|
|
| Line 14: |
Line 14: |
| * ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]] | | * ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]] |
| * ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]] | | * ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]] |
| | * [[Ennealimmal]] (+2401/2400) → [[Tritrizo clan #Ennealimmal|Tritrizo clan]] |
| * ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]] | | * ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]] |
| * [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]] | | * [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]] |
| Line 28: |
Line 29: |
| * ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]] | | * ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]] |
| * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]] | | * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]] |
|
| |
| == 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, 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.
| |
|
| |
| 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]].
| |
|
| |
| 7-limit ennealimmal's S-expression-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}. Interestingly, the [[landscape comma]] is equal to [[2401/2400|S49]]/([[4375/4374|S25/S27]]) while the [[wizma]] is equal to [[2401/2400|S49]]*[[4375/4374|S25/S27]].
| |
|
| |
| ''For the 5-limit temperament, see [[Ennealimma#Ennealimmal]].''
| |
|
| |
| [[Subgroup]]: 2.3.5.7
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|
| |
| [[Comma list]]: 2401/2400, 4375/4374
| |
|
| |
| {{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
| |
|
| |
| {{Multival|legend=1| 18 27 18 1 -22 -34 }}
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|
| |
| : mapping generators: ~27/25, ~5/3
| |
|
| |
| [[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205)
| |
|
| |
| [[Tuning ranges]]:
| |
| * 7-odd-limit [[diamond monotone]]: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
| |
| * 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
| |
| * 7- and 9-odd-limit [[diamond tradeoff]]: ~36/35 = [48.920, 49.179]
| |
| * 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]
| |
|
| |
| {{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
| |
|
| |
| [[Badness]]: 0.003610
| |
|
| |
| === 11-limit ===
| |
| The ennealimmal temperament can be described as 99e & 171e, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).
| |
|
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 2401/2400, 4375/4374, 5632/5625
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
| |
|
| |
| Badness: 0.027332
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
| |
|
| |
| Badness: 0.029404
| |
|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
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|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
| |
|
| |
| ==== Ennealimmalis ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
| |
|
| |
| Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
| |
|
| |
| Badness: 0.022068
| |
|
| |
| === Ennealimmia ===
| |
| The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).
| |
|
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 2401/2400, 4375/4374, 131072/130977
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1251, 2232e }}
| |
|
| |
| Badness: 0.026463
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1692e, 2673e }}
| |
|
| |
| Badness: 0.016607
| |
|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99, 171, 270 }}
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
| |
|
| |
| {{Optimal ET sequence|legend=1| 99, 171, 270 }}
| |
|
| |
| === Ennealimnic ===
| |
| Ennealimnic (72 & 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
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|
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| Subgroup: 2.3.5.7.11
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|
| |
| Comma list: 243/242, 441/440, 4375/4356
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|
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| Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}
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|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)
| |
|
| |
| Tuning ranges:
| |
| * 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
| |
| * 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]
| |
| * 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 171, 243 }}
| |
|
| |
| Badness: 0.020347
| |
|
| |
| See also: [[Chords of ennealimnic]]
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 243/242, 364/363, 441/440, 625/624
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}
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|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414)
| |
|
| |
| Tuning ranges:
| |
| * 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
| |
| * 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]
| |
| * 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 171, 243 }}
| |
|
| |
| Badness: 0.023250
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|
| |
| ===== 17-limit =====
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| Subgroup: 2.3.5.7.11.13.17
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|
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| Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
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|
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| Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}
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|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353)
| |
|
| |
| Tuning ranges:
| |
| * 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
| |
| * 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]
| |
| * 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 171, 243 }}
| |
|
| |
| Badness: 0.014602
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
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|
| |
| Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78 | 0 2 3 2 5 10 6 -6 }}
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 171, 243 }}
| |
|
| |
| ==== Ennealim ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 169/168, 243/242, 325/324, 441/440
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}
| |
|
| |
| Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27e, 45ef, 72 }}
| |
|
| |
| Badness: 0.020697
| |
|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}
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|
| |
| Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}
| |
|
| |
| Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
| |
|
| |
| === Ennealiminal ===
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 385/384, 1375/1372, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}
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|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27, 45, 72, 171e, 243e, 315e }}
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|
| |
| Badness: 0.031123
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 169/168, 325/324, 385/384, 1375/1372
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}
| |
|
| |
| Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27, 45f, 72, 171ef, 243eff }}
| |
|
| |
| Badness: 0.030325
| |
|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}
| |
|
| |
| Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27, 45f, 72 }}
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}
| |
|
| |
| Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
| |
|
| |
| {{Optimal ET sequence|legend=1| 27, 45f, 72 }}
| |
|
| |
| === Hemiennealimmal ===
| |
| Hemiennealimmal (72 & 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 to an octave split into two equal parts. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.
| |
|
| |
| Its S-expression-based comma list is {([[3025/3024|S22/S24 = S55 = S25/S27 * S99]],) [[4375/4374|S25/S27]], [[2401/2400|S49]], [[9801/9800|S33/S35 = S99]]}.
| |
|
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 2401/2400, 3025/3024, 4375/4374
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
| |
|
| |
| : mapping generators: ~80/77, ~400/231
| |
|
| |
| Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553
| |
|
| |
| Tuning ranges:
| |
| * 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
| |
| * 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
| |
| * 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1566 }}
| |
|
| |
| Badness: 0.006283
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}
| |
|
| |
| Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
| |
|
| |
| Tuning ranges:
| |
| * 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
| |
| * 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
| |
| * 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
| |
| * 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
| |
| * 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]
| |
| * 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 198, 270 }}
| |
|
| |
| Badness: 0.012505
| |
|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}
| |
|
| |
| Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 198g, 270 }}
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}
| |
|
| |
| Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 198g, 270 }}
| |
|
| |
| ==== Semihemiennealimmal ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}
| |
|
| |
| : mapping generators: ~80/77, ~1053/800
| |
|
| |
| Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
| |
|
| |
| {{Optimal ET sequence|legend=1| 126, 144, 270, 684, 954 }}
| |
|
| |
| Badness: 0.013104
| |
|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
| |
|
| |
| : mapping generators: ~80/77, ~1053/800
| |
|
| |
| Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
| |
|
| |
| {{Optimal ET sequence|legend=1| 270, 684, 954 }}
| |
|
| |
| Badness: 0.013104
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374
| |
|
| |
| Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
| |
|
| |
| : mapping generators: ~80/77, ~1053/800
| |
|
| |
| Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
| |
|
| |
| {{Optimal ET sequence|legend=1| 270, 684h, 954h, 1224 }}
| |
|
| |
| Badness: 0.013104
| |
|
| |
| === Semiennealimmal ===
| |
| Semiennealimmal tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.
| |
|
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 2401/2400, 4000/3993, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
| |
|
| |
| : mapping generators: ~27/25, ~140/121
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 369, 441 }}
| |
|
| |
| Badness: 0.034196
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
| |
|
| |
| Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375
| |
|
| |
| {{Optimal ET sequence|legend=1| 72, 297ef, 369f, 441 }}
| |
|
| |
| Badness: 0.026122
| |
|
| |
| === Quadraennealimmal ===
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 2401/2400, 4375/4374, 234375/234256
| |
|
| |
| Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
| |
|
| |
| : mapping generators: ~27/25, ~25/22
| |
|
| |
| Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717
| |
|
| |
| {{Optimal ET sequence|legend=1| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
| |
|
| |
| Badness: 0.021320
| |
|
| |
| === Trinealimmal ===
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 2401/2400, 4375/4374, 2097152/2096325
| |
|
| |
| Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
| |
|
| |
| : mapping generators: ~2744/2673, ~2352/1375
| |
|
| |
| Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000
| |
|
| |
| {{Optimal ET sequence|legend=1| 27, 243, 270, 783, 1053, 1323 }}
| |
|
| |
| Badness: 0.029812
| |
|
| |
| === Rhodium ===
| |
| {{Main| Rhodium }}
| |
| 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: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
| |
|
| |
| : mapping generators: ~3072/3025, ~55/32
| |
|
| |
| Optimal tunings:
| |
| * CTE: ~3072/3025 = 1\45, ~55/32 = 937.6658 (~385/384 = 4.3325)
| |
| * CWE: ~3072/3025 = 1\45, ~55/32 = 937.6630 (~385/384 = 4.3397)
| |
|
| |
| Optimal ET sequence: {{Optimal ET sequence| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
| |
|
| |
| Badness: 0.0381
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655
| |
|
| |
| Mapping: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}
| |
|
| |
| Optimal tunings:
| |
| * CTE: ~66/65 = 1\45, ~55/32 = 937.6569 (~385/384 = 4.3236)
| |
| * CWE: ~66/65 = 1\45, ~55/32 = 937.6515 (~385/384 = 4.3182)
| |
|
| |
| Optimal ET sequence: {{Optimal ET sequence| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
| |
|
| |
| Badness: 0.0226
| |
|
| |
|
| == Supermajor == | | == Supermajor == |