Garischismic family: Difference between revisions
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{{Mapping|legend=1| 1 0 0 25 | 0 1 0 -14 | 0 0 1 0 }} | {{Mapping|legend=1| 1 0 0 25 | 0 1 0 -14 | 0 0 1 0 }} | ||
: mapping generators: ~2, ~3, ~5 | : mapping generators: ~2, ~3, ~5 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = | * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}} | ||
* [[CWE]]: ~2 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}} | ||
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }} | {{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> | [[Badness]] (Smith): 1.31 × 10<sup>-3</sup> | ||
== Cassaschismic == | == Cassaschismic == | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = | * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}} | ||
* [[CWE]]: ~2 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}} | ||
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }} | {{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }} | ||
[[Badness]]: 1.41 × 10<sup>-3</sup> | [[Badness]] (Smith): 1.41 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}} | ||
* CWE: ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }} | ||
Badness: 0.872 × 10<sup>-3</sup> | Badness (Smith): 0.872 × 10<sup>-3</sup> | ||
=== 2.3.5.7.11.13.19 subgroup === | === 2.3.5.7.11.13.19 subgroup === | ||
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Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079 | Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079 | ||
Subgroup-val 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 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }} | ||
Badness: 0.496 × 10<sup>-3</sup> | Badness (Smith): 0.496 × 10<sup>-3</sup> | ||
== Androschismic == | == Androschismic == | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 151263/151250, | [[Comma list]]: 151263/151250, 200704/200475 | ||
{{Mapping|legend=1| 1 0 0 25 62 | 0 1 0 -14 -34 | 0 0 1 0 -2 }} | {{Mapping|legend=1| 1 0 0 25 62 | 0 1 0 -14 -34 | 0 0 1 0 -2 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* CTE: ~2 = | * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}} | ||
* CWE: ~2 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}} | ||
{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }} | {{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }} | ||
[[Badness]]: 1.64 × 10<sup>-3</sup> | [[Badness]] (Smith): 1.64 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 2080/2079, | Comma list: 2080/2079, 43904/43875, 154880/154791 | ||
Mapping: {{mapping| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }} | Mapping: {{mapping| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}} | ||
* CWE: ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }} | |||
Badness: 1.01 × 10<sup>-3</sup> | Badness (Smith): 1.01 × 10<sup>-3</sup> | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 11:15, 6 February 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The garischismic family of rank-3 temperaments tempers out the garischisma (ratio: 33554432/33480783, monzo: [25 -14 0 -1⟩). The head of this family is garischismic, which is generated by a perfect fifth and an independent generator for 5/4. Two apotomes i.e. 14 fifths octave-reduced make a septimal major second (8/7). Equivalently stated, the harmonic seventh (7/4) is found at the double-diminished octave (C–Cbb).
The best extension to the 11-limit identifies the 11/8 at +23 fifths. This is also the mapping used in cassandra, so we call it cassaschismic. An alternative, supported by andromeda, is androschismic.
Garischismic
Subgroup: 2.3.5.7
Comma list: 33554432/33480783
Mapping: [⟨1 0 0 25], ⟨0 1 0 -14], ⟨0 0 1 0]]
- mapping generators: ~2, ~3, ~5
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 702.2224 ¢, ~5/4 = 386.3137 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2124 ¢, ~5/4 = 386.4496 ¢
Optimal ET sequence: 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd
Badness (Smith): 1.31 × 10-3
Cassaschismic
Subgroup: 2.3.5.7.11
Comma list: 19712/19683, 41503/41472
Mapping: [⟨1 0 0 25 -33], ⟨0 1 0 -14 23], ⟨0 0 1 0 0]]
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 702.2280 ¢, ~5/4 = 386.3137 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2290 ¢, ~5/4 = 386.3819 ¢
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 1121
Badness (Smith): 1.41 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4096/4095, 19712/19683
Mapping: [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]
Optimal tunings:
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 702.2289 ¢, ~5/4 = 386.2869 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2303 ¢, ~5/4 = 386.3031 ¢
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 2283b
Badness (Smith): 0.872 × 10-3
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
Subgroup-val 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 = 1200.0000 ¢, ~3/2 = 702.2293 ¢, ~5/4 = 386.3021 ¢
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851
Badness (Smith): 0.496 × 10-3
Androschismic
Subgroup: 2.3.5.7.11
Comma list: 151263/151250, 200704/200475
Mapping: [⟨1 0 0 25 62], ⟨0 1 0 -14 -34], ⟨0 0 1 0 -2]]
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 702.2308 ¢, ~5/4 = 386.3806 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2178 ¢, ~5/4 = 386.5048 ¢
Optimal ET sequence: 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee
Badness (Smith): 1.64 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 43904/43875, 154880/154791
Mapping: [⟨1 0 0 25 62 82], ⟨0 1 0 -14 -34 -43], ⟨0 0 1 0 -2 -3]]
Optimal tunings:
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 702.2305 ¢, ~5/4 = 386.3748 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2174 ¢, ~5/4 = 386.4968 ¢
Optimal ET sequence: 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff
Badness (Smith): 1.01 × 10-3