20ed5: Difference between revisions

20th root of 5 "Hieronymus' Tuning"

A [[Harmonic_Entropy|harmonic entropy]] minimum, that has better approximations of a variety of [[just_interval|just interval]]s than [[Bohlen_Pierce|Bohlen Pierce]] (of course, not the same intervals) among which are <span style="">13/12, 7/6, 14/11, 11/8, 3/2, 13/8, 7/4, 21/11, 33/32, ~9/4, 39/32, 21/16, 10/7, 20/13, 10/3 ... etc. In terms of strict 5/1 equivalence and high-limit harmony, it also approximates the harmonics and their pentave reductions: ‎8, 12 (or 61), 23, 27, 32, 44, 48, 52, 56, 66, 71, 77, etc. within 20 cents. Note that there are (at most) ~4.3 pentaves within [[human_hearing_range|human hearing range]]; imagine if that were the case with octaves (yes that is a helpful analogy).</span>

<span style="">One way of looking at it comes by constructing it via four tempered 3/2</span> ([[Meantone|meantone]] without octaves) <span style="">each of which is divided</span> into five tones, which in turn approximate 11/8 13/8 7/6 etc., and themselves end up on the "pentave", 5/1, wherein the scale repeats itself. By analogy to common practice, this is familiar extended meantone but ''turned entirely inside-out''. Interestingly, while Hieronymus does not repeat at the octave or even approximate it well, factors of 2 are nevertheless important to its perception and structure; it might even be helpful to think of the 3/2 intervals as a cellular structure of sorts.

Adding octaves makes it [[Meantone_family#Jerome|jerome temperament]], with generator a meantone fifth divided in five, and Hieronymus is the generator chain of that. Jerome/Hieronymus only really comes into its own as a higher limit temperament, as a 13, or even higher limit system. It is related to [[43edo]], and 5\43 can be used as a generator.