Varunismic temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Sort links by ploidacot. Sort main entries by badness
Switch to Sintel's badness, WE & CWE tunings
Line 26: Line 26:
: mapping generators: ~160/147, ~350/243
: mapping generators: ~160/147, ~350/243


[[Optimal tuning]] ([[POTE]]): ~160/147 = 150.000{{c}}, ~350/243 = 633.738{{c}} (~49/48 = 33.738{{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| 32c, 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}


[[Badness]] (Smith): 0.102295
[[Badness]] (Sintel): 2.59


=== 11-limit ===
=== 11-limit ===
Line 39: Line 43:
Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}
Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~121/84 = 633.755{{c}} (~49/48 = 33.755{{c}})
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 }}
{{Optimal ET sequence|legend=0| 32c, 72, 176, 248, 320 }}


Badness (Smith): 0.030159
Badness (Sintel): 0.997


==== 13-limit ====
==== 13-limit ====
Line 52: Line 58:
Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}
Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~75/52 = 633.742{{c}} (~49/48 = 33.742{{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| 72, 176f, 248, 320 }}
{{Optimal ET sequence|legend=0| 72, 176f, 248, 320 }}


Badness (Smith): 0.027632
Badness (Sintel): 1.14


==== Octowerckis ====
==== Octowerckis ====
Line 65: Line 73:
Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}
Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~13/9 = 633.8370{{c}} (~49/48 = 33.8370{{c}})
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 ET sequence|legend=0| 32cf, 72, 176, 248f }}
{{Optimal ET sequence|legend=0| 32cf, 72, 176, 248f }}


Badness (Smith): 0.030192
Badness (Sintel): 1.25


== Essence ==
== Essence ==
Line 78: Line 88:
[[Comma list]]: 321489/320000, 823543/820125
[[Comma list]]: 321489/320000, 823543/820125


{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}
{{Mapping|legend=1| 2 -1 -9 -5 | 0 11 36 28 }}
: mapping generators: ~567/400, ~243/196
: mapping generators: ~567/400, ~343/300


[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~243/196 = 372.597{{c}} (~343/300 = 227.403{{c}})
[[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 }}
{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}


[[Badness]] (Smith): 0.173565
[[Badness]] (Sintel): 4.39


=== 11-limit ===
=== 11-limit ===
Line 92: Line 106:
Comma list: 441/440, 8019/8000, 456533/455625
Comma list: 441/440, 8019/8000, 456533/455625


Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}
Mapping: {{mapping| 2 -1 -9 -5 -9 | 0 11 36 28 42 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~150/121 = 372.599{{c}} (~154/135 = 227.401{{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| 58, 132de, 190, 248, 438d }}
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}


Badness (Smith): 0.046447
Badness (Sintel): 1.54


=== 13-limit ===
=== 13-limit ===
Line 105: Line 121:
Comma list: 441/440, 729/728, 847/845, 1001/1000
Comma list: 441/440, 729/728, 847/845, 1001/1000


Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}
Mapping: {{mapping| 2 -1 -9 -5 -9 -7 | 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 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| 58, 132de, 190, 248, 438d }}
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}


Badness (Smith): 0.026107
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]]

Revision as of 04:15, 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 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

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

Retrieved from "https://en.xen.wiki/index.php?title=Varunismic_temperaments&oldid=234420"