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Kleismic rank-3 family: Difference between revisions

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: mapping generators: ~2, ~6/5, ~7
: mapping generators: ~2, ~6/5, ~7


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0586{{c}}, ~7/4 = 968.8259{{c}}
[[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]]:  
Line 19: Line 23:
{{Optimal ET sequence|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]] (Smith): 0.363 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.64


=== 2.3.5.7.13 subgroup ===
=== 2.3.5.7.13 subgroup ===
Line 28: Line 32:
Subgroup-val mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}
Subgroup-val mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}, ~7/4 = 968.8259{{c}}
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 }}
{{Optimal ET sequence|legend=0| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }}


Badness (Smith): 0.290 × 10<sup>-3</sup>
Badness (Sintel): 0.385


== Enlil ==
== Enlil ==
Line 41: Line 47:
{{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }}
{{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0667{{c}}, ~7/4 = 967.6041{{c}}
[[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 basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7


{{Optimal ET sequence|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]] (Smith): 0.798 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.959


=== 13-limit (sumatra) ===
=== 13-limit (sumatra) ===
Line 59: Line 69:
Mapping: {{mapping| 1 0 1 0 6 0 | 0 6 5 0 1 14 | 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 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.1156{{c}}, ~7/4 = 967.6235{{c}}
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:  
Line 68: Line 80:
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}


Badness (Smith): 0.680 × 10<sup>-3</sup>
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
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
Line 81: Line 93:
{{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 316.7584{{c}}, ~7/4 = 968.8259{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 19, 34e, …, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}


[[Badness]] (Smith): 3.690 × 10<sup>-3</sup>
[[Badness]] (Sintel): 4.43


=== 13-limit ===
=== 13-limit ===
Line 94: Line 110:
Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }}
Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 316.8446{{c}}, ~7/4 = 968.8259{{c}}
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, 303cdeeff }}
{{Optimal ET sequence|legend=0| 19, 34e, …, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeefff }}


Badness (Smith): 2.273 × 10<sup>-3</sup>
Badness (Sintel): 2.13


== Enki ==
== Enki ==
Line 108: Line 126:
: mapping generators: ~2, ~6/5, ~66/25
: mapping generators: ~2, ~6/5, ~66/25


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0396{{c}}, ~33/25 = 484.2358{{c}}
[[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 basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11


{{Optimal ET sequence|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]] (Smith): 1.09 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Line 126: Line 148:
Mapping: {{mapping| 1 0 1 0 1 0 | 0 6 5 0 4 14 | 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 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.1387{{c}}, ~33/25 = 483.3828{{c}}
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:  
Line 133: Line 157:
: unchanged-interval (eigenmonzo) basis: 2.7.13/5
: unchanged-interval (eigenmonzo) basis: 2.7.13/5


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


Badness (Smith): 0.874 × 10<sup>-3</sup>
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|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~6/5, ~11
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{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{{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}}
 
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197ef }}
 
Badness (Sintel): 0.853


[[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]]
Retrieved from "https://en.xen.wiki/w/Kleismic_rank-3_family"