Keemic temperaments: Difference between revisions

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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasitemp]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasitemp]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 2401/2400
[[Comma list]]: 875/864, 2401/2400


[[Mapping]]: [{{val| 1 5 5 5 }}, {{val| 0 -14 -11 -9 }}]
{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}
 
: mapping generators: ~2, ~25/21


{{Multival|legend=1| 14 11 9 -15 -25 -10 }}
{{Multival|legend=1| 14 11 9 -15 -25 -10 }}


[[POTE generator]]: ~25/21 = 292.710
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710


{{Optimal ET sequence|legend=1| 4, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, 37, 41 }}
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Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


Mapping: [{{val| 1 5 5 5 2 }}, {{val| 0 -14 -11 -9 6 }}]
Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}


POTE generator: ~25/21 = 292.547
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547


{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}
{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}
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Comma list: 100/99, 196/195, 275/273, 385/384
Comma list: 100/99, 196/195, 275/273, 385/384


POTE generator: ~13/11 = 292.457
Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
 
Mapping: [{{val| 1 5 5 5 2 2 }}, {{val| 0 -14 -11 -9 6 7 }}]


POTE generator: ~13/11 = 292.457
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457


{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
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Comma list: 243/242, 441/440, 625/616
Comma list: 243/242, 441/440, 625/616


Mapping: [{{val| 1 5 5 5 12 }}, {{val| 0 -14 -11 -9 -35 }}]
Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}


POTE generator: ~25/21 = 292.851
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851


{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}
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Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


Mapping: [{{val| 1 5 5 5 12 12 }}, {{val| 0 -14 -11 -9 -35 -34 }}]
Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}


POTE generator: ~13/11 = 292.928
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928


{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}
{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}
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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Chromo]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Chromo]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 2430/2401
[[Comma list]]: 875/864, 2430/2401


[[Mapping]]: [{{Val|1 1 2 2}}, {{Val|0 13 7 18}}]
{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
 
: mapping generators: ~2, ~25/24


[[POTE generator]]: ~25/24 = 53.816
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816


{{Optimal ET sequence|legend=1| 22, 45, 67c }}
{{Optimal ET sequence|legend=1| 22, 45, 67c }}
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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Undeka]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Undeka]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 3200/3087
[[Comma list]]: 875/864, 3200/3087


[[Mapping]]: [{{Val|11 0 8 31}}, {{Val|0 1 1 0}}]
{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}


{{Multival|legend=1|11 11 0 -8 -31 -31}}
: mapping generators: ~21/20, ~3


[[POTE generator]]: ~3/2 = 708.792
{{Multival|legend=1| 11 11 0 -8 -31 -31 }}
 
[[Optimal tuning]] ([[POTE]]): ~21/20 = 1\11, ~3/2 = 708.792


{{Optimal ET sequence|legend=1| 11c, 22 }}
{{Optimal ET sequence|legend=1| 11c, 22 }}
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Comma list: 100/99, 352/343, 385/384
Comma list: 100/99, 352/343, 385/384


Mapping: [{{Val|11 0 8 31 38}}, {{Val|0 1 1 0 0}}]
Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}


POTE generator: ~3/2 = 706.768
Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 706.768


{{Optimal ET sequence|legend=1| 11c, 22 }}
{{Optimal ET sequence|legend=1| 11c, 22 }}
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Comma list: 65/63, 100/99, 169/165, 352/343
Comma list: 65/63, 100/99, 169/165, 352/343


Mapping: [{{Val|11 0 8 31 38 23}}, {{Val|0 1 1 0 0 1}}]
Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}


POTE generator: ~3/2 = 707.764
Optimal tuning (POTE): ~13/12 = 1\11, ~3/2 = 707.764


{{Optimal ET sequence|legend=1| 11cf, 22 }}
{{Optimal ET sequence|legend=1| 11cf, 22 }}
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== Barbad ==
== Barbad ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 16875/16807
[[Comma list]]: 875/864, 16875/16807


[[Mapping]]: [{{val| 1 9 7 11 }}, {{val| 0 -19 -12 -21 }}]
{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}
 
: mapping generators: ~2, ~98/75


{{Multival|legend=1| 19 12 21 -25 -20 15 }}
{{Multival|legend=1| 19 12 21 -25 -20 15 }}


[[POTE generator]]: ~98/75 = 468.331
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331


{{Optimal ET sequence|legend=1| 18, 23d, 41 }}
{{Optimal ET sequence|legend=1| 18, 23d, 41 }}
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Comma list: 245/242, 540/539, 625/616
Comma list: 245/242, 540/539, 625/616


Mapping: [{{val| 1 9 7 11 14 }}, {{val| 0 -19 -12 -21 -27 }}]
Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}


POTE generator: ~98/75 = 468.367
Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367


{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
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Comma list: 144/143, 196/195, 245/242, 275/273
Comma list: 144/143, 196/195, 245/242, 275/273


Mapping: [{{val| 1 9 7 11 14 8 }}, {{val| 0 -19 -12 -21 -27 -11 }}]
Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}


POTE generator: ~13/10 = 468.270
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270


{{Optimal ET sequence|legend=1| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=1| 18e, 23de, 41 }}
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== Hyperkleismic ==
== Hyperkleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 51200/50421
[[Comma list]]: 875/864, 51200/50421


[[Mapping]]: [{{val|1 -3 -2 2}}, {{val|0 17 16 3}}]
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}


{{Multival|legend=1|17 16 3 -14 -43 -38}}
: mapping generators: ~2, ~6/5


[[POTE generator]]: ~6/5 = 323.780
{{Multival|legend=1| 17 16 3 -14 -43 -38 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
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Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 385/384, 2420/2401<br>
Comma list: 100/99, 385/384, 2420/2401


Mapping: [{{val|1 -3 -2 2 4}}, {{val|0 17 16 3 -2}}]<br>
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}


POTE generator: ~6/5 = 323.796<br>
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
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Comma list: 100/99, 169/168, 275/273, 385/384
Comma list: 100/99, 169/168, 275/273, 385/384


Mapping: [{{val|1 -3 -2 2 4 1}}, {{val|0 17 16 3 -2 10}}]
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}


POTE generator: ~6/5 = 323.790
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
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== Sevond ==
== Sevond ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 327680/321489
[[Comma list]]: 875/864, 327680/321489


[[Mapping]]: [{{val| 7 0 -6 53 }}, {{val| 0 1 2 -3 }}]
{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
 
: mapping generators: ~10/9, ~3


[[POTE generator]]: ~3/2 = 705.613
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
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Comma list: 100/99, 385/384, 6655/6561
Comma list: 100/99, 385/384, 6655/6561


Mapping: [{{val| 7 0 -6 53 2 }}, {{val| 0 1 2 -3 2 }}]
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}


POTE generator: ~3/2 = 705.518
Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
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Comma list: 100/99, 169/168, 352/351, 385/384
Comma list: 100/99, 169/168, 352/351, 385/384


Mapping: [{{val| 7 0 -6 53 2 37 }}, {{val| 0 1 2 -3 2 -1 }}]
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


POTE generator: ~3/2 = 705.344
Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}

Revision as of 15:00, 24 August 2023

These temper out the keema, [-5 -3 3 1⟩ = 875/864. Keemic temperaments include doublewide, flattone, porcupine, superkleismic, magic, keemun, and sycamore. Discussed below are quasitemp and barbad.

Quasitemp

For the 5-limit version of this temperament, see High badness temperaments #Quasitemp.

Subgroup: 2.3.5.7

Comma list: 875/864, 2401/2400

Mapping: [⟨1 5 5 5], ⟨0 -14 -11 -9]]

mapping generators: ~2, ~25/21

Wedgie: ⟨⟨ 14 11 9 -15 -25 -10 ]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.710

Optimal ET sequence: 4, 37, 41

Badness: 0.060269

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1375/1372

Mapping: [⟨1 5 5 5 2], ⟨0 -14 -11 -9 6]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547

Optimal ET sequence: 4, 37, 41, 119

Badness: 0.043209

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 275/273, 385/384

Mapping: [⟨1 5 5 5 2 2], ⟨0 -14 -11 -9 6 7]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457

Optimal ET sequence: 4, 37, 41, 78, 119f

Badness: 0.032913

Quato

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 625/616

Mapping: [⟨1 5 5 5 12], ⟨0 -14 -11 -9 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851

Optimal ET sequence: 41, 127cd, 168cd

Badness: 0.041170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 243/242, 275/273, 325/324

Mapping: [⟨1 5 5 5 12 12], ⟨0 -14 -11 -9 -35 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928

Optimal ET sequence: 41, 86ce, 127cd

Badness: 0.030081

Chromo

For the 5-limit version of this temperament, see High badness temperaments #Chromo.

Subgroup: 2.3.5.7

Comma list: 875/864, 2430/2401

Mapping: [⟨1 1 2 2], ⟨0 13 7 18]]

mapping generators: ~2, ~25/24

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 53.816

Optimal ET sequence: 22, 45, 67c

Badness: 0.090769

Undeka

For the 5-limit version of this temperament, see High badness temperaments #Undeka.

Subgroup: 2.3.5.7

Comma list: 875/864, 3200/3087

Mapping: [⟨11 0 8 31], ⟨0 1 1 0]]

mapping generators: ~21/20, ~3

Wedgie: ⟨⟨ 11 11 0 -8 -31 -31 ]]

Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 708.792

Optimal ET sequence: 11c, 22

Badness: 0.141782

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 352/343, 385/384

Mapping: [⟨11 0 8 31 38], ⟨0 1 1 0 0]]

Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 706.768

Optimal ET sequence: 11c, 22

Badness: 0.068672

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/63, 100/99, 169/165, 352/343

Mapping: [⟨11 0 8 31 38 23], ⟨0 1 1 0 0 1]]

Optimal tuning (POTE): ~13/12 = 1\11, ~3/2 = 707.764

Optimal ET sequence: 11cf, 22

Badness: 0.056528

Barbad

Subgroup: 2.3.5.7

Comma list: 875/864, 16875/16807

Mapping: [⟨1 9 7 11], ⟨0 -19 -12 -21]]

mapping generators: ~2, ~98/75

Wedgie: ⟨⟨ 19 12 21 -25 -20 15 ]]

Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.331

Optimal ET sequence: 18, 23d, 41

Badness: 0.110448

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 540/539, 625/616

Mapping: [⟨1 9 7 11 14], ⟨0 -19 -12 -21 -27]]

Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367

Optimal ET sequence: 18e, 23de, 41, 228ccdd

Badness: 0.050105

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 245/242, 275/273

Mapping: [⟨1 9 7 11 14 8], ⟨0 -19 -12 -21 -27 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270

Optimal ET sequence: 18e, 23de, 41

Badness: 0.039183

Hyperkleismic

Subgroup: 2.3.5.7

Comma list: 875/864, 51200/50421

Mapping: [⟨1 -3 -2 2], ⟨0 17 16 3]]

mapping generators: ~2, ~6/5

Wedgie: ⟨⟨ 17 16 3 -14 -43 -38 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.780

Optimal ET sequence: 26, 37, 63

Badness: 0.157830

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 2420/2401

Mapping: [⟨1 -3 -2 2 4], ⟨0 17 16 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796

Optimal ET sequence: 26, 37, 63

Badness: 0.065356

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 275/273, 385/384

Mapping: [⟨1 -3 -2 2 4 1], ⟨0 17 16 3 -2 10]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790

Optimal ET sequence: 26, 37, 63

Badness: 0.035724

Sevond

Subgroup: 2.3.5.7

Comma list: 875/864, 327680/321489

Mapping: [⟨7 0 -6 53], ⟨0 1 2 -3]]

mapping generators: ~10/9, ~3

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.613

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.206592

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 6655/6561

Mapping: [⟨7 0 -6 53 2], ⟨0 1 2 -3 2]]

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.070437

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 352/351, 385/384

Mapping: [⟨7 0 -6 53 2 37], ⟨0 1 2 -3 2 -1]]

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.041238

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