46edo: Difference between revisions

The '''46 equal divisions of the octave''' ('''46edo'''), or the '''46(-tone) equal temperament''' ('''46tet''', '''46et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 46 [[equal]]ly-sized steps. Each step has a size of about 26.1 [[cent]]s, an interval close in size to [[66/65]], the interval from [[13/11]] to [[6/5]].

46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank two temperament]]s it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[minimax tuning]] for valentine temperament, (11/7)^1/10, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo|41et]]. In fact, while 41 is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral EDO]] but not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap EDO]], 46 is zeta gap but not zeta integral.  

46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.

[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.

 

<nowiki>*</nowiki> Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has the 15th harmonic poorly approximated in general, because, while both the 3rd and 5th harmonics are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46edo is closer to 15/13 by a hair; 10\46edo represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.  

All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:

The following table shows how [[15-odd-limit intervals]] are represented in 46edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

46et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-limit, and others. The next ETs doing better in the aforementioned subgroups are 72, 72, 94, respectively. 46et is even more prominent in the no-19 23-limit, and the next ET doing better in this subgroup is 140.  

46edo represents [[harmonic series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.

* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene Ward Smith]]. The central portion is in [[27edo]], the rest is in 46edo.

[[Category:Chromosounds]]

[[Category:Leapday]]

[[Category:Sensi]]

[[Category:Shrutar]]

[[Category:Valentine]]