Alphatricot family: Difference between revisions
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The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883). | The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883). | ||
Strong 7-limit extensions of this temperament include alphatrimot (53 & 70), alphatrident (53 & 229) and alphatrillium (53 & 441). Tempering out [[5120/5103|hemifamity comma]] (5120/5103) leads to alphatrimot, [[6144/6125|porwell comma]] (6144/6125) leads to alphatrident, and [[4375/4374|ragisma]] (4375/4374) leads to alphatrillium. | Strong 7-limit extensions of this temperament include alphatrimot ({{nowrap| 53 & 70 }}), alphatrident ({{nowrap| 53 & 229 }}) and alphatrillium ({{nowrap| 53 & 441 }}). Tempering out [[5120/5103|hemifamity comma]] (5120/5103) leads to alphatrimot, [[6144/6125|porwell comma]] (6144/6125) leads to alphatrident, and [[4375/4374|ragisma]] (4375/4374) leads to alphatrillium. | ||
== Alphatricot == | == Alphatricot == | ||
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{{Mapping|legend=1| 1 0 -13 | 0 3 29 }} | {{Mapping|legend=1| 1 0 -13 | 0 3 29 }} | ||
: mapping generators: ~2, ~59049/40960 | : mapping generators: ~2, ~59049/40960 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0102 | * [[CTE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0102{{c}} | ||
: [[error map]]: {{val| 0.0000 +0.0757 -0.0168 }} | : [[error map]]: {{val| 0.0000 +0.0757 -0.0168 }} | ||
* [[POTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0124 | * [[POTE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0124{{c}} | ||
: error map: {{val| 0.0000 +0.0821 +0.0454 }} | : error map: {{val| 0.0000 +0.0821 +0.0454 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200. | * CTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0179{{c}} | ||
* POTE: ~2 = 1200. | * POTE: ~2 = 1200.0000{{c}}, ~13/9 = 633.9970{{c}} | ||
{{Optimal ET sequence|legend=0| 17c, 36c, 53 }} | {{Optimal ET sequence|legend=0| 17c, 36c, 53 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~75/52 = 634.009 | * CTE: ~2 = 1200.000{{c}}, ~75/52 = 634.009{{c}} | ||
* POTE: ~2 = 1200. | * POTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0108{{c}} | ||
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }} | {{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }} | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0121 | * [[CTE]]: ~2 = 1200.0000{{c}}, ~23625/16384 = 634.0121{{c}} | ||
: [[error map]]: {{val| 0.0000 +0.0813 +0.0372 +0.0247 }} | : [[error map]]: {{val| 0.0000 +0.0813 +0.0372 +0.0247 }} | ||
* [[POTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0118 | * [[POTE]]: ~2 = 1200.0000{{c}}, ~23625/16384 = 634.0118{{c}} | ||
: error map: {{val| 0.0000 +0.0804 +0.0283 +0.0537 }} | : error map: {{val| 0.0000 +0.0804 +0.0283 +0.0537 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~3888/2695 = 634.0091 | * CTE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0091{{c}} | ||
* POTE: ~2 = 1200.0000, ~3888/2695 = 634.0094 | * POTE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0094{{c}} | ||
{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e }} | {{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~75/52 = 634.0091 | * CTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0091{{c}} | ||
* POTE: ~2 = 1200.0000, ~75/52 = 634.0095 | * POTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0095{{c}} | ||
{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e, 3352de }} | {{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e, 3352de }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~231/160 = 634.0195 | * CTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0195{{c}} | ||
* POTE: ~2 = 1200.0000, ~231/160 = 634.0190 | * POTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0190{{c}} | ||
{{Optimal ET sequence|legend=0| 53, 335, 388 }} | {{Optimal ET sequence|legend=0| 53, 335, 388 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~75/52 = 634.0185 | * CTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0185{{c}} | ||
* POTE: ~2 = 1200.0000, ~75/52 = 634.0181 | * POTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0181{{c}} | ||
{{Optimal ET sequence|legend=0| 53, 335, 388 }} | {{Optimal ET sequence|legend=0| 53, 335, 388 }} | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0484 | * [[CTE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0484{{c}} | ||
: [[error map]]: {{val| 0.0000 +0.1901 +1.0893 +1.1421 }} | : [[error map]]: {{val| 0.0000 +0.1901 +1.0893 +1.1421 }} | ||
* [[POTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0480 | * [[POTE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0480{{c}} | ||
: error map: {{val| 0.0000 +0.1890 +1.0784 +1.1579 }} | : error map: {{val| 0.0000 +0.1890 +1.0784 +1.1579 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~231/160 = 634.0630 | * CTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0630{{c}} | ||
* POTE: ~2 = 1200.0000, ~231/160 = 634.0669 | * POTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0669{{c}} | ||
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }} | {{Optimal ET sequence|legend=0| 53, 123, 176, 229 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~13/9 = 634.0643 | * CTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0643{{c}} | ||
* POTE: ~2 = 1200.0000, ~13/9 = 634.0652 | * POTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0652{{c}} | ||
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }} | {{Optimal ET sequence|legend=0| 53, 123, 176, 229 }} | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~81/56 = 633.9681 | * [[CTE]]: ~2 = 1200.0000{{c}}, ~81/56 = 633.9681{{c}} | ||
: [[error map]]: {{val| 0.0000 -0.0508 -1.2400 +4.8227 }} | : [[error map]]: {{val| 0.0000 -0.0508 -1.2400 +4.8227 }} | ||
* [[POTE]]: ~2 = 1200.0000, ~81/56 = 634.0259 | * [[POTE]]: ~2 = 1200.0000{{c}}, ~81/56 = 634.0259{{c}} | ||
: error map: {{val| 0.0000 +0.1228 +0.4387 +5.4595 }} | : error map: {{val| 0.0000 +0.1228 +0.4387 +5.4595 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~63/44 = 634.0214 | * CTE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0214{{c}} | ||
* POTE: ~2 = 1200.0000, ~63/44 = 634.0273 | * POTE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0273{{c}} | ||
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }} | {{Optimal ET sequence|legend=0| 17c, 36ce, 53 }} | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.0000, ~13/9 = 634.0275 | * CTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0275{{c}} | ||
* POTE: ~2 = 1200.0000, ~13/9 = 634.0115 | * POTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0115{{c}} | ||
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }} | {{Optimal ET sequence|legend=0| 17c, 36ce, 53 }} | ||
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{{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }} | {{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }} | ||
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837 | [[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000{{c}}, ~100352/91125 = 165.9837{{c}} | ||
{{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }} | {{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }} | ||
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Mapping: {{mapping| 3 6 19 30 22 | 0 -3 -29 -52 -28 }} | Mapping: {{mapping| 3 6 19 30 22 | 0 -3 -29 -52 -28 }} | ||
Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9835 | Optimal tuning (POTE): ~63/50 = 400.0000{{c}}, ~11/10 = 165.9835{{c}} | ||
{{Optimal ET sequence|legend=0| 159, 282, 441 }} | {{Optimal ET sequence|legend=0| 159, 282, 441 }} | ||
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Mapping: {{mapping| 3 6 19 30 22 36 | 0 -3 -29 -52 -28 -60 }} | Mapping: {{mapping| 3 6 19 30 22 36 | 0 -3 -29 -52 -28 -60 }} | ||
Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9842 | Optimal tuning (POTE): ~63/50 = 400.0000{{c}}, ~11/10 = 165.9842{{c}} | ||
{{Optimal ET sequence|legend=0| 159, 282, 441 }} | {{Optimal ET sequence|legend=0| 159, 282, 441 }} | ||
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Mapping: {{mapping| 3 6 19 30 22 36 16 | 0 -3 -29 -52 -28 -60 -9 }} | Mapping: {{mapping| 3 6 19 30 22 36 16 | 0 -3 -29 -52 -28 -60 -9 }} | ||
Optimal tuning (POTE): ~34/27 = 400.0000, ~11/10 = 165.9805 | Optimal tuning (POTE): ~34/27 = 400.0000{{c}}, ~11/10 = 165.9805{{c}} | ||
{{Optimal ET sequence|legend=0| 159, 282, 441 }} | {{Optimal ET sequence|legend=0| 159, 282, 441 }} | ||
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: mappin generators: ~63/50, ~1936/1875 | : mappin generators: ~63/50, ~1936/1875 | ||
Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3290 | Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3290{{c}} | ||
{{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }} | {{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }} | ||
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Mapping: {{mapping| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }} | Mapping: {{mapping| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }} | ||
Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3294 | Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3294{{c}} | ||
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | {{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | ||
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Mapping: {{mapping| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }} | Mapping: {{mapping| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }} | ||
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295 | Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~351/340 = 55.3295{{c}} | ||
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | {{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | ||
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Mapping: {{mapping| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }} | Mapping: {{mapping| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }} | ||
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295 | Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~351/340 = 55.3295{{c}} | ||
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | {{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | ||
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Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }} | Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }} | ||
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296 | Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~351/340 = 55.3296{{c}} | ||
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | {{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | ||
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Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }} | Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }} | ||
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296 | Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~351/340 = 55.3296{{c}} | ||
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | {{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }} | ||