Ratios
! Approximate Ratios
|-
|-
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|-
|-
| 1
| 1
| 176.3
| 175.8
| 11/10, 10/9
| 11/10, 10/9
|-
|-
| 2
| 2
| 352.7
| 350.6
| 11/9, '''16/13'''
| 11/9, '''16/13'''
|-
|-
| 3
| 3
| 529.0
| 527.4
| 15/11
| 15/11
|-
|-
| 4
| 4
| 705.3
| 703.3
| '''3/2'''
| '''3/2'''
|-
|-
| 5
| 5
| 881.7
| 879.1
| 5/3
| 5/3
|-
|-
| 6
| 6
| 1058.0
| 1054.9
| 11/6, 24/13
| 11/6, 24/13
|-
|-
| 7
| 7
| 34.4
| 30.7
| 55/54, 45/44, 40/39
| 55/54, 45/44, 40/39
|-
|-
| 8
| 8
| 210.7
| 206.5
| '''9/8'''
| '''9/8'''
|-
|-
| 9
| 9
| 387.0
| 382.3
| '''5/4'''
| '''5/4'''
|-
|-
| 10
| 10
| 563.4
| 558.2
| '''11/8''', 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
| 739.7
| 734.0
| 20/13
| 20/13
|-
|-
| 12
| 12
| 916.0
| 909.8
| 22/13
| 22/13
|-
|-
| 13
| 13
| 1092.4
| 1085.6
| '''15/8'''
| '''15/8'''
|-
|-
| 14
| 14
| 68.7
| 61.4
| 33/32, 27/26, 25/24
| 33/32, 27/26, 25/24
|-
|-
| 15
| 15
| 245.0
| 237.2
| 15/13
| 15/13
|}
|}
: <nowiki>*</nowiki> in 2.3.5.11.13 subgroup CTE tuning
<nowiki />* In 2.3.5.11.13 subgroup CTE tuning


== Scales ==
== Scales ==
* [[Tetracot7]] – [[6L 1s]] scale
* [[Tetracot7]] – [[6L&nbsp;1s]] scale
* [[Tetracot13]] – improper [[7L 6s]]
* [[Tetracot13]] – improper [[7L&nbsp;6s]]
* [[Tetracot20]] – improper [[7L 13s]]
* [[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 ==
== Music ==
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* "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:Temperaments]]
[[Category:Tetracot| ]] <!-- Main article -->
[[Category:Tetracot| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Tetracot family]]
Retrieved from "https://en.xen.wiki/w/Tetracot"