Swetismic temperaments: Difference between revisions

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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]], 540/539 = {{monzo| 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 three family #Terpsichore|Didymus rank-3 family]]
* ''[[Big brother]]'' (+99/98 or 121/120) → [[Nuwell family #Big brother|Nuwell family]]
* ''[[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]]
* ''[[Indra]]'' (+1375/1372) → [[Mirkwai family #Indra|Mirkwai 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]]
* ''[[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 hades, minos, persephone and hypnos.


Generators: 2, 3, 5, 7
== Hades ==
=== 7-limit (stearnsmic) ===
[[Subgroup]]: 2.3.5.7


EDOs: 9, 10, 19, 22, 31, 41, 53, 58, 72, 130, 152, 224, 354, 537, 578
[[Comma list]]: 118098/117649


== Hades  ==
{{Mapping|legend=1| 2 1 0 2 | 0 3 0 5 | 0 0 1 0 }}
Commas: 540/539, 4000/3993


Map: [<2 1 0 2 3|, <0 3 0 5 -1|, <0 0 1 0 1|]
: mapping generators: ~343/243, ~9/7, ~5


Generators: 99/70, 9/7, 5
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}


EDOs: 22, 50, 58, 72, 130, 152, 224, 354, 578
[[Badness]]: 0.748 × 10<sup>-3</sup>


Badness: 0.000361
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11


=== 13-limit  ===
[[Comma list]]: 540/539, 4000/3993
Commas: 540/539, 729/728, 1575/1573


Map: [&lt;2 1 0 2 3 -2 |, &lt;0 3 0 5 -1 13|, &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 }}


Generators: 99/70, 9/7, 5
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee }}


EDOs: 36, 58, 72, 130, 224, 354, 578
[[Badness]]: 0.361 × 10<sup>-3</sup>


Badness: 0.000535
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== Cerberus  ===
Comma list: 540/539, 729/728, 1575/1573
Commas: 351/350, 364/363, 540/539


Map: [&lt;2 1 0 2 3 1|, &lt;0 3 0 5 -1 -4|, &lt;0 0 1 0 1 2|]
Mapping: {{mapping| 2 1 0 2 3 -2 | 0 3 0 5 -1 13 | 0 0 1 0 1 0 }}


Generators: 55/39.9/7.5
{{Optimal ET sequence|legend=1| 22f, 36, 58, 72, 130, 224, 354, 578 }}


EDOs: 22, 50, 58, 72, 130
Badness: 0.535 × 10<sup>-3</sup>


Badness: 0.000766
=== Cerberus ===
Subgroup: 2.3.5.7.11.13


== Minos  ==
Comma list: 351/350, 364/363, 540/539
Commas: 540/539, 12005/11979


Map: [&lt;1 0 2 1 2|, &lt;0 2 4 3 4|, &lt;0 0 -5 -1 -3|]
Mapping: {{mapping| 2 1 0 2 3 1 | 0 3 0 5 -1 -4 | 0 0 1 0 1 2 }}


Generators: 2.121/70.49/33
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130 }}


EDOs: 9, 49, 58, 72, 130, 193, 202, 467
Badness: 0.766 × 10<sup>-3</sup>


Badness: 0.000886
== Minos ==
[[Subgroup]]: 2.3.5.7.11


=== 13-limit  ===
[[Comma list]]: 540/539, 12005/11979
Commas: 364/363, 540/539, 676/675


Map: [&lt;1 0 2 1 2 1|, &lt;0 2 4 3 4 7|, &lt;0 0 -5 -1 -3 -5|]
{{Mapping|legend=1| 1 0 2 1 2 | 0 2 4 3 4 | 0 0 -5 -1 -3 }}


Generators: 2.26/15.49/33
: mapping generators: ~2, ~121/70, ~49/33


EDOs: 9, 54, 58, 72, 121, 130, 193
{{Optimal ET sequence|legend=1| 9, 49, 58, 72, 130, 193, 202, 467 }}


Badness: 0.000724
[[Badness]]: 0.886 × 10<sup>-3</sup>


== Persephone  ==
=== 13-limit ===
Commas: 540/539, 88209/87500
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 4 -2 10|, &lt;0 1 4 0 7|, &lt;0 0 -5 3 -11|]
Comma list: 364/363, 540/539, 676/675


EDOs: 50, 53, 58, 103, 108, 111, 161, 214, 272, 375, 1022bcde, 1236bcde, 1397bcde
Mapping: {{mapping| 1 0 2 1 2 1 | 0 2 4 3 4 7 | 0 0 -5 -1 -3 -5 }}


Badness: 0.002015
{{Optimal ET sequence|legend=1| 9, 54, 58, 72, 121, 130, 193 }}


=== 13-limit  ===
Badness: 0.724 × 10<sup>-3</sup>
Commas: 351/350, 540/539, 847/845


Map: [&lt;1 0 4 -2 10 7|, &lt;0 1 4 0 7 5|, &lt;0 0 -5 3 -11 -7|]
== Persephone ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 50, 53, 58, 103, 108, 111, 161, 214, 375, 700cdef, 861bcdef, 1075bcdef, 1236bcdef, 1450bcdef
[[Comma list]]: 540/539, 88209/87500


Badness: 0.000997
{{Mapping|legend=1| 1 0 4 -2 10 | 0 1 4 0 7 | 0 0 -5 3 -11 }}


== Hypnos  ==
: mapping generators: ~2, ~3, ~100/33
Commas: 540/539, 72171/71680


Map: [&lt;1 0 0 -3 8|, &lt;0 3 0 11 -13|, &lt;0 0 1 0 1|]
{{Optimal ET sequence|legend=1| 50, 53, 58, 103, 108, 111, 161, 214, 272, 375, 1022bcde, 1236bcde, 1397bcde }}


EDOs: 19, 36, 53, 72, 161, 197e, 214, 233, 286, 358, 591be, 663be, 949bce
[[Badness]]: 2.015 × 10<sup>-3</sup>


Badness: 0.001629
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 351/350, 540/539, 847/845
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: {{mapping| 1 0 4 -2 10 7 | 0 1 4 0 7 5 | 0 0 -5 3 -11 -7 }}


EDOs: 19, 36, 53, 72, 125f, 197ef
{{Optimal ET sequence|legend=1| 50, 53, 58, 103, 108, 111, 161, 214, 375, 700cdef, 861bcdef, 1075bcdef, 1236bcdef, 1450bcdef }}


Badness: 0.001439
Badness: 0.997 × 10<sup>-3</sup>


[[Category:Theory]]
== Hypnos ==
[[Category:Temperament]]
[[Subgroup]]: 2.3.5.7.11
[[Category:Swetismic]]
 
[[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 ET sequence|legend=1| 19, 36, 53, 72, 161, 197e, 214, 233, 286, 358, 591be, 663be, 949bce }}
 
[[Badness]]: 1.629 × 10<sup>-3</sup>
 
=== 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 ET sequence|legend=1| 19, 36, 53, 72, 125f, 197ef }}
 
Badness: 1.439 × 10<sup>-3</sup>
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Swetismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:28, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-3 temperaments tempering out the swetisma, 540/539 = [2 3 1 -2 -1⟩. For the rank-4 swetismic temperament, see Catalog of rank-4 temperaments #Swetismic (540/539).

Temperaments discussed elsewhere are:

Considered below are hades, minos, persephone and hypnos.

Hades

7-limit (stearnsmic)

Subgroup: 2.3.5.7

Comma list: 118098/117649

Mapping: [⟨2 1 0 2], ⟨0 3 0 5], ⟨0 0 1 0]]

mapping generators: ~343/243, ~9/7, ~5

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 202, 224, 354

Badness: 0.748 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993

Mapping: [⟨2 1 0 2 3], ⟨0 3 0 5 -1], ⟨0 0 1 0 1]]

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee

Badness: 0.361 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573

Mapping: [⟨2 1 0 2 3 -2], ⟨0 3 0 5 -1 13], ⟨0 0 1 0 1 0]]

Optimal ET sequence: 22f, 36, 58, 72, 130, 224, 354, 578

Badness: 0.535 × 10-3

Cerberus

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539

Mapping: [⟨2 1 0 2 3 1], ⟨0 3 0 5 -1 -4], ⟨0 0 1 0 1 2]]

Optimal ET sequence: 22, 50, 58, 72, 130

Badness: 0.766 × 10-3

Minos

Subgroup: 2.3.5.7.11

Comma list: 540/539, 12005/11979

Mapping: [⟨1 0 2 1 2], ⟨0 2 4 3 4], ⟨0 0 -5 -1 -3]]

mapping generators: ~2, ~121/70, ~49/33

Optimal ET sequence: 9, 49, 58, 72, 130, 193, 202, 467

Badness: 0.886 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 540/539, 676/675

Mapping: [⟨1 0 2 1 2 1], ⟨0 2 4 3 4 7], ⟨0 0 -5 -1 -3 -5]]

Optimal ET sequence: 9, 54, 58, 72, 121, 130, 193

Badness: 0.724 × 10-3

Persephone

Subgroup: 2.3.5.7.11

Comma list: 540/539, 88209/87500

Mapping: [⟨1 0 4 -2 10], ⟨0 1 4 0 7], ⟨0 0 -5 3 -11]]

mapping generators: ~2, ~3, ~100/33

Optimal ET sequence: 50, 53, 58, 103, 108, 111, 161, 214, 272, 375, 1022bcde, 1236bcde, 1397bcde

Badness: 2.015 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845

Mapping: [⟨1 0 4 -2 10 7], ⟨0 1 4 0 7 5], ⟨0 0 -5 3 -11 -7]]

Optimal ET sequence: 50, 53, 58, 103, 108, 111, 161, 214, 375, 700cdef, 861bcdef, 1075bcdef, 1236bcdef, 1450bcdef

Badness: 0.997 × 10-3

Hypnos

Subgroup: 2.3.5.7.11

Comma list: 540/539, 72171/71680

Mapping: [⟨1 0 0 -3 8], ⟨0 3 0 11 -13], ⟨0 0 1 0 1]]

mapping generators: ~2, ~81/56, ~5

Optimal ET sequence: 19, 36, 53, 72, 161, 197e, 214, 233, 286, 358, 591be, 663be, 949bce

Badness: 1.629 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 540/539, 729/728

Mapping: [⟨1 0 0 -3 8 0], ⟨0 3 0 11 -13 7], ⟨0 0 1 0 1 0]]

Optimal ET sequence: 19, 36, 53, 72, 125f, 197ef

Badness: 1.439 × 10-3

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