Varunismic temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament]]s tempering out the [[varunisma]]~~, 321489/320000 =~~ {{monzo| -9 8 -4 2 }}. For the [[rank-3 temperament]] tempering out 321489/320000 in the 7-limit, see [[Miscellaneous 7-limit temperaments #Varuna]].

This is a collection of [[rank-2 temperament|rank-2]] '''varunismic temperaments''', which [[tempering out|temper out]] the [[varunisma]] ({{monzo|legend=1| -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:

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* ''[[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]].

== 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.

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

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

[[Optimal tuning]] ([[POTE]]): ~567/400 = ~~1\2~~, ~243/196 = 372.597 (~343/300 = 227.403)

[[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]]: 0.173565

[[Badness]] (Smith): 0.173565

=== 11-limit ===

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

Optimal tuning (POTE): ~99/70 = ~~1\2~~, ~150/121 = 372.599 (~154/135 = 227.401)

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

Badness: 0.046447

Badness (Smith): 0.046447

=== 13-limit ===

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Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}

Optimal tuning (POTE): ~99/70 = ~~1\2~~, ~26/21 = 372.602 (~154/135 = 227.398)

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

Badness: 0.026107

Badness (Smith): 0.026107

== Octowerck ==

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

[[Optimal tuning]] ([[POTE]]): ~160/147 = ~~1\8~~, ~350/243 = 633.738 (~49/48 = 33.738)

[[Optimal tuning]] ([[POTE]]): ~160/147 = 150.000{{c}}, ~350/243 = 633.738{{c}} (~49/48 = 33.738{{c}})

[[Badness]]: 0.102295

[[Badness]] (Smith): 0.102295

=== 11-limit ===

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Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}

Optimal tuning (POTE): ~12/11 = ~~1\8~~, ~121/84 = 633.755 (~49/48 = 33.755)

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

Badness: 0.030159

Badness (Smith): 0.030159

==== 13-limit ====

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Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}

Optimal tuning (POTE): ~12/11 = ~~1\8~~, ~75/52 = 633.742 (~49/48 = 33.742)

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

Badness: 0.027632

Badness (Smith): 0.027632

==== Octowerckis ====

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Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}

Optimal tuning (POTE): ~12/11 = ~~1\8~~, ~13/9 = 633.8370 (~49/48 = 33.8370)

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

Badness: 0.030192

Badness (Smith): 0.030192

[[Category:Temperament collections]]

[[Category:Varunismic temperaments| ]]

~~[[Category:Varuna| ]] ~~

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This is a collection of [[rank-2 temperament]]s tempering out the [[varunisma]], 321489/320000 = {{monzo| -9 8 -4 2 }}. For the [[rank-3 temperament]] tempering out 321489/320000 in the 7-limit, see [[Miscellaneous 7-limit temperaments #Varuna]].
+This is a collection of [[rank-2 temperament|rank-2]] '''varunismic temperaments''', which [[tempering out|temper out]] the [[varunisma]] ({{monzo|legend=1| -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:
@@ Line 15: / Line 15: @@
 * ''[[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]].
 == 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.
+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
@@ Line 24: / Line 26: @@
 {{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}
 : mapping generators: ~567/400, ~243/196
-[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~243/196 = 372.597 (~343/300 = 227.403)
+[[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]]: 0.173565
+[[Badness]] (Smith): 0.173565
 === 11-limit ===
@@ Line 40: / Line 41: @@
 Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~150/121 = 372.599 (~154/135 = 227.401)
+Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~150/121 = 372.599{{c}} (~154/135 = 227.401{{c}})
-{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }}
+{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
-Badness: 0.046447
+Badness (Smith): 0.046447
 === 13-limit ===
@@ Line 53: / Line 54: @@
 Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~26/21 = 372.602 (~154/135 = 227.398)
+Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~26/21 = 372.602{{c}} (~154/135 = 227.398{{c}})
-{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }}
+{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
-Badness: 0.026107
+Badness (Smith): 0.026107
 == Octowerck ==
@@ Line 65: / Line 66: @@
 {{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}
 : mapping generators: ~160/147, ~350/243
-[[Optimal tuning]] ([[POTE]]): ~160/147 = 1\8, ~350/243 = 633.738 (~49/48 = 33.738)
+[[Optimal tuning]] ([[POTE]]): ~160/147 = 150.000{{c}}, ~350/243 = 633.738{{c}} (~49/48 = 33.738{{c}})
 {{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
-[[Badness]]: 0.102295
+[[Badness]] (Smith): 0.102295
 === 11-limit ===
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 Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~121/84 = 633.755 (~49/48 = 33.755)
+Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~121/84 = 633.755{{c}} (~49/48 = 33.755{{c}})
-{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
+{{Optimal ET sequence|legend=0| 32c, 72, 176, 248, 320 }}
-Badness: 0.030159
+Badness (Smith): 0.030159
 ==== 13-limit ====
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 Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~75/52 = 633.742 (~49/48 = 33.742)
+Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~75/52 = 633.742{{c}} (~49/48 = 33.742{{c}})
-{{Optimal ET sequence|legend=1| 72, 176f, 248, 320 }}
+{{Optimal ET sequence|legend=0| 72, 176f, 248, 320 }}
-Badness: 0.027632
+Badness (Smith): 0.027632
 ==== Octowerckis ====
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 Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~13/9 = 633.8370 (~49/48 = 33.8370)
+Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~13/9 = 633.8370{{c}} (~49/48 = 33.8370{{c}})
-{{Optimal ET sequence|legend=1| 32cf, 72, 176, 248f }}
+{{Optimal ET sequence|legend=0| 32cf, 72, 176, 248f }}
-Badness: 0.030192
+Badness (Smith): 0.030192
 [[Category:Temperament collections]]
 [[Category:Varunismic temperaments| ]] <!-- main article -->
-[[Category:Varuna| ]] <!-- key article -->
 [[Category:Rank 2]]