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, the associated comma being the [[linus comma]] 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]], [[~~Vishnuzmic~~ family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]].

Temperaments discussed elsewhere include: [[Quintosec family #Decoid|decoid]], [[Quintile family #Decile|decile]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], [[Vishnu family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]] (the 15/14 relationship mentioned above).

== Neon ==

Neon tempers out {{monzo| 21 60 -50 }} in the 5-limit. ~~Neon extensions discussed elsewhere include~~ [[~~deca~~]], [[~~calcium~~]], ~~and~~ [[~~zinc~~]].

: ''For extensions, see [[Ragismic microtemperaments #Deca]], [[Landscape microtemperaments #Zinc]], and [[20th-octave temperaments #Calcium]].''

Neon tempers out the [[neon comma]], {{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.

Extending neon with [[ragisma]], or equivalently with the linus comma, produces [[deca]].

[[Subgroup]]: 2.3.5

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: mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}

[[Optimal tuning]] ([[~~CTE~~]]): ~3125/2916 = 1\10, {{monzo| 10 29 -24 }} = 284.~~3888~~

[[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]]: 0.~~207~~

[[Badness]] (Sintel): 4.85

[[Category:10edo]]

@@ Line 1: / Line 1: @@
-{{Fractional-octave navigation|10}}
+{{Technical data page}}
-[[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.
+{{Infobox fractional-octave|10}}
+[[10edo]] is notable for having close approximations of [[15/14]] to one step, the associated comma being the [[linus comma]] 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]], [[Vishnuzmic family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]].
+Temperaments discussed elsewhere include: [[Quintosec family #Decoid|decoid]], [[Quintile family #Decile|decile]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], [[Vishnu family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]] (the 15/14 relationship mentioned above).
 == Neon ==
-Neon tempers out {{monzo| 21 60 -50 }} in the 5-limit. Neon extensions discussed elsewhere include [[deca]], [[calcium]], and [[zinc]].
+: ''For extensions, see [[Ragismic microtemperaments #Deca]], [[Landscape microtemperaments #Zinc]], and [[20th-octave temperaments #Calcium]].''
+Neon tempers out the [[neon comma]], {{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.
+Extending neon with [[ragisma]], or equivalently with the linus comma, produces [[deca]].
 [[Subgroup]]: 2.3.5
@@ Line 12: / Line 17: @@
 {{Mapping|legend=1| 10 4 9 | 0 5 6 }}
 : mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}
-[[Optimal tuning]] ([[CTE]]): ~3125/2916 = 1\10, {{monzo| 10 29 -24 }} = 284.3888
+[[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]]: 0.207
+[[Badness]] (Sintel): 4.85
 {{Navbox fractional-octave}}
 [[Category:10edo]]