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 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 ==
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 = 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.  
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 ===
=== Prime harmonics ===
{{Primes in edo|3125}}
{{Harmonics in equal|3125|columns=11}}
 
[[Category:Equal divisions of the octave|####]]


== Regular temperament properties ==
== 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"
{| 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]]
[[Category:Quartismic]]

Revision as of 17:02, 17 September 2022

Template:EDO intro

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 117440512/117406179 are tempered out – it should be noted this edo is so far the only one 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, described as 1789 & 3125.

Prime harmonics

Approximation of prime harmonics in 3125edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.003 -0.010 +0.006 +0.106 +0.048 -0.123 +0.087 -0.050 -0.073 +0.052
Relative (%) +0.0 -0.8 -2.5 +1.6 +27.6 +12.6 -32.1 +22.7 -13.1 -19.1 +13.7
Steps
(reduced) 		3125
(0) 		4953
(1828) 		7256
(1006) 		8773
(2523) 		10811
(1436) 		11564
(2189) 		12773
(273) 		13275
(775) 		14136
(1636) 		15181
(2681) 		15482
(2982)

Regular temperament properties

3125et is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower relative error.

Rank-2 temperaments

Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 1359\3125 249.00 80275/59392 Estates general
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