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Miscellaneous 5-limit temperaments: Difference between revisions

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* Individual [[temperament families and clans]]
* Individual [[temperament families and clans]]
* Individual [[equivalence continuum|equivalence continua]]
* Individual [[equivalence continuum|equivalence continua]]
* [[Fractional-octave temperaments]]
* [[Very low accuracy temperaments]]
* [[Very low accuracy temperaments]]
* [[Very high accuracy temperaments]]
* [[Very high accuracy temperaments]]
== Ennealimmal ==
: ''For extensions, see [[Septiennealimmal clan #Ennealimmal]].''
The 5-limit version of ennealimmal tempers out the [[ennealimma]], the amount by which a stack of nine [[27/25]]'s falls short of the octave. Since the [[7-limit]] temperament (definable by tempering out both [[2401/2400]] and [[4375/4374]], the two smallest superparticular ratios in the 7-limit) is far more natural to think of than the 5-limit, the 5-limit temperament is only provided below for bookkeeping purposes.
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 1 -27 18 }}
{{Mapping|legend=1| 9 1 1 | 0 2 3 }}
: mapping generators: ~27/25, ~5/3
[[Optimal tuning]]s:
* [[WE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3190{{c}}
: [[error map]]: {{val| -0.000 +0.016 -0.023 }}
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3191{{c}}
: error map: {{val| 0.000 +0.017 -0.023 }}
{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612, 1665, 2277, 2889, 3501, 6390, 16281, 22671, 29061c }}
[[Badness]] (Sintel): 0.403


== Unidecmic ==
== Unidecmic ==
Line 26: Line 47:


[[Badness]] (Sintel): 1.93
[[Badness]] (Sintel): 1.93
== Quasithird ==
: ''For extensions, see [[Ragismic microtemperaments #Quasithird]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 55 -64 20 }}
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
: mapping generators: ~51200000/43046721, ~1594323/1280000
[[Optimal tuning]]s:
* [[WE]]: ~51200000/43046721 = 299.9990{{c}}, ~1594323/1280000 = 380.3934{{c}}
: [[error map]]: {{val| -0.004 +0.012 -0.008 }}
* [[CWE]]: ~51200000/43046721 = 300.0000{{c}}, ~1594323/1280000 = 380.3944{{c}}
: error map: {{val| 0.000 +0.017 -0.004 }}
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
[[Badness]] (Sintel): 2.33


== Supersharp ==
== Supersharp ==
Line 70: Line 111:


== Lafa ==
== Lafa ==
This temperament was named by [[Petr Pařízek]] in 2011, referring to the characteristic that stacking 12 generators makes 6/1 – "l" for 12, "f" for 6<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. Its ploidacot is therefore beta-dodecacot.  
This temperament was named by [[Petr Pařízek]] in 2011, referring to the characteristic that stacking 12 generators makes 6/1 – "l" for 12, "f" for 6<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. Its ploidacot is therefore beta-dodecacot.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 99: Line 140:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~25/24 = 138.722
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/24 = 138.722{{c}}
* [[POTE]]: ~2 = 1200.000, ~25/24 = 140.995
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/24 = 140.995{{c}}


{{Optimal ET sequence|legend=1| 8, 9, 17c }}
{{Optimal ET sequence|legend=1| 8, 9, 17c }}
Line 106: Line 147:
[[Badness]] (Sintel): 4.70
[[Badness]] (Sintel): 4.70


== Stump ==
== Neon ==
: ''For extensions, see [[Sensamagic clan #Pycnic]] and [[Marvel temperaments #Triton]].''
: ''For extensions, see [[Ragismic microtemperaments #Deca]], [[Landscape microtemperaments #Zinc]], and [[20th-octave temperaments #Calcium]].''  
{{See also| Neon comma }}
 
Neon tempers out {{monzo| 21 60 -50 }} in the 5-limit, equating [[3125/2916]] with one step of [[10edo]]. It was named by [[Eliora]] in 2023 after the 10th element.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 273375/262144
[[Comma list]]: {{monzo| 21 60 -50 }}


{{Mapping|legend=1| 1 0 6 | 0 3 -7 }}
{{Mapping|legend=1| 10 4 9 | 0 5 6 }}
: mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~64/45 = 631.504
* [[WE]]: ~3125/2916 = 119.9999{{c}}, {{monzo| 10 29 -24 }} = 284.3890{{c}}
* [[POTE]]: ~2 = 1200.000, ~64/45 = 631.737
: [[error map]]: {{val| -0.001 -0.011 +0.019 }}
* [[CWE]]: ~3125/2916 = 120.0000{{c}}, {{monzo| 10 29 -24 }} = 284.3891{{c}}
: error map: {{val| 0.000 -0.010 +0.021 }}


{{Optimal ET sequence|legend=1| 17, 19, 207bbccc }}
{{Optimal ET sequence|legend=1| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }}


[[Badness]] (Sintel): 4.71
[[Badness]] (Sintel): 4.85


== Dodifo ==
== Dodifo ==
Line 133: Line 180:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~768/625 = 357.087
* [[CTE]]: ~2 = 1200.000{{c}}, ~768/625 = 357.087{{c}}
* [[POTE]]: ~2 = 1200.000, ~768/625 = 357.088
* [[POTE]]: ~2 = 1200.000{{c}}, ~768/625 = 357.088{{c}}


{{Optimal ET sequence|legend=1| 37, 84, 205, 289, 494, 783, 2060, 2843, 3626 }}
{{Optimal ET sequence|legend=1| 37, 84, 205, 289, 494, 783, 2060, 2843, 3626 }}
Line 148: Line 195:


{{Mapping|legend=1| 2 1 3 | 0 4 3 }}
{{Mapping|legend=1| 2 1 3 | 0 4 3 }}
: mapping generators: ~625/432, ~6/5
: mapping generators: ~625/432, ~6/5


Line 171: Line 217:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 321.773
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.773{{c}}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 321.852
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.852{{c}}


{{Optimal ET sequence|legend=1| 15, 26, 41, 179c }}
{{Optimal ET sequence|legend=1| 15, 26, 41, 179c }}
Line 188: Line 234:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~162/125 = 448.449
* [[CTE]]: ~2 = 1200.000{{c}}, ~162/125 = 448.449{{c}}
* [[POTE]]: ~2 = 1200.000, ~162/125 = 448.449
* [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 448.449{{c}}


{{Optimal ET sequence|legend=1| 8, …, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc }}
{{Optimal ET sequence|legend=1| 8, …, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc }}
Line 205: Line 251:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 310.030
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 310.030{{c}}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 310.140
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 310.140{{c}}


{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
Line 222: Line 268:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~45/32 = 638.922
* [[CTE]]: ~2 = 1200.000{{c}}, ~45/32 = 638.922{{c}}
* [[POTE]]: ~2 = 1200.000, ~45/32 = 638.736
* [[POTE]]: ~2 = 1200.000{{c}}, ~45/32 = 638.736{{c}}


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 62bc }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 62bc }}
Line 239: Line 285:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~{{monzo| -12 24 -11 }} = 600.000, ~200000/177147 = 208.907
* [[CTE]]: ~{{monzo| -12 24 -11 }} = 600.000{{c}}, ~200000/177147 = 208.907{{c}}
* [[CWE]]: ~{{monzo| -12 24 -11 }} = 600.000, ~200000/177147 = 208.906
* [[CWE]]: ~{{monzo| -12 24 -11 }} = 600.000{{c}}, ~200000/177147 = 208.906{{c}}


{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 1752, 2246, 2740, 4986, 7726b }}
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 1752, 2246, 2740, 4986, 7726b }}
Line 256: Line 302:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~625/486 = 435.081
* [[CTE]]: ~2 = 1200.000{{c}}, ~625/486 = 435.081{{c}}
* [[CWE]]: ~2 = 1200.000, ~625/486 = 435.081
* [[CWE]]: ~2 = 1200.000{{c}}, ~625/486 = 435.081{{c}}


{{Optimal ET sequence|legend=1| 80, 171, 593, 764, 935, 1106, 3489, 4595, 5701 }}
{{Optimal ET sequence|legend=1| 80, 171, 593, 764, 935, 1106, 3489, 4595, 5701 }}
Line 268: Line 314:
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 12207031250000000000/12157665459056928801
[[Comma list]]: {{monzo| 10 -40 23 }}


{{Mapping|legend=1| 1 6 10 | 0 -23 -40 }}
{{Mapping|legend=1| 1 6 10 | 0 -23 -40 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~2500/2187 = 230.345
* [[CTE]]: ~2 = 1200.000{{c}}, ~2500/2187 = 230.345{{c}}
* [[CWE]]: ~2 = 1200.000, ~2500/2187 = 230.345
* [[CWE]]: ~2 = 1200.000{{c}}, ~2500/2187 = 230.345{{c}}


{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 547, 870 }}
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 547, 870 }}
Line 290: Line 336:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~864/625 = 559.328
* [[CTE]]: ~2 = 1200.000{{c}}, ~864/625 = 559.328{{c}}
* [[POTE]]: ~2 = 1200.000, ~864/625 = 559.332
* [[POTE]]: ~2 = 1200.000{{c}}, ~864/625 = 559.332{{c}}


{{Optimal ET sequence|legend=1| 15, 88, 103, 118, 959, 1077 }}
{{Optimal ET sequence|legend=1| 15, 88, 103, 118, 959, 1077 }}


[[Badness]] (Sintel): 6.65
[[Badness]] (Sintel): 6.65
== Octoid ==
: ''For extensions, see [[Ragismic microtemperaments #Octoid]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -5 -32 24 }}
{{Mapping|legend=1| 8 1 3 | 0 3 4 }}
: Mapping generators: ~2125764/1953125, ~4374/3125
[[Optimal tuning]]s:
* [[WE]]: ~2125764/1953125 = 150.001{{c}}, ~4374/3125 = 584.027{{c}}
* [[CWE]]: ~2125764/1953125 = 150.000{{c}}, ~4374/3125 = 584.025{{c}}
{{Optimal ET sequence|legend=1| 8, 56bcc, 64c, 72, 152, 224, 376, 976 }}
[[Badness]] (Sintel): 6.69


== Kangaroo ==
== Kangaroo ==
Line 307: Line 371:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~40/27 = 637.879
* [[CTE]]: ~2 = 1200.000{{c}}, ~40/27 = 637.879{{c}}
* [[POTE]]: ~2 = 1200.000, ~40/27 = 638.805
* [[POTE]]: ~2 = 1200.000{{c}}, ~40/27 = 638.805{{c}}


{{Optimal ET sequence|legend=1| 15, 47b, 62b }}
{{Optimal ET sequence|legend=1| 15, 47b, 62b }}
Line 324: Line 388:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~17578125/16777216 = 77.186
* [[CTE]]: ~2 = 1200.000{{c}}, ~17578125/16777216 = 77.186{{c}}
* [[POTE]]: ~2 = 1200.000, ~17578125/16777216 = 77.191
* [[POTE]]: ~2 = 1200.000{{c}}, ~17578125/16777216 = 77.191{{c}}


{{Optimal ET sequence|legend=1| 31, 109, 140, 171, 1679c, 1850c, 2021c, 2192c, 2363c, 2534c, 2705c, 2876c, 3047c }}
{{Optimal ET sequence|legend=1| 31, 109, 140, 171, 1679c, 1850c, 2021c, 2192c, 2363c, 2534c, 2705c, 2876c, 3047c }}
Line 341: Line 405:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~45/32 = 588.727
* [[CTE]]: ~2 = 1200.000{{c}}, ~45/32 = 588.727{{c}}
* [[POTE]]: ~2 = 1200.000, ~45/32 = 588.722
* [[POTE]]: ~2 = 1200.000{{c}}, ~45/32 = 588.722{{c}}


{{Optimal ET sequence|legend=1| 51, 53, 267, 320, 373, 426, 479, 532, 2607bc }}
{{Optimal ET sequence|legend=1| 51, 53, 267, 320, 373, 426, 479, 532, 2607bc }}


[[Badness]] (Sintel): 7.20
[[Badness]] (Sintel): 7.20
== Counterhanson ==
: ''For extensions, see [[Ragismic microtemperaments #Counterkleismic]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -20 -24 25 }}
{{Mapping|legend=1| 1 -5 -4 | 0 25 24 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~6/5 = 316.084
* [[POTE]]: ~2 = 1200.000, ~6/5 = 316.081
{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
[[Badness]] (Sintel): 7.45


== Nessafof ==
== Nessafof ==
Line 375: Line 422:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~78125/62208 = 400.000, ~9765625/8957952 = 157.392
* [[CTE]]: ~78125/62208 = 400.000{{c}}, ~9765625/8957952 = 157.392{{c}}
* [[CWE]]: ~78125/62208 = 400.000, ~9765625/8957952 = 157.409
* [[CWE]]: ~78125/62208 = 400.000{{c}}, ~9765625/8957952 = 157.409{{c}}


{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 183, 282, 465, 1212c, 1677cc }}
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 183, 282, 465, 1212c, 1677cc }}
Line 390: Line 437:


{{Mapping|legend=1| 2 2 5 | 0 3 -1 }}
{{Mapping|legend=1| 2 2 5 | 0 3 -1 }}
: Mapping generators: ~375/256, ~75/64
: Mapping generators: ~375/256, ~75/64


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~375/256 = 600.000, ~75/64 = 232.986
* [[CTE]]: ~375/256 = 600.000{{c}}, ~75/64 = 232.986{{c}}
: [[error map]]: {{val| 0.000 -2.998 -19.300 }}
: [[error map]]: {{val| 0.000 -2.998 -19.300 }}
* [[CWE]]: ~375/256 = 600.000, ~75/64 = 231.832
* [[CWE]]: ~375/256 = 600.000{{c}}, ~75/64 = 231.832{{c}}
: error map: {{val| 0.000 -6.458 -18.146 }}
: error map: {{val| 0.000 -6.458 -18.146 }}


Line 413: Line 459:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 284.817
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 284.817{{c}}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 284.293
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 284.293{{c}}


{{Optimal ET sequence|legend=1| 4, 13c, 17c, 21, 38c }}
{{Optimal ET sequence|legend=1| 4, 13c, 17c, 21, 38c }}
Line 430: Line 476:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~125/96 = 459.006
* [[CTE]]: ~2 = 1200.000{{c}}, ~125/96 = 459.006{{c}}
* [[POTE]]: ~2 = 1200.000, ~125/96 = 458.995
* [[POTE]]: ~2 = 1200.000{{c}}, ~125/96 = 458.995{{c}}


{{Optimal ET sequence|legend=1| 34, 115, 149, 183, 217, 400, 617 }}
{{Optimal ET sequence|legend=1| 34, 115, 149, 183, 217, 400, 617 }}
Line 447: Line 493:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~512/405 = 356.492
* [[CTE]]: ~2 = 1200.000{{c}}, ~512/405 = 356.492{{c}}
* [[POTE]]: ~2 = 1200.000, ~512/405 = 355.930
* [[POTE]]: ~2 = 1200.000{{c}}, ~512/405 = 355.930{{c}}


{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
Line 464: Line 510:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 391.068
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 391.068{{c}}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 391.225
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 391.225{{c}}


{{Optimal ET sequence|legend=1| 3, …, 40, 43, 46, 181c, 227c, 273c, 319c }}
{{Optimal ET sequence|legend=1| 3, …, 40, 43, 46, 181c, 227c, 273c, 319c }}


[[Badness]] (Sintel): 8.45
[[Badness]] (Sintel): 8.45
== Countermeantone ==
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 10 23 -20 }}
{{Mapping|legend=1| 1 -10 -11 | 0 20 23 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~78125/52488 = 695.082
* [[POTE]]: ~2 = 1200.000, ~78125/52488 = 695.087
{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}
[[Badness]] (Sintel): 8.76


== Tritonic ==
== Tritonic ==
Line 496: Line 527:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~45/32 = 580.278
* [[CTE]]: ~2 = 1200.000{{c}}, ~45/32 = 580.278{{c}}
* [[POTE]]: ~2 = 1200.000, ~45/32 = 580.219
* [[POTE]]: ~2 = 1200.000{{c}}, ~45/32 = 580.219{{c}}


{{Optimal ET sequence|legend=1| 29, 31, 60, 91 }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91 }}
Line 511: Line 542:


{{Mapping|legend=1| 2 1 5 | 0 6 -1 }}
{{Mapping|legend=1| 2 1 5 | 0 6 -1 }}
: mapping generators: ~42875/32768, ~9375/8192
: mapping generators: ~42875/32768, ~9375/8192


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~42875/32768 = 600.000, ~9375/8192 = 216.950
* [[CTE]]: ~42875/32768 = 600.000{{c}}, ~9375/8192 = 216.950{{c}}
* [[POTE]]: ~42875/32768 = 600.000, ~9375/8192 = 216.788
* [[POTE]]: ~42875/32768 = 600.000{{c}}, ~9375/8192 = 216.788{{c}}


{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 548bcc }}
{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 548bcc }}
Line 523: Line 553:


Scales (Scala files): [[wizard22]]
Scales (Scala files): [[wizard22]]
== Hogzilla ==
: ''For extensions, see [[Semaphoresmic clan #Helayo]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: 4428675/4194304
{{Mapping|legend=1| 1 0 11 | 0 2 -11 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~2048/1215 = 946.982
* [[POTE]]: ~2 = 1200.000, ~2048/1215 = 947.351
{{Optimal ET sequence|legend=1| 14, 19 }}
[[Badness]] (Sintel): 9.96


== Merman ==
== Merman ==
Line 551: Line 564:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~45/32 = 585.653
* [[CTE]]: ~2 = 1200.000{{c}}, ~45/32 = 585.653{{c}}
* [[POTE]]: ~2 = 1200.000, ~45/32 = 585.634
* [[POTE]]: ~2 = 1200.000{{c}}, ~45/32 = 585.634{{c}}


{{Optimal ET sequence|legend=1| 41, 84, 125, 209, 334c, 543bc }}
{{Optimal ET sequence|legend=1| 41, 84, 125, 209, 334c, 543bc }}
Line 558: Line 571:
[[Badness]] (Sintel): 10.6
[[Badness]] (Sintel): 10.6


== Unsmate ==
== Hendecatonic ==
: ''For extensions, see [[Porwell temperaments #Hendecatonic]] and [[Marvel temperaments #Hendeca]].''
 
The 5-limit version of hendecatonic simply equates a stack of eleven [[16/15]]'s with the [[octave]].
 
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 17578125/16777216
[[Comma list]]: 8796093022208/8649755859375


{{Mapping|legend=1| 1 3 2 | 0 -9 2 }}
{{Mapping|legend=1| 11 0 43 | 0 1 -1 }}
: Mapping generators: ~16/15, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~2048/1875 = 188.773
* [[WE]]: ~16/15 = 109.0904{{c}}, ~3/2 = 702.6861{{c}}
* [[POTE]]: ~2 = 1200.000, ~2048/1875 = 189.049
: [[error map]]: {{val| -0.445 +0.286 +0.613 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9082{{c}}
: error map: {{val| 0.000 +0.953 +1.687 }}


{{Optimal ET sequence|legend=1| 13, 19, 89c, 108c, 127c, 146cc }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 869bcc, 968bcc, 1067bccc, 1166bccc }}


[[Badness]] (Sintel): 10.8
[[Badness]] (Sintel): 10.7


== Nusecond ==
== Nusecond ==
Line 583: Line 603:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3125/2916 = 154.577
* [[CTE]]: ~2 = 1200.000{{c}}, ~3125/2916 = 154.577{{c}}
* [[POTE]]: ~2 = 1200.000, ~3125/2916 = 154.523
* [[POTE]]: ~2 = 1200.000{{c}}, ~3125/2916 = 154.523{{c}}


{{Optimal ET sequence|legend=1| 8, 23, 31, 70, 101, 132c }}
{{Optimal ET sequence|legend=1| 8, 23, 31, 70, 101, 132c }}
Line 621: Line 641:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~1594323/1562500 = 26.2323
* [[CTE]]: ~2 = 1200.0000{{c}}, ~1594323/1562500 = 26.2323{{c}}
* [[POTE]]: ~2 = 1200.0000, ~1594323/1562500 = 26.2342
* [[POTE]]: ~2 = 1200.0000{{c}}, ~1594323/1562500 = 26.2342{{c}}


{{Optimal ET sequence|legend=1| 45, 46, 91, 137, 183, 869, 1052, 1235, 1418c }}
{{Optimal ET sequence|legend=1| 45, 46, 91, 137, 183, 869, 1052, 1235, 1418c }}
Line 630: Line 650:
== Thuja ==
== Thuja ==
: ''For extensions, see [[Buzzardsmic clan #Thuja]].''
: ''For extensions, see [[Buzzardsmic clan #Thuja]].''
{{See also| Hypovishnuzma }}


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 638: Line 659:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3125/2304 = 558.404
* [[CTE]]: ~2 = 1200.000{{c}}, ~3125/2304 = 558.404{{c}}
* [[POTE]]: ~2 = 1200.000, ~3125/2304 = 558.592
* [[POTE]]: ~2 = 1200.000{{c}}, ~3125/2304 = 558.592{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=1| 15, 43, 58 }}


[[Badness]] (Sintel): 12.0
[[Badness]] (Sintel): 12.0
== Sesesix ==
: ''For extensions, see [[Horwell temperaments #Fifthplus]].''
Sesesix is the 22 & 171 temperament, named by [[Petr Pařízek]] in 2011 for 36 generators giving a major sixth<ref name="petr's long post"/>.
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -74 13 23 }}
{{Mapping|legend=1| 1 -12 10 | 1 23 -13 }}
: mapping generators: ~2, ~6328125/4094304
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0899{{c}}, ~6328125/4194304 = 708.8258{{c}}
: [[error map]]: {{val| +0.090 -0.040 -0.151 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6328125/4194304 = 708.7736{{c}}
: error map: {{val| 0.000 -0.161 -0.371 }}
{{Optimal ET sequence|legend=1| 22, 105, 127, 149, 171, 4468bcc, 4639bcc, …, 5323bccc }}
[[Badness]] (Sintel): 12.2


== Leapday ==
== Leapday ==
Line 655: Line 698:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 704.100
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.100{{c}}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 704.179
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.179{{c}}


{{Optimal ET sequence|legend=1| 17c, 29, 46, 121, 167, 455bc, 622bbc }}
{{Optimal ET sequence|legend=1| 17c, 29, 46, 121, 167, 455bc, 622bbc }}
Line 672: Line 715:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~390625/279936 = 600.000, ~6/5 = 312.636
* [[CTE]]: ~390625/279936 = 600.000{{c}}, ~6/5 = 312.636{{c}}
* [[POTE]]: ~390625/279936 = 600.000, ~6/5 = 312.713
* [[POTE]]: ~390625/279936 = 600.000{{c}}, ~6/5 = 312.713{{c}}


{{Optimal ET sequence|legend=1| 4, …, 42, 46, 96, 142, 330c, 472cc }}
{{Optimal ET sequence|legend=1| 4, …, 42, 46, 96, 142, 330c, 472cc }}
Line 689: Line 732:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9765625/7077888 = 551.777
* [[CTE]]: ~2 = 1200.000{{c}}, ~9765625/7077888 = 551.777{{c}}
* [[POTE]]: ~2 = 1200.000, ~9765625/7077888 = 551.784
* [[POTE]]: ~2 = 1200.000{{c}}, ~9765625/7077888 = 551.784{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}
Line 706: Line 749:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~16/15 = 113.233
* [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 113.233{{c}}
* [[POTE]]: ~2 = 1200.000, ~16/15 = 113.240
* [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 113.240{{c}}


{{Optimal ET sequence|legend=1| 21, 32, 53, 392, 445, 498, 551, 604, 657c, 710c, 1367bcc }}
{{Optimal ET sequence|legend=1| 21, 32, 53, 392, 445, 498, 551, 604, 657c, 710c, 1367bcc }}
Line 723: Line 766:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~192/125 = 735.203
* [[CTE]]: ~2 = 1200.000{{c}}, ~192/125 = 735.203{{c}}
* [[POTE]]: ~2 = 1200.000, ~192/125 = 735.146
* [[POTE]]: ~2 = 1200.000{{c}}, ~192/125 = 735.146{{c}}


{{Optimal ET sequence|legend=1| 18, 31, 80, 111, 586cc, 697ccc }}
{{Optimal ET sequence|legend=1| 18, 31, 80, 111, 586cc, 697ccc }}


[[Badness]] (Sintel): 14.8
[[Badness]] (Sintel): 14.8
== Mowgli ==
: ''For extensions, see [[Hemimean clan #Mowglic]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: 31381059609/30517578125
{{Mapping|legend=1| 1 0 0 | 0 15 22 }}
[[Optimal tuning]]s:
* [[CTE]] and [[POTE]]: ~2 = 1200.000, ~27/25 = 126.724
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}
[[Badness]] (Sintel): 15.3


== Septimin ==
== Septimin ==
Line 756: Line 783:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~75/64 = 263.571
* [[CTE]]: ~2 = 1200.000{{c}}, ~75/64 = 263.571{{c}}
* [[POTE]]: ~2 = 1200.000, ~75/64 = 263.658
* [[POTE]]: ~2 = 1200.000{{c}}, ~75/64 = 263.658{{c}}


{{Optimal ET sequence|legend=1| 9, 23b, 32, 41, 91, 132, 223c, 355bcc }}
{{Optimal ET sequence|legend=1| 9, 23b, 32, 41, 91, 132, 223c, 355bcc }}
Line 773: Line 800:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~27/25 = 72.659
* [[CTE]]: ~2 = 1200.000{{c}}, ~27/25 = 72.659{{c}}
* [[POTE]]: ~2 = 1200.000, ~27/25 = 72.742
* [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 72.742{{c}}


{{Optimal ET sequence|legend=1| 16, 17c, 33 }}
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}
Line 781: Line 808:


== Pentadecal ==
== Pentadecal ==
: ''For extensions, see [[Cloudy clan #Pentadecal]].''
: ''For extensions, see [[Keemic temperaments #Pentadecal]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 790: Line 817:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~648/625 = 80.000, ~3/2 = 703.340
* [[CTE]]: ~648/625 = 80.000{{c}}, ~3/2 = 703.340{{c}}
* [[POTE]]: ~648/625 = 80.000, ~3/2 = 702.652
* [[POTE]]: ~648/625 = 80.000{{c}}, ~3/2 = 702.652{{c}}


{{Optimal ET sequence|legend=1| 15, 45, 60, 75, 135, 210c }}
{{Optimal ET sequence|legend=1| 15, 45, 60, 75, 135, 210c }}
Line 807: Line 834:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.303, ~5/4 = 385.024
* [[WE]]: ~2 = 1200.303{{c}}, ~5/4 = 385.024{{c}}
* [[CWE]]: ~2 = 1200.000, ~5/4 = 384.920
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 384.920{{c}}


{{Optimal ET sequence|legend=1| 25, 28, 53 }}
{{Optimal ET sequence|legend=1| 25, 28, 53 }}
Line 824: Line 851:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~512/405 = 425.590
* [[CTE]]: ~2 = 1200.000{{c}}, ~512/405 = 425.590{{c}}
* [[POTE]]: ~2 = 1200.000, ~512/405 = 425.474
* [[POTE]]: ~2 = 1200.000{{c}}, ~512/405 = 425.474{{c}}


{{Optimal ET sequence|legend=1| 14, 17, 31, 79, 110, 361bbcc }}
{{Optimal ET sequence|legend=1| 14, 17, 31, 79, 110, 361bbcc }}


[[Badness]] (Sintel): 17.0
[[Badness]] (Sintel): 17.0
== Undeka ==
: ''For extensions, see [[Keemic temperaments #Undeka]].''
Undeka has a period of 1/11 octave, like [[#Hendecatonic|hendecatonic]], but this temperament equates a stack of eleven 6/5's and three [[octave]]s, so that three periods represent [[6/5]].
[[Subgroup]]: 2.3.5
[[Comma list]]: 48828125/45349632
{{Mapping|legend=1| 11 0 8 | 0 1 1 }}
: Mapping generators: ~648/625, ~3
[[Optimal tuning]]s:
* [[WE]]: ~648/625 = 109.1809{{c}}, ~3/2 = 704.4091{{c}}
: [[error map]]: {{val| +0.990 +3.444 -7.468 }}
* [[CWE]]: ~648/625 = 109.0909{{c}}, ~3/2 = 704.6983{{c}}
: error map: {{val| 0.000 +2.743 -8.888 }}
{{Optimal ET sequence|legend=1| 22, 77c, 99c }}
[[Badness]] (Sintel): 17.1


== Kumonga ==
== Kumonga ==
Line 841: Line 890:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~144/125 = 223.076
* [[CTE]]: ~2 = 1200.000{{c}}, ~144/125 = 223.076{{c}}
* [[POTE]]: ~2 = 1200.000, ~144/125 = 222.912
* [[POTE]]: ~2 = 1200.000{{c}}, ~144/125 = 222.912{{c}}


{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 183cc }}
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 183cc }}
Line 858: Line 907:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2000000/1594323 = 400.000, ~10/9 = 183.593
* [[CTE]]: ~2000000/1594323 = 400.000{{c}}, ~10/9 = 183.593{{c}}
* [[CWE]]: ~2000000/1594323 = 400.000, ~10/9 = 183.585
* [[CWE]]: ~2000000/1594323 = 400.000{{c}}, ~10/9 = 183.585{{c}}


{{Optimal ET sequence|legend=1| 33c, 39, 72, 111, 183, 438, 621, 1425b }}
{{Optimal ET sequence|legend=1| 33c, 39, 72, 111, 183, 438, 621, 1425b }}
Line 875: Line 924:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~162/125 = 449.127
* [[CTE]]: ~2 = 1200.000{{c}}, ~162/125 = 449.127{{c}}
* [[POTE]]: ~2 = 1200.000, ~162/125 = 449.127
* [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 449.127{{c}}


{{Optimal ET sequence|legend=1| 8, …, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
{{Optimal ET sequence|legend=1| 8, …, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
Line 892: Line 941:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~16875/16384 = 38.380
* [[CTE]]: ~2 = 1200.000{{c}}, ~16875/16384 = 38.380{{c}}
* [[CWE]]: ~2 = 1200.000, ~16875/16384 = 38.400
* [[CWE]]: ~2 = 1200.000{{c}}, ~16875/16384 = 38.400{{c}}


{{Optimal ET sequence|legend=1| 31, 94, 125, 781bcc }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 781bcc }}
Line 909: Line 958:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3125/2592 = 292.591
* [[CTE]]: ~2 = 1200.000{{c}}, ~3125/2592 = 292.591{{c}}
* [[POTE]]: ~2 = 1200.000, ~3125/2592 = 292.702
* [[POTE]]: ~2 = 1200.000{{c}}, ~3125/2592 = 292.702{{c}}


{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}
Line 926: Line 975:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~4374/3125 = 541.735
* [[CTE]]: ~2 = 1200.000{{c}}, ~4374/3125 = 541.735{{c}}
* [[POTE]]: ~2 = 1200.000, ~4374/3125 = 541.726
* [[POTE]]: ~2 = 1200.000{{c}}, ~4374/3125 = 541.726{{c}}


{{Optimal ET sequence|legend=1| 11c, 20c, 31, 113c, 144c, 175c }}
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 113c, 144c, 175c }}
Line 943: Line 992:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~25/24 = 54.138
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/24 = 54.138{{c}}
* [[POTE]]: ~2 = 1200.000, ~25/24 = 53.922
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/24 = 53.922{{c}}


{{Optimal ET sequence|legend=1| 22, 45, 67c, 89c }}
{{Optimal ET sequence|legend=1| 22, 45, 67c, 89c }}
Line 960: Line 1,009:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~{{monzo| 30 -7 -8 }} = 400.000, ~16/15 = 112.275
* [[CTE]]: ~{{monzo| 30 -7 -8 }} = 400.000{{c}}, ~16/15 = 112.275{{c}}
* [[POTE]]: ~{{monzo| 30 -7 -8 }} = 400.000, ~16/15 = 112.281
* [[POTE]]: ~{{monzo| 30 -7 -8 }} = 400.000{{c}}, ~16/15 = 112.281{{c}}


{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171 }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171 }}
Line 977: Line 1,026:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~64/45 = 577.593
* [[CTE]]: ~2 = 1200.000{{c}}, ~64/45 = 577.593{{c}}
* [[POTE]]: ~2 = 1200.000, ~64/45 = 577.705
* [[POTE]]: ~2 = 1200.000{{c}}, ~64/45 = 577.705{{c}}


{{Optimal ET sequence|legend=1| 2, …, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, …, 25, 27 }}
Line 994: Line 1,043:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]] and [[POTE]]: ~2 = 1200.000, ~27/25 = 146.476
* [[CTE]] and [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 146.476{{c}}


{{Optimal ET sequence|legend=1| 8, …, 41, 131, 172, 213c }}
{{Optimal ET sequence|legend=1| 8, …, 41, 131, 172, 213c }}
Line 1,010: Line 1,059:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6075/4096 = 600.000, ~64/45 = 644.844
* [[CTE]]: ~6075/4096 = 600.000{{c}}, ~64/45 = 644.844{{c}}
* [[POTE]]: ~6075/4096 = 600.000, ~64/45 = 645.454
* [[POTE]]: ~6075/4096 = 600.000{{c}}, ~64/45 = 645.454{{c}}


{{Optimal ET sequence|legend=1| 26, 80bc, 106bbcc, 132bbcc }}
{{Optimal ET sequence|legend=1| 26, 80bc, 106bbcc, 132bbcc }}
Line 1,025: Line 1,074:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~34171875/33554432 = 36.9607
* [[CTE]]: ~2 = 1200.0000{{c}}, ~34171875/33554432 = 36.9607{{c}}
* [[POTE]]: ~2 = 1200.0000, ~34171875/33554432 = 36.9622
* [[POTE]]: ~2 = 1200.0000{{c}}, ~34171875/33554432 = 36.9622{{c}}


{{Optimal ET sequence|legend=1| 32, 65, 292, 357, 422, 487, 1396c }}
{{Optimal ET sequence|legend=1| 32, 65, 292, 357, 422, 487, 1396c }}
Line 1,042: Line 1,091:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~243/200 = 400.000, ~4000/2187 = 955.656 (~10/9 = 155.656)
* [[CTE]]: ~243/200 = 400.000{{c}}, ~4000/2187 = 955.656{{c}} (~10/9 = 155.656{{c}})
* [[POTE]]: ~243/200 = 400.000, ~4000/2187 = 956.861 (~10/9 = 156.861)
* [[POTE]]: ~243/200 = 400.000{{c}}, ~4000/2187 = 956.861{{c}} (~10/9 = 156.861{{c}})


{{Optimal ET sequence|legend=1| 15, 39c, 54, 69b }}
{{Optimal ET sequence|legend=1| 15, 39c, 54, 69b }}
Line 1,059: Line 1,108:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 311.726
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 311.726{{c}}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 311.694
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 311.694{{c}}


{{Optimal ET sequence|legend=1| 23, 27, 50, 77 }}
{{Optimal ET sequence|legend=1| 23, 27, 50, 77 }}
Line 1,076: Line 1,125:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~4096/3645 = 232.385
* [[CTE]]: ~2 = 1200.000{{c}}, ~4096/3645 = 232.385{{c}}
* [[POTE]]: ~2 = 1200.000, ~4096/3645 = 232.521
* [[POTE]]: ~2 = 1200.000{{c}}, ~4096/3645 = 232.521{{c}}


{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc, 289bbcc }}
{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc, 289bbcc }}
Line 1,093: Line 1,142:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~{{monzo| -31 1 13 }} = 928.896
* [[CTE]]: ~2 = 1200.000{{c}}, ~{{monzo| -31 1 13 }} = 928.896{{c}}
* [[CWE]]: ~2 = 1200.000, ~{{monzo| -31 1 13 }} = 928.893
* [[CWE]]: ~2 = 1200.000{{c}}, ~{{monzo| -31 1 13 }} = 928.893{{c}}


{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 841, 1111, 1381, 2492c }}
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 841, 1111, 1381, 2492c }}
Line 1,110: Line 1,159:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6561/6250 = 38.550
* [[CTE]]: ~2 = 1200.000{{c}}, ~6561/6250 = 38.550{{c}}
* [[POTE]]: ~2 = 1200.000, ~6561/6250 = 38.571
* [[POTE]]: ~2 = 1200.000{{c}}, ~6561/6250 = 38.571{{c}}


{{Optimal ET sequence|legend=1| 31, 156c }}
{{Optimal ET sequence|legend=1| 31, 156c }}
Line 1,127: Line 1,176:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~27/25 = 137.963
* [[CTE]]: ~2 = 1200.000{{c}}, ~27/25 = 137.963{{c}}
* [[POTE]]: ~2 = 1200.000, ~27/25 = 137.962
* [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 137.962{{c}}


{{Optimal ET sequence|legend=1| 26, 61, 87, 287, 374, 461 }}
{{Optimal ET sequence|legend=1| 26, 61, 87, 287, 374, 461 }}
Line 1,142: Line 1,191:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~768/625 = 210.920
* [[CTE]]: ~2 = 1200.000{{c}}, ~768/625 = 210.920{{c}}
* [[POTE]]: ~2 = 1200.000, ~768/625 = 212.444
* [[POTE]]: ~2 = 1200.000{{c}}, ~768/625 = 212.444{{c}}


{{Optimal ET sequence|legend=1| 6b, 11b, 17c }}
{{Optimal ET sequence|legend=1| 6b, 11b, 17c }}
Line 1,159: Line 1,208:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~864/625 = 542.142
* [[CTE]]: ~2 = 1200.000{{c}}, ~864/625 = 542.142{{c}}
* [[POTE]]: ~2 = 1200.000, ~864/625 = 542.208
* [[POTE]]: ~2 = 1200.000{{c}}, ~864/625 = 542.208{{c}}


{{Optimal ET sequence|legend=1| 11b, 20b, 31, 73, 104c, 135c }}
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 73, 104c, 135c }}
Line 1,176: Line 1,225:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1215/1024 = 229.947
* [[CTE]]: ~2 = 1200.000{{c}}, ~1215/1024 = 229.947{{c}}
* [[POTE]]: ~2 = 1200.000, ~1215/1024 = 230.351
* [[POTE]]: ~2 = 1200.000{{c}}, ~1215/1024 = 230.351{{c}}


{{Optimal ET sequence|legend=1| 21, 26, 73bc, 99bc }}
{{Optimal ET sequence|legend=1| 21, 26, 73bc, 99bc }}
Line 1,193: Line 1,242:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~16/15 = 88.7331
* [[CTE]]: ~2 = 1200.0000{{c}}, ~16/15 = 88.7331{{c}}
* [[CWE]]: ~2 = 1200.0000, ~16/15 = 88.8168
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 88.8168{{c}}


{{Optimal ET sequence|legend=1| 13, 14, 27 }}
{{Optimal ET sequence|legend=1| 13, 14, 27 }}
Line 1,210: Line 1,259:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1048576/885735 = 348.462
* [[CTE]]: ~2 = 1200.000{{c}}, ~1048576/885735 = 348.462{{c}}
* [[POTE]]: ~2 = 1200.000, ~1048576/885735 = 348.600
* [[POTE]]: ~2 = 1200.000{{c}}, ~1048576/885735 = 348.600{{c}}


{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
Line 1,227: Line 1,276:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~98415/65536 = 600.000, ~3/2 = 690.708
* [[CTE]]: ~98415/65536 = 600.000{{c}}, ~3/2 = 690.708{{c}}
* [[POTE]]: ~98415/65536 = 600.000, ~3/2 = 691.455
* [[POTE]]: ~98415/65536 = 600.000{{c}}, ~3/2 = 691.455{{c}}


{{Optimal ET sequence|legend=1| 14, 26, 66b, 92bc, 118bbc }}
{{Optimal ET sequence|legend=1| 14, 26, 66b, 92bc, 118bbc }}
Line 1,235: Line 1,284:


== Satin ==
== Satin ==
: ''For extensions, see [[Canousmic temperaments #Satin]].''
: ''For extensions, see [[Garischismic clan #Satin]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 1,242: Line 1,291:


{{Mapping|legend=1| 1 2 12 | 0 -3 -70 }}
{{Mapping|legend=1| 1 2 12 | 0 -3 -70 }}
: mapping generators: ~2, ~{{monzo| -34 23 -1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~{{monzo| -34 23 -1 }} = 165.910
* [[WE]]: ~2 = 1199.9008{{c}}, ~{{monzo| -34 23 -1 }} = 165.8930{{c}}
* [[POTE]]: ~2 = 1200.000, ~{{monzo| -34 23 -1 }} = 165.907
: [[error map]]: {{val| -0.099 +0.168 -0.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -34 23 -1 }} = 165.9079{{c}}
: error map: {{val| 0.000 +0.321 +0.130 }}


{{Optimal ET sequence|legend=1| 94, 217, 528, 745, 1273 }}
{{Optimal ET sequence|legend=1| 94, 217, 528, 745, 1273, 2018b }}


[[Badness]] (Sintel): 66.9
[[Badness]] (Sintel): 66.9
Line 1,259: Line 1,311:


{{Mapping|legend=1| 1 -25 -16 | 0 74 51 }}
{{Mapping|legend=1| 1 -25 -16 | 0 74 51 }}
: mapping generators: ~2, ~{{monzo| 36 20 -29 }}
: mapping generators: ~2, ~{{monzo| 36 20 -29 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~{{monzo| 36 20 -29 }} = 431.1069
* [[CTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 36 20 -29 }} = 431.1069{{c}}
: [[error map]]: {{val| 0.0000 -0.0444 +0.1382 }}
: [[error map]]: {{val| 0.0000 -0.0444 +0.1382 }}
* [[CWE]]: ~2 = 1200.0000, ~{{monzo| 36 20 -29 }} = 431.1073
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 36 20 -29 }} = 431.1073{{c}}
: error map: {{val| 0.0000 -0.0174 +0.1568 }}
: error map: {{val| 0.0000 -0.0174 +0.1568 }}


Line 1,282: Line 1,333:


{{Mapping|legend=1| 1 11 -10 | 0 -13 17 }}
{{Mapping|legend=1| 1 11 -10 | 0 -13 17 }}
: Mapping generators: ~2, ~16384/10125
: Mapping generators: ~2, ~16384/10125


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~16384/10125 = 869.392
* [[CTE]]: ~2 = 1200.000{{c}}, ~16384/10125 = 869.392{{c}}
* [[CWE]]: ~2 = 1200.000, ~16384/10125 = 869.402
* [[CWE]]: ~2 = 1200.000{{c}}, ~16384/10125 = 869.402{{c}}


{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}
{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}
Line 1,312: Line 1,362:


[[Badness]] (Sintel): 203
[[Badness]] (Sintel): 203
== Elven ==
: ''For extensions, see [[Sensibeta temperaments #Elven]].''
Elven has a period of 1/11 octave, like [[#Hendecatonic|hendecatonic]], but this temperament equates a stack of eleven [[3125/2916]]'s with the octave.
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -23 -66 55 }}
{{Mapping|legend=1| 11 2 7 | 0 5 6 }}
: mapping generators: ~3125/2916, ~{{monzo| 10 29 -24 }}
[[Optimal tuning]]s:
* [[WE]]: ~3125/2916 = 109.0998{{c}}, ~{{monzo| 10 29 -24 }} = 336.8921{{c}}
: [[error map]]: {{val| +0.097 +0.705 -1.263 }}
* [[CWE]]: ~3125/2916 = 109.0909{{c}}, ~{{monzo| 10 29 -24 }} = 336.8814{{c}}
: error map: {{val| 0.000 +0.634 -1.389 }}
{{Optimal ET sequence|legend=1| 121, 253, 374, 627c }}
[[Badness]] (Sintel): 429


== References ==
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/Miscellaneous_5-limit_temperaments"