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Tetracot: Difference between revisions

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{{interwiki
{{Interwiki
| en = Tetracot
| de = Tetracot
| de = Tetracot
| en = Tetracot
| es =  
| es =  
| ja =  
| ja =  
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{{Infobox regtemp
{{Infobox regtemp
| Title = Tetracot
| Title = Tetracot
| Subgroups = 2.3.5, 2.3.5.11.13
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13
| Comma basis = [[20000/19683]] (2.3.5); <br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)
| Comma basis = [[20000/19683]] (2.3.5);<br>[[100/99]], [[243/242]] (2.3.5.11)<br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)
| Generator = 3/2
| Edo join 1 = 7 | Edo join 2 = 27e
| Mapping = 1; 4 9 10 -2
| Mapping = 1; 4 9 10 -2
| Generators = 10/9
| Generators tuning = 176.1
| Optimization method = CWE
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]]
| Pergen = (P8, P5/4)
| Pergen = (P8, P5/4)
| Color name = Saquadyo
| Color name = Saquadyo
| Edo join 1 = 7 | Edo join 2 = 27
| Optimization method = CWE
| Generator tuning = 176.1
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]]
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13
| Odd limit 2 = (2.3.5.11.13) 15 | Mistuning 2 = 10.9 | Complexity 2 = 20
| Odd limit 2 = 2.3.5.11.13 15 | Mistuning 2 = 10.9 | Complexity 2 = 20
}}
}}
{{About|the regular temperament|the ploidacot signature|Ploidacot/Tetracot}}
{{About|the regular temperament|the ploidacot signature|Ploidacot/Tetracot}}
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'''Tetracot''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor 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''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor 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 }}.
[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}.
 
Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1|11th]] and [[13/1|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1|7th harmonic]]:
* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>;
* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>;
* [[Modus]] (27e & 34d) obtains the 7th harmonic at 8 generators down, tempering out [[64/63]] and thereby equating 7/4 with [[16/9]];
* [[Wollemia]] (27e & 34) obtains the 7th harmonic at 19 generators up, tempering out [[126/125]] and thereby equating [[7/1]] with ([[5/3]])<sup>3</sup>([[3/2]]).


See [[Tetracot family]] for more technical data.
See [[Tetracot family]] for technical data.


== Interval chain ==
== Intervals ==
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'''.  
=== 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 ==
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== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 176.0283{{c}}
| CWE: ~10/9 = 176.0965{{c}}
| POTE: ~10/9 = 176.1598{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 175.7765{{c}}
| CWE: ~10/9 = 175.8847{{c}}
| POTE: ~10/9 = 175.9849{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11.13-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 175.8150{{c}}
| CWE: ~10/9 = 176.0854{{c}}
| POTE: ~10/9 = 176.1965{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
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| 165.004
| 165.004
|  
|  
|-
|
| 243/200
| 168.574
| 1/2-comma
|-
|-
| 1\7
| 1\7
|  
|  
| 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
|-
|
| 27/20
| 173.184
| 1/3-comma
|-
|-
|  
|  
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| 173.704
| 173.704
|  
|  
|-
|
| 81/80
| 174.501
| 2/7-comma
|-
|-
|  
|  
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| 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
|-
|-
| 6\41
| 6\41
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| 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
|-
|-
|  
|  
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| 5/4
| 5/4
| 176.257
| 176.257
| 5-odd-limit and 5-limit 9-odd-limit minimax
| 5-odd-limit and 5-limit 9-odd-limit minimax, 2/9-comma
|-
|-
|  
|  
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| 5/3
| 5/3
| 176.872
| 176.872
|  
| 1/5-comma
|-
|-
|  
|  
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|  
|  
| 177.778
| 177.778
| Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
| 27e val, 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
|-
|-
|  
|  
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|  
|  
| 180.000
| 180.000
| Upper bound of 2.3.5.11 subgroup 11-odd-limit diamond monotone
| 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
|-
|-
|  
|  
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== Music ==
== Music ==
; [[Flora Canou]]
; [[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) – in [[modus]], 34edo tuning
* [https://soundcloud.com/floracanou/october-dieting-plan?in=floracanou/sets/totmc-suite "October Dieting Plan"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in [[modus]], 34edo tuning


; [[Zhea Erose]]
; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) – in 34edo tuning
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) – in 34edo tuning
; [[Dustin Schallert]]
* [https://web.archive.org/web/20201127015111/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 ''Tetracot Perc-Sitar'']
* [https://web.archive.org/web/20201129105050/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 ''Tetracot Jam'']
* [https://web.archive.org/web/20201127012230/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 ''Tetracot Pump''] – all in modus, 27edo tuning


; [[Xotla]]
; [[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, in Tetracot[13], 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 in Tetracot[13], 34edo tuning


[[Category:Tetracot| ]] <!-- Main article -->
[[Category:Tetracot| ]] <!-- Main article -->
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