10th-octave temperaments: Difference between revisions
declutter miscellaneous 5-limit - neon is of importance to 10edo and fractional octave theory. see discord conversation |
+valerisma is also of importance, 13/8 in 10edo. Adding in the same style as 37-11-comma is documented on 37th-octave temps. |
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{{Technical data page}} | {{Technical data page}} | ||
{{Infobox fractional-octave|10}} | {{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, 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: [[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]]. | 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). | ||
== Valerismic (rank-1) == | |||
Valerismic temperament tempers out the [[valerisma]], 137858491849/137438953472. | |||
Subgroup: 2.13 | |||
Comma list: 2.13 {{monzo|-37 10}} | |||
{{mapping|legend=2| 10 37 }} | |||
: mapping generators: ~2197/2048 = 1\10 | |||
[[Support]]ing [[ET]]s: 10N, N = 1 to 113 | |||
== Neon == | == Neon == | ||
Revision as of 19:33, 1 August 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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: decoid, decile, decimetra, decistearn, decavish, and linus (the 15/14 relationship mentioned above).
Valerismic (rank-1)
Valerismic temperament tempers out the valerisma, 137858491849/137438953472.
Subgroup: 2.13
Comma list: 2.13 [-37 10⟩
Subgroup-val mapping: [⟨10 37]]
- mapping generators: ~2197/2048 = 1\10
Supporting ETs: 10N, N = 1 to 113
Neon
- For extensions, see Ragismic microtemperaments #Deca, Landscape microtemperaments #Zinc, and 20th-octave temperaments #Calcium.
Neon tempers out [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
Comma list: [21 60 -50⟩
Mapping: [⟨10 4 9], ⟨0 5 6]]
- mapping generators: ~3125/2916, [10 29 -24⟩
- WE: ~3125/2916 = 119.9999 ¢, [10 29 -24⟩ = 284.3890 ¢
- error map: ⟨-0.001 -0.011 +0.019]
- CWE: ~3125/2916 = 120.0000 ¢, [10 29 -24⟩ = 284.3891 ¢
- error map: ⟨0.000 -0.010 +0.021]
Optimal ET sequence: 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650
Badness (Sintel): 4.85