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]], ~~and~~ [[~~Vishnuzmic~~ family #Decavish|decavish]].

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).

== ~~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.~~

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

~~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~~ }}

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.

~~=== 13-limit ===~~

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

Subgroup: 2.3.5~~.7.11.13~~

[[Subgroup]]: 2.3.5

Comma list: ~~1001/1000, 4225/4224, 4459/4455~~

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

~~Mapping: [~~{{~~val~~|10 ~~0 0 -11~~ 4 ~~37}}, {{val~~|0 ~~1 0 1 -1 0~~}}, {{~~val~~|~~0 0 1 1 2 0~~}}]

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

: ~~mapping generators~~: ~15/14 = 1\10, ~3/2 = ~~701~~.~~931~~, ~~~5/4~~ = ~~386~~.~~473~~

[[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| ~~10e, 50,~~ 80~~, 130~~, 190, 270, ~~590~~, 730, ~~860~~, ~~1130 }}~~

{{Optimal ET sequence|legend=1| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }}

~~== Neon ==~~

~~Neon tempers out the {{Monzo|21 60 -50~~}} ~~comma in the 5-limit.~~

~~Subgroup~~: 2.~~3.5~~

[[Badness]] (Sintel): 4.85

~~Comma list:~~ {{~~Monzo|21 60~~ -50}}

~~{{Mapping|10 4 9|0 5 6|legend=1}}<blockquote>mapping generators: ~3125/2916 = 1\10, 70277506421640192/59604644775390625 = 284.389</blockquote>~~

[[Category:10edo]]

~~[[Category:Temperament collections]]~~

@@ Line 1: / Line 1: @@
-[[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.
+{{Technical data page}}
+{{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]], and [[Vishnuzmic family #Decavish|decavish]].
+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).
-== 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.
+: ''For extensions, see [[Ragismic microtemperaments #Deca]], [[Landscape microtemperaments #Zinc]], and [[20th-octave temperaments #Calcium]].''
-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 }}
+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.
-=== 13-limit ===
+Extending neon with [[ragisma]], or equivalently with the linus comma, produces [[deca]].
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5
-Comma list: 1001/1000, 4225/4224, 4459/4455
+[[Comma list]]: {{monzo| 21 60 -50 }}
-Mapping: [{{val|10 0 0 -11 4 37}}, {{val|0 1 0 1 -1 0}}, {{val|0 0 1 1 2 0}}]
+{{Mapping|legend=1| 10 4 9 | 0 5 6 }}
+: mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}
-: mapping generators: ~15/14 = 1\10, ~3/2 = 701.931, ~5/4 = 386.473
+[[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| 10e, 50, 80, 130, 190, 270, 590, 730, 860, 1130 }}
+{{Optimal ET sequence|legend=1| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }}
-== Neon ==
-Neon tempers out the {{Monzo|21 60 -50}} comma in the 5-limit.
-Subgroup: 2.3.5
+[[Badness]] (Sintel): 4.85
-Comma list: {{Monzo|21 60 -50}}
+{{Navbox fractional-octave}}
-{{Mapping|10 4 9|0 5 6|legend=1}}<blockquote>mapping generators: ~3125/2916 = 1\10, 70277506421640192/59604644775390625 = 284.389</blockquote>
 [[Category:10edo]]
-[[Category:Temperament collections]]