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10th-octave temperaments: Difference between revisions

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{{Technical data page}}
{{Infobox fractional-octave|10}}
[[10edo]] is notable for having close approximations of [[15/14]] to one step, and [[13/8]] to 7 steps. 10th-octave temperaments naturally occur between any equal divisions of the octave whose greatest common divisor is 10.
[[10edo]] is notable for having close approximations of [[15/14]] to one step, and [[13/8]] to 7 steps. 10th-octave temperaments naturally occur between any equal divisions of the octave whose greatest common divisor is 10.


Temperaments discussed elsewhere include: [[Breedsmic temperaments #Decoid|decoid]], [[Ragismic microtemperaments #Deca|deca]], [[Pental family #Decal|decal]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], and [[Vishnuzmic family #Decavish|decavish]].
Temperaments discussed elsewhere include: [[Quintosec family #Decoid|decoid]], [[Ragismic microtemperaments #Deca|deca]], [[Quintile family #Decile|decile]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], [[Vishnuzmic family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]].  


== Linus (rank-3) ==
== Neon ==
Tempering out the '''linus comma''', 578509309952 / 576650390625 = {{monzo|11 -10 -10 10}} leads a number of regular temperaments, some of which are listed above. Linus rank three temperament can be described as the 130 & 190 & 270 temperament, which tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.
Neon tempers out {{monzo| 21 60 -50 }} in the 5-limit, equating [[3125/2916]] with one step of 10edo. Neon extensions discussed elsewhere include [[deca]], [[calcium]], and [[zinc]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5


Comma list: 578509309952/576650390625
[[Comma list]]: {{monzo| 21 60 -50 }}


Mapping: [{{val|10 0 0 -11}}, {{val|0 1 0 1}}, {{val|0 0 1 1}}]
{{Mapping|legend=1| 10 4 9 | 0 5 6 }}


: mapping generators: ~15/14 = 1\10, ~3/2 = 702.095, ~5/4 = 386.574
: Mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}


{{Optimal ET sequence|legend=1| 10, 40, 50, 80, 130, 140, 190, 270, 1270, 1400, 1540, 1670, 1810, 1940 }}
[[Optimal tuning]] ([[CTE]]): ~3125/2916 = 1\10, {{monzo| 10 29 -24 }} = 284.3888


=== 11-limit ===
{{Optimal ET sequence|legend=1| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }}
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 391314/390625
[[Badness]]: 0.206596


Mapping: [{{val|10 0 0 -11 4}}, {{val|0 1 0 1 -1}}, {{val|0 0 1 1 2}}]
{{Navbox fractional-octave}}
 
: mapping generators: ~15/14 = 1\10, ~3/2 = 702.034, ~5/4 = 386.648
 
{{Optimal ET sequence|legend=1| 10e, 50, 80, 130, 190, 270, 860, 940, 1130, 1400, 1670 }}
 
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 4225/4224, 4459/4455
 
Mapping: [{{val|10 0 0 -11 4 37}}, {{val|0 1 0 1 -1 0}}, {{val|0 0 1 1 2 0}}]
 
: mapping generators: ~15/14 = 1\10, ~3/2 = 701.931, ~5/4 = 386.473
 
{{Optimal ET sequence|legend=1| 10e, 50, 80, 130, 190, 270, 590, 730, 860, 1130 }}


[[Category:10edo]]
[[Category:10edo]]
[[Category:Temperament collections]]
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