No-twos subgroup temperaments: Difference between revisions

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As for extensions of this temperament that include the prime 2, see [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Crepuscular|crepuscular]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Bunya|bunya]], [[Sensamagic clan #Bohpier|bohpier]], and [[~~Shibboleth family~~ #Superkleismic|superkleismic]].

As for extensions of this temperament that include the prime 2, see [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Crepuscular|crepuscular]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Bunya|bunya]], [[Sensamagic clan #Bohpier|bohpier]], and [[Gamelismic clan #Superkleismic|superkleismic]].

Subgroup: 3.5.7

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Sval mapping generators: ~3, ~5

[[~~POTE generator~~]]: ~5/3 = 878.~~042~~

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1903.863¢, ~5/3 = 878.923¢

* [[CWE]]: ~3 = 1901.955¢, ~5/3 = 878.291¢

[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]

Badness (~~Dirichlet~~): 0.535

[[Badness]] (Sintel): 0.535

=== Polturus ===

This extension of Arcturus adds [[Polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.

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[[EDT]]s: 15, 13e, 28e, 43dee

Badness (~~Dirichlet~~): 2.507

Badness (Sintel): 2.507

== BPS ==

For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.

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Sval mapping generators: ~3, ~9/7

[[Optimal tuning]] ([[~~POTE~~]]): ~3 = ~~1\1edt~~, ~9/7 = 440.~~4881~~

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1903.740¢, ~9/7 = 440.901¢

* [[CWE]]: ~3 = 1901.955¢, ~9/7 = 440.665¢

Badness (~~Dirichlet~~): 0.066

[[Badness]] (Sintel): 0.066

=== Alhena ===

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(Note that é is used as the wart for 11/2.)

Badness (~~Dirichlet~~): 0.187

Badness (Sintel): 0.187

=== Mintra ===

''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''

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[[Support]]ing [[ET]]s: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90

Badness (~~Dirichlet~~): 0.302

Badness (Sintel): 0.302

==== Tridecimal Mintra ====

This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].

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[[Support]]ing [[ET]]s: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef

Badness (~~Dirichlet~~): 0.373

Badness (Sintel): 0.373

=== Dubhe ===

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[[Support]]ing [[ET]]s: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44

Badness (~~Dirichlet~~): 0.177

Badness (Sintel): 0.177

== Canopus ==

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Sval mapping generators: ~[[3/1|3]], ~[[7/5]]

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~3 = ~~1\1edt~~, ~7/5 = ~~584~~.~~017~~

* [[WE]]: ~3 = 1901.783¢, ~7/5 = 583.905¢

* [[~~POTE|PETE~~]]: ~3 = ~~1\1edt~~, ~7/5 = 583.~~9584~~

* [[CWE]]: ~3 = 1901.955¢, ~7/5 = 583.986¢

Badness (~~Dirichlet~~): 0.100

Badness (Sintel): 0.100

=== Suhail ===

Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].

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Badness (~~Dirichlet~~): 0.330

Badness (Sintel): 0.330

== Izar ==

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Sval mapping generators: ~3, ~16807/10125

[[Optimal tuning]] ~~(CTE)~~: ~3 = ~~1\1edt~~, ~16807/10125 = 877.~~280~~

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1901.958¢, ~16807/10125 = 877.283¢

* [[CWE]]: ~3 = 1901.955¢, ~16807/10125 = 877.281¢

Badness (~~Dirichlet~~): 0.017

[[Badness]] (Sintel): 0.017

== Nekkar ==

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Sval mapping generators: ~3, ~16807/10935

[[Optimal tuning]] ([[CWE]]): ~3 = ~~1\1edt~~, ~16807/10935 = 776.~~767~~

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1900.155¢, ~16807/10935 = 775.963¢

* [[CWE]]: ~3 = 1901.955¢, ~16807/10935 = 776.767¢

[[~~Support~~]]~~ing~~ [[ET]]~~s: 22~~, 49, 5c, 71, ~~27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d~~

Badness (~~Dirichlet~~): 17.120

[[Badness]] (Sintel): 17.120

=== 3.5.7.11 subgroup ===

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[[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd

Badness (~~Dirichlet~~): 1.375

Badness (Sintel): 1.375

=== 3.5.7.11.13 subgroup ===

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[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf

Badness (~~Dirichlet~~): 1.723

Badness (Sintel): 1.723

== Procyon ==

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: sval mapping generators: ~3, ~17/9

[[Optimal tuning]] ([[~~CTE~~]]): ~3 = ~~1\1edt~~, ~49/45 = 145.~~333~~

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1902.198¢, ~49/45 = 145.412¢

* [[CWE]]: ~3 = 1901.955¢, ~49/45 = 145.368¢

[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79

Badness (~~Dirichlet~~): 0.200

[[Badness]] (Sintel): 0.200

=== Erigone ===

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: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[49/33]]

[[Optimal tuning]] ([[WE]]): ~3 = 1901.9699, ~49/33 = 682.4486

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1901.9699, ~49/33 = 682.4486

~~[[Optimal tuning]] (~~[[CWE]]): ~3 = 1\1edt, ~49/33 = 682.4427

* [[CWE]]: ~3 = 1\1edt, ~49/33 = 682.4427

[[Badness]] (~~Smith~~): 0.~~00271~~

[[Badness]] (Sintel): 0.21396

~~[[Badness]] (Dirichlet): 0.21396~~

==== Hemigone ====

=== Hemigone ===

By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).

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[[Badness]] (~~Smith~~): 0.~~00501~~

[[Badness]] (Sintel): 0.45479

~~[[Badness]] (Dirichlet): 0.45479~~

==== (no-2s) 23-limit ====

=== ~~~~(no-2s)~~~~ 23-limit ===

[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone|erigone]] to the 3.5.7.11.13.23 subgroup.

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[[Badness]] (~~Smith): 0.00539~~

[[Badness]] (Sintel): 0.54174

~~[[Badness]] (Dirichlet~~): 0.54174

== Sirius ==

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This tempers out the [[Don Page comma]] between [[5/3]] and [[7/5]], allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in Procyon.

For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan#Overview to extensions]].

For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].

[[Subgroup]]: 3.5.7

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: sval mapping generators: ~3, ~25/21

[[Optimal tuning]] ([[~~POTE~~]]): ~3 = ~~1\1edt~~, ~25/21 = 293.~~740~~

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1902.445¢, ~25/21 = 293.739¢

* [[CWE]]: ~3 = 1901.955¢, ~25/21 = 293.759¢

Badness (~~Dirichlet~~): 0.213

[[Badness]] (Sintel): 0.213

=== Remus ===

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[[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181

Badness (~~Dirichlet~~): 0.286

Badness (Sintel): 0.286

=== Mizar ===

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[[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd

Badness (~~Dirichlet~~): 0.383

Badness (Sintel): 0.383

==== Full no-twos 17-limit ====

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[[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d

Badness (~~Dirichlet~~): 0.841

Badness (Sintel): 0.841

== Bohlenic ==

This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.

[[Subgroup]]: 3.5.7.11

[[Comma list]]: 245/243, 3125/3087

: sval mapping generators: ~27/25, ~11

[[Optimal tuning]]s:

* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4151.318¢

* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4147.705¢

[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]

[[Badness]] (Sintel): 0.499

=== Full no-twos 13-limit ===

[[Subgroup]]: 3.5.7.11.13

[[Comma list]]: 245/243, 275/273, 847/845

[[Optimal tuning]]s:

* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4149.733¢

* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4146.033¢

[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]

[[Badness]] (Sintel): 0.365

== Tuning diagrams ==

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[[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -15 12 22 38 -21}}]

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 3}}, {{val| 0 -15 12 22 38 -21}}]

Sval mapping generators: ~3, ~119/117

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[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -15 -7 12 22 38 -21}}]

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 3}}, {{val| 0 -15 -7 12 22 38 -21}}]

Sval mapping generators: ~3, ~64/63

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==== 3.5.7.8.11.13.17.19.23 subgroup ====

At ~~slightly~~ lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.

At the cost of lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.

[[Subgroup]]: 3.5.7.8.11.13.17.19.23

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[[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}]

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 2 3 4}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}]

Sval mapping generators: ~3, ~64/63

@@ Line 23: / Line 23: @@
 {{main|Arcturus}}
-As for extensions of this temperament that include the prime 2, see [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Crepuscular|crepuscular]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Bunya|bunya]], [[Sensamagic clan #Bohpier|bohpier]], and [[Shibboleth family #Superkleismic|superkleismic]].
+As for extensions of this temperament that include the prime 2, see [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Crepuscular|crepuscular]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Bunya|bunya]], [[Sensamagic clan #Bohpier|bohpier]], and [[Gamelismic clan #Superkleismic|superkleismic]].
 Subgroup: 3.5.7
@@ Line 33: / Line 33: @@
 Sval mapping generators: ~3, ~5
-[[POTE generator]]: ~5/3 = 878.042
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1903.863¢, ~5/3 = 878.923¢
+* [[CWE]]: ~3 = 1901.955¢, ~5/3 = 878.291¢
 [[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
-Badness (Dirichlet): 0.535
+[[Badness]] (Sintel): 0.535
 === Polturus ===
 This extension of Arcturus adds [[Polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
@@ Line 55: / Line 56: @@
 [[EDT]]s: 15, 13e, 28e, 43dee
-Badness (Dirichlet): 2.507
+Badness (Sintel): 2.507
 == BPS ==
-{{main|Bohlen–Pierce–Stearns}}
+{{main|BPS}}
 For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.
@@ Line 70: / Line 71: @@
 Sval mapping generators: ~3, ~9/7
-[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1903.740¢, ~9/7 = 440.901¢
+* [[CWE]]: ~3 = 1901.955¢, ~9/7 = 440.665¢
 [[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
-Badness (Dirichlet): 0.066
+[[Badness]] (Sintel): 0.066
 === Alhena ===
@@ Line 92: / Line 95: @@
 (Note that é is used as the wart for 11/2.)
-Badness (Dirichlet): 0.187
+Badness (Sintel): 0.187
 === Mintra ===
 ''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''
@@ Line 112: / Line 114: @@
 [[Support]]ing [[ET]]s: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90
-Badness (Dirichlet): 0.302
+Badness (Sintel): 0.302
 ==== Tridecimal Mintra ====
 This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].
@@ Line 130: / Line 131: @@
 [[Support]]ing [[ET]]s: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef
-Badness (Dirichlet): 0.373
+Badness (Sintel): 0.373
 === Dubhe ===
@@ Line 147: / Line 148: @@
 [[Support]]ing [[ET]]s: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44
-Badness (Dirichlet): 0.177
+Badness (Sintel): 0.177
 == Canopus ==
@@ Line 162: / Line 163: @@
 Sval mapping generators: ~[[3/1|3]], ~[[7/5]]
 [[Optimal tuning]]s:
-* [[CTE]]: ~3 = 1\1edt, ~7/5 = 584.017
+* [[WE]]: ~3 = 1901.783¢, ~7/5 = 583.905¢
-* [[POTE|PETE]]: ~3 = 1\1edt, ~7/5 = 583.9584
+* [[CWE]]: ~3 = 1901.955¢, ~7/5 = 583.986¢
 [[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
-Badness (Dirichlet): 0.100
+Badness (Sintel): 0.100
 === Suhail ===
 Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
@@ Line 205: / Line 205: @@
 [[Optimal ET sequence]]: [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]] <!-- b804cff is not a GPV -->
-Badness (Dirichlet): 0.330
+Badness (Sintel): 0.330
 == Izar ==
@@ Line 216: / Line 216: @@
 Sval mapping generators: ~3, ~16807/10125
-[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1901.958¢, ~16807/10125 = 877.283¢
+* [[CWE]]: ~3 = 1901.955¢, ~16807/10125 = 877.281¢
 [[Optimal ET sequence]]: [[13edt|b13]], [[141edt|b141]], [[154edt|b154]], ... [[258edt|b258]], [[271edt|b271]], [[800edt|b800]], [[1071edt|b1071]], [[1342edt|b1342]], [[1613edt|b1613]], [[4568edt|b4568]], [[6181edt|b6181]]
-Badness (Dirichlet): 0.017
+[[Badness]] (Sintel): 0.017
 == Nekkar ==
@@ Line 233: / Line 235: @@
 Sval mapping generators: ~3, ~16807/10935
-[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~16807/10935 = 776.767
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1900.155¢, ~16807/10935 = 775.963¢
+* [[CWE]]: ~3 = 1901.955¢, ~16807/10935 = 776.767¢
-[[Support]]ing [[ET]]s: 22, 49, 5c, 71, 27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d
+[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]]
-Badness (Dirichlet): 17.120
+[[Badness]] (Sintel): 17.120
 === 3.5.7.11 subgroup ===
@@ Line 256: / Line 260: @@
 [[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
-Badness (Dirichlet): 1.375
+Badness (Sintel): 1.375
 === 3.5.7.11.13 subgroup ===
@@ Line 273: / Line 277: @@
 [[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf
-Badness (Dirichlet): 1.723
+Badness (Sintel): 1.723
 == Procyon ==
@@ Line 286: / Line 290: @@
 : sval mapping generators: ~3, ~17/9
-[[Optimal tuning]] ([[CTE]]): ~3 = 1\1edt, ~49/45 = 145.333
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1902.198¢, ~49/45 = 145.412¢
+* [[CWE]]: ~3 = 1901.955¢, ~49/45 = 145.368¢
 [[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79
-Badness (Dirichlet): 0.200
+[[Badness]] (Sintel): 0.200
 === Erigone ===
@@ Line 303: / Line 309: @@
 : [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[49/33]]
-[[Optimal tuning]] ([[WE]]): ~3 = 1901.9699, ~49/33 = 682.4486 <!-- IDK why the second rounded value disagrees with Sintel's calculator, I got it from FloraC's -->
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1901.9699, ~49/33 = 682.4486 <!-- IDK why the second rounded value disagrees with Sintel's calculator, I got it from FloraC's -->
-[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~49/33 = 682.4427
+* [[CWE]]: ~3 = 1\1edt, ~49/33 = 682.4427
 [[Optimal ET sequence]]: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]
-[[Badness]] (Smith): 0.00271
+[[Badness]] (Sintel): 0.21396
-[[Badness]] (Dirichlet): 0.21396
+==== Hemigone ====
-=== Hemigone ===
 By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).
@@ Line 330: / Line 334: @@
 [[Optimal ET sequence]]: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]
-[[Badness]] (Smith): 0.00501
+[[Badness]] (Sintel): 0.45479
-[[Badness]] (Dirichlet): 0.45479
+==== <small>(no-2s) </small>23-limit ====
-=== <small><small>(no-2s)</small> </small>23-limit ===
 [[2277/2275]] may be used in the same way to extend the simpler [[#Erigone|erigone]] to the 3.5.7.11.13.23 subgroup.
@@ Line 351: / Line 353: @@
 [[Optimal ET sequence]]: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
-[[Badness]] (Smith): 0.00539
+[[Badness]] (Sintel): 0.54174
-[[Badness]] (Dirichlet): 0.54174
 == Sirius ==
@@ Line 360: / Line 360: @@
 This tempers out the [[Don Page comma]] between [[5/3]] and [[7/5]], allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in Procyon.
-For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan#Overview to extensions]].
+For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
 [[Subgroup]]: 3.5.7
@@ Line 370: / Line 370: @@
 : sval mapping generators: ~3, ~25/21
-[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~25/21 = 293.740
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1902.445¢, ~25/21 = 293.739¢
+* [[CWE]]: ~3 = 1901.955¢, ~25/21 = 293.759¢
 [[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]], [[123edt|b123]], [[136edt|b136]]
-Badness (Dirichlet): 0.213
+[[Badness]] (Sintel): 0.213
 === Remus ===
@@ Line 395: / Line 397: @@
 [[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181
-Badness (Dirichlet): 0.286
+Badness (Sintel): 0.286
 === Mizar ===
@@ Line 412: / Line 414: @@
 [[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd
-Badness (Dirichlet): 0.383
+Badness (Sintel): 0.383
 ==== Full no-twos 17-limit ====
@@ Line 429: / Line 431: @@
 [[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d
-Badness (Dirichlet): 0.841
+Badness (Sintel): 0.841
+== Bohlenic ==
+This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.
+[[Subgroup]]: 3.5.7.11
+[[Comma list]]: 245/243, 3125/3087
+{{Mapping|legend=2| 13 19 23 0 | 0 0 0 1 }}
+: sval mapping generators: ~27/25, ~11
+[[Optimal tuning]]s:
+* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4151.318¢
+* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4147.705¢
+[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
+[[Badness]] (Sintel): 0.499
+=== Full no-twos 13-limit ===
+[[Subgroup]]: 3.5.7.11.13
+[[Comma list]]: 245/243, 275/273, 847/845
+{{Mapping|legend=2| 13 19 23 0 2 | 0 0 0 1 1 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4149.733¢
+* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4146.033¢
+[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
+[[Badness]] (Sintel): 0.365
 == Tuning diagrams ==
@@ Line 737: / Line 773: @@
 [[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503
-[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -15 12 22 38 -21}}]
+[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 3}}, {{val| 0 -15 12 22 38 -21}}]
 Sval mapping generators: ~3, ~119/117
@@ Line 754: / Line 790: @@
 [[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969
-[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -15 -7 12 22 38 -21}}]
+[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 3}}, {{val| 0 -15 -7 12 22 38 -21}}]
 Sval mapping generators: ~3, ~64/63
@@ Line 766: / Line 802: @@
 ==== 3.5.7.8.11.13.17.19.23 subgroup ====
-At slightly lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.
+At the cost of lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.
 [[Subgroup]]: 3.5.7.8.11.13.17.19.23
@@ Line 772: / Line 808: @@
 [[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125
-[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}]
+[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 2 3 4}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}]
 Sval mapping generators: ~3, ~64/63