Xenharmonic Wiki

Miscellaneous 5-limit temperaments: Difference between revisions

← Older edit
VisualWikitext
m Update
Move aluminium, tridecatonic, and triskaidekic here
 
(6 intermediate revisions by the same user not shown)
Line 5: Line 5:
* [[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 67: 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 124: 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 297: 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 490: 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 507: 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 772: 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 1,175: 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 ==
Line 1,255: 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]]
Retrieved from "https://en.xen.wiki/w/Miscellaneous_5-limit_temperaments"