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{{interwiki
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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}}


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


See [[Tetracot family]] or [[No-sevens subgroup temperaments #Tetracot]] for more technical data.
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]]).


== Interval chain ==
See [[Tetracot family]] for technical data.
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'''.  
 
== Intervals ==
=== 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
| 176.3
| 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
 
=== 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 ==
* [[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 ==
== 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"
|-
|-
! Edo<br>Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator<br>(¢)
! Generator (¢)
! Comments
! Comments
|-
|-
Line 106: Line 179:
| 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
|-
|  
|  
| 27/20
| 173.184
| 1/3-comma
|-
|-
|  
|  
Line 116: Line 199:
| 173.704
| 173.704
|  
|  
|-
|
| 81/80
| 174.501
| 2/7-comma
|-
|-
|  
|  
Line 130: Line 218:
| 11/8
| 11/8
| 175.132
| 175.132
|  
| 2.3.5.11-subgroup 11-odd-limit minimax
|-
|-
|  
|  
| 3/2
| 3/2
| 175.489
| 175.489
|  
| 1/4-comma
|-
|-
| 6\41
| 6\41
Line 145: Line 233:
| 13/11
| 13/11
| 175.899
| 175.899
|  
| 2.3.5.11.13-subgroup 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 155: Line 243:
| 5/4
| 5/4
| 176.257
| 176.257
| 5-odd-limit minimax
| 5-odd-limit and 5-limit 9-odd-limit minimax, 2/9-comma
|-
|-
|  
|  
Line 175: Line 263:
| 5/3
| 5/3
| 176.872
| 176.872
|  
| 1/5-comma
|-
|-
|  
|  
Line 190: Line 278:
|  
|  
| 177.778
| 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
|-
|-
|  
|  
Line 205: Line 303:
|  
|  
| 180.000
| 180.000
|  
| 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 212: Line 310:
|  
|  
|}
|}
<nowiki/>* Besides the octave


== 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) – [[modus]] in 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, 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 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]]

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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