Kleismic rank three family: Difference between revisions
Cmloegcmluin (talk | contribs) unchanged interval → unchanged-interval |
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
||
| Line 15: | Line 15: | ||
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6, 4/3 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6, 4/3 | ||
{{ | {{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }} | ||
== Enlil == | == Enlil == | ||
| Line 31: | Line 31: | ||
: [[Eigenmonzo]]s (unchanged-intervals): 2, 4/3, 14/11 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 4/3, 14/11 | ||
{{ | {{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]]: 0.798 × 10<sup>-3</sup> | ||
| Line 49: | Line 49: | ||
: [[Eigenmonzo]]s (unchanged-intervals): 2, 13/11, 14/11 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 13/11, 14/11 | ||
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }} | |||
Badness: 0.680 × 10<sup>-3</sup> | Badness: 0.680 × 10<sup>-3</sup> | ||
| Line 69: | Line 69: | ||
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/8, 7/5 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 11/8, 7/5 | ||
{{ | {{Optimal ET sequence|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }} | ||
[[Badness]]: 1.09 × 10<sup>-3</sup> | [[Badness]]: 1.09 × 10<sup>-3</sup> | ||
| Line 87: | Line 87: | ||
: Eigenmonzos (unchanged-intervals): 2, 8/7 13/10 | : Eigenmonzos (unchanged-intervals): 2, 8/7 13/10 | ||
{{Optimal ET sequence|legend=1| 34d, 49f, 72, 87, 159, 193, 280 }} | |||
Badness: 0.874 × 10<sup>-3</sup> | Badness: 0.874 × 10<sup>-3</sup> | ||
| Line 98: | Line 98: | ||
[[Mapping]]: [{{val| 1 0 1 0 9 }}, {{val| 0 6 5 0 -21 }}, {{val| 0 0 0 1 0 }}] | [[Mapping]]: [{{val| 1 0 1 0 9 }}, {{val| 0 6 5 0 -21 }}, {{val| 0 0 0 1 0 }}] | ||
{{ | {{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }} | ||
[[Badness]]: 3.690 × 10<sup>-3</sup> | [[Badness]]: 3.690 × 10<sup>-3</sup> | ||
| Line 109: | Line 109: | ||
Mapping: [{{val| 1 0 1 0 9 0 }}, {{val| 0 6 5 0 -21 14 }}, {{val| 0 0 0 1 0 0 }}] | Mapping: [{{val| 1 0 1 0 9 0 }}, {{val| 0 6 5 0 -21 14 }}, {{val| 0 0 0 1 0 0 }}] | ||
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }} | |||
Badness: 2.273 × 10<sup>-3</sup> | Badness: 2.273 × 10<sup>-3</sup> | ||
Revision as of 18:25, 7 May 2023
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
- 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
Optimal ET sequence: 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c
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
Optimal ET sequence: 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee
Badness: 0.798 × 10-3
13-limit (aka 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]]
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
Optimal ET sequence: 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef
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
Optimal ET sequence: 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424
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
Optimal ET sequence: 34d, 49f, 72, 87, 159, 193, 280
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]]
Optimal ET sequence: 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee
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]]
Optimal ET sequence: 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff
Badness: 2.273 × 10-3