Didymus rank-3 family: Difference between revisions
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The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth. | |||
= Euterpe = | |||
[[Comma list]]: 81/80, 99/98 | |||
[|1 0 0 0 0 | [[Minimax tuning]]: | ||
|0 0 0 1 0 | * [[11-odd-limit]]: | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}] | |||
: [[Eigenmonzo]]s: 2, 5, 7 | |||
[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}] | |||
{{Val list|legend=1| 5, 12, 26, 31, 88 }} | |||
[[Badness]]: 0.536 × 10<sup>-3</sup> | |||
= Calliope = | |||
[[Comma list]]: 81/80, 45/44 | |||
[[Minimax tuning]]: | |||
* [[11-odd-limit]]: | |||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}] | |||
: [[Eigenmonzo]]s: 2, 7/3, 11 | |||
[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}] | |||
{{Val list|legend=1| 7, 12, 19, 26, 45 }} | |||
[[Badness]]: 0.530 × 10<sup>-3</sup> | |||
= Erato = | |||
[[Comma list]]: 81/80, 126/125 | |||
[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}] | |||
{{Val list|legend=1| 12, 19, 31, 50, 81 }} | |||
[[Badness]]: 0.558 × 10<sup>-3</sup> | |||
== 13-limit == | |||
Comma list: 81/80, 105/104, 126/125 | |||
Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}] | |||
{{Val list|legend=1| 12f, 19, 31, 50, 81 }} | |||
= | = Clio = | ||
[[Comma list]]: 81/80, 176/175 | |||
[[Minimax tuning]] | |||
* [[11-odd-limit]]: | |||
: {{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}] | |||
: [[Eigenmonzo]]s: 2, 5, 7 | |||
Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}] | |||
= | {{Val list|legend=1| 7, 12, 24, 31, 105, 129 }} | ||
Badness: 0.738 × 10<sup>-3</sup> | |||
= Polyhymnia = | |||
[[Comma list]]: 81/80, 385/384 | |||
[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}] | |||
{{Val list|legend=1| 7, 19, 24, 26, 31, 38, 45 }} | |||
= Thalia = | |||
[[Comma list]]: 81/80, 55/54 | |||
[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}] | |||
{{Val list|legend=1| 5, 7, 12e, 19e }} | |||
= Melpomene = | |||
[[Comma list]]: 81/80, 56/55 | |||
[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}] | |||
{{Val list|legend=1| 5, 12, 19, 24, 36 }} | |||
= Urania = | |||
[[Comma list]]: 81/80, 121/120 | |||
[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}] | |||
Lattice basis: | |||
: 11/9 length = 0.2536, 8/7 length = 2.807 | |||
: Angle (11/9, 8/7) = 90 degrees | |||
Map to lattice: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}] | |||
= | {{Val list|legend=1| 7, 14c, 24, 31, 100de, 131bde, 162bde }} | ||
[[Badness]]: 0.842 × 10<sup>-3</sup> | |||
[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5 | |||
Scale: [[urania24]] | Scale: [[urania24]] | ||
=Terpsichore= | = Terpsichore = | ||
[[Comma list]]: 81/80, 540/539 | |||
[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}] | |||
{{Val list|legend=1| 19, 31, 81, 112b }} | |||
Badness: 0. | [[Badness]]: 0.850 × 10<sup>-3</sup> | ||
[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11 | |||
[[Category:Theory]] | [[Category:Theory]] |
Revision as of 07:42, 20 March 2021
The didymus rank-3 family are rank-3 temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
Euterpe
Comma list: 81/80, 99/98
- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-1 0 -1/2 2 0⟩]
- Eigenmonzos: 2, 5, 7
Mapping: [⟨1 0 -4 0 1], ⟨0 1 4 0 -2], ⟨0 0 0 1 2]]
Badness: 0.536 × 10-3
Calliope
Comma list: 81/80, 45/44
- [[1 0 0 0 0⟩, [1 0 0 0 1/6⟩, [0 0 0 0 2/3⟩, [1 -1 0 1 1/6⟩, [0 0 0 0 1⟩]
- Eigenmonzos: 2, 7/3, 11
Mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6], ⟨0 0 0 1 0]]
Badness: 0.530 × 10-3
Erato
Comma list: 81/80, 126/125
Mapping: [⟨1 0 -4 -13 0], ⟨0 1 4 10 0], ⟨0 0 0 0 1]]
Badness: 0.558 × 10-3
13-limit
Comma list: 81/80, 105/104, 126/125
Mapping: [⟨1 0 -4 -13 0 -20], ⟨0 1 4 10 0 15], ⟨0 0 0 0 1 0]]
Clio
Comma list: 81/80, 176/175
- [1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-4 0 2 1 0⟩]
- Eigenmonzos: 2, 5, 7
Mapping: [⟨1 0 -4 0 -12], ⟨0 1 4 0 8], ⟨0 0 0 1 1]]
Badness: 0.738 × 10-3
Polyhymnia
Comma list: 81/80, 385/384
Mapping: [⟨1 0 -4 0 11], ⟨0 1 4 0 -3], ⟨0 0 0 1 -1]]
Thalia
Comma list: 81/80, 55/54
Mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1], ⟨0 0 0 1 0]]
Melpomene
Comma list: 81/80, 56/55
Mapping: [⟨1 0 -4 0 7], ⟨0 1 4 0 -4], ⟨0 0 0 1 1]]
Urania
Comma list: 81/80, 121/120
Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]
Lattice basis:
- 11/9 length = 0.2536, 8/7 length = 2.807
- Angle (11/9, 8/7) = 90 degrees
Map to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]
Badness: 0.842 × 10-3
Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
Scale: urania24
Terpsichore
Comma list: 81/80, 540/539
Mapping: [⟨1 0 -4 0 -2], ⟨0 1 4 0 7], ⟨0 0 0 1 -2]]
Badness: 0.850 × 10-3
Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11