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Werckismic temperaments: Difference between revisions

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This is a collection of rank-3 temperaments tempering out [[441/440]]
{{Technical data page}}
This is a collection of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out]] the [[werckisma]] ([[ratio]]: 441/440, {{monzo|legend=1| -3 2 -1 2 -1 }}). For the rank-4 werckismic temperament, see [[Catalog of rank-4 temperaments #Werckismic (441/440)]].


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Festival]]'' (+45/44 or 50/49) → [[Jubilismic family #Festival|Jubilismic family]]
* ''[[Festival]]'' (+45/44 or 50/49) → [[Jubilismic family #Festival|Jubilismic family]]
* ''[[Vulcan]]'' (+56/55 or 64/63) → [[Archytas family #Vulcan|Archytas family]]
* ''[[Vulcan]]'' (+56/55 or 64/63) → [[Archytas family #Vulcan|Archytas family]]
* ''[[Euterpe]]'' (+81/80 or 99/98) → [[Didymus rank three family #Euterpe|Didymus rank-3 family]]
* ''[[Euterpe]]'' (+81/80 or 99/98) → [[Didymus rank-3 family #Euterpe|Didymus rank-3 family]]
* ''[[Nakika]]'' (+100/99 or 245/242) → [[Octagar family #Nakika|Octagar family]]
* ''[[Nakika]]'' (+100/99 or 245/242) → [[Octagar family #Nakika|Octagar family]]
* ''[[Aphrodite]]'' (+121/120) → [[Biyatismic clan #Aphrodite|Biyatismic clan]]
* ''[[Aphrodite]]'' (+121/120) → [[Biyatismic clan #Aphrodite|Biyatismic clan]]
Line 14: Line 15:
* ''[[Demeter]]'' (+686/675) → [[Sengic family #Demeter|Sengic family]]
* ''[[Demeter]]'' (+686/675) → [[Sengic family #Demeter|Sengic family]]
* ''[[Pele]]'' (+896/891) → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[Pele]]'' (+896/891) → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[Tiberius]]'' (+2048/2025) → [[Diaschismic rank three family #Tiberius|Diaschismic rank-3 family]]
* ''[[Tiberius]]'' (+2048/2025) → [[Diaschismic rank-3 family #Tiberius|Diaschismic rank-3 family]]
* ''[[Belobog]]'' (+3136/3125) → [[Hemimean family #Belobog|Hemimean family]]
* ''[[Belobog]]'' (+3136/3125) → [[Hemimean family #Belobog|Hemimean family]]
* ''[[Landy]]'' (+4375/4356) → [[Landscape family #Landy|Landscape family]]
* ''[[Turan]]'' (+10976/10935) → [[Hemimage family #Turan|Hemimage family]]
* ''[[Turan]]'' (+10976/10935) → [[Hemimage family #Turan|Hemimage family]]
* ''[[Snape]]'' (+19683/19600) → [[Cataharry family #Snape|Cataharry family]]
* ''[[Snape]]'' (+19683/19600) → [[Cataharry family #Snape|Cataharry family]]


Considered below are history, varuna, and trimyna, but we can begin by looking at the rank-4 temperament.  
Considered below are history, varuna, and trimyna.  


== Werckismic ==
== History ==
[[Subgroup]]: 2.3.5.7.11
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #History]].''
 
[[Comma list]]: [[441/440]]
 
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 0 0 2 | 0 0 1 0 -1 | 0 0 0 1 2 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/2 2/3 1/6 -1/3 1/6 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 1 -2/3 1/3 1/3 1/3 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7.11


{{Optimal ET sequence|legend=1| 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}
History, named by [[Graham Breed]] some time before 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96550.html#96556 Yahoo! Tuning Group | ''Madagascar<nowiki>[</nowiki>19<nowiki>]</nowiki>'']</ref>, splits the perfect fourth into six, each for [[21/20]][[~]][[22/21]]. A recommendable tuning is given by [[289edo]].  
 
== History ==
History splits the perfect fourth into six, each for [[21/20]]~[[22/21]]. A recommendable tuning is given by [[289edo]].  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 43: Line 33:


{{Mapping|legend=1| 1 2 0 0 1 | 0 -6 0 7 2 | 0 0 1 1 1 }}
{{Mapping|legend=1| 1 2 0 0 1 | 0 -6 0 7 2 | 0 0 1 1 1 }}
: mapping generators: ~2, ~21/20, ~5


: mapping generators: ~2, ~21/20, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0906{{c}}, ~21/20 = 83.0793{{c}}, ~5/4 = 386.1759{{c}}
: [[error map]]: {{val| +0.091 -0.250 +0.043 -0.914 +1.288 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0654{{c}}, ~5/4 = 386.2710{{c}}
: error map: {{val| 0.000 -0.348 -0.043 -1.097 +1.084 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7.11
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11


{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }}


[[Badness]]: 0.604 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.725


=== 13-limit ===
=== 13-limit ===
Line 62: Line 57:
Mapping: {{mapping| 1 2 0 0 1 2 | 0 -6 0 7 2 -9 | 0 0 1 1 1 1 }}
Mapping: {{mapping| 1 2 0 0 1 2 | 0 -6 0 7 2 -9 | 0 0 1 1 1 1 }}


: mapping generators: ~2, ~21/20, ~5
Optimal tunings:  
* WE: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}, ~5/4 = 386.6582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}, ~5/4 = 386.6802{{c}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit
* 13-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 0 0 -3/8 0 3/8 }}, {{monzo| 0 0 1 0 0 0 }}, {{monzo| 7/8 0 1 7/16 0 -7/16 }}, {{monzo| 5/4 0 1 1/8 0 -1/8 }}, {{monzo| 7/8 0 1 -9/16 0 9/16 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 0 0 -3/8 0 3/8 }}, {{monzo| 0 0 1 0 0 0 }}, {{monzo| 7/8 0 1 7/16 0 -7/16 }}, {{monzo| 5/4 0 1 1/8 0 -1/8 }}, {{monzo| 7/8 0 1 -9/16 0 9/16 }}]
: eigenmonzo (unchanged-interval) basis: 2.5.13/7
: unchanged-interval (eigenmonzo) basis: 2.5.13/7


{{Optimal ET sequence|legend=1| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}


Badness: 0.540 × 10<sup>-3</sup>
Badness (Sintel): 0.505


Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7


== Varuna ==
== Varuna ==
=== 7-limit ===
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Varuna]].''
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 321489/320000
 
{{Mapping|legend=1| 2 0 0 9 | 0 1 0 -4 | 0 0 1 2 }}
 
: mapping generators: ~567/400, ~3, ~5
 
{{Optimal ET sequence|legend=1| 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd }}
 
[[Badness]]: 0.501 × 10<sup>-3</sup>


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 95: Line 80:


{{Mapping|legend=1| 2 0 0 9 12 | 0 1 0 -4 -6 | 0 0 1 2 3 }}
{{Mapping|legend=1| 2 0 0 9 12 | 0 1 0 -4 -6 | 0 0 1 2 3 }}
: mapping generators: ~99/70, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.1002{{c}}, ~3/2 = 701.3812{{c}}, ~5/4 = 386.3126{{c}}
: [[error map]]: {{val| +0.200 -0.373 +0.400 -0.823 +0.536 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.3457{{c}}, ~5/4 = 386.5123{{c}}
: error map: {{val| 0.000 -0.609 +0.199 -1.184 +0.144 }}


{{Optimal ET sequence|legend=1| 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320 }}
{{Optimal ET sequence|legend=1| 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320 }}


[[Badness]]: 0.419 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.503


=== 13-limit ===
=== 13-limit ===
Line 107: Line 99:
Mapping: {{mapping| 2 0 0 9 12 11 | 0 1 0 -4 -6 -7 | 0 0 1 2 3 4 }}
Mapping: {{mapping| 2 0 0 9 12 11 | 0 1 0 -4 -6 -7 | 0 0 1 2 3 4 }}


{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 46, 58, 72, 130 }}
Optimal tunings:
* WE: ~55/39 = 600.0052{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 387.1861{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.4418{{c}}, ~5/4 = 387.1949{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 14cf, 26, 46, 58, 72, 130 }}


Badness: 0.636 × 10<sup>-3</sup>
Badness (Sintel): 0.595


== Trimyna ==
== Trimyna ==
=== 7-limit ===
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Trimyna]].''
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50421/50000
 
{{Mapping|legend=1| 1 4 0 0 | 0 -5 0 1 | 0 0 1 1 }}
 
: mapping generators: ~2, ~7/5, ~5


{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} <nowiki>*</nowiki>
Trimyna can be described as the {{nowrap| 27e & 29 & 31 }} temperament. All three equal temperaments have a fairly accurate tritone for ~10/7, and trimyna takes it as one of its generators, five of which get us to the [[6/1|6th]] [[harmonic]]. The name ''trimyna'' is a contraction of ''[[tritonic]]'' and ''[[myna]]''.


<nowiki>*</nowiki> [[optimal patent val]]: [[294edo|294]]
[[Badness]]: 0.870 × 10<sup>-3</sup>
[[Projection pair]]: 3 50000/16807 to 2.5.7
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 3388/3375
[[Comma list]]: 441/440, 3388/3375


{{Mapping|legend=1| 1 4 0 0 5 | 0 -5 0 1 -8 | 0 0 1 1 1 }}
{{Mapping|legend=1| 1 -1 0 1 -3 | 0 5 0 -1 8 | 0 0 1 1 1 }}
: mapping generators: ~2, ~10/7, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0909{{c}}, ~10/7 = 620.4267{{c}}, ~5/4 = 387.4364{{c}}
: [[error map]]: {{val| +0.091 +0.088 +1.304 -1.544 -0.559 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3901{{c}}, ~5/4 = 387.5065{{c}}
: error map: {{val| 0.000 -0.004 +1.193 -1.710 -0.691 }}


{{Optimal ET sequence|legend=1| 27e, 31, 58, 87, 118, 294, 412d }}
{{Optimal ET sequence|legend=1| 27e, 31, 58, 87, 118, 294, 412d }}


[[Badness]]: 0.512 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.711


==== 13-limit ====
==== 13-limit ====
Line 145: Line 134:
Comma list: 196/195, 352/351, 1001/1000
Comma list: 196/195, 352/351, 1001/1000


Mapping: {{mapping| 1 4 0 0 5 -2 | 0 -5 0 1 -8 7 | 0 0 1 1 1 1 }}
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 5 0 -1 8 -7 | 0 0 1 1 1 1 }}


{{Optimal ET sequence|legend=1| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}
Optimal tunings:
* WE: ~2 = 1199.6799{{c}}, ~10/7 = 620.4057{{c}}, ~5/4 = 387.7593{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 620.5510{{c}}, ~5/4 = 387.5132{{c}}


Badness: 0.814 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}
 
Badness (Sintel): 0.761


==== Trimy ====
==== Trimy ====
Line 156: Line 149:
Comma list: 441/440, 676/675, 847/845
Comma list: 441/440, 676/675, 847/845


Mapping: {{mapping| 1 4 0 0 5 5 | 0 -10 0 2 -16 -15 | 0 0 1 1 1 1 }}
Mapping: {{mapping| 1 -6 0 2 -11 -10 | 0 10 0 -2 16 15 | 0 0 1 1 1 1 }}
: mapping generators: ~2, ~22/13, ~5


: mapping generators: ~2, ~13/11, ~5
Optimal tunings:  
* WE: ~2 = 1200.0924{{c}}, ~22/13 = 910.2591{{c}}, ~5/4 = 387.4264{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 910.1939{{c}}, ~5/4 = 387.4946{{c}}


{{Optimal ET sequence|legend=1| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}
{{Optimal ET sequence|legend=0| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}


Badness: 1.478 × 10<sup>-3</sup>
Badness (Sintel): 1.38


== Seminaiad ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Seminaiad]].''
Named by [[User:Xenllium|Xenllium]] in 2026, seminaiad is a weak extension of [[No-threes subgroup temperaments #No-threes naiad|no-threes naiad]]. It splits ~13/5 in two to represent ~21/13.
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 441/440, 1127357/1125000
{{Mapping|legend=1| 1 5 0 -2 3 | 0 -11 0 8 -6 | 0 0 1 1 1 }}
: mapping generators: ~2, ~150/121, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.203{{c}}, ~150/121 = 372.665{{c}}, ~5/4 = 386.499{{c}}
: [[error map]]: {{val| +0.203 -0.257 +0.592 -1.004 +0.206 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~150/121 = 372.591{{c}}, ~5/4 = 386.702{{c}}
: error map: {{val| 0.000 -0.453 +0.388 -1.399 -0.161 }}
{{Optimal ET sequence|legend=1| 16, 29, 45e, 58, 87, 103, 145, 161, 190 }}
[[Badness]] (Sintel): 2.582
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 847/845, 1001/1000
Mapping: {{mapping| 1 5 0 -2 3 2 | 0 -11 0 8 -6 -2 | 0 0 1 1 1 1 }}
Optimal tunings:
* WE: ~2 = 1200.161{{c}}, ~26/21 = 372.652{{c}}, ~5/4 = 386.381{{c}}
* CWE: ~2 = 1200.000{{c}}, ~26/21 = 372.593{{c}}, ~5/4 = 386.546{{c}}
{{Optimal ET sequence|legend=0| 16, 29, 45ef, 58, 87, 103, 145, 161, 190 }}
Badness (Sintel): 0.944
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 221/220, 441/440, 847/845
Mapping: {{mapping| 1 5 0 -2 3 2 3 | 0 -11 0 8 -6 -2 -4 | 0 0 1 1 1 1 1 }}
Optimal tunings:
* WE: ~2 = 1200.375{{c}}, ~21/17 = 372.711{{c}}, ~5/4 = 386.747{{c}}
* CWE: ~2 = 1200.000{{c}}, ~21/17 = 372.573{{c}}, ~5/4 = 387.138{{c}}
{{Optimal ET sequence|legend=0| 16, 29g, 45efg, 58, 87g }}
Badness (Sintel): 1.748
==== Seminaiadia ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 833/832, 847/845, 1001/1000
Mapping: {{mapping| 1 5 0 -2 3 2 12 | 0 -11 0 8 -6 -2 -18 | 0 0 1 1 1 1 -1 }}
Optimal tunings:
* WE: ~2 = 1199.925{{c}}, ~26/21 = 372.566{{c}}, ~5/4 = 387.117{{c}}
* CWE: ~2 = 1200.000{{c}}, ~26/21 = 372.594{{c}}, ~5/4 = 387.080{{c}}
{{Optimal ET sequence|legend=0| 16, 29, 45ef, 58, 87, 103, 145, 161, 190 }}
Badness (Sintel): 1.284
== References ==
[[Category:Werckismic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Werckismic temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
[[Category:Varuna]]
[[Category:Trimyna]]
Retrieved from "https://en.xen.wiki/w/Werckismic_temperaments"