Olympic clan: Difference between revisions
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+orthoschismic, +cassaschismic, +lif |
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=== Overview to extensions === | === Overview to extensions === | ||
The second comma in the comma list determines how we extend olympian to include the harmonic 5. Akea adds [[385/384]], and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Pessoal adds [[9801/9800]], splitting the octave into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. Finally, sophia adds 42875/42768, splitting the ~8/7 into three. These all have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]]. | The second comma in the comma list determines how we extend olympian to include the harmonic 5. Akea adds [[385/384]], and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds the harmonic 5 on the chain of fifths. Cassaschismic adds [[19712/19683]] with an independent generator for harmonic 5. Pessoal adds [[9801/9800]], splitting the octave into two. Lif adds [[2401/2400]], splitting the perfect fifth into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. Finally, sophia adds 42875/42768, splitting the ~8/7 into three. These all have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]]. | ||
Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
* ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]] | * ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]] | ||
* ''[[Pessoal]]'' (+9801/9800) → [[Kalismic temperaments #Pessoal|Kalismic temperaments]] | * ''[[Pessoal]]'' (+9801/9800) → [[Kalismic temperaments #Pessoal|Kalismic temperaments]] | ||
* ''[[Lif]]'' (+2401/2400) → [[Breed family #Lif|Breed family]] | |||
* ''[[Lux]]'' (+3025/3024) → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]] | * ''[[Lux]]'' (+3025/3024) → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]] | ||
* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]] | * ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]] | ||
Considered below are baffin and sophia. | Considered below are orthoschismic, cassaschismic, baffin and sophia. | ||
== Orthoschismic == | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 540/539, 32805/32768 | |||
{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 2 }} | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7405, ~7/4 = 969.6950 | |||
{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }} | |||
[[Badness]]: 1.18 × 10<sup>-3</sup> | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 540/539, 729/728, 4096/4095 | |||
Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 2 -1 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7333, ~7/4 = 969.7085 | |||
{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }} | |||
Badness: 0.833 × 10<sup>-3</sup> | |||
== Cassaschismic == | |||
=== 7-limit (garischismic) === | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 33554432/33480783 | |||
{{Mapping|legend=1| 1 0 0 25 | 0 1 0 -14 | 0 0 1 0 }} | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2079, ~5/4 = 386.5100 | |||
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }} | |||
[[Badness]]: 1.31 × 10<sup>-3</sup> | |||
=== 11-limit === | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 19712/19683, 41503/41472 | |||
{{Mapping|legend=1| 1 0 0 25 -33 | 0 1 0 -14 23 | 0 0 1 0 0 }} | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2280, ~5/4 = 386.3137 | |||
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }} | |||
[[Badness]]: 1.41 × 10<sup>-3</sup> | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 2080/2079, 4096/4095, 19712/19683 | |||
Mapping: {{mapping| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2289, ~5/4 = 386.2869 | |||
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }} | |||
Badness: 0.872 × 10<sup>-3</sup> | |||
=== 2.3.5.7.11.13.19 subgroup === | |||
Subgroup: 2.3.5.7.11.13.19 | |||
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079 | |||
Sval mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2293, ~5/4 = 386.3021 | |||
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851 }} | |||
Badness: 0.496 × 10<sup>-3</sup> | |||
== Baffin == | == Baffin == | ||
| Line 39: | Line 119: | ||
: mapping generators: ~2, ~8192/4725, ~5 | : mapping generators: ~2, ~8192/4725, ~5 | ||
[[Optimal tuning]] ([[POTE]]): ~8192/4725 = 951.0868, ~5/4 = 386.6183 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8192/4725 = 951.0868, ~5/4 = 386.6183 | ||
{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }} | {{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }} | ||
| Line 52: | Line 132: | ||
{{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 0 0 1 -2 4 }} | {{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 0 0 1 -2 4 }} | ||
[[Optimal tuning]] ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~400/231 = 951.0585, ~5/4 = 386.7912 | ||
{{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }} | {{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }} | ||
| Line 65: | Line 145: | ||
Mapping: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 0 0 1 -2 4 1 }} | Mapping: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 0 0 1 -2 4 1 }} | ||
Optimal tuning (POTE): ~26/15 = 951.0882, ~5/4 = 386.7507 | Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.0882, ~5/4 = 386.7507 | ||
{{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }} | {{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }} | ||
| Line 80: | Line 160: | ||
{{Mapping|legend=1| 1 0 2 3 11 | 0 1 0 0 -5 | 0 0 5 -3 6 }} | {{Mapping|legend=1| 1 0 2 3 11 | 0 1 0 0 -5 | 0 0 5 -3 6 }} | ||
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.3024, ~256/245 = 77.1952 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.3024, ~256/245 = 77.1952 | ||
{{Optimal ET sequence|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }} | {{Optimal ET sequence|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }} | ||
| Line 93: | Line 173: | ||
Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }} | Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }} | ||
Optimal tuning (POTE): ~3/2 = 702.3319, ~117/112 = 77.2152 | Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3319, ~117/112 = 77.2152 | ||
{{Optimal ET sequence|legend=1| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }} | {{Optimal ET sequence|legend=1| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }} | ||
| Line 106: | Line 186: | ||
Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }} | Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }} | ||
Optimal tuning (POTE): ~3/2 = 702.3205, ~68/65 = 77.2255 | Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3205, ~68/65 = 77.2255 | ||
{{Optimal ET sequence|legend=1| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }} | {{Optimal ET sequence|legend=1| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }} | ||