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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 (Sintel): 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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== 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.
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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 (Sintel): 0.066
[[Badness]] (Sintel): 0.066


=== Alhena ===
=== Alhena ===
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=== 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]].''


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==== 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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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]]
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=== 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]].


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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 (Sintel): 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 (Sintel): 17.120
[[Badness]] (Sintel): 17.120


=== 3.5.7.11 subgroup ===
=== 3.5.7.11 subgroup ===
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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 (Sintel): 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]] (Sintel): 0.21396
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[[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]] (Sintel): 0.45479


==== <small><small>(no-2s)</small> </small>23-limit ====
==== <small>(no-2s) </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.


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[[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]] (Sintel): 0.54174
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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.
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
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: 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 (Sintel): 0.213
[[Badness]] (Sintel): 0.213


=== Remus ===
=== Remus ===
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Badness (Sintel): 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 ==
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Non-octave temperaments]]
[[Category:Non-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/No-twos_subgroup_temperaments"