User:VIxen/Temperament proposals: Difference between revisions

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=== Kibi ===

== 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 & ~~103p~~ & 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 ~~103p~~ val.

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.~~012~~ cents/octave - but the complexity of most intervals involving powers of 2 is high like in ennealimmal, which makes the badness high.

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]]: 1275/1274, ~~1575/1573, 2601~~/~~2600~~, ~~2720~~/~~2717~~, ~~4394~~/~~4389~~

[[Comma list]]: 969/968, 1275/1274, 1521/1520, 4200/4199, 6175/6171

[[Mapping]]: [{{val| 1 0 23 29 26 23 36 15 }}, {{val| 0 1 0 0 1 0 -1 -2 }}, {{val| 0 0 -30 -38 -35 -28 -44 -11 }}]

[[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, ~~~931~~/~~578~~

Mapping generators: ~2, ~3, ~266/165

{{~~Val list~~|legend=1| 29g, ~~74gh~~, ~~103~~, ~~132deg~~, ~~251e~~, ~~325deg~~, 354, 737 }}

{{Optimal ET sequence|legend=1| 29g, 74g, 103h, 132degh, 222cdeefgh, 251eh, 354, 486g, 737, 840 }}

[[Badness]]: 3.19 × 10-3

[[Badness]]: 1.78 × 10-3

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-=== Kibi ===
+== 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)<sup>-4</sup>. 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 & 103p & 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 103p val.
+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 - 0.012 cents/octave - but the complexity of most intervals involving powers of 2 is high like in ennealimmal, which makes the badness high.
+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]]: 1275/1274, 1575/1573, 2601/2600, 2720/2717, 4394/4389
+[[Comma list]]: 969/968, 1275/1274, 1521/1520, 4200/4199, 6175/6171
-[[Mapping]]: [{{val| 1 0 23 29 26 23 36 15 }}, {{val| 0 1 0 0 1 0 -1 -2 }}, {{val| 0 0 -30 -38 -35 -28 -44 -11 }}]
+[[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, ~931/578
+Mapping generators: ~2, ~3, ~266/165
-{{Val list|legend=1| 29g, 74gh, 103, 132deg, 251e, 325deg, 354, 737 }}
+{{Optimal ET sequence|legend=1| 29g, 74g, 103h, 132degh, 222cdeefgh, 251eh, 354, 486g, 737, 840 }}
-[[Badness]]: 3.19 × 10<sup>-3</sup>
+[[Badness]]: 1.78 × 10<sup>-3</sup>
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