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Catalog of rank-4 temperaments: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 694.823{{c}}, ~5/4 = 378.660{{c}}, ~7/4 = 968.826{{c}}
* [[WE]]: ~2 = 1202.4372{{c}}, ~3/2 = 693.2306{{c}}, ~5/4 = 374.6677{{c}}, ~7/4 = 963.8387{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.197{{c}}, ~5/4 = 376.082{{c}}, ~7/4 = 965.060{{c}}
: [[error map]]: {{val| +2.437 -6.287 -6.772 -0.113 +14.685 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.1966{{c}}, ~5/4 = 376.0821{{c}}, ~7/4 = 965.0600{{c}}
: error map: {{val| 0.000 -8.758 -10.232 -3.766 +11.157 }}


{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 26, 45 }}


[[Badness]] (Smith): 0.0219 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.400


== Mothwellsmic (99/98) ==
== Mothwellsmic (99/98) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.306{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 974.000{{c}}
* [[WE]]: ~2 = 1200.3175{{c}}, ~3/2 = 700.0109{{c}}, ~5/4 = 385.6677{{c}}, ~7/4 = 973.2571{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.060{{c}}, ~5/4 = 385.931{{c}}, ~7/4 = 973.388{{c}}
: [[error map]]: {{val| +0.318 -1.627 -0.011 +5.066 -3.873 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.0599{{c}}, ~5/4 = 385.9307{{c}}, ~7/4 = 973.4880{{c}}
: error map: {{val| 0.000 -1.895 -0.383 +4.662 -4.462 }}


{{Optimal ET sequence|legend=1| 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127 }}
{{Optimal ET sequence|legend=1| 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127 }}


[[Badness]] (Smith): 0.0299 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.545


== Ptolemismic (100/99) ==
== Ptolemismic (100/99) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.961{{c}}, ~5/4 = 382.009{{c}}, ~7/4 = 968.826{{c}}
* [[WE]]: ~2 = 1199.2633{{c}}, ~3/2 = 704.5205{{c}}, ~5/4 = 383.8317{{c}}, ~7/4 = 970.2843{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.475{{c}}, ~5/4 = 383.076{{c}}, ~7/4 = 969.890{{c}}
: [[error map]]: {{val| -0.737 +1.829 -3.955 -0.015 +4.358 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.4751{{c}}, ~5/4 = 383.0756{{c}}, ~7/4 = 969.8904{{c}}
: error map: {{val| 0.000 +2.520 -3.238 +1.065 +5.883 }}


{{Optimal ET sequence|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
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<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]


[[Badness]] (Smith): 0.0225 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.411


== Biyatismic (121/120) ==
== Biyatismic (121/120) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.602{{c}}, ~11/10 = 157.450{{c}}, ~7/4 = 968.826{{c}}
* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}, ~7/4 = 967.4929{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.985{{c}}, ~11/10 = 157.610{{c}}, ~7/4 = 967.819{{c}}
: [[error map]]: {{val| +0.663 +0.553 +1.190 -0.007 -5.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~11/10 = 157.6099{{c}}, ~7/4 = 967.8187{{c}}
: error map: {{val| 0.000 +0.030 +0.451 -1.007 -6.943 }}


{{Optimal ET sequence|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
{{Optimal ET sequence|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}


[[Badness]] (Smith): 0.0345 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.629


== Valinorsmic (176/175) ==
== Valinorsmic (176/175) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~5/4 = 388.882{{c}}, ~7/4 = 970.703{{c}}
* [[WE]]: ~2 = 1199.3113{{c}}, ~3/2 = 702.6414{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.587{{c}}, ~5/4 = 389.393{{c}}, ~7/4 = 971.520{{c}}
: [[error map]]: {{val| -0.689 -0.022 +1.849 +1.350 -2.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5869{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
: error map: {{val| 0.000 +0.632 +3.080 +2.694 -1.011 }}


{{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
{{Optimal ET sequence|legend=1| 22, 27e, 31, 46, 53, 58, 80, 111, 237cd, 268cd }}


[[Badness]] (Smith): 0.0186 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.339


=== Tridecimal valinorsmic ===
=== Tridecimal valinorsmic ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.304{{c}}, ~5/4 = 388.738{{c}}, ~7/4 = 970.598{{c}}
* WE: ~2 = 1199.3397{{c}}, ~3/2 = 702.5360{{c}}, ~5/4 = 389.5451{{c}}, ~7/4 = 971.5415{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.694{{c}}, ~5/4 = 389.382{{c}}, ~7/4 = 971.530{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6943{{c}}, ~5/4 = 389.3823{{c}}, ~7/4 = 971.5305{{c}}


{{Optimal ET sequence|legend=0| }}
{{Optimal ET sequence|legend=0| 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 268cd }}


Badness (Smith):
Badness (Sintel): 0.603
 
{{Todo|complete temperament data|inline=1}}


== Rastmic (243/242) ==
== Rastmic (243/242) ==
Rastmic halves 3/2 into two 11/9s, 32/27 into two 12/11s, and 25/24 into two 45/44~55/54s.
Rastmic halves [[3/2]] into two [[11/9]]'s, [[32/27]] into two [[12/11]]'s, and [[25/24]] into two [[45/44]][[~]][[55/54]]'s.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.572{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 968.826{{c}}
* [[WE]]: ~2 = 1200.0635{{c}}, ~11/9 = 350.5439{{c}}, ~5/4 = 386.1857{{c}}, ~7/4 = 968.6977{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.549{{c}}, ~5/4 = 386.239{{c}}, ~7/4 = 968.735{{c}}
: [[error map]]: {{val| +0.064 -0.804 -0.001 -0.001 +1.529 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5486{{c}}, ~5/4 = 386.2387{{c}}, ~7/4 = 968.7352{{c}}
: error map: {{val| 0.000 -0.858 -0.075 -0.091 +1.425 }}


{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}


[[Badness]] (Smith): 0.0509 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.928


== Frostmic (245/242) ==
== Frostmic (245/242) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~11/7 = 792.479{{c}}, ~7/4 = 964.861{{c}}
* [[WE]]: ~2 = 1200.2442{{c}}, ~3/2 = 701.7038{{c}}, ~11/7 = 792.3594{{c}}, ~7/4 = 964.4060{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.753{{c}}, ~11/7 = 792.335{{c}}, ~7/4 = 964.527{{c}}
: [[error map]]: {{val| +0.244 -0.007 -1.351 -3.931 +5.936 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7533{{c}}, ~11/7 = 792.3351{{c}}, ~7/4 = 964.5274{{c}}
: error map: {{val| 0.000 -0.202 -1.644 -4.298 +5.545 }}


{{Optimal ET sequence|legend=1| 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d }}
{{Optimal ET sequence|legend=1| 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d }}


[[Badness]] (Smith): 0.0897 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.64


== Akua (352/351, 847/845) ==
== Akua (352/351, 847/845) ==
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