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]]
* ''[[Fifives]]'' (+875/864) → [[Fifive family #Fifives|Fifive family]]
* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]]
* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Decile]]'' (+235298/234375) → [[Quintile family #Decile|Quintile 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) → [[Vishnu family #Narayana|Vishnu 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]]


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


== Octowerck ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 321489/320000, 823543/820125
[[Comma list]]: 321489/320000, 420175/419904


{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}
{{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}
: mapping generators: ~567/400, ~243/196
: mapping generators: ~160/147, ~350/243


[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~243/196 = 372.597{{c}} (~343/300 = 227.403{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~160/147 = 150.0317{{c}}, ~350/243 = 633.8714{{c}} (~49/48 = 33.7447{{c}})
: [[error map]]: {{val| +0.253 -0.341 +0.438 -0.640 }}
* [[CWE]]: ~160/147 = 150.0000{{c}}, ~350/243 = 633.7791{{c}} (~49/48 = 33.7791{{c}})
: error map: {{val| 0.000 -0.618 +0.140 -1.268 }}


{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}
{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}


[[Badness]] (Smith): 0.173565
[[Badness]] (Sintel): 2.59


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 8019/8000, 456533/455625
Comma list: 441/440, 8019/8000, 41503/41472
 
Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0233{{c}}, ~121/84 = 633.8533{{c}} (~49/48 = 33.7602{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~121/84 = 633.7841{{c}} (~49/48 = 33.7841{{c}})
 
{{Optimal ET sequence|legend=0| 32c, 72, 176, 248, 320 }}
 
Badness (Sintel): 0.997
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 729/728, 1001/1000, 6656/6655


Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}
Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~150/121 = 372.599{{c}} (~154/135 = 227.401{{c}})
Optimal tunings:
* WE: ~12/11 = 150.0247{{c}}, ~75/52 = 633.8469{{c}} (~49/48 = 33.7480{{c}})
* CWE: ~12/11 = 150.000{{c}}, ~75/52 = 633.7616{{c}} (~49/48 = 33.7616{{c}})


{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
{{Optimal ET sequence|legend=0| 72, 176f, 248, 320 }}


Badness (Smith): 0.046447
Badness (Sintel): 1.14


=== 13-limit ===
==== Octowerckis ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 729/728, 847/845, 1001/1000
Comma list: 351/350, 364/363, 441/440, 2197/2187
 
Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}


Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}
Optimal tunings:  
* WE: ~12/11 = 150.0289{{c}}, ~13/9 = 633.9590{{c}} (~49/48 = 33.8435{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~13/9 = 633.8739{{c}} (~49/48 = 33.8739{{c}})


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~26/21 = 372.602{{c}} (~154/135 = 227.398{{c}})
{{Optimal ET sequence|legend=0| 32cf, 72, 176, 248f }}


{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
Badness (Sintel): 1.25


Badness (Smith): 0.026107
== Bosonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''


== Octowerck ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 321489/320000, 420175/419904
[[Comma list]]: 321489/320000, 589824/588245


{{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}
{{Mapping|legend=1| 26 0 -22 73 | 0 1 2 0 }}
: mapping generators: ~160/147, ~350/243
: mapping generators: ~36/35, ~3


[[Optimal tuning]] ([[POTE]]): ~160/147 = 150.000{{c}}, ~350/243 = 633.738{{c}} (~49/48 = 33.738{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 46.1534{{c}}, ~3/2 = 701.2433{{c}}
: [[error map]]: {{val| -0.012 -0.723 +0.775 +0.372 }}
* [[CWE]]: ~36/35 = 46.1538{{c}}, ~3/2 = 701.2451{{c}}
: error map: {{val| 0.000 -0.710 +0.792 +0.405 }}


{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=1| 26, …, 104c, 130, 546, 676, 806c }}


[[Badness]] (Smith): 0.102295
[[Badness]] (Sintel): 3.94


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 8019/8000, 41503/41472
Comma list: 441/440, 8019/8000, 65536/65219
 
Mapping: {{mapping| 26 0 -22 73 90 | 0 1 2 0 0 }}
 
Optimal tunings:
* WE: ~36/35 = 46.1443{{c}}, ~3/2 = 701.4143{{c}}
* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.4791{{c}}
 
{{Optimal ET sequence|legend=0| 26, …, 104c, 130 }}
 
Badness (Sintel): 2.16
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440, 15379/15360
 
Mapping: {{mapping| 26 0 -22 73 90 55 | 0 1 2 0 0 1 }}
 
Optimal tunings:
* WE: ~36/35 = 46.1469{{c}}, ~3/2 = 701.4410{{c}}
* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.4873{{c}}
 
{{Optimal ET sequence|legend=0| 26, …, 104c, 130 }}
 
Badness (Sintel): 1.36
 
=== Fermionic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 78408/78125, 177147/176000


Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}
Mapping: {{mapping| 26 0 -22 73 -116 | 0 1 2 0 5 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~121/84 = 633.755{{c}} (~49/48 = 33.755{{c}})
Optimal tunings:
* WE: ~36/35 = 46.1550{{c}}, ~3/2 = 701.0945{{c}}
* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.0853{{c}}


{{Optimal ET sequence|legend=0| 32c, 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=0| 130, 286, 416, 546 }}


Badness (Smith): 0.030159
Badness (Sintel): 3.00


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 729/728, 1001/1000, 6656/6655
Comma list: 351/350, 540/539, 40656/40625, 142884/142805
 
Mapping: {{mapping| 26 0 -22 73 -116 55 | 0 1 2 0 5 1 }}
 
Optimal tunings:
* WE: ~36/35 = 46.1573{{c}}, ~3/2 = 701.0904{{c}}
* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.0603{{c}}
 
{{Optimal ET sequence|legend=0| 130, 286, 416, 546 }}
 
Badness (Sintel): 1.80
 
== 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 -1 -9 -5 | 0 11 36 28 }}
: mapping generators: ~567/400, ~343/300
 
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.0984{{c}}, ~343/300 = 227.4405{{c}}
: [[error map]]: {{val| +0.197 -0.208 +0.659 -0.984 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~343/300 = 227.4104{{c}}
: error map: {{val| 0.000 -0.441 +0.460 -1.335 }}
 
{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}
 
[[Badness]] (Sintel): 4.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 8019/8000, 456533/455625


Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}
Mapping: {{mapping| 2 -1 -9 -5 -9 | 0 11 36 28 42 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~75/52 = 633.742{{c}} (~49/48 = 33.742{{c}})
Optimal tunings:
* WE: ~99/70 = 600.0988{{c}}, ~154/135 = 227.4386{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~154/135 = 227.4084{{c}}


{{Optimal ET sequence|legend=0| 72, 176f, 248, 320 }}
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}


Badness (Smith): 0.027632
Badness (Sintel): 1.54


==== Octowerckis ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 364/363, 441/440, 2197/2187
Comma list: 441/440, 729/728, 847/845, 1001/1000


Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}
Mapping: {{mapping| 2 -1 -9 -5 -9 -7 | 0 11 36 28 42 38 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~13/9 = 633.8370{{c}} (~49/48 = 33.8370{{c}})
Optimal tunings:
* WE: ~99/70 = 600.0805{{c}}, ~154/135 = 227.4285{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~154/135 = 227.4040{{c}}


{{Optimal ET sequence|legend=0| 32cf, 72, 176, 248f }}
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}


Badness (Smith): 0.030192
Badness (Sintel): 1.08


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

Latest revision as of 04:28, 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, bosonic, 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 tunings:

  • WE: ~160/147 = 150.0317 ¢, ~350/243 = 633.8714 ¢ (~49/48 = 33.7447 ¢)
error map: ⟨+0.253 -0.341 +0.438 -0.640]
  • CWE: ~160/147 = 150.0000 ¢, ~350/243 = 633.7791 ¢ (~49/48 = 33.7791 ¢)
error map: ⟨0.000 -0.618 +0.140 -1.268]

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

Badness (Sintel): 2.59

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 tunings:

  • WE: ~12/11 = 150.0233 ¢, ~121/84 = 633.8533 ¢ (~49/48 = 33.7602 ¢)
  • CWE: ~12/11 = 150.0000 ¢, ~121/84 = 633.7841 ¢ (~49/48 = 33.7841 ¢)

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

Badness (Sintel): 0.997

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 tunings:

  • WE: ~12/11 = 150.0247 ¢, ~75/52 = 633.8469 ¢ (~49/48 = 33.7480 ¢)
  • CWE: ~12/11 = 150.000 ¢, ~75/52 = 633.7616 ¢ (~49/48 = 33.7616 ¢)

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

Badness (Sintel): 1.14

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 tunings:

  • WE: ~12/11 = 150.0289 ¢, ~13/9 = 633.9590 ¢ (~49/48 = 33.8435 ¢)
  • CWE: ~12/11 = 150.0000 ¢, ~13/9 = 633.8739 ¢ (~49/48 = 33.8739 ¢)

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

Badness (Sintel): 1.25

Bosonic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tridecatonic.

Subgroup: 2.3.5.7

Comma list: 321489/320000, 589824/588245

Mapping: [⟨26 0 -22 73], ⟨0 1 2 0]]

mapping generators: ~36/35, ~3

Optimal tunings:

  • WE: ~36/35 = 46.1534 ¢, ~3/2 = 701.2433 ¢
error map: ⟨-0.012 -0.723 +0.775 +0.372]
  • CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.2451 ¢
error map: ⟨0.000 -0.710 +0.792 +0.405]

Optimal ET sequence: 26, …, 104c, 130, 546, 676, 806c

Badness (Sintel): 3.94

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [⟨26 0 -22 73 90], ⟨0 1 2 0 0]]

Optimal tunings:

  • WE: ~36/35 = 46.1443 ¢, ~3/2 = 701.4143 ¢
  • CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.4791 ¢

Optimal ET sequence: 26, …, 104c, 130

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [⟨26 0 -22 73 90 55], ⟨0 1 2 0 0 1]]

Optimal tunings:

  • WE: ~36/35 = 46.1469 ¢, ~3/2 = 701.4410 ¢
  • CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.4873 ¢

Optimal ET sequence: 26, …, 104c, 130

Badness (Sintel): 1.36

Fermionic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [⟨26 0 -22 73 -116], ⟨0 1 2 0 5]]

Optimal tunings:

  • WE: ~36/35 = 46.1550 ¢, ~3/2 = 701.0945 ¢
  • CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.0853 ¢

Optimal ET sequence: 130, 286, 416, 546

Badness (Sintel): 3.00

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [⟨26 0 -22 73 -116 55], ⟨0 1 2 0 5 1]]

Optimal tunings:

  • WE: ~36/35 = 46.1573 ¢, ~3/2 = 701.0904 ¢
  • CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.0603 ¢

Optimal ET sequence: 130, 286, 416, 546

Badness (Sintel): 1.80

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 -1 -9 -5], ⟨0 11 36 28]]

mapping generators: ~567/400, ~343/300

Optimal tunings:

  • WE: ~567/400 = 600.0984 ¢, ~343/300 = 227.4405 ¢
error map: ⟨+0.197 -0.208 +0.659 -0.984]
  • CWE: ~567/400 = 600.0000 ¢, ~343/300 = 227.4104 ¢
error map: ⟨0.000 -0.441 +0.460 -1.335]

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

Badness (Sintel): 4.39

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -1 -9 -5 -9], ⟨0 11 36 28 42]]

Optimal tunings:

  • WE: ~99/70 = 600.0988 ¢, ~154/135 = 227.4386 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~154/135 = 227.4084 ¢

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

Badness (Sintel): 1.54

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -1 -9 -5 -9 -7], ⟨0 11 36 28 42 38]]

Optimal tunings:

  • WE: ~99/70 = 600.0805 ¢, ~154/135 = 227.4285 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~154/135 = 227.4040 ¢

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

Badness (Sintel): 1.08

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