|
|
| Line 1: |
Line 1: |
| {{Breadcrumb|Tetracot}}
| | #redirect [[Tetracot family #Overview to extensions]] |
| | |
| [[Tetracot]] has multiple competing [[extension]]s to the [[7-limit]]. The simplest 2.3.5.11 [[subgroup]] [[comma]]s of tetracot are [[100/99|ptolemisma (100/99)]] and [[243/242|rastma (243/242)]], and it can be viewed as the merge of the two corresponding [[rank-3 temperament]]s. 100/99 is particularly tricky as it can suggest tempering out either [[225/224]] (leading to [[apollo]]) or [[385/384]] (leading to [[keemic]]), resulting in distinct extensions of tetracot: | |
| * '''Monkey''' (34 & 41) – tempering out 100/99, 243/242, 385/384
| |
| * '''Bunya''' (34d & 41) – tempering out 100/99, 225/224, 243/242
| |
| | |
| The two unite in [[41edo|41et]], where both 225/224 and 385/384 are tempered out.
| |
| | |
| In addition, there are two simpler entries:
| |
| * '''Modus''' (27e & 34d) – tempering out 64/63, 100/99, 243/242
| |
| * '''Wollemia''' (27e & 34) – tempering out 56/55, 100/99, 243/242
| |
| | |
| The two unite in [[27edo|27et]], where both 56/55 and 64/63 are tempered out.
| |
| | |
| Another possible path, which is a weak extension, is to temper out [[441/440]], [[540/539]], or [[2401/2400]], leading to [[octacot]], discussed separately.
| |
| | |
| == Interval chain ==
| |
| === 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
| |
| | |
| |
| |}
| |
| | |
| {{Todo|split page|comment=split to monkey, bunya, modus, and wollemia, as each of them is a temperament in its own rigth. }}
| |
| [[Category:Temperament extensions]]
| |
| [[Category:Rank-2 temperaments]]
| |
| [[Category:Tetracot]]
| |
| [[Category:Monkey]]
| |
| [[Category:Bunya]]
| |
| [[Category:Modus]]
| |
| [[Category:Wollemia]]
| |
| [[Category:Tetracot family]]
| |
| [[Category:Pages with mostly numerical content]]
| |