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Didymus rank-3 family: Difference between revisions

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== Euterpe ==
== Euterpe ==
Euterpe tempers out 99/98 and is related to [[huygens]].
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 46: Line 48:
[[Badness]] (Sintel): 0.644
[[Badness]] (Sintel): 0.644


== Calliope ==
== Clio ==
[[Subgroup]]: 2.3.5.7.11
Clio tempers out 176/175. It is another temperament related to huygens.  
 
[[Comma list]]: 45/44, 81/80
 
{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}, ~7/4 = 964.5866{{c}}
: [[error map]]: {{val| +2.062 -5.348 -8.135 -0.115 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}, ~7/4 = 965.3621{{c}}
: error map: {{val| 0.000 -8.047 -10.680 -3.464 +12.133 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
: [[eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3.11


{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
[[Badness]] (Sintel): 0.637
== Erato ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 81/80, 126/125
{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}, ~11/8 = 547.5886{{c}}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 -0.022 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}, ~11/8 = 548.5396{{c}}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 -2.778 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81, 112b }}
[[Badness]] (Sintel): 0.670
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125
Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}
Optimal tunings:
* WE: ~2 = 1201.6213{{c}}, ~3/2 = 696.9287{{c}}, ~11/8 = 546.4190{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1573{{c}}, ~11/8 = 547.5865{{c}}
{{Optimal ET sequence|legend=0| 12f, 19, 31, 50, 69de, 81, 119bdde }}
Badness (Sintel): 0.823
== Clio ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 121: Line 71:


[[Badness]] (Sintel): 0.886
[[Badness]] (Sintel): 0.886
== Terpsichore ==
Terpsichore tempers out 540/539 and is related to [[meanpop]].
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 81/80, 540/539
{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5055{{c}}, ~3/2 = 697.1093{{c}}, ~7/4 = 965.2101{{c}}
: [[error map]]: {{val| +1.506 -3.340 +2.123 -0.605 -0.468 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.7166{{c}}, ~7/4 = 964.9479{{c}}
: error map: {{val| 0.000 -5.238 +0.553 -3.878 -4.197 }}
{{Optimal ET sequence|legend=1| 12e, 14c, 17c, 19, 31, 81, 112b }}
[[Badness]] (Sintel): 1.02
[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11


== Polyhymnia ==
== Polyhymnia ==
Polyhymnia tempers out 385/38. It is another temperament related to meanpop.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 140: Line 113:


Scales: [[polyhymnia12]]
Scales: [[polyhymnia12]]
== Calliope ==
Calliope tempers out 45/44 and is related to [[flattone]].
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 45/44, 81/80
{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}, ~7/4 = 964.5866{{c}}
: [[error map]]: {{val| +2.062 -5.348 -8.135 -0.115 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}, ~7/4 = 965.3621{{c}}
: error map: {{val| 0.000 -8.047 -10.680 -3.464 +12.133 }}
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
: [[eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3.11
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
[[Badness]] (Sintel): 0.637


== Thalia ==
== Thalia ==
Line 175: Line 172:
[[Badness]] (Sintel): 0.929
[[Badness]] (Sintel): 0.929


== Terpsichore ==
== Erato ==
Erato uses the same mapping for the 7-limit as septimal meantone, but adds an independent generator for prime 11.
 
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 81/80, 540/539
[[Comma list]]: 81/80, 126/125


{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.5055{{c}}, ~3/2 = 697.1093{{c}}, ~7/4 = 965.2101{{c}}
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}, ~11/8 = 547.5886{{c}}
: [[error map]]: {{val| +1.506 -3.340 +2.123 -0.605 -0.468 }}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 -0.022 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.7166{{c}}, ~7/4 = 964.9479{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}, ~11/8 = 548.5396{{c}}
: error map: {{val| 0.000 -5.238 +0.553 -3.878 -4.197 }}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 -2.778 }}
 
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81, 112b }}
 
[[Badness]] (Sintel): 0.670
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 105/104, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}


{{Optimal ET sequence|legend=1| 12e, 14c, 17c, 19, 31, 81, 112b }}
Optimal tunings:
* WE: ~2 = 1201.6213{{c}}, ~3/2 = 696.9287{{c}}, ~11/8 = 546.4190{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1573{{c}}, ~11/8 = 547.5865{{c}}


[[Badness]] (Sintel): 1.02
{{Optimal ET sequence|legend=0| 12f, 19, 31, 50, 69de, 81, 119bdde }}


[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
Badness (Sintel): 0.823


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Didymus_rank-3_family"