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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|harmonic entropy]] minimum, that has better approximations of a variety of [[just_interval|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|human hearing range]]; imagine if that were the case with octaves (yes that is a helpful analogy).
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|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.


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. It is related to [[43edo|43EDO]], and 5\43 can be used as a generator.
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. It is related to [[43edo|43EDO]], and 5\43 can be used as a generator.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! degree
! | cents value
! cents value
! | corresponding <br>JI intervals
! corresponding <br>JI intervals
! | comments
! comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0000
| | '''exact [[1/1]]'''
| '''exact [[1/1]]'''
| |  
|  
|-
|-
| | 1
| 1
| | 139.3157
| 139.3157
| | [[13/12]]
| [[13/12]]
| |  
|  
|-
|-
| | 2
| 2
| | 278.6314
| 278.6314
| | [[20/17]], 27/23
| [[20/17]], 27/23
| |  
|  
|-
|-
| | 3
| 3
| | 417.9471
| 417.9471
| | [[14/11]]
| [[14/11]]
| |  
|  
|-
|-
| | 4
| 4
| | 557.2627
| 557.2627
| | 29/21, 40/29
| 29/21, 40/29
| |  
|  
|-
|-
| | 5
| 5
| | 696.5784
| 696.5784
| |  
|  
| | meantone fifth
| meantone fifth
|-
|-
| | 6
| 6
| | 835.8941
| 835.8941
| | [[13/8]], [[34/21]]
| [[13/8]], [[34/21]]
| |  
|  
|-
|-
| | 7
| 7
| | 975.2098
| 975.2098
| | 58/33, 65/37, 72/41
| 58/33, 65/37, 72/41
| |  
|  
|-
|-
| | 8
| 8
| | 1114.5255
| 1114.5255
| | [[40/21]]
| [[40/21]]
| |  
|  
|-
|-
| | 9
| 9
| | 1253.8412
| 1253.8412
| | [[33/32|33/16]]
| [[33/32|33/16]]
| |  
|  
|-
|-
| | 10
| 10
| | 1393.1569
| 1393.1569
| | [[19/17|38/17]], 85/38
| [[19/17|38/17]], 85/38
| | meantone major second plus an octave
| meantone major second plus an octave
|-
|-
| | 11
| 11
| | 1532.4725
| 1532.4725
| | [[40/33|80/33]]
| [[40/33|80/33]]
| |  
|  
|-
|-
| | 12
| 12
| | 1671.7882
| 1671.7882
| | [[21/16|21/8]]
| [[21/16|21/8]]
| |  
|  
|-
|-
| | 13
| 13
| | 1811.1039
| 1811.1039
| | 37/13
| 37/13
| |  
|  
|-
|-
| | 14
| 14
| | 1950.4196
| 1950.4196
| | [[17/11|34/11]], 37/12, [[20/13|40/13]]
| [[17/11|34/11]], 37/12, [[20/13|40/13]]
| |  
|  
|-
|-
| | 15
| 15
| | 2089.7353
| 2089.7353
| |  
|  
| | meantone major sixth plus an octave
| meantone major sixth plus an octave
|-
|-
| | 16
| 16
| | 2229.0510
| 2229.0510
| | [[29/16|29/8]]
| [[29/16|29/8]]
| |  
|  
|-
|-
| | 17
| 17
| | 2368.3667
| 2368.3667
| | 55/14
| 55/14
| |  
|  
|-
|-
| | 18
| 18
| | 2507.6823
| 2507.6823
| | [[17/16|17/4]]
| [[17/16|17/4]]
| |  
|  
|-
|-
| | 19
| 19
| | 2646.9980
| 2646.9980
| | [[15/13|60/13]]
| [[15/13|60/13]]
| |  
|  
|-
|-
| | 20
| 20
| | 2786.3137
| 2786.3137
| | '''exact [[5/1]]'''
| '''exact [[5/1]]'''
| | just major third plus two octaves
| just major third plus two octaves
|}
|}


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[[Category:Edonoi]]
[[Category:Edonoi]]


[[Category:Todo:add sound example]]
{{todo|add sound example}}

Revision as of 07:17, 24 March 2022

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
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