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No-twos subgroup temperaments: Difference between revisions

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Sval mapping generators: ~3, ~5
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]]
[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]


Badness (Dirichlet): 0.535
[[Badness]] (Sintel): 0.535


=== Polturus ===
=== Polturus ===
This extension of Arcturus adds [[Polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
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
[[EDT]]s: 15, 13e, 28e, 43dee


Badness (Dirichlet): 2.507
Badness (Sintel): 2.507


== BPS ==
== 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.
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
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]]
[[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 ===
=== Alhena ===
Line 92: Line 95:
(Note that é is used as the wart for 11/2.)
(Note that é is used as the wart for 11/2.)


Badness (Dirichlet): 0.187
Badness (Sintel): 0.187


=== Mintra ===
=== Mintra ===
''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''
''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
[[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 ====
==== Tridecimal Mintra ====
This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].
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
[[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 ===
=== Dubhe ===
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[[Support]]ing [[ET]]s: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44
[[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 ==
== Canopus ==
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Sval mapping generators: ~[[3/1|3]], ~[[7/5]]
Sval mapping generators: ~[[3/1|3]], ~[[7/5]]


[[Optimal tuning]]s:
[[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]]
[[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 ===
=== Suhail ===
Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
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 -->
[[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 ==
== Izar ==
Line 216: Line 216:
Sval mapping generators: ~3, ~16807/10125
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]]
[[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 ==
== Nekkar ==
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Sval mapping generators: ~3, ~16807/10935
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 ===
=== 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
[[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 ===
=== 3.5.7.11.13 subgroup ===
Line 273: Line 277:
[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf
[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf


Badness (Dirichlet): 1.723
Badness (Sintel): 1.723


== Procyon ==
== Procyon ==
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: sval mapping generators: ~3, ~17/9
: 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
[[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 ===
=== Erigone ===
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: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[49/33]]
: [[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]]
[[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 ====
Line 330: Line 334:
[[Optimal ET sequence]]: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]
[[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.
[[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]]
[[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 ==
== 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.
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
[[Subgroup]]: 3.5.7
Line 370: Line 370:
: sval mapping generators: ~3, ~25/21
: 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]]
[[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 ===
=== Remus ===
Line 395: Line 397:
[[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181
[[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181


Badness (Dirichlet): 0.286
Badness (Sintel): 0.286


=== Mizar ===
=== Mizar ===
Line 412: Line 414:
[[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd
[[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 ====
==== Full no-twos 17-limit ====
Line 429: Line 431:
[[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d
[[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 ==
== Tuning diagrams ==
Line 737: Line 773:
[[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503
[[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
Sval mapping generators: ~3, ~119/117
Line 754: Line 790:
[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969
[[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
Sval mapping generators: ~3, ~64/63
Line 766: Line 802:


==== 3.5.7.8.11.13.17.19.23 subgroup ====
==== 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
[[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
[[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
Sval mapping generators: ~3, ~64/63
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