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Swetismic temperaments: Difference between revisions

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This is a collection of rank-3 temperaments tempering out [[540/539]].  
{{Technical data page}}
This is a collection of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out]] the [[swetisma]] ([[ratio]]: 540/539, {{monzo|legend=1| 2 3 1 -2 -1 }}). For the rank-4 swetismic temperament, see [[Catalog of rank-4 temperaments #Swetismic (540/539)]].  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Terpsichore]]'' (+81/80) → [[Didymus rank three family #Terpsichore|Didymus rank-3 family]]
* ''[[Terpsichore]]'' (+81/80) → [[Didymus rank-3 family #Terpsichore|Didymus rank-3 family]]
* ''[[Big brother]]'' (+99/98 or 121/120) → [[Nuwell family #Big brother|Nuwell family]]
* ''[[Big brother]]'' (+99/98 or 121/120) → [[Biyatismic clan #Big brother|Biyatismic clan]]
* ''[[Octarod]]'' (+100/99 or 245/243) → [[Sensamagic family #Octarod|Sensamagic family]]
* ''[[Octarod]]'' (+100/99 or 245/243) → [[Sensamagic family #Octarod|Sensamagic family]]
* ''[[Aplonis]]'' (+126/125) → [[Starling family #Aplonis|Starling family]]
* ''[[Aplonis]]'' (+126/125) → [[Starling family #Aplonis|Starling family]]
Line 11: Line 12:
* ''[[Uni]]'' (+896/891) → [[Hemimage family #Uni|Hemimage family]]
* ''[[Uni]]'' (+896/891) → [[Hemimage family #Uni|Hemimage family]]
* ''[[Gamel]]'' (+1029/1024) → [[Gamelismic family #Gamel|Gamelismic family]]
* ''[[Gamel]]'' (+1029/1024) → [[Gamelismic family #Gamel|Gamelismic family]]
* ''[[Indra]]'' (+1375/1372) → [[Mirkwai family #Indra|Mirkwai family]]
* ''[[Claudius]]'' (+2048/2025) → [[Diaschismic rank-3 family #Claudius|Diaschismic rank-3 family]]
* ''[[Claudius]]'' (+2048/2025) → [[Diaschismic rank three family #Claudius|Diaschismic rank-3 family]]
* ''[[Octasand]]'' (+2200/2187) → [[Octagar family #Octasand|Octagar family]]
* ''[[Gariboh]]'' (+3125/3087) → [[Gariboh family #Undecimal gariboh|Gariboh family]]
* ''[[Gariboh]]'' (+3125/3087) → [[Gariboh family #Undecimal gariboh|Gariboh family]]
* ''[[Siebog]]'' (+3136/3125) → [[Hemimean family #Siebog|Hemimean family]]
* ''[[Siebog]]'' (+3136/3125) → [[Hemimean family #Siebog|Hemimean family]]
Line 19: Line 20:
* ''[[Albus]]'' (+8019/8000) → [[Cataharry family #Albus|Cataharry family]]
* ''[[Albus]]'' (+8019/8000) → [[Cataharry family #Albus|Cataharry family]]


Considered below are hades, minos, persephone and hypnos, but we can begin by looking at the rank-4 temperament.  
Considered below are indra, hades, minos, persephone and hypnos, in the order of increasing [[badness]].
 
== Indra ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Canopic]].''


== Swetismic ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539
[[Comma list]]: 540/539, 1375/1372
 
{{Mapping|legend=1| 1 0 -5 -4 5 | 0 1 3 3 0 | 0 0 5 4 -3 }}
: Mapping generators: ~2, ~3, ~10/7
 
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]
 
Lattice basis:
: 15/14 length = 0.6184, 55/36 length = 0.7585
: Angle (15/14, 55/36) = 68.1048 deg
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9693{{c}}, ~3/2 = 701.7444{{c}}, ~10/7 = 616.1152{{c}}
: [[error map]]: {{val| -0.031 -0.241 -0.443 +0.899 +0.183 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7342{{c}}, ~10/7 = 616.1422{{c}}
: error map: {{val| 0.000 -0.221 -0.400 +0.946 +0.256 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10 10 -3 0 -5 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 6 -1 0 -3 }}, {{monzo| 20 18 -6 0 -9 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.11/9
 
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}
 
[[Badness]] (Sintel): 0.320
 
[[Projection pair]]s: <code>5 84375/16807 7 16875/2401 11 1372/125</code> to 2.3.7/5
 
Scales: [[indra31]]
 
=== Shibi ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372
 
Mapping: {{mapping| 1 0 -5 -4 5 1 | 0 1 3 3 0 3 | 0 0 5 4 -3 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.9762{{c}}, ~3/2 = 701.7232{{c}}, ~10/7 = 616.1317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7261{{c}}, ~10/7 = 616.1467{{c}}
 
{{Optimal ET sequence|legend=0| 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d }}
 
Badness (Sintel): 0.674
 
=== Yajna ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 1375/1372
 
Mapping: {{mapping| 1 0 -5 -4 5 -24 | 0 1 3 3 0 11 | 0 0 5 4 -3 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.0073{{c}}, ~3/2 = 701.7067{{c}}, ~10/7 = 616.1052{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7089{{c}}, ~10/7 = 616.0986{{c}}
 
Minimax tuning:
* 13-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/5 6/5 -4/5 0 0 1/5 }}, {{monzo| 8/5 2/5 -3/5 0 0 2/5 }}, {{monzo| 44/25 26/25 -24/25 0 0 11/25 }}, {{monzo| 62/25 48/25 -27/25 0 0 3/25 }}, {{monzo| 8/5 2/5 -8/5 0 0 7/5 }}]
: eigenmonzo (unchanged-interval) basis: 2.9/5.13/5
* 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 5/4 1/2 0 -5/16 0 5/16 }}, {{monzo| 1 1 0 -1/4 0 1/4 }}, {{monzo| 5/4 3/2 0 3/16 0 -3/16 }}, {{monzo| 1 1 0 -5/4 0 5/4 }}]
: eigenmonzo (unchanged-interval) basis: 2.3.13/7
 
{{Optimal ET sequence|legend=0| 31, 49f, 72, 103, 121, 152f, 193, 224 }}
 
Badness (Sintel): 0.655


{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 3 | 0 0 0 1 -2 }}
=== Vibhu ===
Subgroup: 2.3.5.7.11.13


: mapping generators: ~2, ~3, ~5, ~7
Comma list: 351/350, 540/539, 1375/1372


{{Optimal ET sequence|legend=1| 9, 10, 19, 22, 31, 41, 53, 58, 72, 130, 152, 224, 354, 537, 578 }}
Mapping: {{mapping| 1 0 -5 -4 5 -13 | 0 1 3 3 0 6 | 0 0 5 4 -3 14 }}


== Hades ==
Optimal tunings:
=== 7-limit (stearnsmic) ===
* WE: ~2 = 1200.1587{{c}}, ~3/2 = 701.2394{{c}}, ~10/7 = 616.6507{{c}}
[[Subgroup]]: 2.3.5.7
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2689{{c}}, ~10/7 = 616.5306{{c}}
 
{{Optimal ET sequence|legend=0| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}
 
Badness (Sintel): 0.772
 
=== Vish ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 118098/117649
Comma list: 325/324, 364/363, 540/539


{{Mapping|legend=1| 2 1 0 2 | 0 3 0 5 | 0 0 1 0 }}
Mapping: {{mapping| 1 0 0 0 2 2 | 0 1 3 3 0 -2 | 0 0 5 4 -3 -10 }}


: mapping generators: ~343/243, ~9/7, ~5
Optimal tunings:  
* WE: ~2 = 1199.9463{{c}}, ~3/2 = 702.3912{{c}}, ~10/7 = 615.5733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3742{{c}}, ~10/7 = 615.6198{{c}}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}
{{Optimal ET sequence|legend=0| 31f, 41, 72, 121, 193 }}


[[Badness]]: 0.748 × 10<sup>-3</sup>
Badness (Sintel): 0.870
 
== Hades ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Stearnsmic]].''


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


Line 52: Line 133:


{{Mapping|legend=1| 2 1 0 2 3 | 0 3 0 5 -1 | 0 0 1 0 1 }}
{{Mapping|legend=1| 2 1 0 2 3 | 0 3 0 5 -1 | 0 0 1 0 1 }}
: mapping generators: ~99/70, ~9/7, ~5
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9486{{c}}, ~9/7 = 433.9084{{c}}, ~5/4 = 386.2290{{c}}
: [[error map]]: {{val| -0.103 -0.281 -0.290 +0.613 +0.643 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9204{{c}}, ~5/4 = 386.1272{{c}}
: error map: {{val| 0.000 -0.194 -0.187 +0.776 +0.889 }}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee }}
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee }}


[[Badness]]: 0.361 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.434


=== 13-limit ===
=== 13-limit ===
Line 64: Line 152:
Mapping: {{mapping| 2 1 0 2 3 -2 | 0 3 0 5 -1 13 | 0 0 1 0 1 0 }}
Mapping: {{mapping| 2 1 0 2 3 -2 | 0 3 0 5 -1 13 | 0 0 1 0 1 0 }}


{{Optimal ET sequence|legend=1| 22f, 36, 58, 72, 130, 224, 354, 578 }}
Optimal tunings:
* WE: ~99/70 = 599.9598{{c}}, ~9/7 = 433.8891{{c}}, ~5/4 = 386.1680{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9059{{c}}, ~5/4 = 386.1014{{c}}


Badness: 0.535 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 22f, 36ce, 58, 72, 130, 224, 354, 578 }}
 
Badness (Sintel): 0.501


=== Cerberus ===
=== Cerberus ===
Line 75: Line 167:
Mapping: {{mapping| 2 1 0 2 3 1 | 0 3 0 5 -1 -4 | 0 0 1 0 1 2 }}
Mapping: {{mapping| 2 1 0 2 3 1 | 0 3 0 5 -1 -4 | 0 0 1 0 1 2 }}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130 }}
Optimal tunings:
* WE: ~55/39 = 599.9565{{c}}, ~9/7 = 433.8372{{c}}, ~5/4 = 387.1905{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 433.8475{{c}}, ~5/4 = 387.1030{{c}}


Badness: 0.766 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 8d, 14cf, 22, 36f, 44ceff, 50, 58, 72, 130 }}
 
Badness (Sintel): 0.716


== Minos ==
== Minos ==
Line 84: Line 180:
[[Comma list]]: 540/539, 12005/11979
[[Comma list]]: 540/539, 12005/11979


{{Mapping|legend=1| 1 0 2 1 2 | 0 2 4 3 4 | 0 0 -5 -1 -3 }}
{{Mapping|legend=1| 1 0 -3 0 -1 | 0 2 4 3 4 | 0 0 5 1 3 }}
: mapping generators: ~2, ~121/70, ~66/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8923{{c}}, ~121/70 = 950.8327{{c}}, ~66/49 = 516.4492{{c}}
: [[error map]]: {{val| -0.108 -0.290 -0.414 +0.121 +1.468 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~121/70 = 950.8712{{c}}, ~66/49 = 516.5067{{c}}
: error map: {{val| 0.000 -0.213 -0.295 +0.294 +1.687 }}


: mapping generators: ~2, ~121/70, ~49/33
{{Optimal ET sequence|legend=1| 9, …, 49, 58, 72, 130, 193, 202, 323e, 395e, 525e }} *


{{Optimal ET sequence|legend=1| 9, 49, 58, 72, 130, 193, 202, 467 }}
<nowiki>*</nowiki> [[optimal patent val]]: [[467edo|467]]


[[Badness]]: 0.886 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.06


=== 13-limit ===
=== 13-limit ===
Line 97: Line 200:
Comma list: 364/363, 540/539, 676/675
Comma list: 364/363, 540/539, 676/675


Mapping: {{mapping| 1 0 2 1 2 1 | 0 2 4 3 4 7 | 0 0 -5 -1 -3 -5 }}
Mapping: {{mapping| 1 0 -3 0 -1 -4 | 0 2 4 3 4 7 | 0 0 5 1 3 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.7669{{c}}, ~26/15 = 950.9360{{c}}, ~66/49 = 516.3652{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0324{{c}}, ~66/49 = 516.4883{{c}}


{{Optimal ET sequence|legend=1| 9, 54, 58, 72, 121, 130, 193 }}
{{Optimal ET sequence|legend=0| 9, …, 49f, 58, 72, 121, 130, 193, 251e, 323e }}


Badness: 0.724 × 10<sup>-3</sup>
Badness (Sintel): 0.677


== Persephone ==
== Persephone ==
Line 108: Line 215:
[[Comma list]]: 540/539, 88209/87500
[[Comma list]]: 540/539, 88209/87500


{{Mapping|legend=1| 1 0 4 -2 10 | 0 1 4 0 7 | 0 0 -5 3 -11 }}
{{Mapping|legend=1| 1 0 -6 4 -12 | 0 1 4 0 7 | 0 0 5 -3 11 }}
: mapping generators: ~2, ~3, ~33/25


: mapping generators: ~2, ~3, ~100/33
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0185{{c}}, ~3/2 = 700.9179{{c}}, ~33/25 = 476.7538{{c}}
: [[error map]]: {{val| +0.019 -1.019 +1.090 +0.987 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.9220{{c}}, ~33/25 = 476.7393{{c}}
: error map: {{val| 0.000 -1.033 +1.071 +0.956 -0.732 }}


{{Optimal ET sequence|legend=1| 50, 53, 58, 103, 108, 111, 161, 214, 272, 375, 1022bcde, 1236bcde, 1397bcde }}
{{Optimal ET sequence|legend=1| 8d, …, 45e, 50, 53, 58, 103, 161, 214, 272, 375 }}


[[Badness]]: 2.015 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.42


=== 13-limit ===
=== 13-limit ===
Line 121: Line 233:
Comma list: 351/350, 540/539, 847/845
Comma list: 351/350, 540/539, 847/845


Mapping: {{mapping| 1 0 4 -2 10 7 | 0 1 4 0 7 5 | 0 0 -5 3 -11 -7 }}
Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 1 4 0 7 5 | 0 0 5 -3 11 7 }}


{{Optimal ET sequence|legend=1| 50, 53, 58, 103, 108, 111, 161, 214, 375, 700cdef, 861bcdef, 1075bcdef, 1236bcdef, 1450bcdef }}
Optimal tunings:
* WE: ~2 = 1199.9553{{c}}, ~3/2 = 700.9477{{c}}, ~33/25 = 476.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9378{{c}}, ~33/25 = 476.7054{{c}}


Badness: 0.997 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 8d, …, 45ef, 50, 53, 58, 103, 111, 161, 214, 375 }}
 
Badness (Sintel): 0.933


== Hypnos ==
== Hypnos ==
Line 133: Line 249:


{{Mapping|legend=1| 1 0 0 -3 8 | 0 3 0 11 -13 | 0 0 1 0 1 }}
{{Mapping|legend=1| 1 0 0 -3 8 | 0 3 0 11 -13 | 0 0 1 0 1 }}
: mapping generators: ~2, ~81/56, ~5


: mapping generators: ~2, ~81/56, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3393{{c}}, ~81/56 = 633.6837{{c}}, ~5/4 = 385.6892{{c}}
: [[error map]]: {{val| +0.339 -0.904 +0.054 +0.677 -0.124 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 633.5228{{c}}, ~5/4 = 385.9359{{c}}
: error map: {{val| 0.000 -1.387 -0.378 -0.075 -1.178 }}


{{Optimal ET sequence|legend=1| 19, 36, 53, 72, 161, 197e, 214, 233, 286, 358, 591be, 663be, 949bce }}
{{Optimal ET sequence|legend=1| 17c, 19, …, 36, 53, 72, 161, 214, 233, 286, 358, 591be }}


[[Badness]]: 1.629 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.96


=== 13-limit ===
=== 13-limit ===
Line 147: Line 268:
Mapping: {{mapping| 1 0 0 -3 8 0 | 0 3 0 11 -13 7 | 0 0 1 0 1 0 }}
Mapping: {{mapping| 1 0 0 -3 8 0 | 0 3 0 11 -13 7 | 0 0 1 0 1 0 }}


{{Optimal ET sequence|legend=1| 19, 36, 53, 72, 125f, 197ef }}
Optimal tunings:
* WE: ~2 = 1200.6306{{c}}, ~81/56 = 633.8505{{c}}, ~5/4 = 385.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~81/56 = 633.5546{{c}}, ~5/4 = 385.4668{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 19, 36ce, 53, 72, 125f, 197ef }}


Badness: 1.439 × 10<sup>-3</sup>
Badness (Sintel): 1.35


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Swetismic temperaments| ]] <!-- main article -->
[[Category:Swetismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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