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]] for the 2.3.5.11.13 [[subgroup]] defined by [[tempering out]] [[100/99]], [[144/143]], and [[243/242]].


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 }}.
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 extensions for 7, 11 and 13-limit include monkey (34 & 41), bunya (34d & 41), modus (27e & 34d) and wollemia (27e & 34).
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.


== Intervals ==
== 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, prime harmonics 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 ~ 121/120 all tempered together. In the following table, prime harmonics are in '''bold'''.  


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|}
|}
: <nowiki>*</nowiki> in 2.3.5.11.13 POTE tuning
: <nowiki>*</nowiki> in 2.3.5.11.13 POTE tuning
=== Monkey ===
{| class="wikitable right-1 right-2"
|-
! #
! Cents<nowiki>*</nowiki>
! Approximate Ratios
|-
| 0
| 0.00
| 1/1
|-
| 1
| 175.62
| 11/10, 10/9
|-
| 2
| 351.24
| 11/9, '''16/13'''
|-
| 3
| 526.87
| 15/11
|-
| 4
| 702.49
| '''3/2'''
|-
| 5
| 878.11
| 5/3
|-
| 6
| 1053.73
| 11/6, 24/13
|-
| 7
| 29.36
| 55/54, 45/44, 40/39
|-
| 8
| 204.98
| 9/8
|-
| 9
| 380.60
| '''5/4'''
|-
| 10
| 556.22
| '''11/8''', 18/13
|-
| 11
| 731.85
| 20/13
|-
| 12
| 907.47
| 22/13
|-
| 13
| 1083.09
| 13/7, 15/8
|-
| 14
| 58.71
| 33/32, 27/26, 25/24
|-
| 15
| 234.34
| '''8/7''', 15/13
|-
| 16
| 409.96
|
|-
| 17
| 585.58
| 45/32
|-
| 18
| 761.20
|
|-
| 19
| 936.83
| 12/7
|-
| 20
| 1112.45
|
|-
| 21
| 88.07
|
|-
| 22
| 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
|-
| 10
| 558.86
| '''11/8''', 18/13
|-
| 11
| 734.74
| 20/13
|-
| 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
|-
| 16
| 414.17
| 14/11
|-
| 17
| 590.06
| 7/5
|-
| 18
| 765.94
| 14/9
|-
| 19
| 941.83
|
|-
| 20
| 1117.72
| 21/11
|-
| 21
| 93.60
| 21/20
|-
| 22
| 269.49
| 7/6
|-
| 23
| 445.37
|
|-
| 24
| 621.26
|
|-
| 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
|-
| 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
|
|-
| 20
| 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
| 612.93
|
|-
| 18
| 790.16
|
|-
| 19
| 967.39
| '''7/4'''
|-
| 20
| 1144.62
|
|-
| 21
| 121.86
| 14/13
|}
: <nowiki>*</nowiki> in 13-limit POTE tuning
== Tuning spectra ==
=== Monkey ===
Gencom: [2 10/9; 100/99 105/104 144/143 243/242]
Gencom mapping: [{{val|1 1 1 5 2 4}}, {{val|0 4 9 -15 10 -2}}]
{| class="wikitable center-all"
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! 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 ==
== Scales ==
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[[Category:Tetracot| ]] <!-- main article -->
[[Category:Tetracot| ]] <!-- main article -->
[[Category:Tetracot family]]
[[Category:Tetracot family]]
{{Todo| cleanup |comment=Move 7-limit extensions to their own pages. }}

Revision as of 03:10, 31 December 2023

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.

It can be seen as implying a rank-2 tuning which is generated by a sub-major second of about 176 cents 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 27, 34, and 41.

Tetracot has many extensions for the 7-, 11- and 13-limit. See Tetracot extensions.

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 ~ 121/120 all tempered together. In the following table, prime harmonics are in bold.

# Cents* Approximate Ratios
0 0.00 1/1
1 176.20 11/10, 10/9
2 352.39 11/9, 16/13
3 528.59 15/11
4 704.79 3/2
5 880.98 5/3
6 1057.18 11/6, 24/13
7 33.38 55/54, 45/44, 40/39
8 209.57 9/8
9 385.77 5/4
10 561.96 11/8, 18/13
11 738.16 20/13
12 914.36 22/13
13 1090.55 15/8
14 66.75 33/32, 27/26, 25/24
15 242.95 15/13
* in 2.3.5.11.13 POTE tuning

Scales

Music

Flora Canou
Zhea Erose
Xotla
  • "Electrostat" from Lesser Groove (2020) – Spotify | Bandcamp | YouTube – ambient electro, tetracot[13] in 34edo
Retrieved from "https://en.xen.wiki/index.php?title=Tetracot&oldid=130124"