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Amity family: Difference between revisions

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=== Overview to extensions ===
=== Overview to extensions ===
The second comma to extend the 5-limit amity include 4375/4374 for amity, 225/224 for houborizic, 65625/65536 for paramity, 126/125 for accord, 245/243 for bamity, 2430/2401 for hamity, 1029/1024 for gamity, and 16875/16807 for familia.  
The second comma to extend the 5-limit amity include 4375/4374 for amity, 225/224 for houborizic, 65625/65536 for paramity, 126/125 for accord, 245/243 for bamity, 2430/2401 for hamity, 1029/1024 for gamity, 2401/2400 for amicable, and 16875/16807 for familia.  


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable]]
* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]
* ''[[Witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]
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{{Multival|legend=1| 10 26 12 18 -9 -45 }}
{{Multival|legend=1| 10 26 12 18 -9 -45 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 260.402
[[Optimal tuning]] ([[POTE]]): ~343/240 = 1\2, ~7/6 = 260.402


{{Val list|legend=1| 14c, 32c, 46, 60, 106d }}
{{Val list|legend=1| 14c, 32c, 46, 60, 106d }}
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Mapping: [{{val| 2 1 -1 3 3 }}, {{val| 0 5 13 6 9 }}]
Mapping: [{{val| 2 1 -1 3 3 }}, {{val| 0 5 13 6 9 }}]


Mapping generators: ~99/70, ~7/6
Optimal tuning (POTE): ~99/70 = 1\2, ~7/6 = 260.393
 
POTE generator: ~7/6 = 260.393


Optimal GPV sequence: {{Val list| 14c, 32c, 46, 60e, 106de }}
Optimal GPV sequence: {{Val list| 14c, 32c, 46, 60e, 106de }}
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Mapping: [{{val| 2 1 -1 3 3 0 }}, {{val| 0 5 13 6 9 17 }}]
Mapping: [{{val| 2 1 -1 3 3 0 }}, {{val| 0 5 13 6 9 17 }}]


Mapping generators: ~55/39, ~7/6
Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 260.618
 
POTE generator: ~7/6 = 260.618


Optimal GPV sequence: {{Val list| 14cf, 32cf, 46, 106def, 152def }}
Optimal GPV sequence: {{Val list| 14cf, 32cf, 46, 106def, 152def }}
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Badness: 0.012038
Badness: 0.012038
== Amicable ==
The amicable temperament tempers out the [[amity comma]] and the [[canousma]] in addition to the breedsma, and is closely associated with the canou temperament.
While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.
Subgroup: 2.3.5.7
[[Comma list]]: 2401/2400, 1600000/1594323
[[Mapping]]: [{{val|1 3 6 5}}, {{val|0 -20 -52 -31}}]
{{Multival|legend=1|20 52 31 36 -7 -74}}
[[POTE generator]]: ~21/20 = 84.880
{{Val list|legend=1| 99, 212, 311, 410, 1131, 1541b }}
[[Badness]]: 0.045473
=== Amical ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 131072/130977, 1600000/1594323
Mapping: [{{val| 1 3 6 5 -8 }}, {{val| 0 -20 -52 -31 162 }}]
POTE generator: ~21/20 = 84.8843
Optimal GPV sequence: {{val list| 99, 212e, 311, 410, 721, 1032, 1343 }}
Badness: 0.100668
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
Mapping: [{{val| 1 3 6 5 -8 -5 }}, {{val| 0 -20 -52 -31 162 123 }}]
POTE generator: ~21/20 = 84.8838
Optimal GPV sequence: {{val list| 99, 212ef, 311, 410, 721, 1032 }}
Badness: 0.049893
=== Amorous ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 6250/6237, 19712/19683
Mapping: [{{val| 1 3 6 5 14 }}, {{val| 0 -20 -52 -31 -149 }}]
POTE generator: ~21/20 = 84.8896
Optimal GPV sequence: {{val list| 99e, 212, 311, 1145c, 1456cd }}
Badness: 0.048924
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Mapping: [{{val| 1 3 6 5 14 17 }}, {{val| 0 -20 -52 -31 -149 -188 }}]
POTE generator: ~21/20 = 84.8910
Optimal GPV sequence: {{val list| 99ef, 212, 311, 834, 1145c }}
Badness: 0.034681
=== Pseudoamical ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 1600000/1594323
Mapping: [{{val| 1 3 6 5 -1 }}, {{val| 0 -20 -52 -31 63 }}]
POTE generator: ~21/20 = 84.9091
Optimal GPV sequence: {{val list| 99, 113, 212, 961ccdeee }}
Badness: 0.085837
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Mapping: [{{val| 1 3 6 5 -1 2 }}, {{val| 0 -20 -52 -31 63 24 }}]
POTE generator: ~21/20 = 84.9127
Optimal GPV sequence: {{val list| 99, 113, 212, 537cdeff, 749ccdeefff }}
Badness: 0.047025
=== Pseudoamorous ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 980000/970299
Mapping: [{{val| 1 3 6 5 7 }}, {{val| 0 -20 -52 -31 -50 }}]
POTE generator: ~21/20 = 84.8917
Optimal GPV sequence: {{val list| 99e, 212e }}
Badness: 0.056583
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 1875/1859
Mapping: [{{val| 1 3 6 5 7 10 }}, {{val| 0 -20 -52 -31 -50 -89 }}]
POTE generator: ~21/20 = 84.9164
Optimal GPV sequence: {{val list| 99ef, 113, 212ef }}
Badness: 0.042826
=== Floral ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 14641/14580
Mapping: [{{val| 2 6 12 10 13 }}, {{val| 0 -20 -52 -31 -43 }}]
POTE generator: ~21/20 = 84.8788
Optimal GPV sequence: {{val list| 198, 212, 410 }}
Badness: 0.065110
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Mapping: [{{val| 2 6 12 10 13 19 }}, {{val| 0 -20 -52 -31 -43 -82 }}]
POTE generator: ~21/20 = 84.8750
Optimal GPV sequence: {{val list| 198, 410 }}
Badness: 0.037013
=== Humorous ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 1600000/1594323
Mapping: [{{val| 1 3 6 5 3 }}, {{val| 0 -40 -104 -62 13 }}]
POTE generator: ~4096/3993 = 42.4391
Optimal GPV sequence: {{val list| 85c, 113, 198, 311, 509, 820 }}
Badness: 0.058249
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
Mapping: [{{val| 1 3 6 5 3 6 }}, {{val| 0 -40 -104 -62 13 -65 }}]
POTE generator: ~40/39 = 42.4391
Optimal GPV sequence: {{val list| 85c, 113, 198, 311, 509, 820f }}
Badness: 0.028267


== Familia ==
== Familia ==
Retrieved from "https://en.xen.wiki/w/Amity_family"