Archytas family: Difference between revisions
m Removing from Category:Regular temperament theory using Cat-a-lot |
Update keys and OPGV sequences |
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== Archytas == | == Archytas == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: [[64/63]] | [[Comma list]]: [[64/63]] | ||
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: Angle (3/2, 5/4) = 90 degrees | : Angle (3/2, 5/4) = 90 degrees | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~3/2 = 709.3213, ~5/4 = 393.3747 | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] | * [[7-odd-limit]] | ||
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}] | ||
: [[Eigenmonzo]] | : [[Eigenmonzo basis]] (unchanged intervals): 2.5/3.7/3 | ||
* [[9-odd-limit]] | * [[9-odd-limit]] | ||
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}] | ||
: [[Eigenmonzo]] | : [[Eigenmonzo basis]] (unchanged intervals): 2.5/3.9/7 | ||
{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }} | {{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }} | ||
Scales: [[archytas12]], [[archytas12synch]] | Scales: [[archytas12]], [[archytas12synch]] | ||
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{{Main| Ares }} | {{Main| Ares }} | ||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 64/63, 100/99 | [[Comma list]]: 64/63, 100/99 | ||
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: Angle (3/2, 6/5) = 75.0475 degrees | : Angle (3/2, 6/5) = 75.0475 degrees | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~3/2 = 709.6869, ~5/4 = 391.4507 | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 0 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 -1 0 1/2 0 }}, {{monzo| 3 -1 0 -1/2 1 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 0 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 -1 0 1/2 0 }}, {{monzo| 3 -1 0 -1/2 1 }}] | ||
: [[Eigenmonzo]] | : [[Eigenmonzo basis]] (unchanged intervals): 2.11/9.9/7 | ||
{{Val list|legend=1| 12, 15, 22, 27e, 34d, 49 }} | {{Val list|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }} | ||
[[Badness]]: 0.469 × 10<sup>-3</sup> | [[Badness]]: 0.469 × 10<sup>-3</sup> | ||
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== Vulcan == | == Vulcan == | ||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 56/55, 64/63 | [[Comma list]]: 56/55, 64/63 | ||
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Mapping generators: ~2, ~3, ~5 | Mapping generators: ~2, ~3, ~5 | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~3/2 = 710.9720, ~5/4 = 401.2699 | ||
{{Val list|legend=1| 5, 10, 12, 15, 27e }} | {{Val list|legend=1| 5, 10, 12, 15, 27e }} | ||
Revision as of 07:35, 9 October 2022
The archytas family of temperaments tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord, which is familiar from 12edo and also a feature of 22edo and 27edo.
Archytas
Subgroup: 2.3.5.7
Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]
Mapping generators: ~2, ~3, ~5
Mapping to lattice: [⟨0 1 0 -2], ⟨0 0 1 0]]
Lattice basis:
- 3/2 length = 1.0508, 5/4 length = 2.3219
- Angle (3/2, 5/4) = 90 degrees
Optimal tuning (POTE): ~3/2 = 709.3213, ~5/4 = 393.3747
- [[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
- Eigenmonzo basis (unchanged intervals): 2.5/3.7/3
- [[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]
- Eigenmonzo basis (unchanged intervals): 2.5/3.9/7
Scales: archytas12, archytas12synch
- Music
- Chris Vaisvil – 12 archytas hobbits with ale
- Joel Taylor – Portrait of a Tragedy (Death of a Son) - Archytas12 sync Study 3
Ares
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99
Mapping: [⟨1 0 0 6 2], ⟨0 1 0 -2 -2], ⟨0 0 1 0 2]]
Mapping generators: ~2, ~3, ~5
Mapping to lattice: [⟨0 1 1 -2 0], ⟨0 0 -1 0 -2]]
Lattice basis:
- 3/2 length = 0.9878, 6/5 length = 1.4312
- Angle (3/2, 6/5) = 75.0475 degrees
Optimal tuning (POTE): ~3/2 = 709.6869, ~5/4 = 391.4507
- [[1 0 0 0 0⟩, [3/2 1/2 0 -1/4 0⟩, [2 0 0 -1/2 1/2⟩, [3 -1 0 1/2 0⟩, [3 -1 0 -1/2 1⟩]
- Eigenmonzo basis (unchanged intervals): 2.11/9.9/7
Badness: 0.469 × 10-3
Scales: ares12
Vulcan
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63
Mapping: [⟨1 0 0 6 9], ⟨0 1 0 -2 -2], ⟨0 0 1 0 -1]]
Mapping generators: ~2, ~3, ~5
Optimal tuning (POTE): ~3/2 = 710.9720, ~5/4 = 401.2699
Badness: 0.577 × 10-3