Aberschismic temperaments: Difference between revisions

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The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal diesis, {{monzo| 2 2 -1 -1 }} = [[36/35]] into two equal steps.


The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal diesis, {{monzo| 2 2 -1 -1 }} = [[36/35]] into two equal steps. Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are [[Meantone family #Dominant|dominant]], [[Schismatic family #Garibaldi|garibaldi]], [[Breedsmic temperaments #Hemififths|hemififths]], [[Ragismic microtemperaments #Amity|amity]], [[Hemimean clan #Misty|misty]], [[Gamelismic clan #Rodan|rodan]], [[Kleismic family #Countercata|countercata]] and [[Mirkwai clan #Kwai|kwai]].
Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are [[Meantone family #Dominant|dominant]], [[Schismatic family #Garibaldi|garibaldi]], [[Breedsmic temperaments #Hemififths|hemififths]], [[Ragismic microtemperaments #Amity|amity]], [[Hemimean clan #Misty|misty]], [[Gamelismic clan #Rodan|rodan]], [[Kleismic family #Countercata|countercata]] and [[Mirkwai clan #Kwai|kwai]].


= Buzzard =
= Buzzard =
{{see also| Vulture family }}
Subgroup: 2.3.5.7
[[Comma list]]: 1728/1715, 5120/5103
[[Comma list]]: 1728/1715, 5120/5103


[[Mapping]]: [<1 0 -6 4|, <0 4 21 -3|]
[[Mapping]]: [{{val| 1 0 -6 4 }}, {{val| 0 4 21 -3 }}]


[[Wedgie]]: <<4 21 -3 24 -16 -66||
{{Multival|legend=1| 4 21 -3 24 -16 -66 }}


[[POTE_tuning|POTE generator]]: ~21/16 = 475.636
[[POTE generator]]: ~21/16 = 475.636


{{Val list|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
{{Val list|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
Line 17: Line 21:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 5120/5103
Comma list: 176/175, 540/539, 5120/5103


Mapping: [<1 0 -6 4 -12|, <0 4 21 -3 39|]
Mapping: [{{val| 1 0 -6 4 -12 }}, {{val| 0 4 21 -3 39 }}]


POTE generator: ~21/16 = 475.700
POTE generator: ~21/16 = 475.700
2, 3, 5, 53, 58, 111, 391,
 
Vals: {{Val list| 53, 58, 111, 280cd, 391cd }}
Vals: {{Val list| 53, 58, 111, 280cd, 391cd }}


Badness: 0.034484
Badness: 0.034484


== 13-limit ==
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 540/539, 676/675
Comma list: 176/175, 351/350, 540/539, 676/675


Mapping: [<1 0 -6 4 -12 -7|, <0 4 21 -3 39 27|]
Mapping: [{{val| 1 0 -6 4 -12 -7 }}, {{val| 0 4 21 -3 39 27 }}]


POTE generator: ~21/16 = 475.697
POTE generator: ~21/16 = 475.697
Line 38: Line 48:
Badness: 0.018842
Badness: 0.018842


== 17-limit ==
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
Comma list: 176/175, 256/255, 351/350, 442/441, 540/539


Mapping: [<1 0 -6 4 -12 -7 14|, <0 4 21 -3 39 27 -25|]
Mapping: [{{val| 1 0 -6 4 -12 -7 14 }}, {{val| 0 4 21 -3 39 27 -25 }}]


POTE generator: ~21/16 = 475.692
POTE generator: ~21/16 = 475.692
Line 49: Line 62:
Badness: 0.018403
Badness: 0.018403


== 19-limit ==
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539


Mapping: [<1 0 -6 4 -12 -7 14 -12|, <0 4 21 -3 39 27 -25 41|]
Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]


POTE generator: ~21/16 = 475.679
POTE generator: ~21/16 = 475.679
Line 61: Line 77:


== Buteo ==
== Buteo ==
Subgroup: 2.3.5.7.11
Comma list: 99/98, 385/384, 2200/2187
Comma list: 99/98, 385/384, 2200/2187


Mapping: [<1 0 -6 4 9|, <0 4 21 -3 -14|]
Mapping: [{{val| 1 0 -6 4 9 }}, {{val| 0 4 21 -3 -14 }}]


POTE generator: ~21/16 = 475.436
POTE generator: ~21/16 = 475.436
Line 72: Line 91:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 275/273, 385/384, 572/567
Comma list: 99/98, 275/273, 385/384, 572/567


Mapping: [<1 0 -6 4 9 -7|, <0 4 21 -3 -14 27|]
Mapping: [{{val| 1 0 -6 4 9 -7 }}, {{val| 0 4 21 -3 -14 27 }}]


POTE generator: ~21/16 = 475.464
POTE generator: ~21/16 = 475.464
Line 83: Line 105:


= Undecental =
= Undecental =
Subgroup: 2.3.5.7
[[Comma list]]: 5120/5103, 235298/234375
[[Comma list]]: 5120/5103, 235298/234375


[[Mapping]]: [<1 0 61 71|, <0 1 -37 -43|]
[[Mapping]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]


[[Wedgie]]: <<1 -37 -43 -61 -71 4||
{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}


[[POTE_tuning|POTE generator]]: ~3/2 = 703.039
[[POTE generator]]: ~3/2 = 703.039


{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}


Badness: 0.094603
[[Badness]]: 0.094603


= Leapday =
= Leapday =
[[Comma]]: 10737418240/10460353203 (trisayo comma)


[[Mapping]]: [<1 0 -31|, <0 1 21|]
Subgroup: 2.3.5


[[POTE tuning|POTE generator]]: ~3/2 = 704.179
[[Comma list]]: 10737418240/10460353203
 
[[Mapping]]: [{{val| 1 0 -31 }}, {{val| 0 1 21 }}]
 
[[POTE generator]]: ~3/2 = 704.179


{{Val list|legend=1| 29, 46, 121, 167, 455bc, 622bbc }}
{{Val list|legend=1| 29, 46, 121, 167, 455bc, 622bbc }}
Line 107: Line 135:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
[[Comma list]]: 686/675, 5120/5103
[[Comma list]]: 686/675, 5120/5103


[[Mapping]]: [<1 0 -31 -21|, <0 1 21 15|]
[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]


[[Wedgie]]: <<1 21 15 31 21 -24||
{{Multival|legend=1| 1 21 15 31 21 -24 }}


[[POTE tuning|POTE generator]]: ~3/2 = 704.263
[[POTE generator]]: ~3/2 = 704.263


{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
Line 120: Line 151:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440, 686/675
Comma list: 121/120, 441/440, 686/675


Mapping: [<1 0 -31 -21 -14|, <0 1 21 15 11|]
Mapping: [{{val| 1 0 -31 -21 -14 }}, {{val| 0 1 21 15 11 }}]


POTE generator: ~3/2 = 704.250
POTE generator: ~3/2 = 704.250
Line 131: Line 165:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 169/168, 352/351
Comma list: 91/90, 121/120, 169/168, 352/351


Mapping: [<1 0 -31 -21 -14 -9|, <0 1 21 15 11 8|]
Mapping: [{{val| 1 0 -31 -21 -14 -9 }}, {{val| 0 1 21 15 11 8 }}]


POTE generator: ~3/2 = 704.214
POTE generator: ~3/2 = 704.214
Line 142: Line 179:


== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168


Mapping: [<1 0 -31 -21 -14 -9 -34|, <0 1 21 15 11 8 24|]
Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 }}, {{val| 0 1 21 15 11 8 24 }}]


POTE generator: ~3/2 = 704.229
POTE generator: ~3/2 = 704.229
Line 152: Line 192:
Badness: 0.017863
Badness: 0.017863


== 19-limit ==
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168


Mapping: [<1 0 -31 -21 -14 -9 -34 9|, <0 1 21 15 11 8 24 -3|]
Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 }}, {{val| 0 1 21 15 11 8 24 -3 }}]


POTE generator: ~3/2 = 704.135
POTE generator: ~3/2 = 704.135
Line 163: Line 206:
Badness: 0.017356
Badness: 0.017356


== Leapling ==
=== Leapling ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168


Revision as of 07:00, 22 April 2021

The hemifamity temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103, dividing an exact or approximate septimal diesis, [2 2 -1 -1⟩ = 36/35 into two equal steps.

Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are dominant, garibaldi, hemififths, amity, misty, rodan, countercata and kwai.

Buzzard

See also: Vulture family

Subgroup: 2.3.5.7

Comma list: 1728/1715, 5120/5103

Mapping: [⟨1 0 -6 4], ⟨0 4 21 -3]]

Wedgie: ⟨⟨ 4 21 -3 24 -16 -66 ]]

POTE generator: ~21/16 = 475.636

Template:Val list

Badness: 0.047963

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 5120/5103

Mapping: [⟨1 0 -6 4 -12], ⟨0 4 21 -3 39]]

POTE generator: ~21/16 = 475.700

Vals: Template:Val list

Badness: 0.034484

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 676/675

Mapping: [⟨1 0 -6 4 -12 -7], ⟨0 4 21 -3 39 27]]

POTE generator: ~21/16 = 475.697

Vals: Template:Val list

Badness: 0.018842

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 442/441, 540/539

Mapping: [⟨1 0 -6 4 -12 -7 14], ⟨0 4 21 -3 39 27 -25]]

POTE generator: ~21/16 = 475.692

Vals: Template:Val list

Badness: 0.018403

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539

Mapping: [⟨1 0 -6 4 -12 -7 14 -12], ⟨0 4 21 -3 39 27 -25 41]]

POTE generator: ~21/16 = 475.679

Vals: Template:Val list

Badness: 0.015649

Buteo

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 2200/2187

Mapping: [⟨1 0 -6 4 9], ⟨0 4 21 -3 -14]]

POTE generator: ~21/16 = 475.436

Vals: Template:Val list

Badness: 0.060238

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 275/273, 385/384, 572/567

Mapping: [⟨1 0 -6 4 9 -7], ⟨0 4 21 -3 -14 27]]

POTE generator: ~21/16 = 475.464

Vals: Template:Val list

Badness: 0.039854

Undecental

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

Wedgie: ⟨⟨ 1 -37 -43 -61 -71 4 ]]

POTE generator: ~3/2 = 703.039

Template:Val list

Badness: 0.094603

Leapday

Subgroup: 2.3.5

Comma list: 10737418240/10460353203

Mapping: [⟨1 0 -31], ⟨0 1 21]]

POTE generator: ~3/2 = 704.179

Template:Val list

Badness: 0.523182

7-limit

Subgroup: 2.3.5.7

Comma list: 686/675, 5120/5103

Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]

Wedgie: ⟨⟨ 1 21 15 31 21 -24 ]]

POTE generator: ~3/2 = 704.263

Template:Val list

Badness: 0.096123

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

POTE generator: ~3/2 = 704.250

Vals: Template:Val list

Badness: 0.038624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

POTE generator: ~3/2 = 704.214

Vals: Template:Val list

Badness: 0.024732

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

POTE generator: ~3/2 = 704.229

Vals: Template:Val list

Badness: 0.017863

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9], ⟨0 1 21 15 11 8 24 -3]]

POTE generator: ~3/2 = 704.135

Vals: Template:Val list

Badness: 0.017356

Leapling

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168

Mapping: [<1 0 -31 -21 -14 -9 -34 -37|, <0 1 21 15 11 8 24 26|]

POTE generator: ~3/2 = 704.123

Vals: Template:Val list

Badness: 0.019065

Mystery

Commas: 5120/5103, 50421/50000

POTE generator: ~5/4 = 388.646

Map: [<29 46 0 14|, <0 0 1 1|]

Wedgie: <<0 29 29 46 46 -14||

EDOs: 29, 58, 87, 145

Badness: 0.1037

11-limit

Commas: 441/440, 896/891, 3388/3375

POTE generator: ~5/4 = 388.460

Map: [<29 46 0 14 33|, <0 0 1 1 1|]

EDOs: 29, 58, 87, 145

Badness: 0.0343

13-limit

Commas: 196/195, 352/351, 364/363, 676/675

POTE generator: ~5/4 = 388.354

Map: [<29 46 0 14 33 40|, <0 0 1 1 1 1|]

EDOs: 29, 58, 87, 145, 232, 377

Badness: 0.0186

Quanic

Commas: 5120/5103, 5832000/5764801

POTE generator: ~160/147 = 140.493

Map: [<1 1 -4 0|, <0 5 54 24|]

EDOs: 94, 111, 205

Badness: 0.1795

11-limit

Commas: 540/539, 1331/1323, 5120/5103

POTE generator: ~88/81 = 140.489

Map: [<1 1 -4 0 1|, <0 5 54 24 21|]

EDOs: 94, 111, 205

Badness: 0.0587

13-limit

Commas: 352/351, 540/539, 729/728, 1331/1323

POTE generator: ~13/12 = 140.496

Map: [<1 1 -4 0 1 3|, <0 5 54 24 21 6|]

EDOs: 94, 111, 205

Badness: 0.0325

17-limit

Commas: 352/351, 442/441, 540/539, 715/714, 847/845

POTE generator: ~13/12 = 140.497

Map: [<1 1 -4 0 1 3 -2|, <0 5 54 24 21 6 52|]

EDOs: 94, 111, 205

Badness: 0.0211

19-limit

Commas: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

POTE generator: ~13/12 = 140.496

Map: [<1 1 -4 0 1 3 -2 -5|, <0 5 54 24 21 6 52 79|]

EDOs: 94, 111, 205

Badness: 0.0173

Supers

Commas: 5120/5103, 118098/117649

POTE generator: ~9/7 = 434.218

Map: [<2 1 -12 2|, <0 3 23 5|]

Wedgie: <<6 46 10 59 -1 -106||

EDOs: 58, 94, 152

Badness: 0.092748

11-limit

Commas: 540/539, 4000/3993, 5120/5103

POTE generator: ~9/7 = 434.217

Map: [<2 1 -12 2 -9|, <0 3 23 5 22|]

EDOs: 58, 94, 152

Badness: 0.028240

13-limit

Commas: 352/351, 540/539, 729/728, 1575/1573

POTE generator: ~9/7 = 434.221

Map: [<2 1 -12 2 -9 -2|, <0 3 23 5 22 13|]

EDOs: 58, 94, 152f

Badness: 0.021645

17-limit

Commas: 170/169, 289/288, 352/351, 442/441, 561/560

POTE generator: ~9/7 = 434.181

Map: [<2 1 -12 2 -9 -2 6|, <0 3 23 5 22 13 3|]

EDOs: 58, 94, 152f

Badness: 0.021316

Alphaquarter

Commas: 5120/5103, 29360128/29296875

POTE generator: ~16128/15625 = 55.243

Map: [<1 2 2 0|, <0 -9 7 61|]

Wedgie: <<9 -7 -61 -32 -122 -122||

EDOs: 87, 152, 239, 391

Badness: 0.1166

11-limit

Commas: 3025/3024, 4000/3993, 5120/5103

POTE generator: ~33/32 = 55.243

Map: [<1 2 2 0 3|, <0 -9 7 61 10|]

EDOs: 87, 152, 239, 391

Badness: 0.0296

Septiquarter

Commas: 5120/5103, 420175/419904

POTE generator: ~147/128 = 242.453

Map: [<1 3 10 2|, <0 -7 -38 4|]

Wedgie: <<7 38 -4 44 -26 -116||

EDOs: 94, 99, 292, 391, 881bd, 1272bcd

Badness: 0.0538

Semiseptiquarter

Commas: 5120/5103, 9801/9800, 14641/14580

POTE generators: ~121/105 = 242.4511

Map: [<2 6 20 4 15|, <0 -7 -38 4 -20|]

EDOs: 94, 198, 292, 490

Badness: 0.0642

13-limit

Commas: 352/351, 847/845, 1716/1715, 14641/14580

POTE generators: ~121/105 = 242.4448

Map: [<2 6 20 4 15 7|, <0 -7 -38 4 -20 1|]

EDOs: 94, 198, 490f

Badness: 0.0348

Tricot (aka Trimot)

See also: Tricot family

The generator for tricot is the real cube root of third harmonic, 31/3, tuned between 63/44 and 13/9. Tricot can be described as 53&70 temperament (also known as trimot), tempering out the tricot comma, [39 -29 3⟩ in the 5-limit, 2430/2401 (nuwell comma) and 5120/5103 in the 7-limit, 99/98 and 121/120 in the 11-limit, 169/168, 352/351, 640/637, and 729/728 in the 13-limit.

Comma list: 2430/2401, 5120/5103

Mapping: [<1 0 -13 -3|, <0 3 29 11|]

Wedgie: <<3 29 11 39 9 -56||

POTE generator: ~81/56 = 634.026

Template:Val list

Badness: 0.100127

11-limit

Comma list: 99/98, 121/120, 5120/5103

Mapping: [<1 0 -13 -3 -5|, <0 3 29 11 16|]

POTE generator: ~63/44 = 634.027

Vals: Template:Val list

Badness: 0.056134

13-limit

Comma list: 99/98, 121/120, 169/168, 352/351

Mapping: [<1 0 -13 -3 -5 0|, <0 3 29 11 16 7|]

POTE generator: ~13/9 = 634.012

Vals: Template:Val list

Badness: 0.032102

Ketchup

Comma list: 5120/5103, 1071875/1062882

Mapping: [<2 3 4 6|, <0 4 15 -9|]

Wedgie: <<8 30 -18 29 -51 -126||

POTE generator: ~64/63 = ~81/80 = 25.719

Template:Val list

Badness: 0.084538

11-limit

Comma list: 385/384, 1331/1323, 2200/2187

Mapping: [<2 3 4 6 7|, <0 4 15 -9 -2|]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693

Vals: Template:Val list

Badness: 0.039555

13-limit

Comma list: 325/324, 352/351, 847/845, 1331/1323

Mapping: [<2 3 4 6 7 8|, <0 4 15 -9 -2 -14|]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697

Vals: Template:Val list

Badness: 0.024824

17-limit

Comma list: 289/288, 325/324, 352/351, 385/384, 561/560

Map: [<2 3 4 6 7 8 8|, <0 4 15 -9 -2 -14 4|]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701

Vals: Template:Val list

Badness: 0.016591

19-limit

Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560

Mapping: [<2 3 4 6 7 8 8 9|, <0 4 15 -9 -2 -14 4 -12|]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660

Vals: Template:Val list

Badness: 0.018170

23-limit

Commas: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384

Mapping: [<2 3 4 6 7 8 8 9 9|, <0 4 15 -9 -2 -14 4 -12 1|]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661

Vals: Template:Val list

Badness: 0.014033

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