Sensamagic family: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The '''sensamagic family''' of [[rank-3 temperament]] | The '''sensamagic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the sensamagic comma, [[245/243]]. | ||
For a list of rank-2 temperaments, see [[Sensamagic clan]]. | |||
== Sensamagic == | == Sensamagic == | ||
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: Angle (3/2, 9/7) = 86.5288° | : Angle (3/2, 9/7) = 86.5288° | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.7424{{c}}, ~9/7 = 440.9020{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }} | {{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }} | ||
[[Badness]]: 0.129 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.129 × 10<sup>-3</sup> | ||
[[Projection pair]]: 5 243/49 to 2.3.7 | [[Projection pair]]: 5 243/49 to 2.3.7 | ||
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{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }} | {{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.8004{{c}}, ~9/7 = 440.9178{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }} | {{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }} | ||
[[Badness]]: 0.722 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.722 × 10<sup>-3</sup> | ||
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7 | [[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7 | ||
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Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }} | Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }} | ||
Badness: 1.20 × 10<sup>-3</sup> | Badness (Smith): 1.20 × 10<sup>-3</sup> | ||
== Sensawer == | == Sensawer == | ||
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{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }} | {{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.1900{{c}}, ~9/7 = 441.1359{{c}} | ||
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }} | {{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }} | ||
[[Badness]]: 0.796 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.796 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }} | Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }} | ||
Badness: 0.928 × 10<sup>-3</sup> | Badness (Smith): 0.928 × 10<sup>-3</sup> | ||
== Octarod == | == Octarod == | ||
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{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }} | {{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 705.0464{{c}}, ~9/7 = 439.5050{{c}} | ||
{{Optimal ET sequence|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }} | {{Optimal ET sequence|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }} | ||
[[Badness]]: 0.581 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.581 × 10<sup>-3</sup> | ||
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]] | Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]] | ||
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{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }} | {{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 706.3702{{c}}, ~9/7 = 442.1147{{c}} | ||
{{Optimal ET sequence|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }} | {{Optimal ET sequence|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }} | ||
[[Badness]]: 0.877 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.877 × 10<sup>-3</sup> | ||
=== Shrusic === | === Shrusic === | ||
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Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }} | Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }} | ||
Badness: 1.125 × 10<sup>-3</sup> | Badness (Smith): 1.125 × 10<sup>-3</sup> | ||
== Bisector == | == Bisector == | ||
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: mapping generators: ~77/54, ~3, ~9/7 | : mapping generators: ~77/54, ~3, ~9/7 | ||
[[Optimal tuning]] ([[POTE]]): ~77/54 = | [[Optimal tuning]] ([[POTE]]): ~77/54 = 600.0000{{c}}, ~3/2 = 703.0884{{c}}, ~9/7 = 441.1060{{c}} | ||
{{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }} | {{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }} | ||
[[Badness]]: 1.089 × 10<sup>-3</sup> | [[Badness]] (Smith): 1.089 × 10<sup>-3</sup> | ||
== Sensigh == | == Sensigh == | ||
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: mapping generators: ~2, ~9/7, ~11 | : mapping generators: ~2, ~9/7, ~11 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 27, 46, 111df, 157df }} | ||
Badness: 0.939 × 10<sup>-3</sup> | Badness (Smith): 0.939 × 10<sup>-3</sup> | ||
=== 17-limit === | === 17-limit === | ||
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Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }} | Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 27, 46 }} | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 14:07, 11 October 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 sensamagic family of rank-3 temperaments tempers out the sensamagic comma, 245/243.
For a list of rank-2 temperaments, see Sensamagic clan.
Sensamagic
Subgroup: 2.3.5.7
Comma list: 245/243
Mapping: [⟨1 0 0 0], ⟨0 1 1 2], ⟨0 0 2 -1]]
- mapping generators: ~2, ~3, ~9/7
Mapping to lattice: [⟨0 1 1 2], ⟨0 0 2 -1]]
Lattice basis:
- 3/2 length = 0.9644, 9/7 length = 1.0807
- Angle (3/2, 9/7) = 86.5288°
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 703.7424 ¢, ~9/7 = 440.9020 ¢
- [[1 0 0 0⟩, [0 0 1/5 2/5⟩, [0 0 1 0⟩, [0 0 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.5.7
- [[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]
- unchanged-interval (eigenmonzo) basis: 2.3.7/5
Optimal ET sequence: 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283
Badness (Smith): 0.129 × 10-3
Projection pair: 5 243/49 to 2.3.7
2.3.7 subgroup
- 12: 729/686, 64/63
- 17: 64/63, 19683/19208
- 19: 49/48, 177147/175616
- 22: 64/63, 537824/531441
- 24: 64/63, 15059072/14348907
Overview to extensions
Temperaments discussed elsewhere include supernatural (→ Keemic family). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
Undecimal sensamagic
Subgroup: 2.3.5.7.11
Comma list: 245/243, 385/384
Mapping: [⟨1 0 0 0 7], ⟨0 1 1 2 -2], ⟨0 0 2 -1 -1]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 703.8004 ¢, ~9/7 = 440.9178 ¢
- [[1 0 0 0 0⟩, [21/13 6/13 -1/13 1/13 -3/13⟩, [35/13 10/13 7/13 -7/13 -5/13⟩, [35/13 10/13 -6/13 6/13 -5/13⟩, [42/13 -14/13 -2/13 2/13 7/13⟩]
- unchanged-interval (eigenmonzo) basis: 2.7/5.11/9
Optimal ET sequence: 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd
Badness (Smith): 0.722 × 10-3
Projection pairs: 5 243/49 11 896/81 to 2.3.7
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 245/243, 352/351, 364/363
Mapping: [⟨1 0 0 0 7 12], ⟨0 1 1 2 -2 -5], ⟨0 0 2 -1 -1 -1]]
Optimal ET sequence: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
Badness (Smith): 1.20 × 10-3
Sensawer
Subgroup: 2.3.5.7.11
Comma list: 245/243, 441/440
Mapping: [⟨1 0 0 0 -3], ⟨0 1 1 2 5], ⟨0 0 2 -1 -4]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 703.1900 ¢, ~9/7 = 441.1359 ¢
Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd
Badness (Smith): 0.796 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 245/243, 352/351
Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 1 2 5 2], ⟨0 0 2 -1 -4 -4]]
Optimal ET sequence: 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd
Badness (Smith): 0.928 × 10-3
Octarod
Subgroup: 2.3.5.7.11
Comma list: 100/99, 245/243
Mapping: [⟨1 0 0 0 2], ⟨0 1 1 2 0], ⟨0 0 2 -1 4]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 705.0464 ¢, ~9/7 = 439.5050 ¢
Optimal ET sequence: 8d, 14c, 19, 22, 27e, 41, 104, 131e
Badness (Smith): 0.581 × 10-3
Scales: octarod1, octarod2, octarod3, octarod4, octarod5
Shrusus
Subgroup: 2.3.5.7.11
Comma list: 176/175, 245/243
Mapping: [⟨1 0 0 0 -4], ⟨0 1 1 2 4], ⟨0 0 2 -1 3]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 706.3702 ¢, ~9/7 = 442.1147 ¢
Optimal ET sequence: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
Badness (Smith): 0.877 × 10-3
Shrusic
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 176/175, 245/243
Mapping: [⟨1 0 0 0 -4 1], ⟨0 1 1 2 4 1], ⟨0 0 2 -1 3 3]]
Optimal ET sequence: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
Badness (Smith): 1.125 × 10-3
Bisector
Subgroup: 2.3.5.7.11
Comma list: 121/120, 245/243
Mapping: [⟨2 0 0 0 3], ⟨0 1 1 2 1], ⟨0 0 2 -1 1]]
- mapping generators: ~77/54, ~3, ~9/7
Optimal tuning (POTE): ~77/54 = 600.0000 ¢, ~3/2 = 703.0884 ¢, ~9/7 = 441.1060 ¢
Optimal ET sequence: 22, 46, 68, 82e, 106de, 114, 128e
Badness (Smith): 1.089 × 10-3
Sensigh
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168
Mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10], ⟨0 0 0 0 1 0]]
- mapping generators: ~2, ~9/7, ~11
Optimal ET sequence: 19, 27, 46, 111df, 157df
Badness (Smith): 0.939 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 126/125, 154/153, 169/168
Mapping: [⟨1 6 8 11 0 10 0], ⟨0 -7 -9 -13 0 -10 1], ⟨0 0 0 0 1 0 1]]