113edo: Difference between revisions
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'''113edo''' is the [[EDO|equal division of the octave]] into 113 parts of 10.6195 [[cent]]s each. | '''113edo''' is the [[EDO|equal division of the octave]] into 113 parts of 10.6195 [[cent]]s each. | ||
== Theory == | |||
113edo [[tempers out]] [[1600000/1594323]] and [[34171875/33554432]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], and [[441/440]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It supports the 5-limit [[Amity family|amity temperament]], 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle temperament]], and 13-limit [[manna]] temperament. | |||
113edo is the 30th [[prime EDO]]. | 113edo is the 30th [[prime EDO]]. | ||
Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament. | === Prime harmonics === | ||
{{Primes in edo|113|prec=2}} | |||
== Regular temperament properties == | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 6\113 | |||
| 63.717 | |||
| 28/27 | |||
| [[Sycamore]] / [[betic]] | |||
|- | |||
| 1 | |||
| 8\113 | |||
| 84.956 | |||
| 21/20 | |||
| [[Amicable]] | |||
|- | |||
| 1 | |||
| 11\113 | |||
| 116.814 | |||
| 15/14~16/15 | |||
| [[Miracle]] / [[manna]] | |||
|- | |||
| 1 | |||
| 13\113 | |||
| 138.053 | |||
| 27/25 | |||
| [[Quartemka]] | |||
|- | |||
| 1 | |||
| 22\113 | |||
| 233.628 | |||
| 8/7 | |||
| [[Slendric]] | |||
|- | |||
| 1 | |||
| 27\113 | |||
| 286.726 | |||
| 13/11 | |||
| [[Gamity]] | |||
|- | |||
| 1 | |||
| 29\113 | |||
| 307.965 | |||
| 3200/2673 | |||
| [[Familia]] | |||
|- | |||
| 1 | |||
| 32\113 | |||
| 339.823 | |||
| 243/200 | |||
| [[Amity]] / [[houborizic]] | |||
|- | |||
| 1 | |||
| 34\113 | |||
| 360.062 | |||
| 16/13 | |||
| [[Phicordial]] | |||
|- | |||
| 1 | |||
| 37\113 | |||
| 392.920 | |||
| 2744/2187 | |||
| [[Emmthird]] | |||
|- | |||
| 1 | |||
| 47\113 | |||
| 499.115 | |||
| 4/3 | |||
| [[Gracecordial]] | |||
|} | |||
== Scales == | |||
Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Circulating temperaments in 113edo | |+Circulating temperaments in 113edo | ||