Trienstonic clan: Difference between revisions

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__FORCETOC__
The '''trienstonic clan''' of temperaments tempers out [[28/27]], the septimal third-tone or trienstonic comma.
Adding 16/15 to the trienstonic comma of 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharptone, sharp, and blacksmith.
 
Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharptone, sharp, and blacksmith.
 
= Father =
{{see also| Father family #Father }}


=Father=
Commas: 16/15, 28/27
Commas: 16/15, 28/27


Line 9: Line 12:
[<1 0 0 0|, <3/2 0 -1/4 1/4|, <5/2 0 1/4 -1/4|, <5/2 0 -3/4 3/4|]
[<1 0 0 0|, <3/2 0 -1/4 1/4|, <5/2 0 1/4 -1/4|, <5/2 0 -3/4 3/4|]


[[Eigenmonzo_subgroup|Eigenmonzo subgroup]]: 2.7/5
[[Eigenmonzo subgroup]]: 2.7/5


9-limit eigenmonzo subgroup: 2.9/5
9-limit eigenmonzo subgroup: 2.9/5


[[POTE_tuning|POTE generator]]: 742.002
[[POTE generator]]: 742.002


Map: [<1 0 4 -2|, <0 1 -1 3|]
Map: [<1 0 4 -2|, <0 1 -1 3|]
Line 23: Line 26:
Badness: 0.0213
Badness: 0.0213


=Uncle=
= Uncle =
Commas: 28/27, 256/245
Commas: 28/27, 256/245


Line 38: Line 41:
Badness: 0.0727
Badness: 0.0727


=Octokaidecal=
= Octokaidecal =
Commas: 28/27, 50/49
Commas: 28/27, 50/49


Line 45: Line 48:
9-limit eigenmonzo subgroup: 2.5
9-limit eigenmonzo subgroup: 2.5


[[POTE_tuning|POTE generator]]: 728.874
[[POTE generator]]: 728.874


Map: [<2 0 -5 -4|, <0 1 3 3|]
Map: [<2 0 -5 -4|, <0 1 3 3|]
Line 55: Line 58:
Badness: 0.0367
Badness: 0.0367


==11-limit==
== 11-limit ==
Commas: 28/27, 50/49, 55/54
Commas: 28/27, 50/49, 55/54


Line 66: Line 69:
Badness: 0.0302
Badness: 0.0302


=Opossum=
= Opossum =
Commas: 28/27, 126/125
Commas: 28/27, 126/125


Line 73: Line 76:
9-limit eigenmonzo subgroup: 2.7
9-limit eigenmonzo subgroup: 2.7


[[POTE_tuning|POTE generator]]: ~10/9 = 159.691
[[POTE generator]]: ~10/9 = 159.691


Map: [<1 2 3 4|, <0 -3 -5 -9|]
Map: [<1 2 3 4|, <0 -3 -5 -9|]
Line 83: Line 86:
Badness: 0.0407
Badness: 0.0407


==11-limit==
== 11-limit ==
Commas: 28/27, 55/54, 77/75
Commas: 28/27, 55/54, 77/75


11-limit eigenmonzo subgroup: 2.7
11-limit eigenmonzo subgroup: 2.7


[[POTE_tuning|POTE generator]]: ~10/9 = 159.807
[[POTE generator]]: ~10/9 = 159.807


Map: [<1 2 3 4 4|, <0 -3 -5 -9 -4|]
Map: [<1 2 3 4 4|, <0 -3 -5 -9 -4|]
Line 96: Line 99:
Badness: 0.0223
Badness: 0.0223


==13-limit==
== 13-limit ==
Commas: 28/27, 40/39, 55/54, 66/65
Commas: 28/27, 40/39, 55/54, 66/65


Line 103: Line 106:
15-limit eigenmonzo subgroup: 2.7
15-limit eigenmonzo subgroup: 2.7


[[POTE_tuning|POTE generator]]: ~10/9 = 158.805
[[POTE generator]]: ~10/9 = 158.805


Map: [<1 2 3 4 4 4|, <0 -3 -5 -9 -4 -2|]
Map: [<1 2 3 4 4 4|, <0 -3 -5 -9 -4 -2|]
Line 111: Line 114:
Badness: 0.0194
Badness: 0.0194


=Wallaby=
= Wallaby =
Commas: 28/27, 35/32
Commas: 28/27, 35/32


Line 124: Line 127:
Badness: 0.0585
Badness: 0.0585


=Kangaroo=
= Kangaroo =
== 5-limit ==
Comma: 64000/59049
Comma: 64000/59049


Line 135: Line 139:
Badness: 0.2958
Badness: 0.2958


==7-limit==
== 7-limit ==
Commas: 28/27, 1029/1000
Commas: 28/27, 1029/1000


Line 148: Line 152:
Badness: 0.0779
Badness: 0.0779


==11-limit==
== 11-limit ==
Commas: 28/27, 77/75, 245/242
Commas: 28/27, 77/75, 245/242


Line 159: Line 163:
Badness: 0.0432
Badness: 0.0432


==13-limit==
== 13-limit ==
Commas: 28/27, 40/39, 77/75, 147/143
Commas: 28/27, 40/39, 77/75, 147/143


Line 170: Line 174:
Badness: 0.0327
Badness: 0.0327


=Quindecic=
= Quindecic =
Commas: 28/27, 49/48, 55/54, 77/75
Commas: 28/27, 49/48, 55/54, 77/75


Line 180: Line 184:


Badness: 0.0289
Badness: 0.0289
[[Category:Theory]]
[[Category:Temperament clan]]
[[Category:Trienstonic]]
[[Category:Rank 2]]

Revision as of 15:23, 9 January 2021

The trienstonic clan of temperaments tempers out 28/27, the septimal third-tone or trienstonic comma.

Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharptone, sharp, and blacksmith.

Father

See also: Father family § Father

Commas: 16/15, 28/27

7-limit minimax

[<1 0 0 0|, <3/2 0 -1/4 1/4|, <5/2 0 1/4 -1/4|, <5/2 0 -3/4 3/4|]

Eigenmonzo subgroup: 2.7/5

9-limit eigenmonzo subgroup: 2.9/5

POTE generator: 742.002

Map: [<1 0 4 -2|, <0 1 -1 3|]

Wedgie: <<1 -1 3 -4 2 10||

EDOs: 5, 8d, 13cd, 21bcd

Badness: 0.0213

Uncle

Commas: 28/27, 256/245

7-limit eigenmonzo subgroup: 2.5/3

9-limit eigenmonzo subgroup: 2.9/5

Map: [<1 0 12 -2|, <0 1 -6 3|]

Wedgie: <<1 -6 3 -12 2 24||

EDOs: 13d, 18, 23bc, 41bcd

Badness: 0.0727

Octokaidecal

Commas: 28/27, 50/49

7-limit eigenmonzo subgroup: 2.5

9-limit eigenmonzo subgroup: 2.5

POTE generator: 728.874

Map: [<2 0 -5 -4|, <0 1 3 3|]

Wedgie: <<2 6 6 5 4 -2||

EDOs: 10, 18, 28b

Badness: 0.0367

11-limit

Commas: 28/27, 50/49, 55/54

POTE generator: ~3/2 = 732.330

Map: [<2 0 -5 -4 7|, <0 1 3 3 0|]

EDOs: 10, 18e

Badness: 0.0302

Opossum

Commas: 28/27, 126/125

7-limit eigenmonzo subgroup: 2.7

9-limit eigenmonzo subgroup: 2.7

POTE generator: ~10/9 = 159.691

Map: [<1 2 3 4|, <0 -3 -5 -9|]

Wedgie: <<3 5 9 1 6 7||

EDOs: 7d, 8d, 15

Badness: 0.0407

11-limit

Commas: 28/27, 55/54, 77/75

11-limit eigenmonzo subgroup: 2.7

POTE generator: ~10/9 = 159.807

Map: [<1 2 3 4 4|, <0 -3 -5 -9 -4|]

EDOs: 7d, 8d, 15

Badness: 0.0223

13-limit

Commas: 28/27, 40/39, 55/54, 66/65

13-limit eigenmonzo subgroup: 2.7

15-limit eigenmonzo subgroup: 2.7

POTE generator: ~10/9 = 158.805

Map: [<1 2 3 4 4 4|, <0 -3 -5 -9 -4 -2|]

EDOs: 7d, 8d, 15, 38bcef

Badness: 0.0194

Wallaby

Commas: 28/27, 35/32

POTE generator: ~3/2 = 691.351

Map: [<1 0 7 -2|, <0 1 -3 3|]

Wedgie: <<1 -3 3 -7 2 15||

EDOs: 2d, 5c, 19ccdd

Badness: 0.0585

Kangaroo

5-limit

Comma: 64000/59049

POTE generator: ~27/20 = 561.195

Map: [<1 0 -3|, <0 3 10|]

EDOs: 15, 47b, 62b

Badness: 0.2958

7-limit

Commas: 28/27, 1029/1000

POTE generator: ~7/5 = 560.328

Map: [<1 0 -3 -2|, <0 3 10 9|]

Wedgie: <<3 10 9 9 6 -7||

EDOs: 15

Badness: 0.0779

11-limit

Commas: 28/27, 77/75, 245/242

POTE generator: ~7/5 = 560.155

Map: [<1 0 -3 -2 -4|, <0 3 10 9 14|]

EDOs: 15

Badness: 0.0432

13-limit

Commas: 28/27, 40/39, 77/75, 147/143

POTE generator: ~7/5 = 559.770

Map: [<1 0 -3 -2 -4 0|, <0 3 10 9 14 7|]

EDOs: 15

Badness: 0.0327

Quindecic

Commas: 28/27, 49/48, 55/54, 77/75

POTE generator: ~13/8 = 852.924

Map: [<15 24 35 42 52 0|, <0 0 0 0 0 1|]

EDOs: 15, 30, 45bc

Badness: 0.0289

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