Kleismic family: Difference between revisions

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The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 6 5 -6 }} for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 6 5 -6 }} for the [[wedgie]]. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].


= Hanson =
= Hanson =
Subgroup: 2.3.5


[[Comma list]]: 15625/15552
[[Comma list]]: 15625/15552
[[Mapping]]: [{{val| 1 0 1 }}, {{val|0 6 5 }}]


[[POTE generator]]: ~6/5 = 317.007
[[POTE generator]]: ~6/5 = 317.007
[[Mapping]]: [{{val| 1 0 1 }}, {{val|0 6 5 }}]


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 33: Line 35:


= Keemun =
= Keemun =
Subgroup: 2.3.5.7
[[Comma list]]: 49/48, 126/125
[[Comma list]]: 49/48, 126/125
[[POTE generator]]: ~6/5 = 316.473


[[Mapping]]: [{{val| 1 0 1 2 }}, {{val| 0 6 5 3 }}]
[[Mapping]]: [{{val| 1 0 1 2 }}, {{val| 0 6 5 3 }}]


{{Multival|legend=1| 6 5 3 -6 -12 -7 }}
{{Multival|legend=1| 6 5 3 -6 -12 -7 }}
[[POTE generator]]: ~6/5 = 316.473


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [300.000, 327.273] (4 to 11b)
* valid range: [300.000, 327.273] (1\4 to 3\11)
* nice range: [308.744, 322.942]
* nice range: [308.744, 322.942]
* strict range: [308.744, 322.942]
* strict range: [308.744, 322.942]
Line 51: Line 56:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 100/99
Comma list: 49/48, 56/55, 100/99
Mapping: [{{val| 1 0 1 2 4 }}, {{val| 0 6 5 3 -2 }}]


POTE generator: ~6/5 = 317.576
POTE generator: ~6/5 = 317.576
Mapping: [{{val| 1 0 1 2 4 }}, {{val| 0 6 5 3 -2 }}]


Tuning ranges:  
Tuning ranges:  
* valid range: [315.789, 320.000] (19 to 15)
* valid range: [315.789, 320.000] (5\19 to 4\15)
* nice range: [308.744, 324.341]
* nice range: [308.744, 324.341]
* strict range: [315.789, 320.000]
* strict range: [315.789, 320.000]
Line 67: Line 75:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 78/77, 100/99
Comma list: 49/48, 56/55, 78/77, 100/99
Mapping: [{{val| 1 0 1 2 4 5 }}, {{val| 0 6 5 3 -2 -5 }}]


POTE generator: ~6/5 = 316.611
POTE generator: ~6/5 = 316.611
Mapping: [{{val| 1 0 1 2 4 5 }}, {{val| 0 6 5 3 -2 -5 }}]


Tuning ranges:  
Tuning ranges:  
* valid range: 315.789 (19)
* valid range: 315.789 (5\19)
* nice range: [303.597, 324.341]
* nice range: [303.597, 324.341]
* strict range: 315.789
* strict range: 315.789
Line 83: Line 94:


=== Kema ===
=== Kema ===
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 91/90, 100/99
Comma list: 49/48, 56/55, 91/90, 100/99
Mapping: [{{val| 1 0 1 2 4 0 }}, {{val| 0 6 5 3 -2 14 }}]


POTE generator: ~6/5 = 317.423
POTE generator: ~6/5 = 317.423
Mapping: [{{val| 1 0 1 2 4 0 }}, {{val| 0 6 5 3 -2 14 }}]


Tuning ranges:  
Tuning ranges:  
* valid range: [315.789, 320.000] (19 to 15)
* valid range: [315.789, 320.000] (5\19 to 4\15)
* nice range: [308.744, 324.341]
* nice range: [308.744, 324.341]
* strict range: [315.789, 320.000]
* strict range: [315.789, 320.000]
Line 99: Line 113:


=== Kumbaya ===
=== Kumbaya ===
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 49/48, 56/55, 66/65
Comma list: 40/39, 49/48, 56/55, 66/65
Mapping: [{{val| 1 0 1 2 4 4 }}, {{val| 0 6 5 3 -2 -1 }}]


POTE generator: ~6/5 = 318.595
POTE generator: ~6/5 = 318.595
Mapping: [{{val| 1 0 1 2 4 4 }}, {{val| 0 6 5 3 -2 -1 }}]


{{Val list|legend=1| 4, 15, 19f, 34ff }}
{{Val list|legend=1| 4, 15, 19f, 34ff }}
Line 110: Line 127:


== Qeema ==
== Qeema ==
Subgroup: 2.3.5.7.11
Comma list: 45/44, 49/48, 126/125
Comma list: 45/44, 49/48, 126/125
Mapping: [{{val| 1 0 1 2 -1 }}, {{val| 0 6 5 3 17 }}]


POTE generator: ~6/5 = 314.730
POTE generator: ~6/5 = 314.730
Mapping: [{{val| 1 0 1 2 -1 }}, {{val| 0 6 5 3 17 }}]


{{Val list|legend=1| 4e, 19, 42bcd, 61bcdd }}
{{Val list|legend=1| 4e, 19, 42bcd, 61bcdd }}
Line 121: Line 141:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 49/48, 78/77, 126/125
Comma list: 45/44, 49/48, 78/77, 126/125
Mapping: [{{val| 1 0 1 2 -1 0 }}, {{val| 0 6 5 3 17 14 }}]


POTE generator: ~6/5 = 315.044
POTE generator: ~6/5 = 315.044
Mapping: [{{val| 1 0 1 2 -1 0 }}, {{val| 0 6 5 3 17 14 }}]


{{Val list|legend=1| 4ef, 19 }}
{{Val list|legend=1| 4ef, 19 }}
Line 132: Line 155:


== Darjeeling ==
== Darjeeling ==
Subgroup: 2.3.5.7.11
Comma list: 49/48, 55/54, 77/75
Comma list: 49/48, 55/54, 77/75
Mapping: [{{val| 1 0 1 2 0 }}, {{val| 0 6 5 3 13 }}]


POTE generator: ~6/5 = 317.656
POTE generator: ~6/5 = 317.656
Mapping: [{{val| 1 0 1 2 0 }}, {{val| 0 6 5 3 13 }}]


{{Val list|legend=1| 15, 19e, 34e }}
{{Val list|legend=1| 15, 19e, 34e }}
Line 143: Line 169:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 55/54, 66/65, 77/75
Comma list: 49/48, 55/54, 66/65, 77/75
Mapping: [{{val| 1 0 1 2 0 0 }}, {{val| 0 6 5 3 13 14 }}]


POTE generator: ~6/5 = 317.298
POTE generator: ~6/5 = 317.298
Mapping: [{{val| 1 0 1 2 0 0 }}, {{val| 0 6 5 3 13 14 }}]


{{Val list|legend=1| 15, 19e, 34e, 53dee }}
{{Val list|legend=1| 15, 19e, 34e, 53dee }}
Line 154: Line 183:


= Catalan =
= Catalan =
Comma list: 64/63, 15625/15552


POTE generator: ~6/5 = 318.267
Subgroup: 2.3.5.7


Mapping: [{{val| 1 0 1 6 }}, {{val| 0 6 5 -12 }}]
[[Comma list]]: 64/63, 15625/15552
 
[[Mapping]]: [{{val| 1 0 1 6 }}, {{val| 0 6 5 -12 }}]


{{Multival|legend=1| 6 5 -12 -6 -36 -42 }}
{{Multival|legend=1| 6 5 -12 -6 -36 -42 }}
[[POTE generator]]: ~6/5 = 318.267


{{Val list|legend=1| 15, 34d, 49, 132bcd }}
{{Val list|legend=1| 15, 34d, 49, 132bcd }}


Badness: 0.0949
[[Badness]]: 0.0949


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 1331/1323
Comma list: 64/63, 100/99, 1331/1323
Mapping: [{{val| 1 0 1 6 4 }}, {{val| 0 6 5 -12 -2 }}]


POTE generator: ~6/5 = 318.282
POTE generator: ~6/5 = 318.282
Mapping: [{{val| 1 0 1 6 4 }}, {{val| 0 6 5 -12 -2 }}]


{{Val list|legend=1| 15, 34d, 49 }}
{{Val list|legend=1| 15, 34d, 49 }}
Line 178: Line 213:


= Catakleismic =
= Catakleismic =
Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 4375/4374
[[Comma list]]: 225/224, 4375/4374
[[POTE generator]]: ~6/5 = 316.732


[[Mapping]]: [{{val| 1 0 1 -3 }}, {{val| 0 6 5 22 }}]
[[Mapping]]: [{{val| 1 0 1 -3 }}, {{val| 0 6 5 22 }}]


{{Multival|legend=1| 6 5 22 -6 18 37 }}
{{Multival|legend=1| 6 5 22 -6 18 37 }}
[[POTE generator]]: ~6/5 = 316.732


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [315.789, 317.647] (19 to 34)
* valid range: [315.789, 317.647] (5\19 to 9\34)
* nice range: [315.641, 317.263]
* nice range: [315.641, 317.263]
* strict range: [315.789, 317.263]
* strict range: [315.789, 317.263]


{{Val list|legend=1| 19, 53, 72, 197, 269c }}
{{Val list|legend=1| 19, 34d, 53, 72, 197, 269c }}


[[Badness]]: 0.0215
[[Badness]]: 0.0215


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 4375/4374
Comma list: 225/224, 385/384, 4375/4374
Mapping: [{{val| 1 0 1 -3 9 }}, {{val| 0 6 5 22 -21 }}]


POTE generator: ~6/5 = 316.719
POTE generator: ~6/5 = 316.719
Mapping: [{{val| 1 0 1 -3 9 }}, {{val| 0 6 5 22 -21 }}]


Tuning ranges:  
Tuning ranges:  
* valid range: [315.789, 316.981] (19 to 53)
* valid range: [315.789, 316.981] (5\19 to 14\53)
* nice range: [315.641, 317.263]
* nice range: [315.641, 317.263]
* strict range: [315.789, 316.981]
* strict range: [315.789, 316.981]


{{Val list|legend=1| 19, 53, 72, 197e, 269ce, 341ce, 610bce }}
{{Val list|legend=1| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}


Badness: 0.0218
Badness: 0.0218


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 325/324, 540/539
Comma list: 169/168, 225/224, 325/324, 540/539
Mapping: [{{val| 1 0 1 -3 9 0 }}, {{val| 0 6 5 22 -21 14 }}]


POTE generator: ~6/5 = 316.738
POTE generator: ~6/5 = 316.738
Mapping: [{{val| 1 0 1 -3 9 0 }}, {{val| 0 6 5 22 -21 14 }}]


Tuning ranges:  
Tuning ranges:  
* valid range: [315.789, 316.981] (19 to 53)
* valid range: [315.789, 316.981] (5\19 to 14\53)
* nice range: [315.641, 318.309]
* nice range: [315.641, 318.309]
* strict range: [315.789, 316.981]
* strict range: [315.789, 316.981]


{{Val list|legend=1| 19, 53, 72, 125f, 197ef, 269cef }}
{{Val list|legend=1| 19, 34de, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.0169
Badness: 0.0169


== Cataclysmic ==
== Cataclysmic ==
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 2200/2187
Comma list: 99/98, 176/175, 2200/2187
Mapping: [{{val| 1 0 1 -3 -5 }}, {{val| 0 6 5 22 32 }}]


POTE generator: ~6/5 = 317.042
POTE generator: ~6/5 = 317.042


Mapping: [{{val| 1 0 1 -3 -5 }}, {{val| 0 6 5 22 32 }}]
{{Val list|legend=1| 19e, 34d, 53, 193dde, …, 352… }}
 
{{Val list|legend=1| 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de }}


Badness: 0.0400
Badness: 0.0400


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 169/168, 176/175, 275/273
Comma list: 99/98, 169/168, 176/175, 275/273
Mapping: [{{val| 1 0 1 -3 -5 0 }}, {{val| 0 6 5 22 32 14 }}]


POTE generator: ~6/5 = 317.036
POTE generator: ~6/5 = 317.036


Mapping: [{{val| 1 0 1 -3 -5 0 }}, {{val| 0 6 5 22 32 14 }}]
{{Val list|legend=1| 19e, 34d, 53, 246ddee, …, 405… }}
 
{{Val list|legend=1| 53, 87d, 140d, 193de, 246de }}


Badness: 0.0226
Badness: 0.0226


== Catalytic ==
== Catalytic ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 4375/4374
Comma list: 225/224, 441/440, 4375/4374
Mapping: [{{val| 1 0 1 -3 -10 }}, {{val| 0 6 5 22 51 }}]


POTE generator: ~6/5 = 316.653
POTE generator: ~6/5 = 316.653


Mapping: [{{val| 1 0 1 -3 -10 }}, {{val| 0 6 5 22 51 }}]
{{Val list|legend=1| 19e, 53e, 72 }}
 
{{Val list|legend=1| 53e, 72 }}


Badness: 0.0304
Badness: 0.0304


=== 13-limit ===
=== 13-limit ===
Comma list: 169/168 225/224 325/324 1716/1715


POTE generator: ~6/5 = 316.639
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 325/324, 1716/1715


Mapping: [{{val| 1 0 1 -3 -10 0 }}, {{val| 0 6 5 22 51 14 }}]
Mapping: [{{val| 1 0 1 -3 -10 0 }}, {{val| 0 6 5 22 51 14 }}]
POTE generator: ~6/5 = 316.639


{{Val list|legend=1| 19e, 53e, 72 }}
{{Val list|legend=1| 19e, 53e, 72 }}
Line 272: Line 328:


== Cataleptic ==
== Cataleptic ==
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 864/847
Comma list: 100/99, 225/224, 864/847
Mapping: [{{val| 1 0 1 -3 4 }}, {{val| 0 6 5 22 -2 }}]


POTE generator: ~6/5 = 317.083
POTE generator: ~6/5 = 317.083
Mapping: [{{val| 1 0 1 -3 4 }}, {{val| 0 6 5 22 -2 }}]


{{Val list|legend=1| 19, 34d, 53e }}
{{Val list|legend=1| 19, 34d, 53e }}
Line 283: Line 342:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 100/99, 144/143, 676/675
Comma list: 78/77, 100/99, 144/143, 676/675
Mapping: [{{val| 1 0 1 -3 4 0 }}, {{val| 0 6 5 22 -2 14 }}]


POTE generator: ~6/5 = 317.118
POTE generator: ~6/5 = 317.118


Mapping: [{{val| 1 0 1 -3 4 0 }}, {{val| 0 6 5 22 -2 14 }}]
{{Val list|legend=1| 19, 34d, 53e, 87dee }}
 
{{Val list|legend=1| 19, 34d, 53e, 87de }}


Badness: 0.0273
Badness: 0.0273


== Bikleismic ==
== Bikleismic ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 4375/4356
Comma list: 225/224, 243/242, 4375/4356
Mapping: [{{val| 2 0 2 -6 -1 }}, {{val| 0 6 5 22 15 }}]


POTE generator: ~6/5 = 316.721
POTE generator: ~6/5 = 316.721


Mapping: [{{val| 2 0 2 -6 -1 }}, {{val| 0 6 5 22 15 }}]
{{Val list|legend=1| 34d, 72, 322c, …, 610bcc }}
 
{{Val list|legend=1| 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc }}


Badness: 0.0293
Badness: 0.0293


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 243/242, 325/324
Comma list: 169/168, 225/224, 243/242, 325/324
Mapping: [{{val| 2 0 2 -6 -1 0 }}, {{val| 0 6 5 22 15 14 }}]


POTE generator: ~6/5 = 316.726
POTE generator: ~6/5 = 316.726


Mapping: [{{val| 2 0 2 -6 -1 0 }}, {{val| 0 6 5 22 15 14 }}]
{{Val list|legend=1| 34d, 72, 322cfff, …, 538… }}
 
{{Val list|legend=1| 72, 106, 322cff, 394cff, 466bcff, 538bcfff }}


Badness: 0.0218
Badness: 0.0218


= Countercata =
= Countercata =
Subgroup: 2.3.5.7
[[Comma list]]: 5120/5103, 15625/15552
[[Comma list]]: 5120/5103, 15625/15552
[[POTE generator]]: ~6/5 = 317.121


[[Mapping]]: [{{val| 1 0 1 11 }}, {{val| 0 6 5 -31 }}]
[[Mapping]]: [{{val| 1 0 1 11 }}, {{val| 0 6 5 -31 }}]


{{Multival|legend=1| 6 5 -31 -6 -66 -86 }}
{{Multival|legend=1| 6 5 -31 -6 -66 -86 }}
[[POTE generator]]: ~6/5 = 317.121


{{Val list|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
{{Val list|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
Line 329: Line 400:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 385/384, 2200/2187, 3388/3375
Comma list: 385/384, 2200/2187, 3388/3375
Mapping: [{{val| 1 0 1 11 -5 }}, {{val| 0 6 5 -31 32 }}]


POTE generator: ~6/5 = 317.162
POTE generator: ~6/5 = 317.162


Mapping: [{{val| 1 0 1 11 -5 }}, {{val| 0 6 5 -31 32 }}]
{{Val list|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
 
{{Val list|legend=1| 34, 53, 87, 140, 227 }}


Badness: 0.0398
Badness: 0.0398


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 385/384, 625/624
Comma list: 325/324, 352/351, 385/384, 625/624
Mapping: [{{val| 1 0 1 11 -5 0 }}, {{val| 0 6 5 -31 32 14 }}]


POTE generator: ~6/5 = 317.162
POTE generator: ~6/5 = 317.162
Mapping: [{{val| 1 0 1 11 -5 0 }}, {{val| 0 6 5 -31 32 14 }}]


{{Val list|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
{{Val list|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
Line 351: Line 428:


= Metakleismic =
= Metakleismic =
Subgroup: 2.3.5.7
[[Comma list]]: 15625/15552, 179200/177147
[[Comma list]]: 15625/15552, 179200/177147
[[POTE generator]]: ~6/5 = 317.314


[[Mapping]]: [{{val| 1 0 1 -12 }}, {{val| 0 6 5 56 }}]
[[Mapping]]: [{{val| 1 0 1 -12 }}, {{val| 0 6 5 56 }}]
Line 359: Line 437:
{{Multival|legend=1| 6 5 56 -6 72 116 }}
{{Multival|legend=1| 6 5 56 -6 72 116 }}


{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
[[POTE generator]]: ~6/5 = 317.314
 
{{Val list|legend=1| 34d, 87, 121, 208 }}


[[Badness]]: 0.1635
[[Badness]]: 0.1635


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 896/891, 2200/2187, 14700/14641
Comma list: 896/891, 2200/2187, 14700/14641
Mapping: [{{val| 1 0 1 -12 -5 }}, {{val| 0 6 5 56 32 }}]


POTE generator: ~6/5 = 317.311
POTE generator: ~6/5 = 317.311
Mapping: [{{val| 1 0 1 -12 -5 }}, {{val| 0 6 5 56 32 }}]


{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
Line 375: Line 458:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 364/363, 625/624
Comma list: 325/324, 352/351, 364/363, 625/624
Mapping: [{{val| 1 0 1 -12 -5 0 }}, {{val| 0 6 5 56 32 14 }}]


POTE generator: ~6/5 = 317.311
POTE generator: ~6/5 = 317.311
Mapping: [{{val| 1 0 1 -12 -5 0 }}, {{val| 0 6 5 56 32 14 }}]


{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
Line 386: Line 472:


= Hemikleismic =
= Hemikleismic =
Subgroup: 2.3.5.7
[[Comma list]]: 4000/3969, 6144/6125
[[Comma list]]: 4000/3969, 6144/6125


[[POTE generator]]: 158.649
[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 12 10 -9 }}]


Mapping: [{{val| 1 0 1 4 }}, {{val| 0 12 10 -9 }}]
[[POTE generator]]: ~35/32 = 158.649


{{Val list|legend=1| 15, 53, 121 }}
{{Val list|legend=1| 15, 38, 53, 121 }}


Badness: 0.0521
[[Badness]]: 0.0521


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175, 4000/3969
Comma list: 121/120, 176/175, 4000/3969
Mapping: [{{val| 1 0 1 4 2 }}, {{val| 0 12 10 -9 11 }}]


POTE generator: ~11/10 = 158.677
POTE generator: ~11/10 = 158.677
Mapping: [{{val| 1 0 1 4 2 }}, {{val| 0 12 10 -9 11 }}]


{{Val list|legend=1| 15, 38, 53, 68, 121e }}
{{Val list|legend=1| 15, 38, 53, 68, 121e }}
Line 408: Line 500:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 176/175, 275/273, 325/324
Comma list: 121/120, 176/175, 275/273, 325/324


Line 414: Line 509:
Mapping: [{{val| 1 0 1 4 2 0 }}, {{val| 0 12 10 -9 11 28 }}]
Mapping: [{{val| 1 0 1 4 2 0 }}, {{val| 0 12 10 -9 11 28 }}]


{{Val list|legend=1| 15, 53, 121e }}
{{Val list|legend=1| 15, 38f, 53, 121e }}


Badness: 0.0260
Badness: 0.0260


= Clyde =
= Clyde =
Subgroup: 2.3.5.7
[[Comma list]]: 245/243, 3136/3125
[[Comma list]]: 245/243, 3136/3125


[[POTE generator]]: ~9/7 = 441.335
[[Mapping]]: [{{val| 1 6 6 12 }}, {{val| 0 -12 -10 -25 }}]


[[Mapping]]: [{{val| 1 6 6 12 }}, {{val| 0 -12 -10 -25 }}]
Mapping generators: ~2, ~9/7


Mapping [[generator]]s: ~2, ~9/7
[[POTE generator]]: ~9/7 = 441.335


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 6/25 0 0 12/25 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 6/25 0 0 12/25 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s: 2, 7
: [[Eigenmonzo]]s: 2, 7
Line 436: Line 534:
{{Val list|legend=1| 19, 49, 68, 87, 155 }}
{{Val list|legend=1| 19, 49, 68, 87, 155 }}


Badness: 0.0473
[[Badness]]: 0.0473


== 11-limit ==
== 11-limit ==
Comma list: 245/243, 3136/3125, 385/384


POTE generator: ~9/7 = 441.355
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 3136/3125


Mapping: [{{val| 1 6 6 12 -5 }}, {{val| 0 -12 -10 -25 23 }}]
Mapping: [{{val| 1 6 6 12 -5 }}, {{val| 0 -12 -10 -25 23 }}]


{{Val list|legend=1| 19, 68, 87, 329bd, 419bd, 503bd, 590bd }}
POTE generator: ~9/7 = 441.355
 
{{Val list|legend=1| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}


Badness: 0.0474
Badness: 0.0474


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 245/243, 385/384, 625/624
Comma list: 196/195, 245/243, 385/384, 625/624


Line 456: Line 560:
Mapping: [{{val| 1 6 6 12 -5 14 }}, {{val| 0 -12 -10 -25 23 -28 }}]
Mapping: [{{val| 1 6 6 12 -5 14 }}, {{val| 0 -12 -10 -25 23 -28 }}]


{{Val list|legend=1| 19, 68, 87, 503bdf, 590bdf }}
{{Val list|legend=1| 19, 49ef, 68, 87, 503bdf, 590bdf }}


Badness: 0.0268
Badness: 0.0268
Line 462: Line 566:
= Tritikleismic =
= Tritikleismic =
{{see also| Gamelismic clan #Tritikleismic }}
{{see also| Gamelismic clan #Tritikleismic }}
Subgroup: 2.3.5.7


[[Comma list]]: 1029/1024, 15625/15552
[[Comma list]]: 1029/1024, 15625/15552
[[POTE generator]]: 316.872


[[Mapping]]: [{{val| 3 0 3 10 }}, {{val| 0 6 5 -2 }}]
[[Mapping]]: [{{val| 3 0 3 10 }}, {{val| 0 6 5 -2 }}]
Line 471: Line 575:
{{Multival|legend=1| 18 15 -6 -18 -60 -56 }}
{{Multival|legend=1| 18 15 -6 -18 -60 -56 }}


{{Val list|legend=1| 15, 57, 72, 159, 231 }}
[[POTE generator]]: 316.872
 
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [[Eigenmonzo]]s: 2, 7/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s: 2, 9/7
 
{{Val list|legend=1| 15, 42bc, 57, 72, 159, 231 }}


[[Badness]]: 0.0563
[[Badness]]: 0.0563


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 4000/3993
Comma list: 385/384, 441/440, 4000/3993
Mapping: [{{val| 3 0 3 10 8 }}, {{val| 0 6 5 -2 3 }}]
Mapping generators: ~44/35, ~6/5


POTE generator: ~6/5 = 316.881
POTE generator: ~6/5 = 316.881


Mapping: [{{val| 3 0 3 10 8 }}, {{val| 0 6 5 -2 3 }}]
Minimax tuning:
* 11-odd-limit
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 46/21 5/7 0 -5/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 71/21 3/7 0 -3/14 0 }}]
: Eigenmonzos: 2, 9/7


{{Val list|legend=1| 15, 57, 72, 159, 231 }}
{{Val list|legend=1| 15, 42bc, 57, 72, 159, 231 }}


Badness: 0.0193
Badness: 0.0193
Line 488: Line 612:
== 13-limit ==
== 13-limit ==
Comma list: 325/324, 364/363, 441/440, 625/624
Comma list: 325/324, 364/363, 441/440, 625/624
Mapping: [{{val| 3 0 3 10 8 0 }}, {{val| 0 6 5 -2 3 14 }}]
Mapping generators: ~44/35, ~6/5


POTE generator: ~6/5 = 316.959
POTE generator: ~6/5 = 316.959


Mapping: [{{val| 3 0 3 10 8 0 }}, {{val| 0 6 5 -2 3 14 }}]
{{Val list|legend=1| 72, 87, 159 }}
 
Badness: 0.0157
 
== 17-limit ==
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 325/324, 364/363, 375/374, 385/384
 
Mapping: [{{val| 3 0 3 10 8 0 -2 }}, {{val| 0 6 5 -2 3 14 18 }}]
 
Mapping generators: ~34/27, ~6/5


{{Val list|legend=1| 15, 72, 87, 159, 867, 1026 }}
POTE generator: ~6/5 = 316.9082


[[Badness]]: 0.0157
{{Val list|legend=1| 72, 159, 231f }}


= Quadritikleismic =
= Quadritikleismic =
Subgroup: 2.3.5.7
[[Comma list]]: 2401/2400, 15625/15552
[[Comma list]]: 2401/2400, 15625/15552
[[POTE generator]]: ~6/5 = 316.9999


[[Mapping]]: [{{val| 4 0 4 7 }}, {{val| 0 6 5 4 }}]
[[Mapping]]: [{{val| 4 0 4 7 }}, {{val| 0 6 5 4 }}]
Line 506: Line 647:
{{Multival|legend=1| 24 20 16 -24 -42 -19 }}
{{Multival|legend=1| 24 20 16 -24 -42 -19 }}


{{Val list|legend=1| 68, 72, 140, 212, 1200 }}
[[POTE generator]]: ~6/5 = 316.9999
 
{{Val list|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}


[[Badness]]: 0.0392
[[Badness]]: 0.0392


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 6250/6237
Comma list: 385/384, 1375/1372, 6250/6237
Mapping: [{{val| 4 0 4 7 17 }}, {{val| 0 6 5 4 -3 }}]


POTE generator: ~6/5 = 316.925
POTE generator: ~6/5 = 316.925
Mapping: [{{val| 4 0 4 7 17 }}, {{val| 0 6 5 4 -3 }}]


{{Val list|legend=1| 68, 72, 140, 212, 284, 496, 780 }}
{{Val list|legend=1| 68, 72, 140, 212, 284, 496, 780 }}
Line 522: Line 668:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 625/624, 1573/1568
Comma list: 325/324, 385/384, 625/624, 1573/1568
Mapping: [{{val| 4 0 4 7 17 0 }}, {{val| 0 6 5 4 -3 14 }}]


POTE generator: ~6/5 = 316.989
POTE generator: ~6/5 = 316.989
Mapping: [{{val| 4 0 4 7 17 0 }}, {{val| 0 6 5 4 -3 14 }}]


{{Val list|legend=1| 68, 72, 140, 212 }}
{{Val list|legend=1| 68, 72, 140, 212 }}
Line 533: Line 682:


= Kleiboh =
= Kleiboh =
Subgroup: 2.3.5.7
[[Comma list]]: 1728/1715, 3125/3087
[[Comma list]]: 1728/1715, 3125/3087
[[Mapping]]: [{{val| 1 6 6 6 }}, {{val| 0 -18 -15 -13 }}]


[[POTE generator]]: ~25/21 = 294.303
[[POTE generator]]: ~25/21 = 294.303
[[Mapping]]: [{{val| 1 6 6 6 }}, {{val| 0 -18 -15 -13 }}]


{{Multival|legend=1| 18 15 13 -18 -30 -12 }}
{{Multival|legend=1| 18 15 13 -18 -30 -12 }}
Line 543: Line 695:
{{Val list|legend=1| 49, 53, 314d }}
{{Val list|legend=1| 49, 53, 314d }}


Badness: 0.0765
[[Badness]]: 0.0765


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 3125/3087
Comma list: 176/175, 540/539, 3125/3087
Mapping: [{{val| 1 6 6 6 14 }}, {{val| 0 -18 -15 -13 -43 }}]


POTE generator: ~25/21 = 294.181
POTE generator: ~25/21 = 294.181
Mapping: [{{val| 1 6 6 6 14 }}, {{val| 0 -18 -15 -13 -43 }}]


{{Val list|legend=1| 49, 53, 102d, 155d }}
{{Val list|legend=1| 49, 53, 102d, 155d }}
Line 557: Line 712:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 275/273, 325/324, 540/539
Comma list: 176/175, 275/273, 325/324, 540/539
Mapping: [{{val| 1 6 6 6 14 14 }}, {{val| 0 -18 -15 -13 -43 -42 }}]


POTE generator: ~13/11 = 294.187
POTE generator: ~13/11 = 294.187


Mapping: [{{val| 1 6 6 6 14 14 }}, {{val| 0 -18 -15 -13 -43 -42 }}]
{{Val list|legend=1| 49f, 53, 102df, 155d }}
 
{{Val list|legend=1| 53, 102df, 155d }}


Badness: 0.0311
Badness: 0.0311


= Novemkleismic =
= Novemkleismic =
Subgroup: 2.3.5.7
[[Comma list]]: 15625/15552, 40353607/40310784
[[Comma list]]: 15625/15552, 40353607/40310784
[[POTE generator]]: ~6/5 = 317.005


[[Mapping]]: [{{val| 9 0 9 11 }}, {{val| 0 6 5 6 }}]
[[Mapping]]: [{{val| 9 0 9 11 }}, {{val| 0 6 5 6 }}]


{{Multival|legend=1| 54 45 54 -54 -66 -1 }}
{{Multival|legend=1| 54 45 54 -54 -66 -1 }}
[[POTE generator]]: ~6/5 = 317.005


{{Val list|legend=1| 72, 261, 333, 405, 477c, 882c }}
{{Val list|legend=1| 72, 261, 333, 405, 477c, 882c }}
Line 581: Line 742:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4000/3993, 15625/15552
Comma list: 1375/1372, 4000/3993, 15625/15552
Mapping: [{{val| 9 0 9 11 24 }}, {{val| 0 6 5 6 3 }}]


POTE generator: ~6/5 = 317.010
POTE generator: ~6/5 = 317.010
Mapping: [{{val| 9 0 9 11 24 }}, {{val| 0 6 5 6 3 }}]


{{Val list|legend=1| 72, 261, 333, 405, 882c }}
{{Val list|legend=1| 72, 261, 333, 405, 882c }}
Line 592: Line 756:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Mapping: [{{val| 9 0 9 11 24 0 }}, {{val| 0 6 5 6 3 14 }}]


POTE generator: ~6/5 = 317.086
POTE generator: ~6/5 = 317.086


Mapping: [{{val| 9 0 9 11 24 0 }}, {{val| 0 6 5 6 3 14 }}]
{{Val list|legend=1| 72, 261, 333, 738cf, 1071bcff }}
 
{{Val list|legend=1| 72, 261, 333, 738cf, 1071bcf }}


Badness: 0.0391
Badness: 0.0391
Line 605: Line 772:


The just value of sqrt (φ) is 416.545 cents.  
The just value of sqrt (φ) is 416.545 cents.  
Subgroup: 2.3.5.7


[[Comma list]]: 15625/15552, 16875/16807
[[Comma list]]: 15625/15552, 16875/16807


[[POTE tuning|POTE generator]]: ~125/98 = 416.603 cents
[[Mapping]]: [{{val| 1 12 11 16 }}, {{val| 0 -30 -25 -38 }}]


[[Mapping]]: [{{val| 1 12 11 16 }}, {{val| 0 -30 -25 -38 }}]
[[POTE generator]]: ~125/98 = 416.603


{{Val list|legend=1| 49, 72, 193, 265 }}
{{Val list|legend=1| 49, 72, 193, 265 }}
Line 617: Line 786:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 4375/4356
Comma list: 540/539, 1375/1372, 4375/4356
Mapping: [{{val| 1 12 11 16 17 }}, {{val| 0 -30 -25 -38 -39 }}]


POTE generator: ~14/11 = 416.604
POTE generator: ~14/11 = 416.604
Mapping: [{{val| 1 12 11 16 17 }}, {{val| 0 -30 -25 -38 -39 }}]


{{Val list|legend=1| 49, 72, 193, 265 }}
{{Val list|legend=1| 49, 72, 193, 265 }}
Line 628: Line 800:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 364/363, 625/624, 1375/1372
Comma list: 325/324, 364/363, 625/624, 1375/1372
Mapping: [{{val| 1 12 11 16 17 28 }}, {{val| 0 -30 -25 -38 -39 -70 }}]


POTE generator: ~14/11 = 416.585
POTE generator: ~14/11 = 416.585
Mapping: [{{val| 1 12 11 16 17 28 }}, {{val| 0 -30 -25 -38 -39 -70 }}]


{{Val list|legend=1| 72, 121, 193 }}
{{Val list|legend=1| 72, 121, 193 }}
Line 639: Line 814:


== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
Mapping: [{{val| 1 12 11 16 17 28 27 }}, {{val| 0 -30 -25 -38 -39 -70 -66 }}]


POTE generator: ~14/11 = 416.585
POTE generator: ~14/11 = 416.585
Mapping: [{{val| 1 12 11 16 17 28 27 }}, {{val| 0 -30 -25 -38 -39 -70 -66 }}]


{{Val list|legend=1| 72, 121, 193 }}
{{Val list|legend=1| 72, 121, 193 }}
Line 650: Line 828:


== 19-limit ==
== 19-limit ==
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
Mapping: [{{val| 1 12 11 16 17 28 27 -2 }}, {{val| 0 -30 -25 -38 -39 -70 -66 18 }}]


POTE generator: ~14/11 = 416.580
POTE generator: ~14/11 = 416.580
Mapping: [{{val| 1 12 11 16 17 28 27 -2 }}, {{val| 0 -30 -25 -38 -39 -70 -66 18 }}]


{{Val list|legend=1| 72, 121, 193 }}
{{Val list|legend=1| 72, 121, 193 }}
Line 674: Line 855:
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
{{todo| review | improve readability }}

Revision as of 07:25, 30 March 2021

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is [-6 -5 6⟩, and flipping that yields ⟨⟨ 6 5 -6 ]] for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)5 = 5/2 × 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called Hanson, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

Hanson

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

POTE generator: ~6/5 = 317.007

Tuning ranges:

  • valid range: [300.000, 327.273] (4 to 11b)
  • nice range: [315.641, 317.263]
  • strict range: [315.641, 317.263]

Template:Val list

Music
Analysis and diagrams

Seven limit extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.

Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Wedgie: ⟨⟨ 6 5 3 -6 -12 -7 ]]

POTE generator: ~6/5 = 316.473

Tuning ranges:

  • valid range: [300.000, 327.273] (1\4 to 3\11)
  • nice range: [308.744, 322.942]
  • strict range: [308.744, 322.942]

Template:Val list

Badness: 0.0274

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 100/99

Mapping: [⟨1 0 1 2 4], ⟨0 6 5 3 -2]]

POTE generator: ~6/5 = 317.576

Tuning ranges:

  • valid range: [315.789, 320.000] (5\19 to 4\15)
  • nice range: [308.744, 324.341]
  • strict range: [315.789, 320.000]

Template:Val list

Badness: 0.0274

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 100/99

Mapping: [⟨1 0 1 2 4 5], ⟨0 6 5 3 -2 -5]]

POTE generator: ~6/5 = 316.611

Tuning ranges:

  • valid range: 315.789 (5\19)
  • nice range: [303.597, 324.341]
  • strict range: 315.789

Template:Val list

Badness: 0.0297

Kema

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 100/99

Mapping: [⟨1 0 1 2 4 0], ⟨0 6 5 3 -2 14]]

POTE generator: ~6/5 = 317.423

Tuning ranges:

  • valid range: [315.789, 320.000] (5\19 to 4\15)
  • nice range: [308.744, 324.341]
  • strict range: [315.789, 320.000]

Template:Val list

Badness: 0.0227

Kumbaya

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 56/55, 66/65

Mapping: [⟨1 0 1 2 4 4], ⟨0 6 5 3 -2 -1]]

POTE generator: ~6/5 = 318.595

Template:Val list

Badness: 0.0316

Qeema

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 126/125

Mapping: [⟨1 0 1 2 -1], ⟨0 6 5 3 17]]

POTE generator: ~6/5 = 314.730

Template:Val list

Badness: 0.0401

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 126/125

Mapping: [⟨1 0 1 2 -1 0], ⟨0 6 5 3 17 14]]

POTE generator: ~6/5 = 315.044

Template:Val list

Badness: 0.0294

Darjeeling

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 77/75

Mapping: [⟨1 0 1 2 0], ⟨0 6 5 3 13]]

POTE generator: ~6/5 = 317.656

Template:Val list

Badness: 0.0276

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 66/65, 77/75

Mapping: [⟨1 0 1 2 0 0], ⟨0 6 5 3 13 14]]

POTE generator: ~6/5 = 317.298

Template:Val list

Badness: 0.02144

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Wedgie: ⟨⟨ 6 5 -12 -6 -36 -42 ]]

POTE generator: ~6/5 = 318.267

Template:Val list

Badness: 0.0949

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1331/1323

Mapping: [⟨1 0 1 6 4], ⟨0 6 5 -12 -2]]

POTE generator: ~6/5 = 318.282

Template:Val list

Badness: 0.0369

Catakleismic

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3], ⟨0 6 5 22]]

Wedgie: ⟨⟨ 6 5 22 -6 18 37 ]]

POTE generator: ~6/5 = 316.732

Tuning ranges:

  • valid range: [315.789, 317.647] (5\19 to 9\34)
  • nice range: [315.641, 317.263]
  • strict range: [315.789, 317.263]

Template:Val list

Badness: 0.0215

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4375/4374

Mapping: [⟨1 0 1 -3 9], ⟨0 6 5 22 -21]]

POTE generator: ~6/5 = 316.719

Tuning ranges:

  • valid range: [315.789, 316.981] (5\19 to 14\53)
  • nice range: [315.641, 317.263]
  • strict range: [315.789, 316.981]

Template:Val list

Badness: 0.0218

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 540/539

Mapping: [⟨1 0 1 -3 9 0], ⟨0 6 5 22 -21 14]]

POTE generator: ~6/5 = 316.738

Tuning ranges:

  • valid range: [315.789, 316.981] (5\19 to 14\53)
  • nice range: [315.641, 318.309]
  • strict range: [315.789, 316.981]

Template:Val list

Badness: 0.0169

Cataclysmic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2200/2187

Mapping: [⟨1 0 1 -3 -5], ⟨0 6 5 22 32]]

POTE generator: ~6/5 = 317.042

Template:Val list

Badness: 0.0400

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 275/273

Mapping: [⟨1 0 1 -3 -5 0], ⟨0 6 5 22 32 14]]

POTE generator: ~6/5 = 317.036

Template:Val list

Badness: 0.0226

Catalytic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4374

Mapping: [⟨1 0 1 -3 -10], ⟨0 6 5 22 51]]

POTE generator: ~6/5 = 316.653

Template:Val list

Badness: 0.0304

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 1716/1715

Mapping: [⟨1 0 1 -3 -10 0], ⟨0 6 5 22 51 14]]

POTE generator: ~6/5 = 316.639

Template:Val list

Badness: 0.0223

Cataleptic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 864/847

Mapping: [⟨1 0 1 -3 4], ⟨0 6 5 22 -2]]

POTE generator: ~6/5 = 317.083

Template:Val list

Badness: 0.0443

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 676/675

Mapping: [⟨1 0 1 -3 4 0], ⟨0 6 5 22 -2 14]]

POTE generator: ~6/5 = 317.118

Template:Val list

Badness: 0.0273

Bikleismic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4375/4356

Mapping: [⟨2 0 2 -6 -1], ⟨0 6 5 22 15]]

POTE generator: ~6/5 = 316.721

Template:Val list

Badness: 0.0293

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0], ⟨0 6 5 22 15 14]]

POTE generator: ~6/5 = 316.726

Template:Val list

Badness: 0.0218

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Wedgie: ⟨⟨ 6 5 -31 -6 -66 -86 ]]

POTE generator: ~6/5 = 317.121

Template:Val list

Badness: 0.0521

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3388/3375

Mapping: [⟨1 0 1 11 -5], ⟨0 6 5 -31 32]]

POTE generator: ~6/5 = 317.162

Template:Val list

Badness: 0.0398

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 625/624

Mapping: [⟨1 0 1 11 -5 0], ⟨0 6 5 -31 32 14]]

POTE generator: ~6/5 = 317.162

Template:Val list

Badness: 0.0202

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Wedgie: ⟨⟨ 6 5 56 -6 72 116 ]]

POTE generator: ~6/5 = 317.314

Template:Val list

Badness: 0.1635

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 14700/14641

Mapping: [⟨1 0 1 -12 -5], ⟨0 6 5 56 32]]

POTE generator: ~6/5 = 317.311

Template:Val list

Badness: 0.0486

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 625/624

Mapping: [⟨1 0 1 -12 -5 0], ⟨0 6 5 56 32 14]]

POTE generator: ~6/5 = 317.311

Template:Val list

Badness: 0.0244

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

POTE generator: ~35/32 = 158.649

Template:Val list

Badness: 0.0521

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 4000/3969

Mapping: [⟨1 0 1 4 2], ⟨0 12 10 -9 11]]

POTE generator: ~11/10 = 158.677

Template:Val list

Badness: 0.0380

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 275/273, 325/324

POTE generator: ~11/10 = 158.655

Mapping: [⟨1 0 1 4 2 0], ⟨0 12 10 -9 11 28]]

Template:Val list

Badness: 0.0260

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping: [⟨1 6 6 12], ⟨0 -12 -10 -25]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 441.335

Minimax tuning:

[[1 0 0 0⟩, [6/25 0 0 12/25⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Eigenmonzos: 2, 7

Algebraic generator: real root of 5x3 - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

Template:Val list

Badness: 0.0473

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3136/3125

Mapping: [⟨1 6 6 12 -5], ⟨0 -12 -10 -25 23]]

POTE generator: ~9/7 = 441.355

Template:Val list

Badness: 0.0474

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 625/624

POTE generator: ~9/7 = 441.363

Mapping: [⟨1 6 6 12 -5 14], ⟨0 -12 -10 -25 23 -28]]

Template:Val list

Badness: 0.0268

Tritikleismic

See also: Gamelismic clan § Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

Wedgie: ⟨⟨ 18 15 -6 -18 -60 -56 ]]

POTE generator: 316.872

Minimax tuning:

[[1 0 0 0⟩, [2 0 6/7 -6/7⟩, [8/3 0 5/7 -5/7⟩, [8/3 0 -2/7 2/7⟩]
Eigenmonzos: 2, 7/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [46/21 5/7 0 -5/14⟩, [20/7 -2/7 0 1/7⟩]
Eigenmonzos: 2, 9/7

Template:Val list

Badness: 0.0563

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Mapping generators: ~44/35, ~6/5

POTE generator: ~6/5 = 316.881

Minimax tuning:

  • 11-odd-limit
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [46/21 5/7 0 -5/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [71/21 3/7 0 -3/14 0⟩]
Eigenmonzos: 2, 9/7

Template:Val list

Badness: 0.0193

13-limit

Comma list: 325/324, 364/363, 441/440, 625/624

Mapping: [⟨3 0 3 10 8 0], ⟨0 6 5 -2 3 14]]

Mapping generators: ~44/35, ~6/5

POTE generator: ~6/5 = 316.959

Template:Val list

Badness: 0.0157

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [⟨3 0 3 10 8 0 -2], ⟨0 6 5 -2 3 14 18]]

Mapping generators: ~34/27, ~6/5

POTE generator: ~6/5 = 316.9082

Template:Val list

Quadritikleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping: [⟨4 0 4 7], ⟨0 6 5 4]]

Wedgie: ⟨⟨ 24 20 16 -24 -42 -19 ]]

POTE generator: ~6/5 = 316.9999

Template:Val list

Badness: 0.0392

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 6250/6237

Mapping: [⟨4 0 4 7 17], ⟨0 6 5 4 -3]]

POTE generator: ~6/5 = 316.925

Template:Val list

Badness: 0.0234

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 1573/1568

Mapping: [⟨4 0 4 7 17 0], ⟨0 6 5 4 -3 14]]

POTE generator: ~6/5 = 316.989

Template:Val list

Badness: 0.0187

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [⟨1 6 6 6], ⟨0 -18 -15 -13]]

POTE generator: ~25/21 = 294.303

Wedgie: ⟨⟨ 18 15 13 -18 -30 -12 ]]

Template:Val list

Badness: 0.0765

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 3125/3087

Mapping: [⟨1 6 6 6 14], ⟨0 -18 -15 -13 -43]]

POTE generator: ~25/21 = 294.181

Template:Val list

Badness: 0.0528

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 275/273, 325/324, 540/539

Mapping: [⟨1 6 6 6 14 14], ⟨0 -18 -15 -13 -43 -42]]

POTE generator: ~13/11 = 294.187

Template:Val list

Badness: 0.0311

Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping: [⟨9 0 9 11], ⟨0 6 5 6]]

Wedgie: ⟨⟨ 54 45 54 -54 -66 -1 ]]

POTE generator: ~6/5 = 317.005

Template:Val list

Badness: 0.1934

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4000/3993, 15625/15552

Mapping: [⟨9 0 9 11 24], ⟨0 6 5 6 3]]

POTE generator: ~6/5 = 317.010

Template:Val list

Badness: 0.05172

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 4000/3993

Mapping: [⟨9 0 9 11 24 0], ⟨0 6 5 6 3 14]]

POTE generator: ~6/5 = 317.086

Template:Val list

Badness: 0.0391

Sqrtphi

The just value of sqrt (φ) is 416.545 cents.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 16875/16807

Mapping: [⟨1 12 11 16], ⟨0 -30 -25 -38]]

POTE generator: ~125/98 = 416.603

Template:Val list

Badness: 0.0704

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4375/4356

Mapping: [⟨1 12 11 16 17], ⟨0 -30 -25 -38 -39]]

POTE generator: ~14/11 = 416.604

Template:Val list

Badness: 0.0255

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624, 1375/1372

Mapping: [⟨1 12 11 16 17 28], ⟨0 -30 -25 -38 -39 -70]]

POTE generator: ~14/11 = 416.585

Template:Val list

Badness: 0.0200

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

Mapping: [⟨1 12 11 16 17 28 27], ⟨0 -30 -25 -38 -39 -70 -66]]

POTE generator: ~14/11 = 416.585

Template:Val list

Badness: 0.0130

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

Mapping: [⟨1 12 11 16 17 28 27 -2], ⟨0 -30 -25 -38 -39 -70 -66 18]]

POTE generator: ~14/11 = 416.580

Template:Val list

Badness: 0.0147

Scales

Music

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