Garischismic clan: Difference between revisions
Expand on the intro and update keys |
Move cotoneum here as it's a strong extension; normalize the mapping of gary |
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* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+225/224) | * [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+225/224) | ||
* ''[[Newt]]'' → [[Breedsmic temperaments #Newt|Breedsmic temperaments]] (+2401/2400) | * ''[[Newt]]'' → [[Breedsmic temperaments #Newt|Breedsmic temperaments]] (+2401/2400) | ||
* ''[[Sextile]]'' → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]] (+250047/250000) | * ''[[Sextile]]'' → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]] (+250047/250000) | ||
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[[Comma list]]: 33554432/33480783 | [[Comma list]]: 33554432/33480783 | ||
{{Mapping|legend=2| 1 | {{Mapping|legend=2| 1 0 25 | 0 1 -14 }} | ||
: sval mapping generators: ~2, ~3 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2079 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2079 | ||
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Comma list: 19712/19683, 41503/41472 | Comma list: 19712/19683, 41503/41472 | ||
Sval mapping: {{mapping| 1 | Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }} | ||
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292 | Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292 | ||
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Badness: 0.00731 | Badness: 0.00731 | ||
== Cotoneum == | |||
{{Main| Cotoneum }} | |||
The cotoneum temperament tempers out 10976/10935 ([[hemimage comma]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. This temperament can be described as 41 & 217, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found at the septuple diminished octave (C-Cbbbbbbb) or equivalently at the perfect fourth minus four Pyth. commas. It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 10976/10935, 823543/819200 | |||
{{Mapping|legend=1| 1 0 80 25 | 0 1 -49 -14 }} | |||
{{Multival|legend=1| 1 -49 -14 -80 -25 105 }} | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317 | |||
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 258, 475 }} | |||
[[Badness]]: 0.105632 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 441/440, 10976/10935, 16384/16335 | |||
Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }} | |||
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303 | |||
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217 }} | |||
Badness: 0.050966 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 364/363, 441/440, 3584/3575, 10976/10935 | |||
Mapping: {{mapping| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }} | |||
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306 | |||
{{Optimal ET sequence|legend=1| 41, 176, 217 }} | |||
Badness: 0.036951 | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262 | |||
Mapping: {{mapping| 1 0 80 25 -33 -93 -137 | 0 1 -49 -14 23 61 89 }} | |||
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307 | |||
{{Optimal ET sequence|legend=1| 41, 176, 217 }} | |||
Badness: 0.029495 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728 | |||
Mapping: {{mapping| 1 0 80 25 -33 -93 -137 74 | 0 1 -49 -14 23 61 89 -44 }} | |||
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308 | |||
{{Optimal ET sequence|legend=1| 41, 176, 217 }} | |||
Badness: 0.021811 | |||
== World calendar == | == World calendar == | ||