Tertiaseptal: Difference between revisions

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{{Todo|inline=1|improve synopsis|improve readability|comment= Add infobox. Cut huge tables to reasonable sizes.}}
#redirect [[Breedsmic temperaments #Tertiaseptal]]
 
'''Tertiaseptal''' is a [[regular temperament|temperament]] for the 7, 11, 13, and 17 limit. EDOs that support tertiaseptal include [[31edo]], [[140edo]], [[171edo]] and [[311edo]].
 
See [[Breedsmic temperaments #Tertiaseptal]] for more information.
 
== Interval chain ==
=== Tertiaseptal and tertia ===
 
{| class="wikitable"
|-
! rowspan="2"| generator
! rowspan="2"| cents value <sup>a</sup><br>(octave-reduced)
! colspan="2"| 17-limit ratio<br>(octave-reduced)
|-
! tertiaseptal <br>(31&amp;171)
! tertia <br>(31&amp;140)
|-
| 1
| 77.2
| 117/112, 256/245, 68/65
| 117/112, 256/245, 68/65, [[22/21]]
|-
| 2
| 154.4
| 130/119, [[35/32]]
| '''[[12/11]]''', 130/119, 35/32
|-
| 3
| 231.6
| colspan="2"| '''[[8/7]]'''
|-
| 4
| 308.8
| colspan="2"| 117/98, 140/117
|-
| 5
| 386.0
| colspan="2"| '''[[5/4]]'''
|-
| 6
| 463.1
| colspan="2"| '''[[17/13]]'''
|-
| 7
| 540.3
| 175/128
| '''[[15/11]]''', 175/128
|-
| 8
| 617.5
| colspan="2"| '''[[10/7]]'''
|-
| 9
| 694.7
| colspan="2"| 112/75
|-
| 10
| 771.9
| colspan="2"| [[25/16]]
|-
| 11
| 849.1
| 44/27, 80/49, 49/30, 85/52, '''[[18/11]]'''
| 80/49, 49/30, 85/52
|-
| 12
| 926.3
| colspan="2"| 128/75
|-
| 13
| 1003.5
| colspan="2"| [[25/14]]
|-
| 14
| 1080.7
| colspan="2"| '''[[28/15]]'''
|-
| 15
| 1157.9
| 39/20
| 39/20, 88/45
|-
| 16
| 35.1
| 55/54, 52/51, 51/50, [[50/49]], [[49/48]], [[45/44]]
| 56/55, 52/51, 51/50, 50/49, 49/48
|-
| 17
| 112.3
| colspan="2"| '''[[16/15]]'''
|-
| 18
| 189.4
| colspan="2"| 39/35
|-
| 19
| 266.6
| colspan="2"| '''[[7/6]]'''
|-
| 20
| 343.8
| 39/32
| 39/32, '''[[11/9]]'''
|-
| 21
| 421.0
| [[51/40]]
| '''[[14/11]]''', 51/40
|-
| 22
| 498.2
| colspan="2"| '''[[4/3]]'''
|-
| 23
| 575.4
| colspan="2"| 39/28
|-
| 24
| 652.6
| [[35/24]]
| '''[[16/11]]''', 35/24
|-
| 25
| 729.8
| colspan="2"| [[32/21]]
|-
| 26
| 807.0
| 51/32
| 35/22, 51/32
|-
| 27
| 884.2
| colspan="2"| '''[[5/3]]'''
|-
| 28
| 961.4
| colspan="2"| 68/39
|-
| 29
| 1038.6
| 51/28
| '''[[20/11]]''', 51/28
|-
| 30
| 1115.7
| [[40/21]], [[21/11]]
| 40/21
|-
| 31
| 1192.9
| colspan="2"|
|-
| 32
| 70.1
| colspan="2"| [[26/25]], [[25/24]]
|-
| 33
| 147.3
| 49/45, '''12/11'''
| 49/45
|-
| 34
| 224.5
| 91/80
| [[25/22]], 91/80
|-
| 35
| 301.7
| colspan="2"| [[25/21]]
|-
| 36
| 378.9
| 56/45, 96/77
| 56/45
|-
| 37
| 456.1
| colspan="2"| '''[[13/10]]'''
|-
| 38
| 533.3
| 34/25, '''15/11'''
| 34/25
|-
| 39
| 610.5
| colspan="2"| [[64/45]]
|-
| 40
| 687.7
| colspan="2"| 52/35
|-
| 41
| 764.8
| colspan="2"| '''[[14/9]]'''
|-
| 42
| 842.0
| colspan="2"| '''[[13/8]]'''
|-
| 43
| 919.2
| colspan="2"| '''[[17/10]]'''
|-
| 44
| 996.4
| colspan="2"| '''[[16/9]]'''
|-
| 45
| 1073.6
| colspan="2"| '''[[13/7]]'''
|-
| 46
| 1150.8
| 68/35, [[35/18]]
| 64/33, 68/35, 35/18
|-
| 47
| 28.0
| [[65/64]], [[64/63]], [[56/55]]
| 78/77, 65/64, 64/63, 55/54
|-
| 48
| 105.2
| colspan="2"| '''[[17/16]]'''
|-
| 49
| 182.4
| colspan="2"| '''[[10/9]]'''
|-
| 50
| 259.6
| 65/56, [[64/55]]
| 65/56
|-
| 51
| 336.8
| colspan="2"| '''[[17/14]]'''
|-
| 52
| 414.0
| 80/63, '''14/11'''
| 80/63
|-
| 53
| 491.1
| colspan="2"| 65/49
|-
| 54
| 568.3
| colspan="2"| [[25/18]]
|-
| 55
| 645.5
| '''16/11'''
|
|-
| 56
| 722.7
| 85/56
| 50/33, 85/56
|-
| 57
| 799.9
| 100/63, 35/22
| 100/63
|-
| 58
| 877.1
| 128/77
|
|-
| 59
| 954.3
| colspan="2"| '''[[26/15]]'''
|-
| 60
| 1031.5
| 136/75, '''20/11'''
| 136/75
|-
| 61
| 1108.7
| colspan="2"| 91/48, 256/135
|-
| 62
| 1185.9
| 208/105
| 196/99, 208/105
|-
| 63
| 63.1
| colspan="2"| [[28/27]]
|-
| 64
| 140.3
| colspan="2"| '''[[13/12]]'''
|-
| 65
| 217.4
| '''[[17/15]]''', 25/22
| '''17/15'''
|-
| 66
| 294.6
| [[32/27]]
| '''[[13/11]]''', 32/27
|-
| 67
| 371.8
| colspan="2"| [[26/21]]
|-
| 68
| 449.0
| colspan="2"| [[35/27]]
|-
| 69
| 526.2
| colspan="2"| 65/48
|-
| 70
| 603.4
| colspan="2"| '''[[17/12]]'''
|-
| 71
| 680.6
| colspan="2"| [[40/27]]
|-
| 72
| 757.8
| 65/42
| '''[[17/11]]''', 65/42
|-
| 73
| 835.0
| colspan="2"| [[34/21]]
|-
| 74
| 912.2
| 56/33
|
|-
| 75
| 989.4
| 39/22
| 136/77, 85/48
|-
| 76
| 1066.6
| colspan="2"| 50/27
|-
| 77
| 1143.7
| 64/33
| 85/44
|-
| 78
| 20.9
| 91/90, 85/84, 78/77
| [[100/99]], 91/90, 85/84
|-
| 79
| 98.1
| 35/33
|
|-
| 80
| 175.3
| 195/176
|
|-
| 81
| 252.5
| colspan="2"| 52/45
|-
| 82
| 329.7
| 40/33
|
|-
| 83
| 406.9
| colspan="2"| 91/72
|-
| 84
| 484.1
| colspan="2"| 119/90
|-
| 85
| 561.3
| colspan="2"| 112/81
|-
| 86
| 638.5
| colspan="2"| '''[[13/9]]'''
|-
| 87
| 715.7
| 68/45, 50/33
| 68/45
|-
| 88
| 792.8
| 128/81
| 52/33, 128/81
|-
| 89
| 870.0
| colspan="2"| 119/72
|-
| 90
| 947.2
| colspan="2"| 140/81
|-
| 91
| 1024.4
| colspan="2"| 65/36
|-
| 92
| 1101.6
| colspan="2"| '''[[17/9]]'''
|-
| 93
| 1178.8
| [[160/81]], 196/99, 240/121
| 65/33, 160/81
|-
| 94
| 56.0
| 91/88
| 34/33
|-
| 95
| 133.2
| colspan="2"| 68/63
|-
| 96
| 210.4
| 112/99
|
|-
| 97
| 287.6
| '''13/11'''
|
|-
| 98
| 364.8
| 68/55
|
|-
| 99
| 442.0
| 128/99
|
|-
| 100
| 519.1
| 104/77
|
|-
| 101
| 596.3
|
|
|-
| 102
| 673.5
|
|
|-
| 103
| 750.7
| '''17/11'''
|
|-
| 104
| 827.9
| 160/99
|
|-
| 105
| 905.1
|
|
|-
| 106
| 982.3
| 136/77
|
|-
| 107
| 1059.5
|
|
|-
| 108
| 1136.7
| 52/27, 85/44
| 52/27
|-
| 109
| 13.9
| 100/99
|
|-
| 110
| 91.1
| 128/121, [[256/243]]
| 104/99, 256/243
|-
| 111
| 168.3
|
|
|-
| 112
| 245.4
|
|
|-
| 113
| 322.6
|
|
|-
| 114
| 399.8
| colspan="2"| 34/27
|-
| 115
| 477.0
|
|
|-
| 116
| 554.2
|
|
|-
| 117
| 631.4
|
|
|-
| 118
| 708.6
|
|
|-
| 119
| 785.8
| 52/33
|
|-
| 120
| 863.0
|
|
|-
| 121
| 940.2
|
|
|-
| 122
| 1017.4
|
|
|-
| 123
| 1094.5
|
|
|-
| 124
| 1171.7
| 65/33
|
|-
| 125
| 48.9
| 34/33
|
|}
<sup>a</sup> in 7-limit POTE tuning
 
=== Hemitert ===
 
{| class="wikitable"
|-
! generator
! cents value <sup>a</sup><br>(octave-reduced)
! 11-limit ratio<br>(octave-reduced)
|-
| 1
| 38.6
| [[45/44]]
|-
| 2
| 77.2
| 256/245
|-
| 3
| 115.8
|
|-
| 4
| 154.4
| [[35/32]]
|-
| 5
| 193.0
|
|-
| 6
| 231.6
| '''[[8/7]]'''
|-
| 7
| 270.2
|
|-
| 8
| 308.8
|
|-
| 9
| 347.4
| '''[[11/9]]'''
|-
| 10
| 386.0
| '''[[5/4]]'''
|-
| 11
| 424.6
|
|-
| 12
| 463.1
| 64/49
|-
| 13
| 501.7
|
|-
| 14
| 540.3
|
|-
| 15
| 578.9
|
|-
| 16
| 617.5
| '''[[10/7]]'''
|-
| 17
| 656.1
|
|-
| 18
| 694.7
|
|-
| 19
| 733.3
|
|-
| 20
| 771.9
| [[25/16]]
|-
| 21
| 810.5
|
|-
| 22
| 849.1
|
|-
| 23
| 887.7
|
|-
| 24
| 926.3
|
|-
| 25
| 964.9
|
|-
| 26
| 1003.5
| [[25/14]]
|-
| 27
| 1042.1
|
|-
| 28
| 1080.7
| '''[[28/15]]'''
|-
| 29
| 1119.3
| [[21/11]]
|-
| 30
| 1157.9
|
|-
| 31
| 1196.5
|
|-
| 32
| 35.1
| [[50/49]], [[49/48]]
|-
| 33
| 73.7
|
|-
| 34
| 112.2
| '''[[16/15]]'''
|-
| 35
| 150.8
| '''[[12/11]]'''
|-
| 36
| 189.4
|
|-
| 37
| 228.0
|
|-
| 38
| 266.6
| '''[[7/6]]'''
|-
| 39
| 305.2
|
|-
| 40
| 343.8
|
|-
| 41
| 382.4
|
|-
| 42
| 421.0
|
|-
| 43
| 459.6
|
|-
| 44
| 498.2
| '''[[4/3]]'''
|-
| 45
| 536.8
| '''[[15/11]]'''
|-
| 46
| 575.4
|
|-
| 47
| 614.0
|
|-
| 48
| 652.6
|
|-
| 49
| 691.2
|
|-
| 50
| 729.8
| 32/21
|-
| 51
| 768.4
|
|-
| 52
| 807.0
|
|-
| 53
| 845.6
|
|-
| 54
| 884.2
| '''[[5/3]]'''
|-
| 55
| 922.8
|
|-
| 56
| 961.4
|
|-
| 57
| 999.9
|
|-
| 58
| 1038.5
|
|-
| 59
| 1077.1
|
|-
| 60
| 1115.7
| 40/21
|-
| 61
| 1154.3
|
|-
| 62
| 1192.9
|
|-
| 63
| 31.5
|
|-
| 64
| 70.1
| [[25/24]]
|-
| 65
| 108.7
|
|-
| 66
| 147.3
|
|-
| 67
| 185.9
|
|-
| 68
| 224.5
|
|-
| 69
| 263.1
|
|-
| 70
| 301.7
|
|-
| 71
| 340.3
|
|-
| 72
| 378.9
|
|-
| 73
| 417.5
| '''[[14/11]]'''
|-
| 74
| 456.1
|
|-
| 75
| 494.7
|
|-
| 76
| 533.3
|
|-
| 77
| 571.9
|
|-
| 78
| 610.5
|
|-
| 79
| 649.1
| '''[[16/11]]'''
|-
| 80
| 687.6
|
|-
| 81
| 726.2
|
|-
| 82
| 764.8
| '''[[14/9]]'''
|-
| 83
| 803.4
|
|-
| 84
| 842.0
|
|-
| 85
| 880.6
|
|-
| 86
| 919.2
|
|-
| 87
| 957.8
|
|-
| 88
| 996.4
| '''[[16/9]]'''
|-
| 89
| 1035.0
| '''[[20/11]]'''
|-
| 90
| 1073.6
|
|-
| 91
| 1112.2
|
|-
| 92
| 1150.8
|
|-
| 93
| 1189.4
|
|-
| 94
| 28.0
| [[64/63]]
|-
| 95
| 66.6
|
|-
| 96
| 105.2
|
|-
| 97
| 143.8
|
|-
| 98
| 182.4
| '''[[10/9]]'''
|}
<sup>a</sup> in 11-limit POTE tuning
 
== Tuning spectra ==
=== Tertiaseptal ===
Gencom: [2 68/65; 243/242 375/374 441/440 625/624 3584/3575]
 
Gencom mapping: [{{val|1 3 2 3 7 1 1}}, {{val|0 -22 5 -3 -55 42 48}}]
 
{| class="wikitable center-all"
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! generator<br>(¢)
! comments
|-
| 8/7
| 77.0580
|
|-
| 13/10
| 77.1409
|
|-
| 14/13
| 77.1489
|
|-
| 16/13
| 77.1554
|
|-
| 15/13
| 77.1569
|
|-
| 16/15
| 77.1607
|
|-
| 13/12
| 77.1652
|
|-
| 18/13
| 77.1700
|
|-
| 20/17
| 77.1777
|
|-
| 17/14
| 77.1790
|
|-
| 17/15
| 77.1798
|
|-
| 15/14
| 77.1827
|
|-
| 4/3
| 77.1839
|
|-
| 18/17
| 77.1853
|
|-
| 24/17
| 77.1857
|
|-
| 7/5
| 77.1860
| 7 and 9-odd-limit minimax
|-
| 17/16
| 77.1866
|
|-
| 10/9
| 77.1919
|
|-
| 9/7
| 77.1931
|
|-
| 6/5
| 77.1985
| 5-odd-limit minimax
|-
| 7/6
| 77.2037
|
|-
| 13/11
| 77.2083
| 13, 15 and 17-odd-limit minimax
|-
| 22/17
| 77.2198
|
|-
| 11/8
| 77.2488
|
|-
| 11/10
| 77.2499
| 11-odd-limit minimax
|-
| 14/11
| 77.2598
|
|-
| 5/4
| 77.2627
|
|-
| 15/11
| 77.2882
|
|-
| 12/11
| 77.2920
|
|-
| 17/13
| 77.4046
|
|-
| 11/9
| 77.5084
|
|}
 
=== Tertia ===
Gencom: [2 22/21; 352/351 385/384 561/560 625/624 715/714]
 
Gencom mapping: [{{val|1 3 2 3 5 1 1}}, {{val|0 -22 5 -3 -24 42 48}}]
 
{| class="wikitable center-all"
|-
! eigenmonzo<br>(unchanged interval)
! generator<br>(¢)
! comments
|-
| 12/11
| 75.3185
|
|-
| 15/11
| 76.7073
|
|-
| 14/11
| 77.0242
|
|-
| 11/8
| 77.0284
|
|-
| 8/7
| 77.0580
|
|-
| 11/10
| 77.0688
|
|-
| 13/11
| 77.1092
|
|-
| 22/17
| 77.1339
|
|-
| 13/10
| 77.1409
|
|-
| 14/13
| 77.1489
|
|-
| 16/13
| 77.1554
|
|-
| 15/13
| 77.1569
|
|-
| 16/15
| 77.1607
|
|-
| 13/12
| 77.1652
|
|-
| 18/13
| 77.1700
| 13, 15 and 17-odd-limit minimax
|-
| 20/17
| 77.1777
|
|-
| 17/14
| 77.1790
|
|-
| 17/15
| 77.1798
|
|-
| 15/14
| 77.1827
|
|-
| 4/3
| 77.1839
| 11-odd-limit minimax
|-
| 18/17
| 77.1853
|
|-
| 24/17
| 77.1857
|
|-
| 7/5
| 77.1860
| 7 and 9-odd-limit minimax
|-
| 17/16
| 77.1866
|
|-
| 10/9
| 77.1919
|
|-
| 9/7
| 77.1931
|
|-
| 6/5
| 77.1985
| 5-odd-limit minimax
|-
| 7/6
| 77.2037
|
|-
| 5/4
| 77.2627
|
|-
| 11/9
| 77.3704
|
|-
| 17/13
| 77.4046
|
|}
 
=== Hemitert ===
Gencom: [2 45/44; 2401/2400 3025/3024 65625/65536]
 
Gencom mapping: [{{val|1 3 2 3 6}}, {{val|0 -44 10 -6 -79}}]
 
{| class="wikitable center-all"
|-
! eigenmonzo<br>(unchanged interval)
! generator<br>(¢)
! comments
|-
| 8/7
| 38.5290
|
|-
| 16/15
| 38.5803
|
|-
| 12/11
| 38.5896
|
|-
| 11/8
| 38.5909
|
|-
| 15/14
| 38.5913
|
|-
| 4/3
| 38.5919
|
|-
| 7/5
| 38.5930
| 7, 9 and 11-odd-limit minimax
|-
| 11/10
| 38.5955
|
|-
| 10/9
| 38.59596
|
|-
| 14/11
| 38.59600
|
|-
| 9/7
| 38.5965
|
|-
| 15/11
| 38.5989
|
|-
| 6/5
| 38.5992
| 5-odd-limit minimax
|-
| 11/9
| 38.6009
|
|-
| 7/6
| 38.6019
|
|-
| 5/4
| 38.6314
|
|}


[[Category:Tertiaseptal| ]] <!-- main article -->
[[Category:Tertiaseptal| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Microtemperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Horwell temperaments]]
[[Category:Horwell temperaments]]
[[Category:Metric microtemperaments]]
[[Category:Metric microtemperaments]]

Latest revision as of 10:26, 30 October 2026

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