Didymus rank-3 family: Difference between revisions

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.

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>.

* ''[[Urania]]'' (+243/242 or 121/120) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80

{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}

[[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 }}

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

[[Badness]] (Sintel): 0.419

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

[[Subgroup]]: 2.3.5.7.11

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

{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}

* [[WE]]: ~2 = 1201.2743{{c}}, ~3/2 = 696.9375{{c}}, ~7/4 = 969.4564{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.5797{{c}}, ~7/4 = 969.7942{{c}}

: [{{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 }}]

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

[[Badness]] (Sintel): 0.644

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

[[Subgroup]]: 2.3.5.7.11

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

 

{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}

 

[[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 }}

 

* [[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 basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7

 

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

 

[[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 tempers out 385/384. It is another temperament related to meanpop.

 

[[Subgroup]]: 2.3.5.7.11

 

[[Comma list]]: 81/80, 385/384

 

{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}

 

* [[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 }}

 

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

 

[[Badness]] (Sintel): 1.07

 

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]]:

: [{{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 ==

[[Subgroup]]: 2.3.5.7.11

 

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

 

{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }}

 

[[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 }}

 

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

 

[[Badness]] (Sintel): 0.898

 

== Melpomene ==

[[Subgroup]]: 2.3.5.7.11

 

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

 

{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }}

 

[[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 }}

 

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

 

 

== 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, 126/125

 

 

* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}, ~11/8 = 547.5886{{c}}

 

 

 

 

 

 

 

 

 

 

[[Category:Didymus| ]] <!-- key article -->