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Ragismic microtemperaments: Difference between revisions

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→Ennealimmal: +rhodium
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{{Main| Ennealimmal }}
{{Main| Ennealimmal }}


[[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.  
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, a stack of 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.  


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.
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.
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 "[[tritave]]s" 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.


Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
Line 49: Line 49:
Mapping generators: ~27/25, ~5/3
Mapping generators: ~27/25, ~5/3


[[Optimal tuning]] ([[POTE]]): ~5/3 = 884.3129 (~36/35 = 49.0205)
[[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205)


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 70: Line 70:
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 }}]


Optimal tuning (POTE): ~5/3 = 884.4679 (~36/35 = 48.8654)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)


Optimal GPV sequence: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
Optimal GPV sequence: {{val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}


Badness: 0.027332
Badness: 0.027332
Line 83: Line 83:
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 }}]


Optimal tuning (POTE): ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)


Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{val list| 99e, 171e, 270 }}


Badness: 0.029404
Badness: 0.029404
Line 96: Line 96:
Mapping: [{{val| 9 1 1 12 -75 93 -3 }}, {{val| 0 2 3 2 16 -9 6 }}]
Mapping: [{{val| 9 1 1 12 -75 93 -3 }}, {{val| 0 2 3 2 16 -9 6 }}]


Optimal tuning (POTE): ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)


Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{val list| 99e, 171e, 270 }}


===== 19-limit =====
===== 19-limit =====
Line 107: Line 107:
Mapping: [{{val| 9 1 1 12 -75 93 -3 -48 }}, {{val| 0 2 3 2 16 -9 6 13 }}]
Mapping: [{{val| 9 1 1 12 -75 93 -3 -48 }}, {{val| 0 2 3 2 16 -9 6 13 }}]


Optimal tuning (POTE): ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)


Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{val list| 99e, 171e, 270 }}


==== Ennealimmalis ====
==== Ennealimmalis ====
Line 118: Line 118:
Mapping: [{{val| 9 1 1 12 -75 -106 }}, {{val| 0 2 3 2 16 21 }}]
Mapping: [{{val| 9 1 1 12 -75 -106 }}, {{val| 0 2 3 2 16 21 }}]


Optimal tuning (CTE): ~5/3 = 884.4560 (~36/35 = 48.8773)
Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)


Optimal GPV sequence: {{Val list| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
Optimal GPV sequence: {{val list| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}


Badness: 0.022068
Badness: 0.022068
Line 133: Line 133:
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 }}]


Optimal tuning (POTE): ~5/3 = 884.4089 (~36/35 = 48.9244)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)


Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
Optimal GPV sequence: {{val list| 99, 171, 270, 711, 981, 1251, 2232e }}


Badness: 0.026463
Badness: 0.026463
Line 146: Line 146:
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 }}]


Optimal tuning (POTE): ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)


Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
Optimal GPV sequence: {{val list| 99, 171, 270, 711, 981, 1692e, 2673e }}


Badness: 0.016607
Badness: 0.016607
Line 159: Line 159:
Mapping: [{{val| 9 1 1 12 124 93 -3 }}, {{val| 0 2 3 2 -14 -9 6 }}]
Mapping: [{{val| 9 1 1 12 124 93 -3 }}, {{val| 0 2 3 2 -14 -9 6 }}]


Optimal tuning (POTE): ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)


Optimal GPV sequence: {{Val list| 99, 171, 270 }}
Optimal GPV sequence: {{val list| 99, 171, 270 }}


===== 19-limit =====
===== 19-limit =====
Line 170: Line 170:
Mapping: [{{val| 9 1 1 12 124 93 -3 -48 }}, {{val| 0 2 3 2 -14 -9 6 13 }}]
Mapping: [{{val| 9 1 1 12 124 93 -3 -48 }}, {{val| 0 2 3 2 -14 -9 6 13 }}]


Optimal tuning (POTE): ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)


Optimal GPV sequence: {{Val list| 99, 171, 270 }}
Optimal GPV sequence: {{val list| 99, 171, 270 }}


=== Ennealimnic ===
=== Ennealimnic ===
Ennealimnic (72&amp;171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
Ennealimnic (72 &amp; 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 183: Line 183:
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 }}]


Optimal tuning (POTE): ~5/3 = 883.9386 (~36/35 = 49.3948)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)


Tuning ranges:  
Tuning ranges:  
Line 190: Line 190:
* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]
* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]


Optimal GPV sequence: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{val list| 72, 171, 243 }}


Badness: 0.020347
Badness: 0.020347
Line 201: Line 201:
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 }}]


Optimal tuning (POTE): ~5/3 = 883.9920 (~36/35 = 49.3414)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414)


Tuning ranges:  
Tuning ranges:  
Line 208: Line 208:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]


Optimal GPV sequence: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{val list| 72, 171, 243 }}


Badness: 0.023250
Badness: 0.023250
Line 219: Line 219:
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 }}]


Optimal tuning (POTE): ~5/3 = 883.9981 (~36/35 = 49.3353)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353)


Tuning ranges:  
Tuning ranges:  
Line 226: Line 226:
* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]
* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]


Optimal GPV sequence: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{val list| 72, 171, 243 }}


Badness: 0.014602
Badness: 0.014602
Line 237: Line 237:
Mapping: [{{val| 9 1 1 12 -2 -33 -3 78  }}, {{val| 0 2 3 2 5 10 6 -6 }}]
Mapping: [{{val| 9 1 1 12 -2 -33 -3 78  }}, {{val| 0 2 3 2 5 10 6 -6 }}]


Optimal GPV sequence: {{Val list| 72, 171, 243 }}
Optimal GPV sequence: {{val list| 72, 171, 243 }}


==== Ennealim ====
==== Ennealim ====
Line 246: Line 246:
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 }}]


Optimal tuning (POTE): ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)


Optimal GPV sequence: {{Val list| 27e, 45ef, 72 }}
Optimal GPV sequence: {{val list| 27e, 45ef, 72 }}


Badness: 0.020697
Badness: 0.020697
Line 259: Line 259:
Mapping: [{{val| 9 1 1 12 -2 20 -3 }}, {{val| 0 2 3 2 5 2 6 }}]
Mapping: [{{val| 9 1 1 12 -2 20 -3 }}, {{val| 0 2 3 2 5 2 6 }}]


Optimal tuning (POTE): ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)


Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
Line 270: Line 270:
Mapping: [{{val| 9 1 1 12 -2 20 -3 25 }}, {{val| 0 2 3 2 5 2 6 2 }}]
Mapping: [{{val| 9 1 1 12 -2 20 -3 25 }}, {{val| 0 2 3 2 5 2 6 2 }}]


Optimal tuning (POTE): ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)


Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
Optimal GPV sequence: {{val list| 27eg, 45efg, 72 }}


=== Ennealiminal ===
=== Ennealiminal ===
Line 281: Line 281:
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 }}]


Optimal tuning (POTE): ~5/3 = 883.8298 (~36/35 = 49.5036)
Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)


Optimal GPV sequence: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
Optimal GPV sequence: {{val list| 27, 45, 72, 171e, 243e, 315e }}


Badness: 0.031123
Badness: 0.031123
Line 294: Line 294:
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 }}]


Optimal tuning (POTE): ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)


Optimal GPV sequence: {{Val list| 27, 45f, 72, 171ef, 243eff }}
Optimal GPV sequence: {{val list| 27, 45f, 72, 171ef, 243eff }}


Badness: 0.030325
Badness: 0.030325
Line 307: Line 307:
Mapping: [{{val| 9 1 1 12 51 20 50 }}, {{val| 0 2 3 2 -3 2 -2 }}]
Mapping: [{{val| 9 1 1 12 51 20 50 }}, {{val| 0 2 3 2 -3 2 -2 }}]


Optimal tuning (POTE): ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)


Optimal GPV sequence: {{Val list| 27, 45f, 72 }}
Optimal GPV sequence: {{val list| 27, 45f, 72 }}


===== 19-limit =====
===== 19-limit =====
Line 318: Line 318:
Mapping: [{{val| 9 1 1 12 51 20 50 25 }}, {{val| 0 2 3 2 -3 2 -2 2 }}]
Mapping: [{{val| 9 1 1 12 51 20 50 25 }}, {{val| 0 2 3 2 -3 2 -2 2 }}]


Optimal tuning (POTE): ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)


Optimal GPV sequence: {{Val list| 27, 45f, 72 }}
Optimal GPV sequence: {{val list| 27, 45f, 72 }}


=== Hemiennealimmal ===
=== Hemiennealimmal ===
Hemiennealimmal (72&amp;198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads an octave split into two equal parts.
Hemiennealimmal (72 &amp; 198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads an octave split into two equal parts.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 333: Line 333:
Mapping generators: ~80/77, ~400/231
Mapping generators: ~80/77, ~400/231


Optimal tuning (POTE): ~400/231 = 950.9553
Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553


Tuning ranges:  
Tuning ranges:  
Line 351: Line 351:
Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]
Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]


Optimal tuning (POTE): ~26/15 = 951.0837
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837


Tuning ranges:  
Tuning ranges:  
Line 372: Line 372:
Mapping: [{{val| 18 0 -1 22 48 -19 -12 }}, {{val| 0 2 3 2 1 6 6 }}]
Mapping: [{{val| 18 0 -1 22 48 -19 -12 }}, {{val| 0 2 3 2 1 6 6 }}]


Optimal tuning (POTE): ~26/15 = 951.0837
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837


Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
Optimal GPV sequence: {{val list| 72, 198g, 270 }}


===== 19-limit =====
===== 19-limit =====
Line 383: Line 383:
Mapping: [{{val| 18 0 -1 22 48 -19 -12 48 105 }}, {{val| 0 2 3 2 1 6 6 -2 }}]
Mapping: [{{val| 18 0 -1 22 48 -19 -12 48 105 }}, {{val| 0 2 3 2 1 6 6 -2 }}]


Optimal tuning (POTE): ~26/15 = 951.0837
Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837


Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
Optimal GPV sequence: {{val list| 72, 198g, 270 }}


==== Semihemiennealimmal ====
==== Semihemiennealimmal ====
Line 396: Line 396:
Mapping generators: ~80/77, ~1053/800
Mapping generators: ~80/77, ~1053/800


Optimal tuning (POTE): ~1053/800 = 475.4727
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727


Optimal GPV sequence: {{Val list| 126, 144, 270, 684, 954 }}
Optimal GPV sequence: {{val list| 126, 144, 270, 684, 954 }}


Badness: 0.013104
Badness: 0.013104
Line 413: Line 413:
Mapping generators: ~27/25, ~140/121
Mapping generators: ~27/25, ~140/121


Optimal tuning (POTE): ~140/121 = 250.3367
Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367


Optimal GPV sequence: {{Val list| 72, 369, 441 }}
Optimal GPV sequence: {{val list| 72, 369, 441 }}


Badness: 0.034196
Badness: 0.034196
Line 426: Line 426:
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 }}]


Optimal tuning (POTE): ~140/121 = 250.3375
Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375


Optimal GPV sequence: {{Val list| 72, 297ef, 369f, 441 }}
Optimal GPV sequence: {{val list| 72, 297ef, 369f, 441 }}


Badness: 0.026122
Badness: 0.026122
Line 441: Line 441:
Mapping generators: ~27/25, ~25/22
Mapping generators: ~27/25, ~25/22


Optimal tuning (POTE): ~25/22 = 221.0717
Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717


Optimal GPV sequence: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
Optimal GPV sequence: {{val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}


Badness: 0.021320
Badness: 0.021320
Line 456: Line 456:
Mapping generators: ~2744/2673, ~2352/1375
Mapping generators: ~2744/2673, ~2352/1375


Optimal tuning (POTE): ~2352/1375 = 928.8000
Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000


Optimal GPV sequence: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
Optimal GPV sequence: {{val list| 27, 243, 270, 783, 1053, 1323 }}


Badness: 0.029812
Badness: 0.029812


=== Rhodium ===
=== Rhodium ===
''Rhodium'' splits the ennealimmal period in five parts and thereby features a period of 9 x 5 = 45, thus the name is given after the 45th element.
Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 4375/4374, 117440512/117406179


Mapping: [{{val| 45 1 -1 56 226 }}, {{val| 0 2 3 2 -2 }}]
Mapping generators: ~3072/3025, ~55/32
Optimal tuning (CTE): ~3072/3025 = 1\45, ~55/32 = 937.6658
Optimal GPV sequence: {{val list| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
Badness: 0.0381
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655
Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655


Mapping: [{{val|45 1 -1 56 226 272}}, {{val| 0 2 3 2 -2 -3}}]
Mapping: [{{val| 45 1 -1 56 226 272 }}, {{val| 0 2 3 2 -2 -3 }}]


Optimal tuning (CTE): ~66/65 = 1\45, ~55/32 = 937.657
Optimal tuning (CTE): ~66/65 = 1\45, ~55/32 = 937.657


Vals: {{EDOs|45, 270, 855, 1125, 1395, 1665}}, ...
Optimal GPV sequence: {{val list| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
 
Badness: 0.0226


== Supermajor ==
== Supermajor ==
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"