10th-octave temperaments: Difference between revisions

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[[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: [[Breedsmic temperaments #Decoid|decoid]], [[Ragismic microtemperaments #Deca|deca]], [[Pental family #Decal|decal]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], [[Vishnuzmic family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]].  
 
== Linus (rank-3) ==
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.
 
Subgroup: 2.3.5.7
 
Comma list: 578509309952/576650390625
 
Mapping: [{{val|10 0 0 -11}}, {{val|0 1 0 1}}, {{val|0 0 1 1}}]
 
: mapping generators: ~15/14 = 1\10, ~3/2 = 702.095, ~5/4 = 386.574
 
{{Optimal ET sequence|legend=1| 10, 40, 50, 80, 130, 140, 190, 270, 1270, 1400, 1540, 1670, 1810, 1940 }}
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 391314/390625
 
Mapping: [{{val|10 0 0 -11 4}}, {{val|0 1 0 1 -1}}, {{val|0 0 1 1 2}}]
 
: 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 }}


== Neon ==
== Neon ==

Revision as of 16:20, 6 January 2024

Template:Fractional-octave navigation 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: decoid, deca, decal, decimetra, decistearn, decavish, and linus.

Neon

Neon tempers out the [21 60 -50⟩ comma in the 5-limit. Neon extensions discussed elsewhere include deca, calcium, and zinc.

Subgroup: 2.3.5

Comma list: [21 60 -50⟩

Mapping: [⟨10 4 9], ⟨0 5 6]]

mapping generators: ~3125/2916 = 1\10, [10 29 -24⟩ = 284.389

Optimal tuning (CTE): ~[10 29 -24⟩ = 284.389

Supporting ETs: 80, 190, 270, 460, 730, 1000, 1190, 1270, 1460, 2460

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