3125edo: Difference between revisions

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 55 = 3125 [[equal]] parts of exactly 384 [[cent|millicents]] each.

==Theory==

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.

== 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.

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.

In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general ~~and defined~~ as 1789 & 3125.

In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general, described as 1789 & 3125.

=== Prime harmonics ===

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

== Regular temperament properties ==

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]].

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]].

=== Rank-2 temperaments ~~by generator~~ ===

=== Rank-2 temperaments ===

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

!Periods

! Periods per Octave

per ~~octave~~

! Generator (Reduced)

!Generator

! Cents (Reduced)

(~~reduced~~)

! Associated Ratio

!Cents

! Temperaments

(~~reduced~~)

!Associated

~~ratio~~

!Temperaments

|-

|1

| 1

|1359\3125

| 1359\3125

|249.00

| 249.00

|80275/59392

| 80275/59392

|[[Estates general]]

| [[Estates general]]

|}~~~~

|}

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

[[Category:Quartismic]]

@@ 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.
+{{EDO intro|3125}}
-==Theory==
-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.
+== 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.
 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.
-In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general and defined as 1789 & 3125.
+In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general, described as 1789 & 3125.
 === Prime harmonics ===
-{{Primes in edo|3125}}
+{{Harmonics in equal|3125|columns=11}}
-[[Category:Equal divisions of the octave|####]]
 == Regular temperament properties ==
-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]].
+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]].
-=== Rank-2 temperaments by generator ===
+=== Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-!Periods
+! Periods<br>per Octave
-per octave
+! Generator<br>(Reduced)
-!Generator
+! Cents<br>(Reduced)
-(reduced)
+! Associated<br>Ratio
-!Cents
+! Temperaments
-(reduced)
-!Associated
-ratio
-!Temperaments
 |-
-|1
+| 1
-|1359\3125
+| 1359\3125
-|249.00
+| 249.00
-|80275/59392
+| 80275/59392
-|[[Estates general]]
+| [[Estates general]]
-|}<!-- 4-digit number -->
+|}
+[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
 [[Category:Quartismic]]