Tetracot: Difference between revisions

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| de = Tetracot
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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]].
{{Infobox regtemp
| Title = Tetracot
| Subgroups = 2.3.5, 2.3.5.11, 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)
| Edo join 1 = 7 | Edo join 2 = 27e
| 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)
| Color name = Saquadyo
| 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
}}
{{About|the regular temperament|the ploidacot signature|Ploidacot/Tetracot}}
 
'''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]].


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 }}.
[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}.  


Tetracot has many extensions for 7, 11 and 13-limit include monkey (34 &amp; 41), bunya (34d &amp; 41), modus (27e &amp; 34d) and wollemia (27e &amp; 34).
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]] or [[No-sevens subgroup temperaments #Tetracot]] for more technical data.
See [[Tetracot family]] for technical data.


== Intervals ==
== 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 ~ 121/120 all tempered together. In the following table, prime harmonics 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"
|-
|-
! #
! #
! Cents<nowiki>*</nowiki>
! Cents*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
| 0.00
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| 176.20
| 175.8
| 11/10, 10/9
| 11/10, 10/9
|-
|-
| 2
| 2
| 352.39
| 350.6
| 11/9, '''16/13'''
| 11/9, '''16/13'''
|-
|-
| 3
| 3
| 528.59
| 527.4
| 15/11
| 15/11
|-
|-
| 4
| 4
| 704.79
| 703.3
| '''3/2'''
| '''3/2'''
|-
|-
| 5
| 5
| 880.98
| 879.1
| 5/3
| 5/3
|-
|-
| 6
| 6
| 1057.18
| 1054.9
| 11/6, 24/13
| 11/6, 24/13
|-
|-
| 7
| 7
| 33.38
| 30.7
| 55/54, 45/44, 40/39
| 55/54, 45/44, 40/39
|-
|-
| 8
| 8
| 209.57
| 206.5
| 9/8
| '''9/8'''
|-
|-
| 9
| 9
| 385.77
| 382.3
| '''5/4'''
| '''5/4'''
|-
|-
| 10
| 10
| 561.96
| 558.2
| '''11/8''', 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
| 738.16
| 734.0
| 20/13
| 20/13
|-
|-
| 12
| 12
| 914.36
| 909.8
| 22/13
| 22/13
|-
|-
| 13
| 13
| 1090.55
| 1085.6
| 15/8
| '''15/8'''
|-
|-
| 14
| 14
| 66.75
| 61.4
| 33/32, 27/26, 25/24
| 33/32, 27/26, 25/24
|-
|-
| 15
| 15
| 242.95
| 237.2
| 15/13
| 15/13
|}
|}
: <nowiki>*</nowiki> in 2.3.5.11.13 POTE tuning
<nowiki/>* In 2.3.5.11.13 subgroup CTE tuning


=== Monkey ===
=== As a detemperament of 7et ===
{| class="wikitable right-1 right-2"
[[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 ==
* [[Tetracot7]] – [[6L&nbsp;1s]] scale
* [[Tetracot13]] – improper [[7L&nbsp;6s]]
* [[Tetracot20]] – improper [[7L&nbsp;13s]]
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! #
! rowspan="2" |
! Cents<nowiki>*</nowiki>
! colspan="3" | Euclidean
! Approximate Ratios
|-
|-
| 0
! Constrained
| 0.00
! Constrained & skewed
| 1/1
! Destretched
|-
|-
| 1
! Tenney
| 175.62
| CTE: ~10/9 = 176.0283{{c}}
| 11/10, 10/9
| 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
|-
|-
| 2
! rowspan="2" |  
| 351.24
! colspan="3" | Euclidean
| 11/9, '''16/13'''
|-
|-
| 3
! Constrained
| 526.87
! Constrained & skewed
| 15/11
! Destretched
|-
|-
| 4
! Tenney
| 702.49
| CTE: ~10/9 = 175.7765{{c}}
| '''3/2'''
| 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
|-
|-
| 5
! rowspan="2" |  
| 878.11
! colspan="3" | Euclidean
| 5/3
|-
|-
| 6
! Constrained
| 1053.73
! Constrained & skewed
| 11/6, 24/13
! Destretched
|-
|-
| 7
! Tenney
| 29.36
| CTE: ~10/9 = 175.8150{{c}}
| 55/54, 45/44, 40/39
| CWE: ~10/9 = 176.0854{{c}}
| POTE: ~10/9 = 176.1965{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
| 8
! Edo<br>generator
| 204.98
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
| 9/8
! Generator (¢)
! Comments
|-
|-
| 9
|  
| 380.60
| 11/10
| '''5/4'''
| 165.004
|  
|-
|-
| 10
|  
| 556.22
| 243/200
| '''11/8''', 18/13
| 168.574
| 1/2-comma
|-
|-
| 11
| 1\7
| 731.85
|
| 20/13
| 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
|-
|-
| 12
|  
| 907.47
| 27/20
| 22/13
| 173.184
| 1/3-comma
|-
|-
| 13
|  
| 1083.09
| 11/9
| 13/7, 15/8
| 173.704
|  
|-
|-
| 14
|  
| 58.71
| 81/80
| 33/32, 27/26, 25/24
| 174.501
| 2/7-comma
|-
|-
| 15
|  
| 234.34
| 11/6
| '''8/7''', 15/13
| 174.894
|  
|-
|-
| 16
| 7\48
| 409.96
|  
| 175.000
|  
|  
|-
|-
| 17
|  
| 585.58
| 11/8
| 45/32
| 175.132
| 2.3.5.11-subgroup 11-odd-limit minimax
|-
|-
| 18
| 761.20
|  
|  
| 3/2
| 175.489
| 1/4-comma
|-
|-
| 19
| 6\41
| 936.83
|  
| 12/7
| 175.610
|  
|-
|-
| 20
| 1112.45
|  
|  
| 13/11
| 175.899
| 2.3.5.11.13-subgroup 13- and 15-odd-limit minimax
|-
|-
| 21
| 88.07
|  
|  
|-
| 15/8
| 22
| 176.021
| 263.69
|  
|  
|-
|-
| 23
| 439.31
| 9/7
|-
| 24
| 614.94
| 10/7
|-
| 25
| 790.56
| 11/7
|-
| 26
| 966.18
|  
|  
|-
| 27
| 1141.80
| 27/14
|-
| 28
| 117.43
| 15/14
|}
: <nowiki>*</nowiki> in 13-limit POTE tuning
=== Bunya ===
{| class="wikitable right-1 right-2"
|-
! #
! Cents<nowiki>*</nowiki>
! Approximate Ratios
|-
| 0
| 0.00
| 1/1
|-
| 1
| 175.89
| 11/10, 10/9
|-
| 2
| 351.77
| 11/9, '''16/13'''
|-
| 3
| 527.66
| 15/11
|-
| 4
| 703.54
| '''3/2'''
|-
| 5
| 879.43
| 5/3
|-
| 6
| 1055.31
| 11/6, 24/13
|-
| 7
| 31.20
| 56/55, 55/54, 45/44, 40/39
|-
| 8
| 207.09
| 9/8
|-
| 9
| 382.97
| 5/4
| 5/4
| 176.257
| 5-odd-limit and 5-limit 9-odd-limit minimax, 2/9-comma
|-
|-
| 10
|  
| 558.86
| 13/9
| '''11/8''', 18/13
| 176.338
|  
|-
|-
| 11
| 5\34
| 734.74
|  
| 20/13
| 176.471
|  
|-
|-
| 12
|  
| 910.63
| 22/13
|-
| 13
| 1086.52
| 28/15, 15/8
|-
| 14
| 62.40
| 33/32, 28/27, 27/26, 25/24
|-
| 15
| 238.29
| 15/13
| 15/13
| 176.516
|
|-
|-
| 16
|  
| 414.17
| 5/3
| 14/11
| 176.872
| 1/5-comma
|-
|-
| 17
|  
| 590.06
| 13/10
| 7/5
| 176.890
|  
|-
|-
| 18
|  
| 765.94
| 13/12
| 14/9
| 176.905
|  
|-
|-
| 19
| 4\27
| 941.83
|  
|  
| 177.778
| 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
|-
|-
| 20
| 1117.72
| 21/11
|-
| 21
| 93.60
| 21/20
|-
| 22
| 269.49
| 7/6
|-
| 23
| 445.37
|  
|  
| 27/25
| 177.794
| 1/6-comma
|-
|-
| 24
| 621.26
|  
|  
| 243/125
| 178.452
| 1/7-comma
|-
|-
| 25
| 797.15
|  
|  
|-
| 26
| 973.03
| '''7/4'''
|-
| 27
| 1148.92
| 35/18
|-
| 28
| 124.80
| 14/13
|}
: <nowiki>*</nowiki> in 13-limit POTE tuning
=== Modus ===
{| class="wikitable right-1 right-2"
|-
! #
! Cents<nowiki>*</nowiki>
! Approximate Ratios
|-
| 0
| 0.00
| 1/1
|-
| 1
| 176.95
| 11/10, 10/9
|-
| 2
| 353.91
| 11/9, '''16/13'''
|-
| 3
| 530.86
| 15/11
| 15/11
| 178.984
|
|-
|-
| 4
| 707.81
| '''3/2'''
|-
| 5
| 884.77
| 5/3
|-
| 6
| 1061.72
| 11/6, 24/13, 13/7
|-
| 7
| 38.67
| 55/54, 45/44, 40/39, 36/35
|-
| 8
| 215.63
| 9/8, '''8/7'''
|-
| 9
| 392.58
| '''5/4'''
|-
| 10
| 569.53
| '''11/8''', 18/13
|-
| 11
| 746.49
| 20/13
|-
| 12
| 923.44
| 22/13, 12/7
|-
| 13
| 1100.39
| 15/8, 40/21
|-
| 14
| 77.35
| 27/26, 25/24, 22/21
|-
| 15
| 254.30
| 15/13
|-
| 16
| 431.25
| 9/7
|-
| 17
| 608.20
| 10/7
|-
| 18
| 785.16
| 11/7
|-
| 19
| 962.11
|  
|  
|-
| 13/8
| 20
| 179.736
| 1139.06
| 27/14
|-
| 21
| 116.02
| 15/14
|}
: <nowiki>*</nowiki> in 13-limit POTE tuning
 
=== Wollemia ===
{| class="wikitable right-1 right-2"
|-
! #
! Cents<nowiki>*</nowiki>
! Approximate Ratios
|-
| 0
| 0.00
| 1/1
|-
| 1
| 177.23
| 11/10, 10/9
|-
| 2
| 354.46
| 11/9, '''16/13'''
|-
| 3
| 531.69
| 15/11
|-
| 4
| 708.92
| '''3/2'''
|-
| 5
| 886.16
| 5/3
|-
| 6
| 1063.39
| 11/6, 24/13, 28/15
|-
| 7
| 40.62
| 55/54, 45/44, 40/39
|-
| 8
| 217.85
| 9/8
|-
| 9
| 395.08
| '''5/4''', 14/11
|-
| 10
| 572.31
| '''11/8''', 18/13, 7/5
|-
| 11
| 749.54
| 20/13, 14/9
|-
| 12
| 926.77
| 22/13
|-
| 13
| 1104.01
| 15/8
|-
| 14
| 81.24
| 27/26, 25/24
|-
| 15
| 258.47
| 15/13, 7/6
|-
| 16
| 435.70
|  
|  
|-
|-
| 17
| 3\20
| 612.93
|  
|  
| 180.000
| 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
|-
|-
| 18
| 790.16
|  
|  
|-
| 9/5
| 19
| 182.404
| 967.39
| '''7/4'''
|-
| 20
| 1144.62
|  
|  
|-
| 21
| 121.86
| 14/13
|}
|}
: <nowiki>*</nowiki> in 13-limit POTE tuning
<nowiki/>* Besides the octave


== Tuning spectra ==
== Music ==
=== Monkey ===
; [[Flora Canou]]
Gencom: [2 10/9; 100/99 105/104 144/143 243/242]
* [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


Gencom mapping: [{{val|1 1 1 5 2 4}}, {{val|0 4 9 -15 10 -2}}]
; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) – in 34edo tuning


{| class="wikitable center-all"
; [[Dustin Schallert]]
|-
* [https://web.archive.org/web/20201127015111/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 ''Tetracot Perc-Sitar'']
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
* [https://web.archive.org/web/20201129105050/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 ''Tetracot Jam'']
! Generator<br>(¢)
* [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
! Comments
|-
| | 11/10
| | 165.004
| |
|-
| | 11/9
| | 173.704
| |
|-
| | 14/13
| | 174.746
| |
|-
| | 12/11
| | 174.894
| |
|-
| | 11/8
| | 175.132
| |
|-
| | 14/11
| | 175.300
| | 11-odd-limit minimax
|-
| | 8/7
| | 175.412
| |
|-
| | 7/6
| | 175.428
| |
|-
| | 9/7
| | 175.438
| |
|-
| | 4/3
| | 175.489
| |
|-
| | 15/14
| | 175.694
| |
|-
| | 7/5
| | 175.729
| | 7, 9, 13 and 15-odd-limit minimax
|-
| | 13/11
| | 175.899
| |
|-
| | 16/15
| | 176.021
| |
|-
| | 5/4
| | 176.257
| | 5-odd-limit minimax
|-
| | 18/13
| | 176.338
| |
|-
| | 15/13
| | 176.516
| |
|-
| | 6/5
| | 176.872
| |
|-
| | 13/10
| | 176.890
| |
|-
| | 13/12
| | 176.905
| |
|-
| | 15/11
| | 178.984
| |
|-
| | 16/13
| | 179.736
| |
|-
| | 10/9
| | 182.404
| |
|}
 
=== Bunya ===
Gencom: [2 10/9; 100/99 144/143 225/224 243/242]
 
Gencom mapping: [{{val|1 1 1 -1 2 4}}, {{val|0 4 9 26 10 -2}}]
 
{| class="wikitable center-all"
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Comments
|-
| | 11/10
| | 165.004
| |
|-
| | 11/9
| | 173.704
| |
|-
| | 12/11
| | 174.894
| |
|-
| | 11/8
| | 175.132
| |
|-
| | 15/14
| | 175.427
| |
|-
| | 7/5
| | 175.442
| | 11-odd-limit minimax
|-
| | 4/3
| | 175.489
| |
|-
| | 8/7
| | 175.724
| |
|-
| | 7/6
| | 175.767
| | 7-odd-limit minimax
|-
| | 9/7
| | 175.829
| | 9-odd-limit minimax
|-
| | 13/11
| | 175.899
| | 13 and 15-odd-limit minimax
|-
| | 14/13
| | 176.011
| |
|-
| | 16/15
| | 176.021
| |
|-
| | 14/11
| | 176.094
| |
|-
| | 5/4
| | 176.257
| | 5-odd-limit minimax
|-
| | 18/13
| | 176.338
| |
|-
| | 15/13
| | 176.516
| |
|-
| | 6/5
| | 176.872
| |
|-
| | 13/10
| | 176.890
| |
|-
| | 13/12
| | 176.905
| |
|-
| | 15/11
| | 178.984
| |
|-
| | 16/13
| | 179.736
| |
|-
| | 10/9
| | 182.404
| |
|}
 
=== Modus ===
Gencom: [2 10/9; 64/63 78/77 100/99 144/143]
 
Gencom mapping: [{{val|1 1 1 4 2 4}}, {{val|0 4 9 -8 10 -2}}]
 
{| class="wikitable center-all"
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Comments
|-
| | 11/10
| | 165.004
| |
|-
| | 11/9
| | 173.704
| |
|-
| | 12/11
| | 174.894
| |
|-
| | 11/8
| | 175.132
| |
|-
| | 4/3
| | 175.489
| |
|-
| | 13/11
| | 175.899
| |
|-
| | 16/15
| | 176.021
| |
|-
| | 5/4
| | 176.257
| | 5-odd-limit minimax
|-
| | 18/13
| | 176.338
| |
|-
| | 15/13
| | 176.516
| |
|-
| | 14/11
| | 176.805
| | 11, 13 and 15-odd-limit minimax
|-
| | 6/5
| | 176.872
| |
|-
| | 13/10
| | 176.890
| |
|-
| | 13/12
| | 176.905
| |
|-
| | 15/14
| | 177.116
| |
|-
| | 9/7
| | 177.193
| | 9-odd-limit minimax
|-
| | 7/5
| | 177.499
| | 7-odd-limit minimax
|-
| | 7/6
| | 177.761
| |
|-
| | 14/13
| | 178.617
| |
|-
| | 8/7
| | 178.897
| |
|-
| | 15/11
| | 178.984
| |
|-
| | 16/13
| | 179.736
| |
|-
| | 10/9
| | 182.404
| |
|}
 
=== Wollemia ===
Gencom: [2 10/9; 56/55 91/90 100/99 243/242]
 
Gencom mapping: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 19 10 -2}}]
 
{| class="wikitable center-all"
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Comments
|-
| | 11/10
| | 165.004
| |
|-
| | 11/9
| | 173.704
| |
|-
| | 12/11
| | 174.894
| |
|-
| | 11/8
| | 175.132
| |
|-
| | 4/3
| | 175.489
| |
|-
| | 13/11
| | 175.899
| |
|-
| | 16/15
| | 176.021
| |
|-
| | 5/4
| | 176.257
| | 5-odd-limit minimax
|-
| | 18/13
| | 176.338
| |
|-
| | 15/13
| | 176.516
| |
|-
| | 6/5
| | 176.872
| |
|-
| | 13/10
| | 176.890
| |
|-
| | 13/12
| | 176.905
| |
|-
| | 8/7
| | 177.307
| | 7, 9, 11, 13 and 15-odd-limit minimax
|-
| | 14/13
| | 177.538
| |
|-
| | 7/6
| | 177.791
| |
|-
| | 7/5
| | 178.251
| |
|-
| | 9/7
| | 178.629
| |
|-
| | 15/11
| | 178.984
| |
|-
| | 14/11
| | 179.723
| |
|-
| | 16/13
| | 179.736
| |
|-
| | 15/14
| | 180.093
| |
|-
| | 10/9
| | 182.404
| |
|}
 
== Scales ==
* [[Tetracot7]] – [[6L 1s]] scale
* [[Tetracot13]] – improper [[7L 6s]]
* [[Tetracot20]] – improper [[7L 13s]]
 
== Music ==
; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) – in 34edo


; [[Xotla]]
; [[Xotla]]
* "Electrostat" from ''Lesser Groove'' (2020) [https://xotla.bandcamp.com/track/electrostat-tetracot-13 Bandcamp] | [https://www.youtube.com/watch?v=5SAuoyDwpgc YouTube] – ambient electro, tetracot[13] in 34edo
* "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:Temperaments]]
[[Category:Tetracot| ]] <!-- Main article -->
[[Category:Tetracot| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Tetracot family]]
 
[[Category:Rastmic clan]]
{{Todo| cleanup |comment=Move 7-limit extensions to their own pages. }}

Latest revision as of 11:25, 18 May 2026

Tetracot
Subgroups 2.3.5, 2.3.5.11, 2.3.5.11.13
Comma basis 20000/19683 (2.3.5);
100/99, 243/242 (2.3.5.11)
100/99, 144/143, 243/242 (2.3.5.11.13)
Reduced mapping ⟨1; 4 9 10 -2]
ET join 7 & 27e
Generators (CWE) ~10/9 = 176.1 ¢
MOS scales 6L 1s, 7L 6s, 7L 13s
Ploidacot tetracot
Pergen (P8, P5/4)
Color name Saquadyo
Minimax error 5-odd-limit: 3.07 ¢;
2.3.5.11.13 15-odd-limit: 10.9 ¢
Target scale size 5-odd-limit: 13 notes;
2.3.5.11.13 15-odd-limit: 20 notes
This page is about the regular temperament. For the ploidacot signature, see Ploidacot/Tetracot.

Tetracot, in this article, is the rank-2 temperament in the 2.3.5.11.13 subgroup generated by a submajor 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.

Equal temperaments that support tetracot include 27, 34, and 41.

Tetracot has four strong extensions for the 7-, 11-, and 13-limit, which use the same methods of obtaining the 11th and 13th harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the 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)3;
  • Bunya (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out 225/224 and thereby equating 7/2 with (15/8)2;
  • 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)3(3/2).

See Tetracot family for technical data.

Intervals

Interval chain

In the following table, odd harmonics and subharmonics 1–15 are in bold.

# Cents* Approximate ratios
0 0.0 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

As a detemperament of 7et

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 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

Tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 176.0283 ¢ CWE: ~10/9 = 176.0965 ¢ POTE: ~10/9 = 176.1598 ¢
2.3.5.11-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 175.7765 ¢ CWE: ~10/9 = 175.8847 ¢ POTE: ~10/9 = 175.9849 ¢
2.3.5.11.13-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 175.8150 ¢ CWE: ~10/9 = 176.0854 ¢ POTE: ~10/9 = 176.1965 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
11/10 165.004
243/200 168.574 1/2-comma
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
27/20 173.184 1/3-comma
11/9 173.704
81/80 174.501 2/7-comma
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 1/4-comma
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, 2/9-comma
13/9 176.338
5\34 176.471
15/13 176.516
5/3 176.872 1/5-comma
13/10 176.890
13/12 176.905
4\27 177.778 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
15/11 178.984
13/8 179.736
3\20 180.000 20ce val, 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
Dustin Schallert
Xotla
  • "Electrostat" from Lesser Groove (2020) – Spotify | Bandcamp | YouTube – ambient electro in Tetracot[13], 34edo tuning
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