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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 3125 [[equal]] parts of exactly 384 [[cent|millicents]] each. | 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. | ||
==Theory== | ==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. | 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. | ||
Revision as of 22:32, 9 May 2022
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 millicents each.
Theory
It is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower 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 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.
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