20ed5
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-09-02 12:53:49 UTC.
- The original revision id was 250338474.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
20th root of 5 "Hieronymus' Tuning" An [[harmonic entropy]] minimum, that has better approximations of a variety of just intervals 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 high-limit harmony it also approximates the harmonics and their pentave reductions:</span> <span class="commentBody">8, 12 (or 61), 23, 27, 32, 44, 48, 52, 56, 66, 71, 77, etc. within 20 cents.</span> <span class="commentBody">One way of looking at it comes by constructing it via four tempered 3/2 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. Adding octaves makes it [[Meantone family#Jerome|jerome temperament]], with generator a meantone fifth divided in five, and Hieronymus is the generators of that. Jerome/Hieronymus only really comes into its own as a higher limit temperament, as a 13, 17, 19 or even 23 limit system.
Original HTML content:
<html><head><title>20ed5</title></head><body>20th root of 5 "Hieronymus' Tuning"<br /> <br /> An <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> minimum, that has better approximations of a variety of just intervals 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 high-limit harmony it also approximates the harmonics and their pentave reductions:</span><br /> <span class="commentBody">8, 12 (or 61), 23, 27, 32, 44, 48, 52, 56, 66, 71, 77, etc. within 20 cents.</span><br /> <br /> <span class="commentBody">One way of looking at it comes by constructing it via four tempered 3/2 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. 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 generators of that. Jerome/Hieronymus only really comes into its own as a higher limit temperament, as a 13, 17, 19 or even 23 limit system.</body></html>