Kleismic rank-3 family: Difference between revisions
m Removing from Category:Regular temperament theory using Cat-a-lot |
Cmloegcmluin (talk | contribs) unchanged interval → unchanged-interval |
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| Line 13: | Line 13: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]] | * [[7-odd-limit|7-]] and [[9-odd-limit]] | ||
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 1 5/6 0 0 }}, {{monzo| 0 0 0 1 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 1 5/6 0 0 }}, {{monzo| 0 0 0 1 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 4/3 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6, 4/3 | ||
{{Val list|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }} | {{Val list|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }} | ||
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* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1 5/6 0 0 0 }}, {{monzo| 3 1/12 0 1/2 -1/2 }}, {{monzo| 3 1/12 0 -1/2 1/2 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1 5/6 0 0 0 }}, {{monzo| 3 1/12 0 1/2 -1/2 }}, {{monzo| 3 1/12 0 -1/2 1/2 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 4/3, 14/11 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 4/3, 14/11 | ||
{{Val list|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }} | {{Val list|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }} | ||
| Line 47: | Line 47: | ||
* 13- and 15-odd-limit | * 13- and 15-odd-limit | ||
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/3 0 0 -2/9 -2/9 4/9 }}, {{monzo| 19/9 0 0 -5/27 -5/27 10/27 }}, {{monzo| 28/9 0 0 13/27 -14/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 13/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 -14/27 28/27 }}] | : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/3 0 0 -2/9 -2/9 4/9 }}, {{monzo| 19/9 0 0 -5/27 -5/27 10/27 }}, {{monzo| 28/9 0 0 13/27 -14/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 13/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 -14/27 28/27 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 13/11, 14/11 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 13/11, 14/11 | ||
Vals: {{Val list| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }} | Vals: {{Val list| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }} | ||
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* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| -18/13 0 6/13 -6/13 12/13 }}, {{monzo| -2/13 0 5/13 -5/13 10/13 }}, {{monzo| -2/13 0 -8/13 8/13 10/13 }}, {{monzo| 0 0 0 0 1 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| -18/13 0 6/13 -6/13 12/13 }}, {{monzo| -2/13 0 5/13 -5/13 10/13 }}, {{monzo| -2/13 0 -8/13 8/13 10/13 }}, {{monzo| 0 0 0 0 1 }}] | ||
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/8, 7/5 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 11/8, 7/5 | ||
{{Val list|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }} | {{Val list|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }} | ||
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* 13- and 15-odd-limit | * 13- and 15-odd-limit | ||
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 2/3 0 -2/3 0 0 2/3 }}, {{monzo| 14/9 0 -5/9 0 0 5/9 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 13/9 0 -4/9 1/2 0 4/9 }}, {{monzo| 14/9 0 -14/9 0 0 14/9 }}] | : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 2/3 0 -2/3 0 0 2/3 }}, {{monzo| 14/9 0 -5/9 0 0 5/9 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 13/9 0 -4/9 1/2 0 4/9 }}, {{monzo| 14/9 0 -14/9 0 0 14/9 }}] | ||
: Eigenmonzos (unchanged intervals): 2, 8/7 13/10 | : Eigenmonzos (unchanged-intervals): 2, 8/7 13/10 | ||
Vals: {{Val list| 34d, 49f, 72, 87, 159, 193, 280 }} | Vals: {{Val list| 34d, 49f, 72, 87, 159, 193, 280 }} | ||
Revision as of 23:45, 5 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
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 (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
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