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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
'''[[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]].
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-09-02 20:46:12 UTC</tt>.<br>
: The original revision id was <tt>250427982</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 &lt;span class="commentBody"&gt;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 only two or three pentaves within human hearing range, imagine if that were the case with octaves (yes that is a helpful analogy).&lt;/span&gt;
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).


&lt;span class="commentBody"&gt;One way of looking at it comes by constructing it via four tempered 3/2 each of which is divided&lt;/span&gt; into five tones ([[meantone]] without octaves), 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.
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.


[As an endorsement, this is the strangest most mind-blowing scale Kosmorsky has yet heard, not as dissonance, but as a supremely peculiar arrangement of consonance.]</pre></div>
== Harmonics ==
<h4>Original HTML content:</h4>
{{Harmonics in equal
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;20ed5&lt;/title&gt;&lt;/head&gt;&lt;body&gt;20th root of 5 &amp;quot;Hieronymus' Tuning&amp;quot;&lt;br /&gt;
| steps = 20
&lt;br /&gt;
| num = 5
An &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; minimum, that has better approximations of a variety of just intervals than Bohlen Pierce (of course, not the same intervals) among which are &lt;span class="commentBody"&gt;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 only two or three pentaves within human hearing range, imagine if that were the case with octaves (yes that is a helpful analogy).&lt;/span&gt;&lt;br /&gt;
| denom = 1
&lt;br /&gt;
}}
&lt;span class="commentBody"&gt;One way of looking at it comes by constructing it via four tempered 3/2 each of which is divided&lt;/span&gt; into five tones (&lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; without octaves), which in turn approximate 11/8 13/8 7/6 etc., and themselves end up on the &amp;quot;pentave&amp;quot;, 5/1, wherein the scale repeats itself. By analogy to common practice, this is familiar extended meantone but &lt;em&gt;turned entirely inside-out&lt;/em&gt;. 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.&lt;br /&gt;
{{Harmonics in equal
&lt;br /&gt;
| steps = 20
Adding octaves makes it &lt;a class="wiki_link" href="/Meantone%20family#Jerome"&gt;jerome temperament&lt;/a&gt;, 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.&lt;br /&gt;
| num = 5
&lt;br /&gt;
| denom = 1
[As an endorsement, this is the strangest most mind-blowing scale Kosmorsky has yet heard, not as dissonance, but as a supremely peculiar arrangement of consonance.]&lt;/body&gt;&lt;/html&gt;</pre></div>
| start = 12
| collapsed = 1
}}
 
== Intervals ==
 
{| class="wikitable"
|-
! degree
! cents value
! corresponding <br>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
| [[33/32|33/16]]
|
|-
| 10
| 1393.1569
| [[19/17|38/17]], 85/38
| meantone major second plus an octave
|-
| 11
| 1532.4725
| [[40/33|80/33]]
|
|-
| 12
| 1671.7882
| [[21/16|21/8]]
|
|-
| 13
| 1811.1039
| 37/13
|
|-
| 14
| 1950.4196
| [[17/11|34/11]], 37/12, [[20/13|40/13]]
|
|-
| 15
| 2089.7353
|
| meantone major sixth plus an octave
|-
| 16
| 2229.0510
| [[29/16|29/8]]
|
|-
| 17
| 2368.3667
| 55/14
|
|-
| 18
| 2507.6823
| [[17/16|17/4]]
|
|-
| 19
| 2646.9980
| [[15/13|60/13]]
|
|-
| 20
| 2786.3137
| '''exact [[5/1]]'''
| just major third plus two octaves
|}
 
{{todo|add sound example}}

Latest revision as of 19:20, 1 August 2025

← 19ed5 20ed5 21ed5 →
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)
Approximation of harmonics in 20ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +17.6 +28.6 +48.5 -63.3 -28.9 +11.5 +57.2 -31.6 +23.2 -57.3 +5.0
Relative (%) +12.6 +20.5 +34.8 -45.4 -20.7 +8.2 +41.0 -22.7 +16.7 -41.1 +3.6
Steps
(reduced) 		32
(12) 		33
(13) 		34
(14) 		34
(14) 		35
(15) 		36
(16) 		37
(17) 		37
(17) 		38
(18) 		38
(18) 		39
(19)

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
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