No-twos subgroup temperaments: Difference between revisions

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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 (Sintel): 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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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 (Sintel): 0.066

[[Badness]] (Sintel): 0.066

=== Alhena ===

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=== Mintra ===

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

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==== 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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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¢

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=== 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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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 (Sintel): 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 (Sintel): 17.120

[[Badness]] (Sintel): 17.120

=== 3.5.7.11 subgroup ===

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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 (Sintel): 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

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~~[[Badness]] (Smith): 0.00501~~

[[Badness]] (Sintel): 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

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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 (Sintel): 0.213

[[Badness]] (Sintel): 0.213

=== Remus ===

@@ 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 (Sintel): 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 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 (Sintel): 0.066
+[[Badness]] (Sintel): 0.066
 === Alhena ===
@@ Line 95: / Line 98: @@
 === Mintra ===
 ''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''
@@ Line 115: / Line 117: @@
 ==== Tridecimal Mintra ====
 This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].
@@ 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]]
@@ Line 171: / Line 172: @@
 === 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 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 (Sintel): 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 (Sintel): 17.120
+[[Badness]] (Sintel): 17.120
 === 3.5.7.11 subgroup ===
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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 (Sintel): 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 <!-- 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
@@ Line 329: / Line 333: @@
 [[Optimal ET sequence]]: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]
-[[Badness]] (Smith): 0.00501
 [[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.
@@ Line 350: / Line 352: @@
 [[Optimal ET sequence]]: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
-[[Badness]] (Smith): 0.00539
 [[Badness]] (Sintel): 0.54174
@@ 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
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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¢
 [[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 ===