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__FORCETOC__
{{Technical data page}}
These are rank three temperaments tempering out [[540/539]], but we can begin by looking at the rank four temperament.
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)]].  


Comma: 540/539
Temperaments discussed elsewhere are:  
* ''[[Terpsichore]]'' (+81/80) → [[Didymus rank-3 family #Terpsichore|Didymus rank-3 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]]
* ''[[Aplonis]]'' (+126/125) → [[Starling family #Aplonis|Starling family]]
* ''[[Guanyin]]'' (+176/175) → [[Orwellismic family #Guanyin|Orwellismic family]]
* [[Marvel]] (+225/224 or 385/384) → [[Marvel family #Undecimal marvel|Marvel family]]
* [[Jove]] (+243/242 or 441/440) → [[Breed family #Jove|Breed family]]
* ''[[Uni]]'' (+896/891) → [[Hemimage family #Uni|Hemimage family]]
* ''[[Gamel]]'' (+1029/1024) → [[Gamelismic family #Gamel|Gamelismic family]]
* ''[[Claudius]]'' (+2048/2025) → [[Diaschismic rank-3 family #Claudius|Diaschismic rank-3 family]]
* ''[[Octasand]]'' (+2200/2187) → [[Octagar family #Octasand|Octagar family]]
* ''[[Gariboh]]'' (+3125/3087) → [[Gariboh family #Undecimal gariboh|Gariboh family]]
* ''[[Siebog]]'' (+3136/3125) → [[Hemimean family #Siebog|Hemimean family]]
* ''[[Laka]]'' (+5120/5103) → [[Hemifamity family #Laka|Hemifamity family]]
* ''[[Jupiter]]'' (+5632/5625) → [[Porwell family #Jupiter|Porwell family]]
* ''[[Albus]]'' (+8019/8000) → [[Cataharry family #Albus|Cataharry family]]


Map: [<1 0 0 0 2|, <0 1 0 0 3|, <0 0 0 1 -2|]
Considered below are indra, hades, minos, persephone and hypnos, in the order of increasing [[badness]].


Generators: 2, 3, 5, 7
== Indra ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Canopic]].''


EDOs: 9, 10, 19, 22, 31, 41, 53, 58, 72, 130, 152, 224, 354, 537, 578
[[Subgroup]]: 2.3.5.7.11


=Hades=
[[Comma list]]: 540/539, 1375/1372
Commas: 540/539, 4000/3993


Map: [<2 1 0 2 3|, <0 3 0 5 -1|, <0 0 1 0 1|]
{{Mapping|legend=1| 1 0 -5 -4 5 | 0 1 3 3 0 | 0 0 5 4 -3 }}
: Mapping generators: ~2, ~3, ~10/7


Generators: 99/70, 9/7, 5
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]


EDOs: 22, 50, 58, 72, 130, 152, 224, 354, 578
Lattice basis:  
: 15/14 length = 0.6184, 55/36 length = 0.7585
: Angle (15/14, 55/36) = 68.1048 deg


Badness: 0.000361
[[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 }}


==13-limit==
[[Minimax tuning]]:
Commas: 540/539, 729/728, 1575/1573
* [[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


Map: [<2 1 0 2 3 -2 |, <0 3 0 5 -1 13|, <0 0 1 0 1 0|]
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}


Generators: 99/70, 9/7, 5
[[Badness]] (Sintel): 0.320


EDOs: 36, 58, 72, 130, 224, 354, 578
[[Projection pair]]s: <code>5 84375/16807 7 16875/2401 11 1372/125</code> to 2.3.7/5


Badness: 0.000535
Scales: [[indra31]]


==Cerberus==
=== Shibi ===
Commas: 351/350, 364/363, 540/539
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 0 2 3 1|, &lt;0 3 0 5 -1 -4|, &lt;0 0 1 0 1 2|]
Comma list: 540/539, 729/728, 1375/1372


Generators: 55/39.9/7.5
Mapping: {{mapping| 1 0 -5 -4 5 1 | 0 1 3 3 0 3 | 0 0 5 4 -3 -4 }}


EDOs: 22, 50, 58, 72, 130
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}}


Badness: 0.000766
{{Optimal ET sequence|legend=0| 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d }}


=Minos=
Badness (Sintel): 0.674
Commas: 540/539, 12005/11979


Map: [&lt;1 0 2 1 2|, &lt;0 2 4 3 4|, &lt;0 0 -5 -1 -3|]
=== Yajna ===
Subgroup: 2.3.5.7.11.13


Generators: 2.121/70.49/33
Comma list: 540/539, 625/624, 1375/1372


EDOs: 9, 49, 58, 72, 130, 193, 202, 467
Mapping: {{mapping| 1 0 -5 -4 5 -24 | 0 1 3 3 0 11 | 0 0 5 4 -3 20 }}


Badness: 0.000886
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}}


==13-limit==
Minimax tuning:
Commas: 364/363, 540/539, 676/675
* 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


Map: [&lt;1 0 2 1 2 1|, &lt;0 2 4 3 4 7|, &lt;0 0 -5 -1 -3 -5|]
{{Optimal ET sequence|legend=0| 31, 49f, 72, 103, 121, 152f, 193, 224 }}


Generators: 2.26/15.49/33
Badness (Sintel): 0.655


EDOs: 9, 54, 58, 72, 121, 130, 193
=== Vibhu ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000724
Comma list: 351/350, 540/539, 1375/1372


=Persephone=
Mapping: {{mapping| 1 0 -5 -4 5 -13 | 0 1 3 3 0 6 | 0 0 5 4 -3 14 }}
Commas: 540/539, 88209/87500


Map: [&lt;1 0 4 -2 10|, &lt;0 1 4 0 7|, &lt;0 0 -5 3 -11|]
Optimal tunings:  
* WE: ~2 = 1200.1587{{c}}, ~3/2 = 701.2394{{c}}, ~10/7 = 616.6507{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2689{{c}}, ~10/7 = 616.5306{{c}}


EDOs: 50, 53, 58, 103, 108, 111, 161, 214, 272, 375, 1022bcde, 1236bcde, 1397bcde
{{Optimal ET sequence|legend=0| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}


Badness: 0.002015
Badness (Sintel): 0.772


==13-limit==
=== Vish ===
Commas: 351/350, 540/539, 847/845
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 4 -2 10 7|, &lt;0 1 4 0 7 5|, &lt;0 0 -5 3 -11 -7|]
Comma list: 325/324, 364/363, 540/539


EDOs: 50, 53, 58, 103, 108, 111, 161, 214, 375, 700cdef, 861bcdef, 1075bcdef, 1236bcdef, 1450bcdef
Mapping: {{mapping| 1 0 0 0 2 2 | 0 1 3 3 0 -2 | 0 0 5 4 -3 -10 }}


Badness: 0.000997
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}}


=Hypnos=
{{Optimal ET sequence|legend=0| 31f, 41, 72, 121, 193 }}
Commas: 540/539, 72171/71680


Map: [&lt;1 0 0 -3 8|, &lt;0 3 0 11 -13|, &lt;0 0 1 0 1|]
Badness (Sintel): 0.870


EDOs: 19, 36, 53, 72, 161, 197e, 214, 233, 286, 358, 591be, 663be, 949bce
== Hades ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Stearnsmic]].''


Badness: 0.001629
[[Subgroup]]: 2.3.5.7.11


==13-limit==
[[Comma list]]: 540/539, 4000/3993
Commas: 169/168, 540/539, 729/728


Map: [&lt;1 0 0 -3 8 0|, &lt;0 3 0 11 -13 7|, &lt;0 0 1 0 1 0|]
{{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


EDOs: 19, 36, 53, 72, 125f, 197ef
[[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 }}


Badness: 0.001439
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee }}


[[Category:Swetismic]]
[[Badness]] (Sintel): 0.434
[[Category:Temperament]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573
 
Mapping: {{mapping| 2 1 0 2 3 -2 | 0 3 0 5 -1 13 | 0 0 1 0 1 0 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 22f, 36ce, 58, 72, 130, 224, 354, 578 }}
 
Badness (Sintel): 0.501
 
=== Cerberus ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 540/539
 
Mapping: {{mapping| 2 1 0 2 3 1 | 0 3 0 5 -1 -4 | 0 0 1 0 1 2 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 8d, 14cf, 22, 36f, 44ceff, 50, 58, 72, 130 }}
 
Badness (Sintel): 0.716
 
== Minos ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 12005/11979
 
{{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 }}
 
{{Optimal ET sequence|legend=1| 9, …, 49, 58, 72, 130, 193, 202, 323e, 395e, 525e }} *
 
<nowiki>*</nowiki> [[optimal patent val]]: [[467edo|467]]
 
[[Badness]] (Sintel): 1.06
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 540/539, 676/675
 
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=0| 9, …, 49f, 58, 72, 121, 130, 193, 251e, 323e }}
 
Badness (Sintel): 0.677
 
== Persephone ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 88209/87500
 
{{Mapping|legend=1| 1 0 -6 4 -12 | 0 1 4 0 7 | 0 0 5 -3 11 }}
: mapping generators: ~2, ~3, ~33/25
 
[[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| 8d, …, 45e, 50, 53, 58, 103, 161, 214, 272, 375 }}
 
[[Badness]] (Sintel): 2.42
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845
 
Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 1 4 0 7 5 | 0 0 5 -3 11 7 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 8d, …, 45ef, 50, 53, 58, 103, 111, 161, 214, 375 }}
 
Badness (Sintel): 0.933
 
== Hypnos ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 72171/71680
 
{{Mapping|legend=1| 1 0 0 -3 8 | 0 3 0 11 -13 | 0 0 1 0 1 }}
: 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| 17c, 19, …, 36, 53, 72, 161, 214, 233, 286, 358, 591be }}
 
[[Badness]] (Sintel): 1.96
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 540/539, 729/728
 
Mapping: {{mapping| 1 0 0 -3 8 0 | 0 3 0 11 -13 7 | 0 0 1 0 1 0 }}
 
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 (Sintel): 1.35
 
[[Category:Temperament collections]]
[[Category:Swetismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
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