Breedsmic temperaments: Difference between revisions

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-This page discusses miscellaneous rank-2 temperaments tempering out the [[breedsma]], {{monzo|-5 -1 -2 4}} = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12EDO, for example) which does not possess a neutral third cannot be tempering out the breedsma.
+This page discusses miscellaneous rank-2 temperaments tempering out the [[breedsma]], {{monzo|-5 -1 -2 4}} = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.
 The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that 49/40 × 10/7 = 7/4 and 49/40 × (10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
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 * [[Harry]] → [[Gravity family #Harry|Gravity family]] ({2401/2400, 19683/19600})
 * ''[[Sesquiquartififths]]'' → [[Schismatic family #Sesquiquartififths|Schismatic family]] ({2401/2400, 32805/32768})
+* ''[[Amicable]]'' → [[Amity family #Amicable|Amity family]] ({2401/2400, 1600000/1594323})
 * ''[[Neptune]]'' → [[Gammic family #Neptune|Gammic family]] ({2401/2400, 48828125/48771072})
 * ''[[Eagle]]'' → [[Vulture family #Eagle|Vulture family]] ({2401/2400, 10485760000/10460353203})
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 Badness: 0.013830
-== Amicable ==
-{{see also| Amity family }}
-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
 == Septidiasemi ==