Archytas family: Difference between revisions
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- music (addressed in the dedicated article) |
Tunings and badnesses |
||
| Line 21: | Line 21: | ||
: Angle (3/2, 5/4) = 90 degrees | : Angle (3/2, 5/4) = 90 degrees | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 709.5949, ~5/4 = 386.3137 | |||
: [[error map]]: {{val| 0.000 +7.640 0.000 +11.984 }} | |||
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 709.3901, ~5/4 = 391.5995 | |||
: error map: {{val| 0.000 +7.435 +5.286 +12.394 }} | |||
<!-- * [[POTE]]: ~2 = 1200.0000, ~3/2 = 709.3213, ~5/4 = 393.3747 | |||
: error map: {{val| 0.000 +7.366 +7.061 +12.532 }} --> | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
| Line 32: | Line 38: | ||
{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }} | {{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }} | ||
[[Badness]] (Sintel): 0.435 | |||
Scales: [[archytas12]], [[archytas12synch]] | Scales: [[archytas12]], [[archytas12synch]] | ||
| Line 45: | Line 53: | ||
{{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }} | {{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }} | ||
Mapping to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}] | [[Mapping to lattice]]: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}] | ||
Lattice basis: | Lattice basis: | ||
| Line 51: | Line 59: | ||
: Angle (3/2, 6/5) = 75.0475 degrees | : Angle (3/2, 6/5) = 75.0475 degrees | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 709.7182, ~5/4 = 385.7114 | |||
: [[error map]]: {{val| 0.000 +7.763 -0.602 +11.738 +0.668 }} | |||
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 709.6947, ~5/4 = 390.0248 | |||
: error map: {{val| 0.000 +7.740 +3.711 +11.785 +9.342 }} | |||
<!-- * [[POTE]]: ~2 = 1200.0000, ~3/2 = 709.6869, ~5/4 = 391.4507 | |||
: error map: {{val| 0.000 +7.732 +5.137 +11.800 +12.210 }} --> | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
| Line 60: | Line 74: | ||
{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }} | {{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.563 | ||
Scales: [[ares12]] | Scales: [[ares12]] | ||
| Line 71: | Line 85: | ||
{{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }} | {{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }} | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 713.9739, ~5/4 = 397.0040 | |||
: [[error map]]: {{val| 0.000 +12.019 +10.690 +3.226 +23.730 }} | |||
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 711.5280, ~5/4 = 400.4798 | |||
: error map: {{val| 0.000 +9.573 +14.166 +8.118 +25.146 }} | |||
<!-- * [[POTE]]: ~2 = 1200.0000, ~3/2 = 710.9720, ~5/4 = 401.2699 | |||
: error map: {{val| 0.000 +9.017 +14.956 +9.230 +25.468 }} --> | |||
{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }} | {{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.693 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 13:14, 17 June 2025
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The archytas family of rank-3 temperaments tempers out 64/63. The head of this family, archytas, tempers out 64/63 alone in the full 7-limit, so it has the same 2.3.7-subgroup structure as archy but giving prime 5 an independent generator.
See Archytas clan for the rank-2 temperament without the last generator of archytas, and its various extensions.
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
- CTE: ~2 = 1200.0000, ~3/2 = 709.5949, ~5/4 = 386.3137
- error map: ⟨0.000 +7.640 0.000 +11.984]
- CWE: ~2 = 1200.0000, ~3/2 = 709.3901, ~5/4 = 391.5995
- error map: ⟨0.000 +7.435 +5.286 +12.394]
- [[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
- eigenmonzo (unchanged-interval) basis: 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 (unchanged-interval) basis: 2.5/3.9/7
Optimal ET sequence: 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd
Badness (Sintel): 0.435
Scales: archytas12, archytas12synch
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 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
- CTE: ~2 = 1200.0000, ~3/2 = 709.7182, ~5/4 = 385.7114
- error map: ⟨0.000 +7.763 -0.602 +11.738 +0.668]
- CWE: ~2 = 1200.0000, ~3/2 = 709.6947, ~5/4 = 390.0248
- error map: ⟨0.000 +7.740 +3.711 +11.785 +9.342]
- [[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 (unchanged-interval) basis: 2.11/9.9/7
Optimal ET sequence: 12, 15, 22, 27e, 34d, 49, 181bbcddeeee
Badness (Sintel): 0.563
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]]
- CTE: ~2 = 1200.0000, ~3/2 = 713.9739, ~5/4 = 397.0040
- error map: ⟨0.000 +12.019 +10.690 +3.226 +23.730]
- CWE: ~2 = 1200.0000, ~3/2 = 711.5280, ~5/4 = 400.4798
- error map: ⟨0.000 +9.573 +14.166 +8.118 +25.146]
Optimal ET sequence: 5, 10, 12, 15, 27e
Badness (Sintel): 0.693