Varunismic temperaments: Difference between revisions

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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Diaschismic]]'' (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Bamity]]'' (+245/243) → [[Amity family #Bamity|Amity family]]
* ''[[Bamity]]'' (+245/243) → [[Amity family #Bamity|Amity family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]]
* ''[[Decile]]'' (+235298/234375) → [[Quintile family #Decile|Quintile family]]
* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]]
* ''[[Bison]]'' (+6144/6125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Bison]]'' (+6144/6125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
* ''[[Narayana]]'' (+65625/65536) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
* ''[[Narayana]]'' (+65625/65536) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
* ''[[Decal]]'' (+235298/234375) → [[Pental family #Decal|Pental family]]
* ''[[Bosonic]]'' (+589824/588245) → [[26th-octave temperaments #Bosonic|26th-octave temperaments]]
* ''[[Bosonic]]'' (+589824/588245) → [[26th-octave temperaments #Bosonic|26th-octave temperaments]]


Considered below are octowerck and essence, in the order of increasing [[badness]].  
Considered below are octowerck and essence, in the order of increasing [[badness]].  
== Essence ==
Essence is of interest because of the abundant supply of [[essentially tempered chord]]s. Specifically, [[werckismic chords]], [[squbemic chords]], [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[nicolic chords]] in the [[15-odd-limit]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 321489/320000, 823543/820125
{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}
: mapping generators: ~567/400, ~243/196
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~243/196 = 372.597{{c}} (~343/300 = 227.403{{c}})
{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}
[[Badness]] (Smith): 0.173565
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 8019/8000, 456533/455625
Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~150/121 = 372.599{{c}} (~154/135 = 227.401{{c}})
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
Badness (Smith): 0.046447
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 847/845, 1001/1000
Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~26/21 = 372.602{{c}} (~154/135 = 227.398{{c}})
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
Badness (Smith): 0.026107


== Octowerck ==
== Octowerck ==
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Badness (Smith): 0.030192
Badness (Smith): 0.030192
== Essence ==
Essence is of interest because of the abundant supply of [[essentially tempered chord]]s. Specifically, [[werckismic chords]], [[squbemic chords]], [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[nicolic chords]] in the [[15-odd-limit]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 321489/320000, 823543/820125
{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}
: mapping generators: ~567/400, ~243/196
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~243/196 = 372.597{{c}} (~343/300 = 227.403{{c}})
{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}
[[Badness]] (Smith): 0.173565
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 8019/8000, 456533/455625
Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~150/121 = 372.599{{c}} (~154/135 = 227.401{{c}})
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
Badness (Smith): 0.046447
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 847/845, 1001/1000
Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~26/21 = 372.602{{c}} (~154/135 = 227.398{{c}})
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
Badness (Smith): 0.026107


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Varunismic temperaments| ]] <!-- main article -->
[[Category:Varunismic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 04:04, 21 July 2026

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-2 varunismic temperaments, which temper out the varunisma (monzo: [-9 8 -4 2⟩, ratio: 321489/320000). For the rank-3 temperament tempering out 321489/320000 in the 7-limit, see Miscellaneous 7-limit temperaments #Varuna.

Temperaments discussed elsewhere are:

Considered below are octowerck and essence, in the order of increasing badness.

Octowerck

Subgroup: 2.3.5.7

Comma list: 321489/320000, 420175/419904

Mapping: [⟨8 0 -11 14], ⟨0 3 7 2]]

mapping generators: ~160/147, ~350/243

Optimal tuning (POTE): ~160/147 = 150.000 ¢, ~350/243 = 633.738 ¢ (~49/48 = 33.738 ¢)

Optimal ET sequence: 32c, 72, 176, 248, 320

Badness (Smith): 0.102295

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 41503/41472

Mapping: [⟨8 0 -11 14 15], ⟨0 3 7 2 3]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~121/84 = 633.755 ¢ (~49/48 = 33.755 ¢)

Optimal ET sequence: 32c, 72, 176, 248, 320

Badness (Smith): 0.030159

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 6656/6655

Mapping: [⟨8 0 -11 14 15 -38], ⟨0 3 7 2 3 16]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~75/52 = 633.742 ¢ (~49/48 = 33.742 ¢)

Optimal ET sequence: 72, 176f, 248, 320

Badness (Smith): 0.027632

Octowerckis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 2197/2187

Mapping: [⟨8 0 -11 14 15 0], ⟨0 3 7 2 3 7]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~13/9 = 633.8370 ¢ (~49/48 = 33.8370 ¢)

Optimal ET sequence: 32cf, 72, 176, 248f

Badness (Smith): 0.030192

Essence

Essence is of interest because of the abundant supply of essentially tempered chords. Specifically, werckismic chords, squbemic chords, cuthbert chords, sinbadmic chords, and petrmic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.

Subgroup: 2.3.5.7

Comma list: 321489/320000, 823543/820125

Mapping: [⟨2 10 27 23], ⟨0 -11 -36 -28]]

mapping generators: ~567/400, ~243/196

Optimal tuning (POTE): ~567/400 = 600.000 ¢, ~243/196 = 372.597 ¢ (~343/300 = 227.403 ¢)

Optimal ET sequence: 58, 132d, 190, 248, 438d, 686d

Badness (Smith): 0.173565

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 456533/455625

Mapping: [⟨2 10 27 23 33], ⟨0 -11 -36 -28 -42]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~150/121 = 372.599 ¢ (~154/135 = 227.401 ¢)

Optimal ET sequence: 58, 132de, 190, 248, 438d

Badness (Smith): 0.046447

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 847/845, 1001/1000

Mapping: [⟨2 10 27 23 33 31], ⟨0 -11 -36 -28 -42 -38]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~26/21 = 372.602 ¢ (~154/135 = 227.398 ¢)

Optimal ET sequence: 58, 132de, 190, 248, 438d

Badness (Smith): 0.026107

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