3125edo: Difference between revisions

Line 1:

The 3125 equal division of the octave divides it into 5^5 = 3125 equal parts of exactly 0.384 cents each. 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 [[~~Quartismic temperaments~~|117440512/117406179]] are tempered out- it should be noted this EDO is so far the only EDO [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 [https://en.xen.wiki/index.php?title=User_talk:CritDeathX&oldid=48622#Unnoticeable_Commas on September 6th, 2020]. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.

The 3125 equal division of the octave divides it into 5^5 = 3125 equal parts of exactly 0.384 cents each. 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 EDO [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 [https://en.xen.wiki/index.php?title=User_talk:CritDeathX&oldid=48622#Unnoticeable_Commas on September 6th, 2020]. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.

The fact that 3125 = 5^5 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}.

Revision as of 06:02, 9 September 2020 view source Aura (talk \| contribs) autopatrolled, emailconfirmed 6,010 edits Added links to back up certain claims ← Older edit		Revision as of 20:24, 13 September 2020 view source Aura (talk \| contribs) autopatrolled, emailconfirmed 6,010 edits No edit summary Newer edit →
Line 1:		Line 1:
	The 3125 equal division of the octave divides it into 5^5 = 3125 equal parts of exactly 0.384 cents each. 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 [[~~Quartismic temperaments~~\|117440512/117406179]] are tempered out- it should be noted this EDO is so far the only EDO [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 [https://en.xen.wiki/index.php?title=User_talk:CritDeathX&oldid=48622#Unnoticeable_Commas on September 6th, 2020]. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.		The 3125 equal division of the octave divides it into 5^5 = 3125 equal parts of exactly 0.384 cents each. 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 EDO [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 [https://en.xen.wiki/index.php?title=User_talk:CritDeathX&oldid=48622#Unnoticeable_Commas on September 6th, 2020]. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.

	The fact that 3125 = 5^5 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}.		The fact that 3125 = 5^5 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}.