3125edo: Difference between revisions

(16 intermediate revisions by 5 users not shown)

Line 1:

== Theory ==

~~3125edo~~ is ~~distinctly~~ [[consistent]] through the [[15-odd-limit]]~~. A basis for its 7-limit commas is 78125000/78121827~~, ~~645700815/645657712 and 281484423828125/281474976710656. In the 11-limit, 151263/151250, 820125/819896, 21437500/21434787~~ and [[~~Quartisma|117440512~~/~~117406179~~]] are tempered out – it should be noted this edo is so far the only one [https://en.xen.wiki/index.php?title=3125edo&oldid=7860 known to have been confirmed] as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five [[33/32]] ~~quartertones~~ and ~~one~~ [[~~7/6~~]] ~~subminor third. In~~ the 13-limit~~, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out~~.

3125et is notable for being an extremely strong [[7-limit]] system. It is also [[consistent]] through the [[15-odd-limit]], and except for [[17/11]], [[19/17]] and their [[octave complement]]s, it is consistent to the [[35-odd-limit]].

In the ~~2.5.~~11.13.19.23.~~29.31 subgroup~~, ~~it supports a temperament called estates general~~, ~~described as 1789 & 3125~~.

A basis for its 7-limit commas is [[78125000/78121827]], [[645700815/645657712]] and 281484423828125/281474976710656. In the [[11-limit]], [[151263/151250]], 820125/819896, 21437500/21434787 and [[quartisma|117440512/117406179]] are tempered out—it should be noted this edo is so far the only one [https://en.xen.wiki/index.php?title=3125edo&oldid=7860 known to have been confirmed] as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five [[33/32]] quartertones and one [[7/6]] subminor third. In the [[13-limit]], [[6656/6655]], [[123201/123200]], [[140625/140608]] and 1399680/1399489 are all tempered out.

=== Prime harmonics ===

{{Harmonics in equal|3125|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 3125edo (continued)}}

=== Subsets and supersets ===

3125 = 55 , and as such it is the 5th edo of the form ~~x^x~~. It ~~hhas~~ subset edos {{EDOs|5, 25, 125, and 625}}.

{{nowrap| 3125 {{=}} 55 }}, and as such 3125edo is the 5th edo of the form ''n''''n''. It has subset edos {{EDOs| 5, 25, 125, and 625 }}.

== Regular temperament properties ==

3125et is ~~notable for being an extremely strong 7-limit system, being~~ the first equal ~~division~~ past [[171edo]] with a lower [[~~Tenney-Euclidean~~ temperament measures #TE simple badness|relative error]].

3125et is the first equal temperament past [[171edo|171]] with a lower [[Tenney–Euclidean temperament measures #TE simple badness|relative error]].

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

Line 59:

Line 62:

| 80275/59392

| [[Estates general]]

|-

| 1

| 1412\3125

| 542.208

| 16807/12288

| [[Revopent]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=mnTF1vBexBg ''Etude for Gamelan in Estates General and Pentonismic''] (2023)

~~[[Category:Equal divisions of the octave|####]] ~~

[[Category:Quartismic]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|3125}}
+{{ED intro}}
 == Theory ==
-edo is distinctly [[consistent]] through the [[15-odd-limit]]. A basis for its 7-limit commas is 78125000/78121827, 645700815/645657712 and 281484423828125/281474976710656. In the 11-limit, 151263/151250, 820125/819896, 21437500/21434787 and [[Quartisma|117440512/117406179]] are tempered out – it should be noted this edo is so far the only one [https://en.xen.wiki/index.php?title=3125edo&oldid=7860 known to have been confirmed] as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five [[33/32]] quartertones and one [[7/6]] subminor third. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.
+et is notable for being an extremely strong [[7-limit]] system. It is also [[consistent]] through the [[15-odd-limit]], and except for [[17/11]], [[19/17]] and their [[octave complement]]s, it is consistent to the [[35-odd-limit]].
-In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general, described as 1789 & 3125.
+A basis for its 7-limit commas is [[78125000/78121827]], [[645700815/645657712]] and 281484423828125/281474976710656. In the [[11-limit]], [[151263/151250]], 820125/819896, 21437500/21434787 and [[quartisma|117440512/117406179]] are tempered out—it should be noted this edo is so far the only one [https://en.xen.wiki/index.php?title=3125edo&oldid=7860 known to have been confirmed] as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five [[33/32]] quartertones and one [[7/6]] subminor third. In the [[13-limit]], [[6656/6655]], [[123201/123200]], [[140625/140608]] and 1399680/1399489 are all tempered out.
 === Prime harmonics ===
 {{Harmonics in equal|3125|columns=11}}
+{{Harmonics in equal|3125|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 3125edo (continued)}}
 === Subsets and supersets ===
-= 5<sup>5</sup> , and as such it is the 5th edo of the form x^x. It hhas subset edos {{EDOs|5, 25, 125, and 625}}.
+{{nowrap| 3125 {{=}} 5<sup>5</sup> }}, and as such 3125edo is the 5th edo of the form ''n''<sup>''n''</sup>. It has subset edos {{EDOs| 5, 25, 125, and 625 }}.
 == Regular temperament properties ==
-et is notable for being an extremely strong 7-limit system, being the first equal division past [[171edo]] with a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative error]].
+et is the first equal temperament past [[171edo|171]] with a lower [[Tenney–Euclidean temperament measures #TE simple badness|relative error]].
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 59: / Line 62: @@
 | 80275/59392
 | [[Estates general]]
+|-
+| 1
+| 1412\3125
+| 542.208
+| 16807/12288
+| [[Revopent]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
+== Music ==
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=mnTF1vBexBg ''Etude for Gamelan in Estates General and Pentonismic''] (2023)
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
 [[Category:Quartismic]]
+[[Category:Listen]]