20ed5
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 intervals 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.
Adding octaves makes it 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.
degree | cents value | corresponding JI intervals |
comments |
---|---|---|---|
0 | 0.0000 | exact 1/1 | |
1 | 139.3157 | 13/12 | |
2 | 278.6314 | 20/17, 27/23 | |
3 | 417.9471 | 14/11 | |
4 | 557.2627 | 29/21, 40/29 | |
5 | 696.5784 | meantone fifth | |
6 | 835.8941 | 13/8, 34/21 | |
7 | 975.2098 | 58/33, 65/37, 72/41 | |
8 | 1114.5255 | 40/21 | |
9 | 1253.8412 | 33/16 | |
10 | 1393.1569 | 38/17, 85/38 | meantone major second plus an octave |
11 | 1532.4725 | 80/33 | |
12 | 1671.7882 | 21/8 | |
13 | 1811.1039 | 37/13 | |
14 | 1950.4196 | 34/11, 37/12, 40/13 | |
15 | 2089.7353 | meantone major sixth plus an octave | |
16 | 2229.0510 | 29/8 | |
17 | 2368.3667 | 55/14 | |
18 | 2507.6823 | 17/4 | |
19 | 2646.9980 | 60/13 | |
20 | 2786.3137 | exact 5/1 | just major third plus two octaves |