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| Line 21: |
Line 21: |
| * ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]] | | * ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]] |
|
| |
|
| The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Kwai tempers out [[5120/5103]], splitting the generator in ten. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen. | | The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Kwai tempers out [[5120/5103]], splitting the generator in ten. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen. |
|
| |
|
| Members of the clan discussed elsewhere are: | | Members of the clan discussed elsewhere are: |
| Line 27: |
Line 27: |
| * ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]] | | * ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]] |
| * [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]] | | * [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]] |
| | * ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]] |
| * ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]] | | * ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]] |
| * ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]] | | * ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]] |
| * ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]] | | * ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]] |
| * [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]] | | * [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]] |
| | * ''[[Kwai]]'' (+16875/16807) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]] |
| * ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]] | | * ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]] |
| * ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]] | | * ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]] |
| Line 38: |
Line 40: |
| For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]]. | | For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]]. |
|
| |
|
| Considered below are grendel, kwai, pluto, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer. | | Considered below are grendel, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer. |
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| |
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| == Grendel == | | == Grendel == |
| Line 119: |
Line 121: |
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| Badness (Sintel): 1.12 | | Badness (Sintel): 1.12 |
|
| |
| == Kwai ==
| |
| : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
| |
|
| |
| Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.
| |
|
| |
| [[Subgroup]]: 2.3.5.7
| |
|
| |
| [[Comma list]]: 5120/5103, 16875/16807
| |
|
| |
| {{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
| |
| : mapping generators: ~2, ~3
| |
|
| |
| [[Optimal tuning]]s:
| |
| * [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
| |
| : [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
| |
| * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
| |
| : error map: {{val| 0.000 +0.653 -0.234 +1.603 }}
| |
|
| |
| {{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
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|
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| [[Badness]] (Sintel): 1.38
| |
|
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| === 11-limit ===
| |
| Subgroup: 2.3.5.7.11
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|
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| Comma list: 540/539, 1375/1372, 5120/5103
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|
| |
| Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
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|
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| Optimal tunings:
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| * WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}
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|
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| {{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}
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|
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| Badness (Sintel): 0.867
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|
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| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
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|
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| Comma list: 352/351, 540/539, 729/728, 1375/1372
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|
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| Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
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|
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| Optimal tunings:
| |
| * WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}
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|
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| {{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}
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|
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| Badness (Sintel): 1.01
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|
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| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
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|
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| Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088
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|
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| Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
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|
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| Optimal tunings:
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| * WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}
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|
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| {{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}
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|
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| Badness (Sintel): 1.12
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|
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| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
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|
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| Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
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|
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| Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
| |
|
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| Optimal tunings:
| |
| * WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}
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|
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| {{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
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|
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| Badness (Sintel): 1.03
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|
| |
| ==== Hemikwai ====
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| Subgroup: 2.3.5.7.11.13
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|
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| Comma list: 540/539, 676/675, 1375/1372, 5120/5103
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|
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| Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
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| : mapping generators: ~2, ~26/15
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|
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| Optimal tunings:
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| * WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}
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|
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| {{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
| |
|
| |
| Badness (Sintel): 1.82
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|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
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|
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| Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
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|
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| Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
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|
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| Optimal tunings:
| |
| * WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
| |
|
| |
| Badness (Sintel): 1.31
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|
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| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
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|
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| Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
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|
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| Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
| |
|
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| Optimal tunings:
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| * WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}
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|
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| {{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
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|
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| Badness (Sintel): 1.16
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|
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| == Pluto ==
| |
| {{Distinguish| Plutus }}
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|
| |
| Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[10/7]], and [[121edo|59\121]] is about perfect as a tuning.
| |
|
| |
| [[Subgroup]]: 2.3.5.7
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|
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| [[Comma list]]: 4000/3969, 10976/10935
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|
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| {{Mapping|legend=1| 1 -2 -11 -10 | 0 7 26 25 }}
| |
| : mapping generators: ~2, ~10/7
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|
| |
| [[Optimal tuning]]s:
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| * [[WE]]: ~2 = 1199.4720{{c}}, ~10/7 = 614.5826{{c}}
| |
| : [[error map]]: {{val| -0.528 +1.179 -1.358 +1.020 }}
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| * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.8282{{c}}
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| : error map: {{val| 0.000 +1.843 -0.780 +1.880 }}
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|
| |
| {{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 283d, 404bd }}
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|
| |
| [[Badness]] (Sintel): 1.46
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|
| |
| === 11-limit ===
| |
| Subgroup: 2.3.5.7.11
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|
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| Comma list: 540/539, 896/891, 1375/1372
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|
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| Mapping: {{mapping| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}
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|
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| Optimal tunings:
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| * WE: ~2 = 1200.1835{{c}}, ~10/7 = 614.4677{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8616{{c}}
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|
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| {{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
| |
|
| |
| Badness (Sintel): 0.987
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|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
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|
| |
| Comma list: 325/324, 352/351, 364/363, 540/539
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|
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| Mapping: {{mapping| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}
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|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.2469{{c}}, ~10/7 = 614.4908{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8684{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
| |
|
| |
| Badness (Sintel): 1.06
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|
| |
| ===== 17-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
| |
|
| |
| Mapping: {{mapping| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.2009{{c}}, ~10/7 = 614.4746{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8813{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
| |
|
| |
| Badness (Sintel): 1.09
| |
|
| |
| ===== 19-limit =====
| |
| Subgroup: 2.3.5.7.11.13.17.19
| |
|
| |
| Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
| |
|
| |
| Mapping: {{mapping| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.2833{{c}}, ~10/7 = 614.5237{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8876{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
| |
|
| |
| Badness (Sintel): 1.07
| |
|
| |
| ==== Orcus ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 144/143, 196/195, 275/273, 896/891
| |
|
| |
| Mapping: {{mapping| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1198.7372{{c}}, ~10/7 = 614.2418{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8502{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 39df, 41, 80f, 121ff }}
| |
|
| |
| Badness (Sintel): 1.38
| |
|
| |
| === Plutino ===
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 100/99, 245/242, 10976/10935
| |
|
| |
| Mapping: {{mapping| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.7793{{c}}, ~10/7 = 614.6040{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7074{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 39dee, 41 }}
| |
|
| |
| Badness (Sintel): 1.92
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 100/99, 196/195, 245/242, 729/728
| |
|
| |
| Mapping: {{mapping| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.0328{{c}}, ~10/7 = 614.2729{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7277{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 39deef, 41 }}
| |
|
| |
| Badness (Sintel): 1.66
| |
|
| |
|
| == Mirkat == | | == Mirkat == |