Keemic family: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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== Supermagic == | == Supermagic == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 875/864 | [[Comma list]]: 875/864 | ||
{{Mapping|legend=1| 1 0 0 5 | 0 1 0 3 | 0 0 1 -3 }} | |||
: mapping generators: ~2, ~3, ~5 | |||
[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}] | [[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}] | ||
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: Angle (6/5, 3/2) = 77.834 | : Angle (6/5, 3/2) = 77.834 | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.8733, ~5/4 = 380.4684 | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* 7- and [[9-odd-limit]] | * [[7-odd-limit|7-]] and [[9-odd-limit]] | ||
: | : {{monzo list| 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 basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7/5 | ||
{{Optimal ET sequence|legend=1| 7, 15, 19, 41 }} | {{Optimal ET sequence|legend=1| 7, 15, 19, 41 }} | ||
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== Undecimal supermagic == | == Undecimal supermagic == | ||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 100/99, 385/384 | [[Comma list]]: 100/99, 385/384 | ||
{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }} | |||
[[POTE | [[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, 145c, 201ce, 264bce, 305bcce }} | {{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104, 145c, 201ce, 264bce, 305bcce }} | ||
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== Supernatural == | == Supernatural == | ||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 225/224, 245/243 | [[Comma list]]: 225/224, 245/243 | ||
{{Mapping|legend=1| 1 0 2 -1 0 | 0 5 1 12 0 | 0 0 0 0 1 }} | |||
[[POTE | : mapping generators: ~2, ~5, ~11 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758 | |||
{{Optimal ET sequence|legend=1| 19, 22, 41, 82e, 101cd, 224ccde, 265ccde * }} | {{Optimal ET sequence|legend=1| 19, 22, 41, 82e, 101cd, 224ccde, 265ccde * }} | ||
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Comma list: 105/104, 196/195, 245/243 | Comma list: 105/104, 196/195, 245/243 | ||
Mapping: | Mapping: {{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|legend=1| 19, 22f, 41, 60e, 101cd }} | {{Optimal ET sequence|legend=1| 19, 22f, 41, 60e, 101cd }} | ||
Revision as of 13:41, 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, 15, 19, 41
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, 145c, 201ce, 264bce, 305bcce
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, 41, 82e, 101cd, 224ccde, 265ccde *
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