20ed5
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- This revision was by author xenwolf and made on 2011-10-25 16:34:23 UTC.
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Original Wikitext content:
20th root of 5 "Hieronymus' Tuning" A [[harmonic entropy]] minimum, that has better approximations of a variety of [[just interval]]s than [[Bohlen Pierce]] (of course, not the same intervals) among which are <span class="commentBody">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]]; imagine if that were the case with octaves (yes that is a helpful analogy).</span> <span class="commentBody">One way of looking at it comes by constructing it via four tempered 3/2</span> ([[meantone]] without octaves) <span class="commentBody">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.
Original HTML content:
<html><head><title>20ed5</title></head><body>20th root of 5 "Hieronymus' Tuning"<br /> <br /> A <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> minimum, that has better approximations of a variety of <a class="wiki_link" href="/just%20interval">just interval</a>s than <a class="wiki_link" href="/Bohlen%20Pierce">Bohlen Pierce</a> (of course, not the same intervals) among which are <span class="commentBody">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 <a class="wiki_link" href="/human%20hearing%20range">human hearing range</a>; imagine if that were the case with octaves (yes that is a helpful analogy).</span><br /> <br /> <span class="commentBody">One way of looking at it comes by constructing it via four tempered 3/2</span> (<a class="wiki_link" href="/meantone">meantone</a> without octaves) <span class="commentBody">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 <em>turned entirely inside-out</em>. 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.<br /> <br /> Adding octaves makes it <a class="wiki_link" href="/Meantone%20family#Jerome">jerome temperament</a>, 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.</body></html>