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Didymus rank-3 family: Difference between revisions

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The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth.
{{Technical data page}}
The '''didymus rank-3 family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] rank-3 temperament, with an [[11-limit]] comma we get an 11-limit temperament, and so forth.


== Didymus ==
Most of these temperaments were named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100647.html#100648 Yahoo! Tuning Group | ''Higher-rank meantone extensions (muse temperaments?)'']</ref>.
Subgroup: 2.3.5.7
 
Temperaments discussed elsewhere include:
* ''[[Urania]]'' (+243/242 or 121/120) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
 
== Didymus ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80
[[Comma list]]: 81/80


[[Mapping]]: [{{val| 1 0 -4 0 }}, {{val| 0 1 4 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}


[[POTE generator]]s: ~3/2 = 696.2387, ~7/4 = 964.9090
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}, ~7/4 = 966.0272{{c}}
: [[error map]]: {{val| +1.391 -3.519 +1.868 -0.018 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}, ~7/4 = 966.5654{{c}}
: error map: {{val| 0.000 -5.304 +0.291 -2.261 }}


{{Val list|legend=1| 12, 19, 31, 81 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Badness]]: 0.095 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.419


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


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 81/80, 99/98
[[Comma list]]: 81/80, 99/98


[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}]
{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}


[[POTE generator]]s: ~3/2 = 696.1982, ~7/4 = 968.4280
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2743{{c}}, ~3/2 = 696.9375{{c}}, ~7/4 = 969.4564{{c}}
: [[error map]]: {{val| +1.274 -3.743 +1.436 +3.179 -2.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.5797{{c}}, ~7/4 = 969.7942{{c}}
: error map: {{val| 0.000 -5.375 +0.005 +0.968 -4.889 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5, 7
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7


{{Val list|legend=1| 12, 17c, 19e, 26, 31, 88 }}
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88, 119be, 150be }}


[[Badness]]: 0.536 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.644


== Calliope  ==
== Clio ==
Clio tempers out 176/175. It is another temperament related to huygens.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 45/44, 81/80
[[Comma list]]: 81/80, 176/175


[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}


[[POTE generator]]s: ~3/2 = 696.1982, ~7/4 = 968.4280
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6376{{c}}, ~3/2 = 697.6207{{c}}, ~7/4 = 969.1442{{c}}
: [[error map]]: {{val| +0.638 -3.697 +4.169 +1.593 -2.483 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3350{{c}}, ~7/4 = 969.0152{{c}}
: error map: {{val| 0.000 -4.620 +3.026 +0.189 -3.622 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[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 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/3, 11
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7


{{Val list|legend=1| 7d, 12, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129, 136b }}


[[Badness]]: 0.530 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.886


== Erato  ==
== Terpsichore ==
Terpsichore tempers out 540/539 and is related to [[meanpop]].


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}


[[POTE generator]]s: ~3/2 = 696.4949, ~11/8 = 547.0252
[[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 }}


{{Val list|legend=1| 12, 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=1| 12e, 14c, 17c, 19, 31, 81, 112b }}


[[Badness]]: 0.558 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.02


=== 13-limit  ===
[[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


Subgroup: 2.3.5.7.11.13
== Polyhymnia ==
Polyhymnia tempers out 385/384. It is another temperament related to meanpop.  


Comma list: 81/80, 105/104, 126/125
[[Subgroup]]: 2.3.5.7.11


Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}]
[[Comma list]]: 81/80, 385/384


POTE generators: ~3/2 = 695.9883, ~11/8 = 545.6817
{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}


{{Val list|legend=1| 12f, 19, 31, 50, 81 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4003{{c}}, ~3/2 = 697.0429{{c}}, ~7/4 = 965.9954{{c}}
: [[error map]]: {{val| +1.400 -3.512 +1.858 -0.030 -0.040 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.3248{{c}}, ~7/4 = 964.8389{{c}}
: error map: {{val| 0.000 -5.630 -1.014 -3.987 -5.131 }}


== Clio  ==
{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31, 81, 112b }}


Subgroup: 2.3.5.7.11
[[Badness]] (Sintel): 1.07


[[Comma list]]: 81/80, 176/175
Scales: [[polyhymnia12]]


Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}]
== Calliope ==
Calliope tempers out 45/44 and is related to [[flattone]].


[[POTE generator]]s: ~3/2 = 697.2502, ~7/4 = 968.6295
[[Subgroup]]: 2.3.5.7.11


[[Minimax tuning]]:  
[[Comma list]]: 45/44, 81/80
* [[11-odd-limit]]
: {{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5, 7


{{Val list|legend=1| 7, 12, 19e, 24, 31, 105, 129 }}
{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}


[[Badness]]: 0.738 × 10<sup>-3</sup>
[[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 }}


== Polyhymnia  ==
[[Minimax tuning]]:  
 
* [[11-odd-limit]]
Subgroup: 2.3.5.7.11
: [{{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
[[Comma list]]: 81/80, 385/384
 
[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}]
 
[[POTE generator]]s: ~3/2 = 696.2305, ~7/4 = 964.8695


{{Val list|legend=1| 7, 12e, 19, 24, 26, 31 }}
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}


== Thalia  ==
[[Badness]] (Sintel): 0.637


Subgroup: 2.3.5.7.11
== Thalia ==
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 33/32, 55/54
[[Comma list]]: 33/32, 55/54


[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }}


[[POTE generator]]s: ~3/2 = 692.0796, ~7/4 = 950.2565
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}, ~7/4 = 955.4695{{c}}
: [[error map]]: {{val| +6.583 +0.504 -2.809 -0.190 -20.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.1206{{c}}, ~7/4 = 955.1393{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -13.687 -44.439 }}


{{Val list|legend=1| 5, 7, 12e, 14c, 19e }}
{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e, 33cdee, 52cdeee }}


== Melpomene  ==
[[Badness]] (Sintel): 0.898


Subgroup: 2.3.5.7.11
== Melpomene ==
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 81/80, 56/55
[[Comma list]]: 56/55, 81/80


[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}]
{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }}


[[POTE generator]]s: ~3/2 = 699.2230, ~7/4 = 964.2363
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.7942{{c}}, ~3/2 = 698.5204{{c}}, ~7/4 = 963.2674{{c}}
: [[error map]]: {{val| -1.206 -4.640 +7.768 -7.970 +11.839 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.9964{{c}}, ~7/4 = 962.5723{{c}}
: error map: {{val| 0.000 -2.959 +9.672 -6.254 +15.269 }}


{{Val list|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}
{{Optimal ET sequence|legend=1| 12, 17c, 19, 24, 31e, 36 }}


== Urania  ==
[[Badness]] (Sintel): 0.929


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


[[Comma list]]: 81/80, 121/120
[[Subgroup]]: 2.3.5.7.11


[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}]
[[Comma list]]: 81/80, 126/125


[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}


Lattice basis:  
[[Optimal tuning]]s:  
: 11/9 length = 0.2536, 8/7 length = 2.807
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}, ~11/8 = 547.5886{{c}}
: Angle (11/9, 8/7) = 90 degrees
: [[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 }}


[[POTE generator]]s: ~11/9 = 348.0938, ~7/4 = 963.6042
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81, 112b }}


{{Val list|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
[[Badness]] (Sintel): 0.670


[[Badness]]: 0.842 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
Comma list: 81/80, 105/104, 126/125
 
Scales: [[urania24]]
 
== Terpsichore  ==
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 81/80, 540/539


[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}]
Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}


[[POTE generator]]s: ~3/2 = 696.2358, ~7/4 = 964.0006
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}}


{{Val list|legend=1| 14c, 17c, 19, 31, 81, 112b }}
{{Optimal ET sequence|legend=0| 12f, 19, 31, 50, 69de, 81, 119bdde }}


[[Badness]]: 0.850 × 10<sup>-3</sup>
Badness (Sintel): 0.823


[[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
== References ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Didymus]]
[[Category:Pages with mostly numerical content]]
[[Category:Didymus rank-3 family| ]] <!-- main article -->
[[Category:Didymus| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Didymus_rank-3_family"