Miscellaneous 5-limit temperaments: Difference between revisions
- hogzilla, stump, unsmate, counterhanson, and countermeantone (addressed in the equivalence continuum of 19et) |
Move aluminium, tridecatonic, and triskaidekic here |
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* [[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 25: | 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 47: | Line 89: | ||
[[Badness]] (Sintel): 2.88 | [[Badness]] (Sintel): 2.88 | ||
== Aluminium == | |||
: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].'' | |||
Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit, and sets [[135/128]] to 1/13 of an [[octave]]. | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo| 92 -39 -13 }} | |||
{{Mapping|legend=1| 13 0 92 | 0 1 -3 }} | |||
: mapping generators: ~135/128, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}} | |||
: [[error map]]: {{val| -0.012 +0.012 +0.009 }} | |||
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 701.9897{{c}} | |||
: error map: {{val| 0.000 +0.032 +0.033 }} | |||
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }} | |||
[[Badness]] (Sintel): 2.89 | |||
== Avila == | == Avila == | ||
| Line 69: | Line 133: | ||
== 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 104: | Line 168: | ||
[[Badness]] (Sintel): 4.70 | [[Badness]] (Sintel): 4.70 | ||
== Neon == | |||
: ''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 | |||
[[Comma list]]: {{monzo| 21 60 -50 }} | |||
{{Mapping|legend=1| 10 4 9 | 0 5 6 }} | |||
: mapping generators: ~3125/2916, {{monzo| 10 29 -24 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~3125/2916 = 119.9999{{c}}, {{monzo| 10 29 -24 }} = 284.3890{{c}} | |||
: [[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| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }} | |||
[[Badness]] (Sintel): 4.85 | |||
== Dodifo == | == Dodifo == | ||
| Line 277: | Line 364: | ||
[[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 470: | Line 575: | ||
Scales (Scala files): [[wizard22]] | Scales (Scala files): [[wizard22]] | ||
== Tridecatonic == | |||
: ''For extensions, see [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[Varunismic temperaments #Bosonic]].'' | |||
Tridecatonic tempers out the [[devil's tridecalimma]], the comma which associates the [[10/9]] minor tone with [[13edo|2\13]]. | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo| -11 26 -13 }} | |||
{{Mapping|legend=1| 13 0 -11 | 0 1 2 }} | |||
: mapping generators: ~531441/500000, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~531441/500000 = 92.3166{{c}}, ~3/2 = 701.2353{{c}} | |||
: [[error map]]: {{val| +0.116 -0.688 +0.737 }} | |||
* [[CWE]]: ~531441/500000 = 92.3077{{c}}, ~3/2 = 701.2353{{c}} | |||
: error map: {{val| 0.000 -0.797 +0.617 }} | |||
{{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }} | |||
[[Badness]] (Sintel): 10.3 | |||
== Merman == | == Merman == | ||
| Line 487: | Line 614: | ||
[[Badness]] (Sintel): 10.6 | [[Badness]] (Sintel): 10.6 | ||
== 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 | |||
[[Comma list]]: 8796093022208/8649755859375 | |||
{{Mapping|legend=1| 11 0 43 | 0 1 -1 }} | |||
: Mapping generators: ~16/15, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~16/15 = 109.0904{{c}}, ~3/2 = 702.6861{{c}} | |||
: [[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| 22, 55, 77, 99, 869bcc, 968bcc, 1067bccc, 1166bccc }} | |||
[[Badness]] (Sintel): 10.7 | |||
== Nusecond == | == Nusecond == | ||
| Line 560: | Line 709: | ||
[[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 681: | Line 852: | ||
== Pentadecal == | == Pentadecal == | ||
: ''For extensions, see [[ | : ''For extensions, see [[Keemic temperaments #Pentadecal]].'' | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 730: | Line 901: | ||
[[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 1,133: | Line 1,326: | ||
[[Badness]] (Sintel): 52.3 | [[Badness]] (Sintel): 52.3 | ||
== Triskaidekic == | |||
: ''For extensions, see [[Orwellismic temperaments #Triskaidekic]].'' | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: 1220703125/1073741824 | |||
{{Mapping|legend=1| 13 0 30 | 0 1 0 }} | |||
: Mapping generators: ~128/125, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~128/125 = 92.589{{c}}, ~3/2 = 698.263{{c}} | |||
: [[error map]]: {{val| +3.656 -0.036 -8.647 }} | |||
* [[CWE]]: ~128/125 = 92.308{{c}}, ~3/2 = 698.068{{c}} | |||
: error map: {{val| 0.000 -3.887 -17.083 }} | |||
{{Optimal ET sequence|legend=1| 13, 26, 91cc }} | |||
[[Badness]] (Sintel): 52.6 | |||
== Satin == | == Satin == | ||
: ''For extensions, see [[ | : ''For extensions, see [[Garischismic clan #Satin]].'' | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 1,142: | Line 1,355: | ||
{{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: | ||
* [[ | * [[WE]]: ~2 = 1199.9008{{c}}, ~{{monzo| -34 23 -1 }} = 165.8930{{c}} | ||
* [[ | : [[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 1,210: | Line 1,426: | ||
[[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]] | ||