Ragismic microtemperaments: Difference between revisions

Line 4:

= 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 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 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||.

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.

Line 23:

Mapping generators: ~27/25, ~5/3

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

[[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~~, 3600~~ }}

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

[[Badness]]: 0.003610

Line 90:

POTE generator: ~140/121 = 250.3375

Vals: {{Val list| 72, 441 }}

Vals: {{Val list| 72, 297ef, 369f, 441 }}

Badness: 0.026122

Line 101:

POTE generator: ~77/75 = 45.595

Vals: {{Val list| 342, 1053, 1395, 1737, ~~4869d~~, ~~6606cd~~ }}

Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}

Badness: 0.021320

@@ Line 4: / Line 4: @@
 = 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 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 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||.
 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.
@@ Line 23: / Line 23: @@
 Mapping generators: ~27/25, ~5/3
-[[POTE Tuning|POTE generators]]: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
+[[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, 3600 }}
+[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171, 441, 612 }}
 [[Badness]]: 0.003610
@@ Line 90: / Line 90: @@
 POTE generator: ~140/121 = 250.3375
-Vals: {{Val list| 72, 441 }}
+Vals: {{Val list| 72, 297ef, 369f, 441 }}
 Badness: 0.026122
@@ Line 101: / Line 101: @@
 POTE generator: ~77/75 = 45.595
-Vals: {{Val list| 342, 1053, 1395, 1737, 4869d, 6606cd }}
+Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
 Badness: 0.021320