20ed5: Difference between revisions
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'''[[Ed5|Division of the 5th harmonic]] into 20 equal parts''' (20ED5) is known as '''Hieronymus' Tuning'''. The step size is about 139.3157 cents, corresponding to 8.6135 [[EDO]]. | '''[[Ed5|Division of the 5th harmonic]] into 20 equal parts''' (20ED5) is known as '''Hieronymus' Tuning'''. The step size is about 139.3157 cents, corresponding to 8.6135 [[EDO]]. | ||
A [[harmonic entropy]] minimum, that has better approximations of a variety of [[just interval]]s than [[ | 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 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). | ||
One way of looking at it comes by constructing it via four tempered 3/2 ([[meantone]] without octaves) each of which is divided 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. | One way of looking at it comes by constructing it via four tempered 3/2 ([[meantone]] without octaves) each of which is divided 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. | ||
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| just major third plus two octaves | | just major third plus two octaves | ||
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{{todo|add sound example}} | {{todo|add sound example}} | ||