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| 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]]. | | 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]]. |
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| [[MOS scale]]s of würschmidt are even more extreme than those of [[magic]]. [[Proper]] scales does not appear until 28, 31 or even 34 notes. | | [[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales does not appear until 28, 31 or even 34 notes, depending on the specific tuning. |
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| The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Würschmidt adds {{monzo| 12 3 -6 -1 }}, worschmidt adds 65625/65536 = {{monzo| -16 1 5 1 }}, whirrschmidt adds 4375/4374 = {{monzo| -1 -7 4 1 }} and hemiwürschmidt adds 6144/6125 = {{monzo| 11 1 -3 -2 }}. | | The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Septimal würschmidt adds {{monzo| 12 3 -6 -1 }}, worschmidt adds 65625/65536 = {{monzo| -16 1 5 1 }}, whirrschmidt adds 4375/4374 = {{monzo| -1 -7 4 1 }}. These all use the same generator as 5-limit würschmidt. |
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| | Hemiwürschmidt adds 6144/6125 = {{monzo| 11 1 -3 -2 }} and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]. |
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| == Würschmidt == | | == Würschmidt == |
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| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799 | | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799 |
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| {{Optimal ET sequence|legend=1|3, 28, 31, 34, 65, 99, 164, 721c, 885c }} | | {{Optimal ET sequence|legend=1| 3, 28, 31, 34, 65, 99, 164, 721c, 885c }} |
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| [[Badness]]: 0.040603 | | [[Badness]]: 0.040603 |
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| Badness: 0.058325 | | Badness: 0.058325 |
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| == Hemiwürschmidt ==
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| {{See also| Hemimean clan #Hemiwürschmidt }}
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| '''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt'''), which splits the major third in two and uses that for a generator, is the most important of these temperaments even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
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| [[Subgroup]]: 2.3.5.7
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| [[Comma list]]: 2401/2400, 3136/3125
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| {{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}
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| {{Multival|legend=1| 16 2 5 -34 -37 6 }}
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| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
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| {{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
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| [[Badness]]: 0.020307
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| === 11-limit ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 243/242, 441/440, 3136/3125
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| Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840
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| {{Optimal ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
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| Badness: 0.021069
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 243/242, 351/350, 441/440, 3584/3575
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| Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
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| {{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
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| Badness: 0.023074
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| ==== Hemithir ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 121/120, 176/175, 196/195, 275/273
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| Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
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| {{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
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| Badness: 0.031199
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| === Hemiwur ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 121/120, 176/175, 1375/1372
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| Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
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| {{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
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| Badness: 0.029270
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 121/120, 176/175, 196/195, 275/273
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| Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
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| {{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
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| Badness: 0.028432
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| ==== Hemiwar ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 66/65, 105/104, 121/120, 1375/1372
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| Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
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| {{Optimal ET sequence|legend=1| 6f, 31 }}
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| Badness: 0.044886
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| === Quadrawürschmidt ===
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| This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.
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| Subgroup: 2.3.5.7.11
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| Comma list: 2401/2400, 3025/3024, 3136/3125
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| Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
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| : mapping generators: ~2, ~147/110
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| Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404
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| {{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
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| Badness: 0.034814
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| === Semihemiwür ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 2401/2400, 3136/3125, 9801/9800
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| Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
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| : mapping generators: ~99/70, ~495/392
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| Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021
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| {{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
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| Badness: 0.044848
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
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| Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
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| Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
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| {{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
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| Badness: 0.023388
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| ===== Semihemiwürat =====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
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| Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
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| Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
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| {{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
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| Badness: 0.028987
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| ====== 19-limit ======
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| Subgroup: 2.3.5.7.11.13.17.19
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| Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
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| Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
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| Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
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| {{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
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| Badness: 0.021707
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| ===== Semihemiwürand =====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
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| Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
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| Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
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| {{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
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| Badness: 0.029718
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| ====== 19-limit ======
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| Subgroup: 2.3.5.7.11.13.17.19
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| Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
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| Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}
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| Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
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| {{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
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| Badness: 0.029545
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| [[Category:Temperament families]] | | [[Category:Temperament families]] |