Ragismic microtemperaments: Difference between revisions

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= Ennealimmal =

[[Ennealimmal]] temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the [[ennealimma|ennealimmal comma]], |1 -27 18>, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two ~~periods~~ equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is <<18 27 18 1 -22 -34||.

[[Ennealimmal]] temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the [[ennealimma|ennealimmal comma]], |1 -27 18>, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two period equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is <<18 27 18 1 -22 -34||.

Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference.

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[[POTE tuning|POTE generators]]: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980

[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171~~, 270~~, 441, 612 }}

[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171, 441, 612 }}

[[Badness]]: 0.003610

== 11-limit ==

Comma list: 2401/2400, 4375/4374, 5632/5625

Mapping: [<9 1 1 12 -75|, <0 2 3 2 16|]

POTE generator: ~36/35 = 48.8654

Vals: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}

Badness: 0.027332

== Hemiennealimmal ==

Comma list: 2401/2400, 4375/4374~~, 3025/3024~~

Comma list: 2401/2400, 3025/3024, 4375/4374

Tuning ranges:

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=== Semihemiennealimmal ===

Comma list: 2401/2400~~, 4375/4374~~, 3025/3024, 4225/4224

Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374

Mapping: [<18 0 -1 22 48 88|, <0 4 6 4 2 -3|]

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== Semiennealimmal ==

Comma list: 2401/2400, 4375/4374~~, 4000/3993~~

Comma list: 2401/2400, 4000/3993, 4375/4374

Mapping: [<9 3 4 14 18|, <0 6 9 6 7|]

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Badness: 0.023250

=== 17-limit ===

==== 17-limit ====

Comma list: 243/242, 364/363, 375/374, 441/440, 595/594

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 = Ennealimmal =
-[[Ennealimmal]] temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the [[ennealimma|ennealimmal comma]], |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.
+[[Ennealimmal]] temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the [[ennealimma|ennealimmal comma]], |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two period equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.
 Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference.
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 [[POTE tuning|POTE generators]]: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
-[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171, 270, 441, 612 }}
+[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171, 441, 612 }}
 [[Badness]]: 0.003610
+== 11-limit ==
+Comma list: 2401/2400, 4375/4374, 5632/5625
+Mapping: [&lt;9 1 1 12 -75|, &lt;0 2 3 2 16|]
+POTE generator: ~36/35 = 48.8654
+Vals: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
+Badness: 0.027332
 == Hemiennealimmal ==
-Comma list: 2401/2400, 4375/4374, 3025/3024
+Comma list: 2401/2400, 3025/3024, 4375/4374
 Tuning ranges:
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 === Semihemiennealimmal ===
-Comma list: 2401/2400, 4375/4374, 3025/3024, 4225/4224
+Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
 Mapping: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
@@ Line 73: / Line 84: @@
 == Semiennealimmal ==
-Comma list: 2401/2400, 4375/4374, 4000/3993
+Comma list: 2401/2400, 4000/3993, 4375/4374
 Mapping: [&lt;9 3 4 14 18|, &lt;0 6 9 6 7|]
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 Badness: 0.023250
-=== 17-limit ===
+==== 17-limit ====
 Comma list: 243/242, 364/363, 375/374, 441/440, 595/594