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Miscellaneous 5-limit temperaments: Difference between revisions

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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 ==
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[[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 ==
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== 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 105: 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{{c}}, ~64/45 = 631.504{{c}}
* [[WE]]: ~3125/2916 = 119.9999{{c}}, {{monzo| 10 29 -24 }} = 284.3890{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~64/45 = 631.737{{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| 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 ==
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[[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 ==
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[[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{{c}}, ~6/5 = 316.084{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.081{{c}}
{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
[[Badness]] (Sintel): 7.45


== Nessafof ==
== Nessafof ==
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[[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{{c}}, ~78125/52488 = 695.082{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~78125/52488 = 695.087{{c}}
{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}
[[Badness]] (Sintel): 8.76


== Tritonic ==
== Tritonic ==
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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{{c}}, ~2048/1215 = 946.982{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~2048/1215 = 947.351{{c}}
{{Optimal ET sequence|legend=1| 14, 19 }}
[[Badness]] (Sintel): 9.96


== Merman ==
== Merman ==
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[[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{{c}}, ~2048/1875 = 188.773{{c}}
* [[WE]]: ~16/15 = 109.0904{{c}}, ~3/2 = 702.6861{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~2048/1875 = 189.049{{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| 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 ==
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[[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 762: 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 811: Line 857:


[[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 1,216: 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 1,223: 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{{c}}, ~{{monzo| -34 23 -1 }} = 165.910{{c}}
* [[WE]]: ~2 = 1199.9008{{c}}, ~{{monzo| -34 23 -1 }} = 165.8930{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~{{monzo| -34 23 -1 }} = 165.907{{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 1,291: 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"