3125edo: Difference between revisions

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The '''3125 equal divisions of the octave''' ('''3125edo'''), or the '''3125-tone equal temperament''' ('''3125tet'''), '''3125 equal temperament''' ('''3125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 55 = 3125 [[equal]] parts of exactly 384 [[cent|millicents]] each.

~~==Theory==~~

It 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]]. It is also 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.

~~The fact that 3125~~ = ~~55 makes curious notations possible based on the symmetric base 5 positional number~~ system~~, by converting~~ the ~~number to base 5 with digits {~~-2, -1, 0, 1, ~~2}. 3125 has subset edos 5, 25, 125, and 625~~.

== Theory ==

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]].

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 ===

{{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 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*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 139\3125

| 53.376

| 33/32

| [[Prequartismic]]

|-

| 1

| 411\3125

| 157.824

| 36756909/33554432

| [[Hemiegads]]

|-

| 1

| 577\3125

| 221.568

| 8388608/7381125

| [[Fortune]]

|-

| 1

| 822\3125

| 315.648

| 6/5

| [[Egads]]

|-

| 1

| 894\3125

| 343.296

| 8000/6561

| [[Raider]]

|-

| 1

| 1359\3125

| 521.856

| 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: @@
-The '''3125 equal divisions of the octave''' ('''3125edo'''), or the '''3125-tone equal temperament''' ('''3125tet'''), '''3125 equal temperament''' ('''3125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 5<sup>5</sup>  = 3125 [[equal]] parts of exactly 384 [[cent|millicents]] each.
+{{Infobox ET}}
-==Theory==
+{{ED intro}}
-It 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]]. It is also 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.
-The fact that 3125 = 5<sup>5</sup> makes curious notations possible based on the symmetric base 5 positional number system, by converting the number to base 5 with digits {-2, -1, 0, 1, 2}. 3125 has subset edos 5, 25, 125, and 625.
+== Theory ==
+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]].
-{{Primes in edo|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 ===
+{{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 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*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 139\3125
+| 53.376
+| 33/32
+| [[Prequartismic]]
+|-
+| 1
+| 411\3125
+| 157.824
+| 36756909/33554432
+| [[Hemiegads]]
+|-
+| 1
+| 577\3125
+| 221.568
+| 8388608/7381125
+| [[Fortune]]
+|-
+| 1
+| 822\3125
+| 315.648
+| 6/5
+| [[Egads]]
+|-
+| 1
+| 894\3125
+| 343.296
+| 8000/6561
+| [[Raider]]
+|-
+| 1
+| 1359\3125
+| 521.856
+| 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]]