User:VIxen/Temperament proposals: Difference between revisions
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Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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In the 33/19.2.3.5.7.11.13.17 basis of the full 19-limit subgroup, it is q23 & q82 & q282 (with equave 2/1, it is 29g & 103h & 354), and a Fokker block of 23 notes per equave is available, corresponding to the [[29edo|29g]] val, although it is complex - perhaps the simplest of the highly available intervals is 7/5 - with a preference for open harmonies, and just about enough to contain the chord of prime harmonics (1:2:3:5:7:11:13:17:19) on one root per equave. | In the 33/19.2.3.5.7.11.13.17 basis of the full 19-limit subgroup, it is q23 & q82 & q282 (with equave 2/1, it is 29g & 103h & 354), and a Fokker block of 23 notes per equave is available, corresponding to the [[29edo|29g]] val, although it is complex - perhaps the simplest of the highly available intervals is 7/5 - with a preference for open harmonies, and just about enough to contain the chord of prime harmonics (1:2:3:5:7:11:13:17:19) on one root per equave. | ||
Extensions to higher limits, perhaps 31-limit at least, seem possible but will require using blocks of 82 notes corresponding to the | Extensions to higher limits, perhaps 31-limit at least, seem possible but will require using blocks of 82 notes corresponding to the 103h val. | ||
Data for equave 2/1 is provided below for compatibility. The TE error is small enough - 0. | Data for equave 2/1 is provided below for compatibility. The TE error is small enough - 0.011 cents/octave - but the complexity of most intervals involving powers of 2 is high like in ennealimmal, which makes the badness high. | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
[[Comma list]]: 969/968, 1275/1274, 1521/1520, | [[Comma list]]: 969/968, 1275/1274, 1521/1520, 4200/4199, 6175/6171 | ||
[[Mapping]]: [{{val| 1 0 23 29 26 23 36 | [[Mapping]]: [{{val| 1 0 23 29 26 23 36 19 }}, {{val| 0 1 0 0 1 0 -1 2 }}, {{val| 0 0 -30 -38 -35 -28 -44 -26 }}] | ||
Mapping generators: ~2, ~3, ~ | Mapping generators: ~2, ~3, ~266/165 | ||
{{ | {{Optimal ET sequence|legend=1| 29g, 74g, 103h, 132degh, 222cdeefgh, 251eh, 354, 486g, 737, 840 }} | ||
[[Badness]]: | [[Badness]]: 1.78 × 10<sup>-3</sup> | ||
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Latest revision as of 18:37, 7 May 2023
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Kibi
This temperament is heavily tailored towards 19-limit use with the 33/19 equave, inspired by that the smallest 19-limit superparticular is (91/10) × (33/19)-4. It was named after minor planet 3319.
In the 33/19.2.3.5.7.11.13.17 basis of the full 19-limit subgroup, it is q23 & q82 & q282 (with equave 2/1, it is 29g & 103h & 354), and a Fokker block of 23 notes per equave is available, corresponding to the 29g val, although it is complex - perhaps the simplest of the highly available intervals is 7/5 - with a preference for open harmonies, and just about enough to contain the chord of prime harmonics (1:2:3:5:7:11:13:17:19) on one root per equave.
Extensions to higher limits, perhaps 31-limit at least, seem possible but will require using blocks of 82 notes corresponding to the 103h val.
Data for equave 2/1 is provided below for compatibility. The TE error is small enough - 0.011 cents/octave - but the complexity of most intervals involving powers of 2 is high like in ennealimmal, which makes the badness high.
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 969/968, 1275/1274, 1521/1520, 4200/4199, 6175/6171
Mapping: [⟨1 0 23 29 26 23 36 19], ⟨0 1 0 0 1 0 -1 2], ⟨0 0 -30 -38 -35 -28 -44 -26]]
Mapping generators: ~2, ~3, ~266/165
Optimal ET sequence: 29g, 74g, 103h, 132degh, 222cdeefgh, 251eh, 354, 486g, 737, 840
Badness: 1.78 × 10-3