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.
{{Technical data page}}
The '''kleismic rank-3 family''' of temperaments [[tempering out|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 ==
== Rank-3 kleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552
[[Comma list]]: 15625/15552


[[Mapping]]: [{{val| 1 0 1 0 }}, {{val| 0 6 5 0 }}, {{val| 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 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[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 list| 1 0 0 0 | 0 1 0 0 | 1 5/6 0 0 | 0 0 0 1 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6, 4/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
 
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
 
[[Badness]]: 0.363 × 10<sup>-3</sup>
 
=== 2.3.5.7.13 subgroup ===
Subgroup: 2.3.5.7.13
 
Comma list: 325/324, 625/624
 
Sval mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1110, ~7/4 = 968.8259


{{Val list|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }}
 
Badness: 0.290 × 10<sup>-3</sup>


== Enlil ==
== Enlil ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


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


Mapping generators: ~2, ~6/5, ~7
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0667, ~7/4 = 967.6041


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[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 basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7


{{Val list|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]]: 0.798 × 10<sup>-3</sup>


=== 13-limit (aka sumatra) ===
=== 13-limit (sumatra) ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 625/624
Comma list: 325/324, 385/384, 625/624


Mapping: [{{val| 1 0 1 0 6 0 }}, {{val| 0 6 5 0 1 14 }}, {{val| 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 }}


Mapping generators: ~2, ~6/5, ~7
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1156, ~7/4 = 967.6235


Minimax tuning:  
Minimax tuning:  
* 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
: unchanged-interval (eigenmonzo) basis: 2.11/7.13/7


Vals: {{Val list| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}
{{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>
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]]
Scales: [[enlil19_13]]
== Isimud ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 4125/4096, 8019/8000
{{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 316.7584, ~7/4 = 968.8259
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}
[[Badness]]: 3.690 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 625/624, 1287/1280
Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }}
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8446, ~7/4 = 968.8259
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }}
Badness: 2.273 × 10<sup>-3</sup>


== Enki ==
== Enki ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 4375/4356
[[Comma list]]: 3025/3024, 4375/4356


[[Mapping]]: [{{val| 1 0 1 0 1 }}, {{val| 0 6 5 0 4 }}, {{val| 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 = 1\1, ~6/5 = 317.0396, ~33/25 = 484.2358


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[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 basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11


{{Val list|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]]: 1.09 × 10<sup>-3</sup>
Line 78: Line 126:
Comma list: 325/324, 364/363, 625/624
Comma list: 325/324, 364/363, 625/624


Mapping: [{{val| 1 0 1 0 1 0 }}, {{val| 0 6 5 0 4 14 }}, {{val| 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 }}


Mapping generators: 2, 6/5, 66/25
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1387, ~33/25 = 483.3828


Minimax tuning:  
Minimax tuning:  
* 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
: unchanged-interval (eigenmonzo) basis: 2.7.13/5


Vals: {{Val list| 34d, 49f, 72, 87, 159, 193, 280 }}
{{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>
== Isimud ==
Subgroup: 2.3.5.7.11
[[Comma list]]: 4125/4096, 8019/8000
[[Mapping]]: [{{val| 1 0 1 0 9 }}, {{val| 0 6 5 0 -21 }}, {{val| 0 0 0 1 0 }}]
{{Val list|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}
[[Badness]]: 3.690 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 625/624, 1287/1280
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, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }}
Badness: 2.273 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Kleismic rank-3 family| ]] <!-- main article -->
[[Category:Kleismic rank-3 family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:30, 24 June 2025

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 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259

Minimax tuning:

[[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: 0.363 × 10-3

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 325/324, 625/624

Sval mapping: [⟨1 0 1 0 0], ⟨0 6 5 0 14], ⟨0 0 0 1 0]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1110, ~7/4 = 968.8259

Optimal ET sequence: 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f

Badness: 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 = 1\1, ~6/5 = 317.0667, ~7/4 = 967.6041

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⟩]
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: 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 = 1\1, ~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: 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 = 1\1, ~6/5 = 316.7584, ~7/4 = 968.8259

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 tuning (CTE): ~2 = 1\1, ~6/5 = 316.8446, ~7/4 = 968.8259

Optimal ET sequence: 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff

Badness: 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 = 1\1, ~6/5 = 317.0396, ~33/25 = 484.2358

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⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11

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]]

Optimal tuning (CTE): ~2 = 1\1, ~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: 0.874 × 10-3

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