Kleismic rank three family: Difference between revisions
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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. | |||
== Kleismic | == Kleismic == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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{{Val list|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212 }} | {{Val list|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212 }} | ||
== Enlil | == Enlil == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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[[Badness]]: 0.798 × 10<sup>-3</sup> | [[Badness]]: 0.798 × 10<sup>-3</sup> | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 58: | Line 58: | ||
Scales: [[enlil19_13]] | Scales: [[enlil19_13]] | ||
== Enki | == Enki == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 77: | Line 77: | ||
[[Badness]]: 1.09 × 10<sup>-3</sup> | [[Badness]]: 1.09 × 10<sup>-3</sup> | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 96: | Line 96: | ||
Badness: 0.874 × 10<sup>-3</sup> | Badness: 0.874 × 10<sup>-3</sup> | ||
== Isimud | == Isimud == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 108: | Line 108: | ||
[[Badness]]: 3.690 × 10<sup>-3</sup> | [[Badness]]: 3.690 × 10<sup>-3</sup> | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Kleismic]] | [[Category:Kleismic rank-3 family| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Revision as of 08:32, 11 July 2021
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.
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
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [0 1 0 0⟩, [1 5/6 0 0⟩, [0 0 0 1⟩]
- Eigenmonzos (unchanged intervals): 2, 7/6, 4/3
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]]
Mapping generators: ~2, ~6/5, ~7
- [[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⟩]
- Eigenmonzos (unchanged intervals): 2, 4/3, 14/11
Badness: 0.798 × 10-3
13-limit
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]]
Mapping generators: 2, 6/5, 7
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⟩]
- Eigenmonzos (unchanged intervals): 2, 13/11, 14/11
Vals: Template:Val list
Badness: 0.680 × 10-3
Scales: enlil19_13
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
- [[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⟩]
- Eigenmonzos (unchanged intervals): 2, 11/8, 7/5
Badness: 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]]
Mapping generators: 2, 6/5, 66/25
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⟩]
- Eigenmonzos (unchanged intervals): 2, 8/7 13/10
Vals: Template:Val list
Badness: 0.874 × 10-3
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]]
Badness: 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]]
Vals: Template:Val list
Badness: 2.273 × 10-3