Kleismic rank-3 family: Difference between revisions
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{{Mapping|legend=1| 1 0 1 0 | 0 6 5 0 | 0 0 0 1 }} | {{Mapping|legend=1| 1 0 1 0 | 0 6 5 0 | 0 0 0 1 }} | ||
: mapping generators: ~2, ~6/5, ~7 | : mapping generators: ~2, ~6/5, ~7 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0586{{c}}, ~7/4 = 968.8259{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }} | {{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }} | ||
[[Badness]]: 0.363 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.363 × 10<sup>-3</sup> | ||
=== 2.3.5.7.13 subgroup === | === 2.3.5.7.13 subgroup === | ||
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Comma list: 325/324, 625/624 | Comma list: 325/324, 625/624 | ||
Subgroup-val mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }} | |||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}, ~7/4 = 968.8259{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }} | ||
Badness: 0.290 × 10<sup>-3</sup> | Badness (Smith): 0.290 × 10<sup>-3</sup> | ||
== Enlil == | == Enlil == | ||
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{{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }} | {{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }} | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0667{{c}}, ~7/4 = 967.6041{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }} | {{Optimal ET sequence|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }} | ||
[[Badness]]: 0.798 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.798 × 10<sup>-3</sup> | ||
=== 13-limit (sumatra) === | === 13-limit (sumatra) === | ||
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Mapping: {{mapping| 1 0 1 0 6 0 | 0 6 5 0 1 14 | 0 0 0 1 -1 0 }} | Mapping: {{mapping| 1 0 1 0 6 0 | 0 6 5 0 1 14 | 0 0 0 1 -1 0 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.1156{{c}}, ~7/4 = 967.6235{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
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: unchanged-interval (eigenmonzo) basis: 2.11/7.13/7 | : unchanged-interval (eigenmonzo) basis: 2.11/7.13/7 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }} | ||
Badness: 0.680 × 10<sup>-3</sup> | Badness (Smith): 0.680 × 10<sup>-3</sup> | ||
Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11 | Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11 | ||
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{{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }} | {{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }} | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 316.7584{{c}}, ~7/4 = 968.8259{{c}} | ||
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }} | {{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }} | ||
[[Badness]]: 3.690 × 10<sup>-3</sup> | [[Badness]] (Smith): 3.690 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }} | Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 316.8446{{c}}, ~7/4 = 968.8259{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }} | ||
Badness: 2.273 × 10<sup>-3</sup> | Badness (Smith): 2.273 × 10<sup>-3</sup> | ||
== Enki == | == Enki == | ||
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{{Mapping|legend=1| 1 0 1 0 1 | 0 6 5 0 4 | 0 0 0 2 1 }} | {{Mapping|legend=1| 1 0 1 0 1 | 0 6 5 0 4 | 0 0 0 2 1 }} | ||
: mapping generators: ~2, ~6/5, ~66/25 | : mapping generators: ~2, ~6/5, ~66/25 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0396{{c}}, ~33/25 = 484.2358{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }} | {{Optimal ET sequence|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }} | ||
[[Badness]]: 1.09 × 10<sup>-3</sup> | [[Badness]] (Smith): 1.09 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 1 0 1 0 | 0 6 5 0 4 14 | 0 0 0 2 1 0 }} | Mapping: {{mapping| 1 0 1 0 1 0 | 0 6 5 0 4 14 | 0 0 0 2 1 0 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.1387{{c}}, ~33/25 = 483.3828{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
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: unchanged-interval (eigenmonzo) basis: 2.7.13/5 | : unchanged-interval (eigenmonzo) basis: 2.7.13/5 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 34d, 49f, 72, 87, 159, 193, 280 }} | ||
Badness: 0.874 × 10<sup>-3</sup> | Badness (Smith): 0.874 × 10<sup>-3</sup> | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Kleismic rank-3 family| ]] <!-- main article --> | [[Category:Kleismic rank-3 family| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Revision as of 09:33, 7 January 2026
- 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 kleismic rank-3 family of temperaments tempers out the kleisma, 15625/15552. If nothing else is tempered out in the 7-limit, we have a planar temperament. With an 11-limit comma we get an 11-limit temperament, and so forth.
Rank-3 kleismic
Subgroup: 2.3.5.7
Comma list: 15625/15552
Mapping: [⟨1 0 1 0], ⟨0 6 5 0], ⟨0 0 0 1]]
- mapping generators: ~2, ~6/5, ~7
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.0586 ¢, ~7/4 = 968.8259 ¢
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [0 1 0 0⟩, [1 5/6 0 0⟩, [0 0 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.3.7
Optimal ET sequence: 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c
Badness (Smith): 0.363 × 10-3
2.3.5.7.13 subgroup
Subgroup: 2.3.5.7.13
Comma list: 325/324, 625/624
Subgroup-val mapping: [⟨1 0 1 0 0], ⟨0 6 5 0 14], ⟨0 0 0 1 0]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.1110 ¢, ~7/4 = 968.8259 ¢
Optimal ET sequence: 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f
Badness (Smith): 0.290 × 10-3
Enlil
Subgroup: 2.3.5.7.11
Comma list: 385/384, 6250/6237
Mapping: [⟨1 0 1 0 6], ⟨0 6 5 0 1], ⟨0 0 0 1 -1]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.0667 ¢, ~7/4 = 967.6041 ¢
- [[1 0 0 0 0⟩, [0 1 0 0 0⟩, [1 5/6 0 0 0⟩, [3 1/12 0 1/2 -1/2⟩, [3 1/12 0 -1/2 1/2⟩]
- unchanged-interval (eigenmonzo) basis: 2.3.11/7
Optimal ET sequence: 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee
Badness (Smith): 0.798 × 10-3
13-limit (sumatra)
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 625/624
Mapping: [⟨1 0 1 0 6 0], ⟨0 6 5 0 1 14], ⟨0 0 0 1 -1 0]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.1156 ¢, ~7/4 = 967.6235 ¢
Minimax tuning:
- 13- and 15-odd-limit
- [[1 0 0 0 0 0⟩, [4/3 0 0 -2/9 -2/9 4/9⟩, [19/9 0 0 -5/27 -5/27 10/27⟩, [28/9 0 0 13/27 -14/27 1/27⟩, [28/9 0 0 -14/27 13/27 1/27⟩, [28/9 0 0 -14/27 -14/27 28/27⟩]
- unchanged-interval (eigenmonzo) basis: 2.11/7.13/7
Optimal ET sequence: 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef
Badness (Smith): 0.680 × 10-3
Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11
Scales: enlil19_13
Isimud
Subgroup: 2.3.5.7.11
Comma list: 4125/4096, 8019/8000
Mapping: [⟨1 0 1 0 9], ⟨0 6 5 0 -21], ⟨0 0 0 1 0]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 316.7584 ¢, ~7/4 = 968.8259 ¢
Optimal ET sequence: 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee
Badness (Smith): 3.690 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 625/624, 1287/1280
Mapping: [⟨1 0 1 0 9 0], ⟨0 6 5 0 -21 14], ⟨0 0 0 1 0 0]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 316.8446 ¢, ~7/4 = 968.8259 ¢
Optimal ET sequence: 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff
Badness (Smith): 2.273 × 10-3
Enki
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4356
Mapping: [⟨1 0 1 0 1], ⟨0 6 5 0 4], ⟨0 0 0 2 1]]
- mapping generators: ~2, ~6/5, ~66/25
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.0396 ¢, ~33/25 = 484.2358 ¢
- [[1 0 0 0 0⟩, [-18/13 0 6/13 -6/13 12/13⟩, [-2/13 0 5/13 -5/13 10/13⟩, [-2/13 0 -8/13 8/13 10/13⟩, [0 0 0 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.7/5.11
Optimal ET sequence: 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424
Badness (Smith): 1.09 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 364/363, 625/624
Mapping: [⟨1 0 1 0 1 0], ⟨0 6 5 0 4 14], ⟨0 0 0 2 1 0]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.1387 ¢, ~33/25 = 483.3828 ¢
Minimax tuning:
- 13- and 15-odd-limit
- [[1 0 0 0 0 0⟩, [2/3 0 -2/3 0 0 2/3⟩, [14/9 0 -5/9 0 0 5/9⟩, [0 0 0 1 0 0⟩, [13/9 0 -4/9 1/2 0 4/9⟩, [14/9 0 -14/9 0 0 14/9⟩]
- unchanged-interval (eigenmonzo) basis: 2.7.13/5
Optimal ET sequence: 34d, 49f, 72, 87, 159, 193, 280
Badness (Smith): 0.874 × 10-3