Kleismic rank-3 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]]s:
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}, ~7/4 = 968.4936{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 -0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}, ~7/4 = 968.5773{{c}}
: error map: {{val| 0.000 +0.230 -1.160 -0.249 }}


[[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


{{Val list|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
 
[[Badness]] (Sintel): 1.64
 
=== 2.3.5.7.13 subgroup ===
Subgroup: 2.3.5.7.13
 
Comma list: 325/324, 625/624
 
Subgroup-val mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}, ~7/4 = 968.5830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}, ~7/4 = 968.6426{{c}}
 
{{Optimal ET sequence|legend=0| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }}
 
Badness (Sintel): 0.385


== 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]]s:
* [[WE]]: ~2 = 1200.3806{{c}}, ~6/5 = 317.0420{{c}}, ~7/4 = 967.7397{{c}}
: [[error map]]: {{val| +0.381 +0.297 -0.723 -0.325 -0.493 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.9734{{c}}, ~7/4 = 967.4765{{c}}
: error map: {{val| 0.000 -0.114 -1.447 -1.349 -1.821 }}


[[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]] (Sintel): 0.959


=== 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 tunings:
* WE: ~2 = 1200.3557{{c}}, ~6/5 = 317.1024{{c}}, ~7/4 = 967.7541{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0355{{c}}, ~7/4 = 967.5068{{c}}


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
 
{{Optimal ET sequence|legend=0| 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 }}
Badness (Sintel): 0.636


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]]s:
* [[WE]]: ~2 = 1200.7273{{c}}, ~6/5 = 316.9521{{c}}, ~7/4 = 967.3697{{c}}
: [[error map]]: {{val| +0.727 -0.242 -0.826 -0.002 -0.767 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7598{{c}}, ~7/4 = 967.1919{{c}}
: error map: {{val| 0.000 -1.396 -2.515 -1.634 -3.273 }}
{{Optimal ET sequence|legend=1| 19, 34e, …, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}
[[Badness]] (Sintel): 4.43
=== 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 tunings:
* WE: ~2 = 1200.8711{{c}}, ~6/5 = 317.0291{{c}}, ~7/4 = 967.0809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.8070{{c}}, ~7/4 = 966.8000{{c}}
{{Optimal ET sequence|legend=0| 19, 34e, …, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeefff }}
Badness (Sintel): 2.13


== 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]]s:  
* [[WE]]: ~2 = 1200.0985{{c}}, ~6/5 = 317.0336{{c}}, ~33/25 = 484.1262{{c}}
: [[error map]]: {{val| +0.099 +0.246 -1.047 -0.376 +1.140 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0222{{c}}, ~33/25 = 484.1546{{c}}
: error map: {{val| 0.000 +0.178 -1.203 -0.517 +0.925 }}


[[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]] (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Line 78: Line 146:
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 tunings:  
* WE: ~2 = 1200.0373{{c}}, ~6/5 = 317.0967{{c}}, ~33/25 = 484.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0919{{c}}, ~33/25 = 484.1707{{c}}


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
 
{{Optimal ET sequence|legend=0| 15, 23deff, 34d, 49f, 72, 87, 121, 159, 193, 280, 352f }}
 
Badness (Sintel): 0.817


Vals: {{Val list| 34d, 49f, 72, 87, 159, 193, 280 }}
== Catakleismoid ==
Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.


Badness: 0.874 × 10<sup>-3</sup>
[[Subgroup]]: 2.3.5.7.11


== Isimud ==
[[Comma list]]: 225/224, 4375/4374
Subgroup: 2.3.5.7.11


[[Comma list]]: 4125/4096, 8019/8000
{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~6/5, ~11


[[Mapping]]: [{{val| 1 0 1 0 9 }}, {{val| 0 6 5 0 -21 }}, {{val| 0 0 0 1 0 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}
: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}


{{Val list|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}


[[Badness]]: 3.690 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.53


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 625/624, 1287/1280
Comma list: 169/168, 225/224, 325/324
 
Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}


Mapping: [{{val| 1 0 1 0 9 0 }}, {{val| 0 6 5 0 -21 14 }}, {{val| 0 0 0 1 0 0 }}]
Optimal tunings:  
* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}


Vals: {{Val list| 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }}
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197ef }}


Badness: 2.273 × 10<sup>-3</sup>
Badness (Sintel): 0.853


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Kleismic rank-3 family| ]] <!-- main article -->
[[Category:Kleismic rank-3 family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 09:42, 11 April 2026

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 tunings:

  • WE: ~2 = 1200.1659 ¢, ~6/5 = 317.0504 ¢, ~7/4 = 968.4936 ¢
error map: ⟨+0.166 +0.347 -0.896 -0.001]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢, ~7/4 = 968.5773 ¢
error map: ⟨0.000 +0.230 -1.160 -0.249]

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 (Sintel): 1.64

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 325/324, 625/624

Subgroup-val mapping: [⟨1 0 1 0 0], ⟨0 6 5 0 14], ⟨0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.1210 ¢, ~6/5 = 317.1076 ¢, ~7/4 = 968.5830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0920 ¢, ~7/4 = 968.6426 ¢

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

Badness (Sintel): 0.385

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 tunings:

  • WE: ~2 = 1200.3806 ¢, ~6/5 = 317.0420 ¢, ~7/4 = 967.7397 ¢
error map: ⟨+0.381 +0.297 -0.723 -0.325 -0.493]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.9734 ¢, ~7/4 = 967.4765 ¢
error map: ⟨0.000 -0.114 -1.447 -1.349 -1.821]

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 (Sintel): 0.959

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 tunings:

  • WE: ~2 = 1200.3557 ¢, ~6/5 = 317.1024 ¢, ~7/4 = 967.7541 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0355 ¢, ~7/4 = 967.5068 ¢

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 (Sintel): 0.636

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 tunings:

  • WE: ~2 = 1200.7273 ¢, ~6/5 = 316.9521 ¢, ~7/4 = 967.3697 ¢
error map: ⟨+0.727 -0.242 -0.826 -0.002 -0.767]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7598 ¢, ~7/4 = 967.1919 ¢
error map: ⟨0.000 -1.396 -2.515 -1.634 -3.273]

Optimal ET sequence: 19, 34e, …, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee

Badness (Sintel): 4.43

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 tunings:

  • WE: ~2 = 1200.8711 ¢, ~6/5 = 317.0291 ¢, ~7/4 = 967.0809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.8070 ¢, ~7/4 = 966.8000 ¢

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

Badness (Sintel): 2.13

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

Optimal tunings:

  • WE: ~2 = 1200.0985 ¢, ~6/5 = 317.0336 ¢, ~33/25 = 484.1262 ¢
error map: ⟨+0.099 +0.246 -1.047 -0.376 +1.140]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0222 ¢, ~33/25 = 484.1546 ¢
error map: ⟨0.000 +0.178 -1.203 -0.517 +0.925]

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 (Sintel): 1.31

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 tunings:

  • WE: ~2 = 1200.0373 ¢, ~6/5 = 317.0967 ¢, ~33/25 = 484.1601 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0919 ¢, ~33/25 = 484.1707 ¢

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: 15, 23deff, 34d, 49f, 72, 87, 121, 159, 193, 280, 352f

Badness (Sintel): 0.817

Catakleismoid

Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 0], ⟨0 0 0 0 1]]

mapping generators: ~2, ~6/5, ~11

Optimal tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢, ~11/8 = 549.5258 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948 -0.003]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢, ~11/8 = 550.0343 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126 -1.284]

Optimal ET sequence: 19, 34d, 53, 72, 197e, 269ce, 341ce

Badness (Sintel): 1.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324

Mapping: [⟨1 0 1 -3 0 0], ⟨0 6 5 22 0 14], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.8238 ¢, ~6/5 = 316.9478 ¢, ~11/8 = 548.9611 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢, ~11/8 = 549.5903 ¢

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197ef

Badness (Sintel): 0.853

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