Keemic family: Difference between revisions
Update keys |
+badnesses; review optimal ET sequences |
||
| Line 23: | Line 23: | ||
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7/5 | : [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7/5 | ||
{{Optimal ET sequence|legend=1| 7, 15, 19, 41 }} | {{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }} | ||
[[Badness]]: 0.212 × 10<sup>-3</sup> | |||
[[Projection pair]]: 7 864/125 | [[Projection pair]]: 7 864/125 | ||
| Line 38: | Line 40: | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394 | ||
{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 | {{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }} | ||
[[Badness]]: 0.641 × 10<sup>-3</sup> | [[Badness]]: 0.641 × 10<sup>-3</sup> | ||
| Line 55: | Line 57: | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758 | ||
{{Optimal ET sequence|legend=1| 19, 22, 41, | {{Optimal ET sequence|legend=1| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde * }} | ||
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]] | <nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]] | ||
| Line 70: | Line 72: | ||
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.0401, ~11/8 = 546.6672 | Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.0401, ~11/8 = 546.6672 | ||
{{Optimal ET sequence|legend=1| 19, 22f, 41, 60e, 101cd }} | {{Optimal ET sequence|legend=1| 19, 22f, 38df, 41, 60e, 79d, 101cd }} | ||
Badness: 1.14 × 10<sup>-3</sup> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 13:52, 24 August 2023
The keemic family of rank-3 temperaments are planar temperaments tempering out the keema, 875/864.
Supermagic
Subgroup: 2.3.5.7
Comma list: 875/864
Mapping: [⟨1 0 0 5], ⟨0 1 0 3], ⟨0 0 1 -3]]
- mapping generators: ~2, ~3, ~5
Mapping to lattice: [⟨0 0 -1 3], ⟨0 1 1 0]]
Lattice basis:
- 6/5 length = 0.8879, 3/2 length = 1.3391
- Angle (6/5, 3/2) = 77.834
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8733, ~5/4 = 380.4684
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [0 1 0 0⟩, [5/4 3/4 1/4 -1/4⟩, [5/4 3/4 -3/4 3/4⟩]
- Eigenmonzo (unchanged-interval) basis: 2.3.7/5
Optimal ET sequence: 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd
Badness: 0.212 × 10-3
Projection pair: 7 864/125
Scales: supermagic15
Undecimal supermagic
Subgroup: 2.3.5.7.11
Comma list: 100/99, 385/384
Mapping: [⟨1 0 0 5 2], ⟨0 1 0 3 -2], ⟨0 0 1 -3 2]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394
Optimal ET sequence: 7, 15, 19, 22, 37, 41, 104
Badness: 0.641 × 10-3
Scales: supermagic15
Supernatural
Subgroup: 2.3.5.7.11
Comma list: 225/224, 245/243
Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 0], ⟨0 0 0 0 1]]
- mapping generators: ~2, ~5, ~11
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758
Optimal ET sequence: 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde *
Badness: 0.888 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 245/243
Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 0 18], ⟨0 0 0 0 1 0]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.0401, ~11/8 = 546.6672
Optimal ET sequence: 19, 22f, 38df, 41, 60e, 79d, 101cd
Badness: 1.14 × 10-3