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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 quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same circle of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  
The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same circle of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  


Belonging to it and considered below are undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are:
Belonging to it and considered below are undecental, leapday, mystery, quanic, septiquarter, countriton and ketchup. Other hemifamity temperaments are:
* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
: +36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3.  
: +36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3.  
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* [[Rodan]] → [[Gamelismic clan #Rodan|Gamelismic clan]]
* [[Rodan]] → [[Gamelismic clan #Rodan|Gamelismic clan]]
: +245/243, the 41 & 46 temperament, fifth sliced into three as does slendric.  
: +245/243, the 41 & 46 temperament, fifth sliced into three as does slendric.  
* ''[[Trimot]]'' → [[Tricot family #Trimot|Tricot family]]
: +2430/2401, the 53 & 70 temperament, twelfth sliced into three as does tricot.
* ''[[Monkey]]'' → [[Tetracot family #Monkey|Tetracot family]]
* ''[[Monkey]]'' → [[Tetracot family #Monkey|Tetracot family]]
: +875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot.  
: +875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot.  
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{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.039
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
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== Mystery ==
== Mystery ==
{{main| Mystery }}
{{Main| Mystery }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mystery]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mystery]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 50421/50000
[[Comma list]]: 5120/5103, 50421/50000
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{{Multival|legend=1| 0 29 29 46 46 -14 }}
{{Multival|legend=1| 0 29 29 46 46 -14 }}


[[POTE generator]]: ~5/4 = 388.646
[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\29, ~5/4 = 388.646


{{Val list|legend=1| 29, 58, 87, 145 }}
{{Val list|legend=1| 29, 58, 87, 145 }}
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Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}]
Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}]


POTE generator: ~5/4 = 388.460
Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.460


Optimal GPV sequence: {{Val list| 29, 58, 87, 145 }}
Optimal GPV sequence: {{Val list| 29, 58, 87, 145 }}
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Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]


POTE generator: ~5/4 = 388.354
Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.354


Optimal GPV sequence: {{Val list| 29, 58, 87, 145, 232, 377cef }}
Optimal GPV sequence: {{Val list| 29, 58, 87, 145, 232, 377cef }}
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== Quanic ==
== Quanic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 5832000/5764801
[[Comma list]]: 5120/5103, 5832000/5764801
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[[Mapping]]: [{{val| 1 1 -4 0 }}, {{val| 0 5 54 24 }}]
[[Mapping]]: [{{val| 1 1 -4 0 }}, {{val| 0 5 54 24 }}]


[[POTE generator]]: ~160/147 = 140.493
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 140.493


{{Val list|legend=1| 94, 111, 205 }}
{{Val list|legend=1| 94, 111, 205 }}
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Mapping: [{{val| 1 1 -4 0 1 }}, {{val| 0 5 54 24 21 }}]
Mapping: [{{val| 1 1 -4 0 1 }}, {{val| 0 5 54 24 21 }}]


POTE generator: ~88/81 = 140.489
Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.489


Optimal GPV sequence: {{Val list| 94, 111, 205 }}
Optimal GPV sequence: {{Val list| 94, 111, 205 }}
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Mapping: [{{val| 1 1 -4 0 1 3 }}, {{val| 0 5 54 24 21 6 }}]
Mapping: [{{val| 1 1 -4 0 1 3 }}, {{val| 0 5 54 24 21 6 }}]


POTE generator: ~13/12 = 140.496
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496


Optimal GPV sequence: {{Val list| 94, 111, 205 }}
Optimal GPV sequence: {{Val list| 94, 111, 205 }}
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Mapping: [{{val| 1 1 -4 0 1 3 -2 }}, {{val| 0 5 54 24 21 6 52 }}]
Mapping: [{{val| 1 1 -4 0 1 3 -2 }}, {{val| 0 5 54 24 21 6 52 }}]


POTE generator: ~13/12 = 140.497
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.497


Optimal GPV sequence: {{Val list| 94, 111, 205 }}
Optimal GPV sequence: {{Val list| 94, 111, 205 }}
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Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]
Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]


POTE generator: ~13/12 = 140.496
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496


Optimal GPV sequence: {{Val list| 94, 111, 205 }}
Optimal GPV sequence: {{Val list| 94, 111, 205 }}
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== Septiquarter ==
== Septiquarter ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 420175/419904
[[Comma list]]: 5120/5103, 420175/419904
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{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
{{Multival|legend=1| 7 38 -4 44 -26 -116 }}


[[POTE generator]]: ~147/128 = 242.453
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.453


{{Val list|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
{{Val list|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
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Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]
Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]


POTE generators: ~121/105 = 242.4511
Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4511


Optimal GPV sequence: {{Val list| 94, 198, 292, 490 }}
Optimal GPV sequence: {{Val list| 94, 198, 292, 490 }}
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Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]
Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]


POTE generators: ~121/105 = 242.4448
Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4448


Optimal GPV sequence: {{Val list| 94, 198, 490f }}
Optimal GPV sequence: {{Val list| 94, 198, 490f }}


Badness: 0.034834
Badness: 0.034834
== Tricot ==
{{see also| Tricot family }}
The generator for tricot is the real cube root of third harmonic, 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Tricot can be described as 53&amp;70 temperament (also called as "trimot", as confirmed by the data from [http://x31eq.com/cgi-bin/rt.cgi?ets=53_17c&limit=7 x31eq]), tempering out the [[tricot comma]], {{monzo| 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.
Subgroup: 2.3.5.7
[[Comma list]]: 2430/2401, 5120/5103
[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
{{Multival|legend=1| 3 29 11 39 9 -56 }}
[[POTE generator]]: ~81/56 = 634.026
{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
[[Badness]]: 0.100127
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 99/98, 121/120, 5120/5103
Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
POTE generator: ~63/44 = 634.027
Optimal GPV sequence: {{Val list| 17c, 36ce, 53, 70, 123de }}
Badness: 0.056134
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 121/120, 169/168, 352/351
Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
POTE generator: ~13/9 = 634.012
Optimal GPV sequence: {{Val list| 17c, 36ce, 53, 70, 123de }}
Badness: 0.032102


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 7558272/7503125
[[Comma list]]: 5120/5103, 7558272/7503125


[[Mapping]]: [{{val|1 6 19 -7}}, {{val|0 -9 -34 20}}]
[[Mapping]]: [{{val| 1 6 19 -7 }}, {{val| 0 -9 -34 20 }}]


{{Multival|legend=1| 9 34 -20 33 -57 -142 }}
{{Multival|legend=1| 9 34 -20 33 -57 -142 }}


[[POTE generator]]: ~1728/1225 = 588.582
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1225 = 588.582


{{Val list|legend=1| 53, 157, 210 }}
{{Val list|legend=1| 53, 157, 210 }}
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Comma list: 176/175, 5120/5103, 41503/41472
Comma list: 176/175, 5120/5103, 41503/41472


Mapping: [{{val|1 6 19 -7 27}}, {{val|0 -9 -34 20 -48}}]
Mapping: [{{val| 1 6 19 -7 27 }}, {{val| 0 -9 -34 20 -48 }}]


POTE generator: ~108/77 = 588.545
Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.545


Optimal GPV sequence: {{Val list| 53, 104c, 157 }}
Optimal GPV sequence: {{Val list| 53, 104c, 157 }}
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Comma list: 176/175, 351/350, 847/845, 2197/2187
Comma list: 176/175, 351/350, 847/845, 2197/2187


Mapping: [{{val|1 6 19 -7 27 14}}, {{val|0 -9 -34 20 -48 -21}}]
Mapping: [{{val| 1 6 19 -7 27 14 }}, {{val| 0 -9 -34 20 -48 -21 }}]


POTE generator: ~108/77 = 588.544
Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.544


Optimal GPV sequence: {{Val list| 53, 104c, 157 }}
Optimal GPV sequence: {{Val list| 53, 104c, 157 }}
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== Ketchup ==
== Ketchup ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 1071875/1062882
[[Comma list]]: 5120/5103, 1071875/1062882
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{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
{{Multival|legend=1| 8 30 -18 29 -51 -126 }}


[[POTE generator]]: ~64/63 = ~81/80 = 25.719
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~64/63 = 25.719


{{Val list|legend=1| 46, 94, 140 }}
{{Val list|legend=1| 46, 94, 140 }}
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Mapping: [{{val| 2 3 4 6 7 }}, {{val| 0 4 15 -9 -2 }}]
Mapping: [{{val| 2 3 4 6 7 }}, {{val| 0 4 15 -9 -2 }}]


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.693


Optimal GPV sequence: {{Val list| 46, 94, 140 }}
Optimal GPV sequence: {{Val list| 46, 94, 140 }}
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Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]
Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.697


Optimal GPV sequence: {{Val list| 46, 94, 140 }}
Optimal GPV sequence: {{Val list| 46, 94, 140 }}
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Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Optimal tuning (POTE): ~17/12 = 1\2, ~64/63 = 25.701


Optimal GPV sequence: {{Val list| 46, 94, 140 }}
Optimal GPV sequence: {{Val list| 46, 94, 140 }}
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Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
Optimal tuning (POTE): ~17/12 = 1\2, ~64/63= 25.660


Optimal GPV sequence: {{Val list| 46, 94, 140h, 234eh }}
Optimal GPV sequence: {{Val list| 46, 94, 140h, 234eh }}
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Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]
Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
Optimal tuning (POTE): ~17/12 = 1\2, ~64/63 = 25.661


Optimal GPV sequence: {{Val list| 46, 94, 140h, 234ehi }}
Optimal GPV sequence: {{Val list| 46, 94, 140h, 234ehi }}
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