Miscellaneous 5-limit temperaments: Difference between revisions

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 * [[Very low 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 ==
@@ Line 124: / Line 146: @@
 [[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 ==
@@ Line 297: / Line 342: @@
 [[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 ==
@@ Line 507: / Line 570: @@
 [[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 ==
@@ Line 723: / Line 808: @@
 == Pentadecal ==
-: ''For extensions, see [[Cloudy clan #Pentadecal]].''
+: ''For extensions, see [[Keemic temperaments #Pentadecal]].''
 [[Subgroup]]: 2.3.5
@@ Line 772: / Line 857: @@
 [[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 ==
@@ Line 1,255: / Line 1,362: @@
 [[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 ==
 [[Category:Temperament collections]]