20ed5

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← 19ed520ed521ed5 →
Prime factorization 22 × 5
Step size 139.316¢ 
Octave 9\20ed5 (1253.84¢)
Twelfth 14\20ed5 (1950.42¢) (→7\10ed5)
Consistency limit 3
Distinct consistency limit 3

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.

Harmonics

Approximation of harmonics in 20ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +53.8 +48.5 -31.6 +0.0 -37.0 -25.2 +22.2 -42.4 +53.8 +28.2 +16.8
Relative (%) +38.6 +34.8 -22.7 +0.0 -26.6 -18.1 +15.9 -30.4 +38.6 +20.2 +12.1
Steps
(reduced)
9
(9)
14
(14)
17
(17)
20
(0)
22
(2)
24
(4)
26
(6)
27
(7)
29
(9)
30
(10)
31
(11)

Intervals

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