Ragismic microtemperaments: Difference between revisions

Line 27:

[[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~~]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). 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 {{multival|18 27 18 1 -22 -34}}~~.

[[Ennealimmal]] 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]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)<sup>9</sup> with the octave, and gives ennealimmal a period of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). 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.

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.

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 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, though its hardly likely anyone could tell the difference.

If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.

Line 40:

[[Comma list]]: 2401/2400, 4375/4374

[[Mapping]]: [{{val|9 1 1 12}}, {{val|0 2 3 2}}]

[[Mapping]]: [{{val| 9 1 1 12 }}, {{val| 0 2 3 2 }}]

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

[[POTE generator]]s: ~~~36/35 = 49.0205; ~10/9 = 182.354; ~6/~~5 ~~= 315.687; ~49~~/40 = ~~350~~.~~980~~

[[POTE generator]]s: ~5/3 = 884.3129

[[Tuning ranges]]:

Line 65:

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

Mapping: [{{val|9 1 1 12 -75}}, {{val|0 2 3 2 16}}]

Mapping: [{{val| 9 1 1 12 -75 }}, {{val| 0 2 3 2 16 }}]

POTE generator: ~36/35 = 48.~~8654~~

POTE generator: ~5/3 = 884.4679

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

Line 78:

Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: [{{val|9 1 1 12 -75 93}}, {{val|0 2 3 2 16 -9}}]

Mapping: [{{val| 9 1 1 12 -75 93 }}, {{val| 0 2 3 2 16 -9 }}]

POTE generator: ~36/35 = 48.~~9030~~

POTE generator: ~5/3 = 884.4304

Vals: {{Val list| 99e, 171e, 270 }}

Line 93:

Comma list: 2401/2400, 4375/4374, 131072/130977

Mapping: [{{val|9 1 1 12 124}}, {{val|0 2 3 2 -14}}]

Mapping: [{{val| 9 1 1 12 124 }}, {{val| 0 2 3 2 -14 }}]

POTE generator: ~36/35 = 48.~~9244~~

POTE generator: ~5/3 = 884.4089

Vals: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}

Line 106:

Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: [{{val|9 1 1 12 124 93}}, {{val|0 2 3 2 -14 -9}}]

Mapping: [{{val| 9 1 1 12 124 93 }}, {{val| 0 2 3 2 -14 -9 }}]

POTE generator: ~36/35 = 48.~~9336~~

POTE generator: ~5/3 = 884.3997

Vals: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}

Line 121:

Comma list: 243/242, 441/440, 4375/4356

Mapping: [{{val|9 1 1 12 -2}}, {{val|0 2 3 2 5}}]

Mapping: [{{val| 9 1 1 12 -2 }}, {{val| 0 2 3 2 5 }}]

POTE generator: ~36/35 = 49.~~395~~

POTE generator: ~5/3 = 883.9386

Tuning ranges:

Line 139:

Comma list: 243/242, 364/363, 441/440, 625/624

Mapping: [{{val|9 1 1 12 -2 -33}}, {{val|0 2 3 2 5 10}}]

Mapping: [{{val| 9 1 1 12 -2 -33 }}, {{val| 0 2 3 2 5 10 }}]

POTE generator: ~36/35 = 49.~~341~~

POTE generator: ~5/3 = 883.9920

Tuning ranges:

Line 157:

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

Mapping: [{{val|9 1 1 12 -2 -33 -3}}, {{val|0 2 3 2 5 10 6}}]

Mapping: [{{val| 9 1 1 12 -2 -33 -3 }}, {{val| 0 2 3 2 5 10 6 }}]

POTE generator: ~36/35 = 49.~~335~~

POTE generator: ~5/3 = 883.9981

Tuning ranges:

Line 171:

==== Ennealim ====

Subgroup: 2.3.5.7.13

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 325/324, 441/440

Mapping: [{{val|9 1 1 12 -2 20}}, {{val|0 2 3 2 5 2}}]

Mapping: [{{val| 9 1 1 12 -2 20 }}, {{val| 0 2 3 2 5 2 }}]

POTE generator: ~36/35 = 49.~~708~~

POTE generator: ~5/3 = 883.6257

Vals: {{Val list| 27e, 45ef, 72 }}

Line 188:

Comma list: 385/384, 1375/1372, 4375/4374

Mapping: [{{val|9 1 1 12 51}}, {{val|0 2 3 2 -3}}]

Mapping: [{{val| 9 1 1 12 51 }}, {{val| 0 2 3 2 -3 }}]

POTE generator: ~36/35 = 49.~~504~~

POTE generator: ~5/3 = 883.8298

Vals: {{Val list| 27, 45, 72, 171e, 243e, 315e }}

Line 201:

Comma list: 169/168, 325/324, 385/384, 1375/1372

Mapping: [{{val|9 1 1 12 51 20}}, {{val|0 2 3 2 -3 2}}]

Mapping: [{{val| 9 1 1 12 51 20 }}, {{val| 0 2 3 2 -3 2 }}]

POTE generator: ~36/35 = 49.~~486~~

POTE generator: ~5/3 = 883.8476

Vals: {{Val list| 27, 45f, 72, 171ef, 243ef }}

Line 220:

Mapping generators: ~80/77, ~400/231

POTE generator: ~99/98 = 17.~~6219~~

POTE generator: ~400/231 = 950.9553

Tuning ranges:

Line 238:

Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]

POTE generator ~99/98 = 17.~~7504~~

POTE generator ~26/15 = 951.0837

Tuning ranges:

Line 261:

Mapping generators: ~80/77, ~1053/800

POTE generator: ~39/32 = ~~342~~.~~139~~

POTE generator: ~1053/800 = 475.4727

Vals: {{Val list| 126, 144, 270, 684, 954 }}

Line 272:

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

Mapping: [{{val|9 3 4 14 18}}, {{val|0 6 9 6 7}}]

Mapping: [{{val| 9 3 4 14 18 }}, {{val| 0 6 9 6 7 }}]

POTE generator: ~140/121 = 250.3367

Line 285:

Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374

Mapping: [{{val|9 3 4 14 18 -8}}, {{val|0 6 9 6 7 22}}]

Mapping: [{{val| 9 3 4 14 18 -8 }}, {{val| 0 6 9 6 7 22 }}]

POTE generator: ~140/121 = 250.3375

Line 298:

Comma list: 2401/2400, 4375/4374, 234375/234256

Mapping: [{{val|9 1 1 12 -7}}, {{val|0 8 12 8 23}}]

Mapping: [{{val| 9 1 1 12 -7 }}, {{val| 0 8 12 8 23 }}]

Mapping generators: ~27/25, ~25/22

POTE generator: ~77/75 = 45.~~595~~

POTE generator: ~25/22 = 221.0717

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

Line 315:

Line 317:

Mapping generators: ~2744/2673, ~2352/1375

POTE generator: ~6/5 = ~~315~~.~~644~~

POTE generator: ~2352/1375 = 928.8000

Vals: {{Val list| 27, 243, 270, 783, 1053, 1323 }}

@@ Line 27: / Line 27: @@
 {{Main| 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]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). 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 {{multival|18 27 18 1 -22 -34}}.
+[[Ennealimmal]] 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]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)<sup>9</sup> with the octave, and gives ennealimmal a period of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). 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.
-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.
+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 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, though its hardly likely anyone could tell the difference.
 If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.
@@ Line 40: / Line 40: @@
 [[Comma list]]: 2401/2400, 4375/4374
-[[Mapping]]: [{{val|9 1 1 12}}, {{val|0 2 3 2}}]
+[[Mapping]]: [{{val| 9 1 1 12 }}, {{val| 0 2 3 2 }}]
-{{Multival|legend=1|18 27 18 1 -22 -34}}
+{{Multival|legend=1| 18 27 18 1 -22 -34 }}
 Mapping generators: ~27/25, ~5/3
-[[POTE generator]]s: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
+[[POTE generator]]s: ~5/3 = 884.3129
 [[Tuning ranges]]:
@@ Line 65: / Line 65: @@
 Comma list: 2401/2400, 4375/4374, 5632/5625
-Mapping: [{{val|9 1 1 12 -75}}, {{val|0 2 3 2 16}}]
+Mapping: [{{val| 9 1 1 12 -75 }}, {{val| 0 2 3 2 16 }}]
-POTE generator: ~36/35 = 48.8654
+POTE generator: ~5/3 = 884.4679
 Vals: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
@@ Line 78: / Line 78: @@
 Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
-Mapping: [{{val|9 1 1 12 -75 93}}, {{val|0 2 3 2 16 -9}}]
+Mapping: [{{val| 9 1 1 12 -75 93 }}, {{val| 0 2 3 2 16 -9 }}]
-POTE generator: ~36/35 = 48.9030
+POTE generator: ~5/3 = 884.4304
 Vals: {{Val list| 99e, 171e, 270 }}
@@ Line 93: / Line 93: @@
 Comma list: 2401/2400, 4375/4374, 131072/130977
-Mapping: [{{val|9 1 1 12 124}}, {{val|0 2 3 2 -14}}]
+Mapping: [{{val| 9 1 1 12 124 }}, {{val| 0 2 3 2 -14 }}]
-POTE generator: ~36/35 = 48.9244
+POTE generator: ~5/3 = 884.4089
 Vals: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
@@ Line 106: / Line 106: @@
 Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
-Mapping: [{{val|9 1 1 12 124 93}}, {{val|0 2 3 2 -14 -9}}]
+Mapping: [{{val| 9 1 1 12 124 93 }}, {{val| 0 2 3 2 -14 -9 }}]
-POTE generator: ~36/35 = 48.9336
+POTE generator: ~5/3 = 884.3997
 Vals: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
@@ Line 121: / Line 121: @@
 Comma list: 243/242, 441/440, 4375/4356
-Mapping: [{{val|9 1 1 12 -2}}, {{val|0 2 3 2 5}}]
+Mapping: [{{val| 9 1 1 12 -2 }}, {{val| 0 2 3 2 5 }}]
-POTE generator: ~36/35 = 49.395
+POTE generator: ~5/3 = 883.9386
 Tuning ranges:
@@ Line 139: / Line 139: @@
 Comma list: 243/242, 364/363, 441/440, 625/624
-Mapping: [{{val|9 1 1 12 -2 -33}}, {{val|0 2 3 2 5 10}}]
+Mapping: [{{val| 9 1 1 12 -2 -33 }}, {{val| 0 2 3 2 5 10 }}]
-POTE generator: ~36/35 = 49.341
+POTE generator: ~5/3 = 883.9920
 Tuning ranges:
@@ Line 157: / Line 157: @@
 Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
-Mapping: [{{val|9 1 1 12 -2 -33 -3}}, {{val|0 2 3 2 5 10 6}}]
+Mapping: [{{val| 9 1 1 12 -2 -33 -3 }}, {{val| 0 2 3 2 5 10 6 }}]
-POTE generator: ~36/35 = 49.335
+POTE generator: ~5/3 = 883.9981
 Tuning ranges:
@@ Line 171: / Line 171: @@
 ==== Ennealim ====
-Subgroup: 2.3.5.7.13
+Subgroup: 2.3.5.7.11.13
 Comma list: 169/168, 243/242, 325/324, 441/440
-Mapping: [{{val|9 1 1 12 -2 20}}, {{val|0 2 3 2 5 2}}]
+Mapping: [{{val| 9 1 1 12 -2 20 }}, {{val| 0 2 3 2 5 2 }}]
-POTE generator: ~36/35 = 49.708
+POTE generator: ~5/3 = 883.6257
 Vals: {{Val list| 27e, 45ef, 72 }}
@@ Line 188: / Line 188: @@
 Comma list: 385/384, 1375/1372, 4375/4374
-Mapping: [{{val|9 1 1 12 51}}, {{val|0 2 3 2 -3}}]
+Mapping: [{{val| 9 1 1 12 51 }}, {{val| 0 2 3 2 -3 }}]
-POTE generator: ~36/35 = 49.504
+POTE generator: ~5/3 = 883.8298
 Vals: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
@@ Line 201: / Line 201: @@
 Comma list: 169/168, 325/324, 385/384, 1375/1372
-Mapping: [{{val|9 1 1 12 51 20}}, {{val|0 2 3 2 -3 2}}]
+Mapping: [{{val| 9 1 1 12 51 20 }}, {{val| 0 2 3 2 -3 2 }}]
-POTE generator: ~36/35 = 49.486
+POTE generator: ~5/3 = 883.8476
 Vals: {{Val list| 27, 45f, 72, 171ef, 243ef }}
@@ Line 220: / Line 220: @@
 Mapping generators: ~80/77, ~400/231
-POTE generator: ~99/98 = 17.6219
+POTE generator: ~400/231 = 950.9553
 Tuning ranges:
@@ Line 238: / Line 238: @@
 Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]
-POTE generator ~99/98 = 17.7504
+POTE generator ~26/15 = 951.0837
 Tuning ranges:
@@ Line 261: / Line 261: @@
 Mapping generators: ~80/77, ~1053/800
-POTE generator: ~39/32 = 342.139
+POTE generator: ~1053/800 = 475.4727
 Vals: {{Val list| 126, 144, 270, 684, 954 }}
@@ Line 272: / Line 272: @@
 Comma list: 2401/2400, 4000/3993, 4375/4374
-Mapping: [{{val|9 3 4 14 18}}, {{val|0 6 9 6 7}}]
+Mapping: [{{val| 9 3 4 14 18 }}, {{val| 0 6 9 6 7 }}]
 POTE generator: ~140/121 = 250.3367
@@ Line 285: / Line 285: @@
 Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
-Mapping: [{{val|9 3 4 14 18 -8}}, {{val|0 6 9 6 7 22}}]
+Mapping: [{{val| 9 3 4 14 18 -8 }}, {{val| 0 6 9 6 7 22 }}]
 POTE generator: ~140/121 = 250.3375
@@ Line 298: / Line 298: @@
 Comma list: 2401/2400, 4375/4374, 234375/234256
-Mapping: [{{val|9 1 1 12 -7}}, {{val|0 8 12 8 23}}]
+Mapping: [{{val| 9 1 1 12 -7 }}, {{val| 0 8 12 8 23 }}]
+Mapping generators: ~27/25, ~25/22
-POTE generator: ~77/75 = 45.595
+POTE generator: ~25/22 = 221.0717
 Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
@@ Line 315: / Line 317: @@
 Mapping generators: ~2744/2673, ~2352/1375
-POTE generator: ~6/5 = 315.644
+POTE generator: ~2352/1375 = 928.8000
 Vals: {{Val list| 27, 243, 270, 783, 1053, 1323 }}