Kleismic rank three family: Difference between revisions

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m Cleanup (1/2)
m Cleanup (2/2)
Line 7: Line 7:
Comma list: 15625/15552
Comma list: 15625/15552


Mapping: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|]
Mapping: [{{val| 1 0 1 0 }}, {{val| 0 6 5 0 }}, {{val| 0 0 0 1 }}]


Mapping generators: ~2, ~6/5, ~7
Mapping generators: ~2, ~6/5, ~7
Line 13: Line 13:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |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: 2, 7/6, 4/3
: [[Eigenmonzo]]s: 2, 7/6, 4/3


Line 24: Line 24:
[[Comma list]]: 385/384, 6250/6237
[[Comma list]]: 385/384, 6250/6237


Mapping: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]
Mapping: [{{val| 1 0 1 0 6 }}, {{val| 0 6 5 0 1 }}, {{val| 0 0 0 1 -1 }}]


Mapping generators: ~2, ~6/5, ~7
Mapping generators: ~2, ~6/5, ~7
Line 30: Line 30:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|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>]
: [{{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: 2, 4/3, 14/11
: [[Eigenmonzo]]s: 2, 4/3, 14/11


{{Val list|legend=1| 53, 68, 72, 140, 159, 212, 284 }}
{{Val list|legend=1| 53, 68, 72, 140, 159, 212, 284 }}


Badness: 0.000798
Badness: 0.798 × 10<sup>-3</sup>


== 13-limit ==
== 13-limit ==
Line 43: Line 43:
Comma list: 325/324, 385/384, 625/624
Comma list: 325/324, 385/384, 625/624


Mapping: [&lt;1 0 1 0 6 0|, &lt;0 6 5 0 1 14|, &lt;0 0 0 1 -1 0|]
Mapping: [{{val| 1 0 1 0 6 0 }}, {{val| 0 6 5 0 1 14 }}, {{val| 0 0 0 1 -1 0 }}]


Mapping generators: 2, 6/5, 7
Mapping generators: 2, 6/5, 7
Line 49: Line 49:
Minimax tuning
Minimax tuning
* 13- and 15-odd-limit
* 13- and 15-odd-limit
: [|1 0 0 0 0 0&gt;, |4/3 0 0 -2/9 -2/9 4/9&gt;, |19/9 0 0 -5/27 -5/27 10/27&gt;, |28/9 0 0 13/27 -14/27 1/27&gt;, |28/9 0 0 -14/27 13/27 1/27&gt;, |28/9 0 0 -14/27 -14/27 28/27&gt;]
: [{{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: 2, 13/11, 14/11
: [[Eigenmonzo]]s: 2, 13/11, 14/11


Vals: {{Val list| 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299 }}
Vals: {{Val list| 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299 }}


Badness: 0.000680
Badness: 0.680 × 10<sup>-3</sup>


Scales: [[enlil19_13]]
Scales: [[enlil19_13]]
Line 64: Line 64:
[[Comma list]]:  3025/3024, 4375/4356
[[Comma list]]:  3025/3024, 4375/4356


Mapping: [&lt;1 0 1 0 1|, &lt;0 6 5 0 4|, &lt;0 0 0 2 1|]
Mapping: [{{val| 1 0 1 0 1 }}, {{val| 0 6 5 0 4 }}, {{val| 0 0 0 2 1 }}]


Mapping generators: 2, 6/5, 66/25
Mapping generators: 2, 6/5, 66/25
Line 70: Line 70:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |-18/13 0 6/13 -6/13 12/13&gt;, |-2/13 0 5/13 -5/13 10/13&gt;, |-2/13 0 -8/13 8/13 10/13&gt;, |0 0 0 0 1&gt;]
: [{{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: 2, 11/8, 7/5
: [[Eigenmonzo]]s: 2, 11/8, 7/5


{{Val list|legend=1| 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439 }}
{{Val list|legend=1| 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439 }}


Badness: 0.00109
Badness: 1.09 × 10<sup>-3</sup>


== 13-limit ==
== 13-limit ==
Line 83: Line 83:
Comma list: 325/324, 364/363, 625/624
Comma list: 325/324, 364/363, 625/624


Mapping: [&lt;1 0 1 0 1 0|, &lt;0 6 5 0 4 14|, &lt;0 0 0 2 1 0|]
Mapping: [{{val| 1 0 1 0 1 0 }}, {{val| 0 6 5 0 4 14 }}, {{val| 0 0 0 2 1 0 }}]


Mapping generators: 2, 6/5, 66/25
Mapping generators: 2, 6/5, 66/25
Line 89: Line 89:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit
* 13- and 15-odd-limit
: [|1 0 0 0 0 0&gt;, |2/3 0 -2/3 0 0 2/3&gt;, |14/9 0 -5/9 0 0 5/9&gt;, |0 0 0 1 0 0&gt;, |13/9 0 -4/9 1/2 0 4/9&gt;, |14/9 0 -14/9 0 0 14/9&gt;]
: [{{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: 2, 8/7 13/10
: Eigenmonzos: 2, 8/7 13/10


Vals: {{Val list| 15, 72, 87, 159, 193, 280, 439 }}
Vals: {{Val list| 15, 72, 87, 159, 193, 280, 439 }}


Badness: 0.000874
Badness: 0.874 × 10<sup>-3</sup>


= Isimud =
= Isimud =
Line 102: Line 102:
[[Comma list]]: 4125/4096, 8019/8000
[[Comma list]]: 4125/4096, 8019/8000


[[Mapping]]: [&lt;1 0 1 0 9|, &lt;0 6 5 0 -21|, &lt;0 0 0 1 0|]
[[Mapping]]: [{{val| 1 0 1 0 9 }}, {{val| 0 6 5 0 -21 }}, {{val| 0 0 0 1 0 }}]


{{Val list|legend=1| 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde }}
{{Val list|legend=1| 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde }}


[[Badness]]: 0.003690
[[Badness]]: 3.690 × 10<sup>-3</sup>


== 13-limit ==
== 13-limit ==
Line 114: Line 114:
Comma list: 325/324, 625/624, 1287/1280
Comma list: 325/324, 625/624, 1287/1280


Mapping: [&lt;1 0 1 0 9 0|, &lt;0 6 5 0 -21 14|, &lt;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 }}]


Vals: {{Val list| 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef }}
Vals: {{Val list| 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef }}


Badness: 0.002273
Badness: 2.273 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 11:44, 23 April 2021

These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit 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

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [1 5/6 0 0⟩, [0 0 0 1⟩]
Eigenmonzos: 2, 7/6, 4/3

Template:Val list

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

Minimax tuning:

[[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: 2, 4/3, 14/11

Template:Val list

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: 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

Minimax tuning:

[[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: 2, 11/8, 7/5

Template:Val list

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: 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]]

Template:Val list

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

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