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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:genewardsmith|genewardsmith]] and made on <tt>2011-09-02 12:52:48 UTC</tt>.<br>
: The original revision id was <tt>250338240</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 high-limit harmony it also approximates the harmonics and their pentave reductions:&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;‎8, 12 (or 61), 23, 27, 32, 44, 48, 52, 56, 66, 71, 77, etc. within 20 cents.&lt;/span&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, which in turn approximate 11/8 13/8 7/6 etc., and themselves end up on the "pentave", 5/1. Adding octaves makes it [[Meantone family#Jerome]] temperament, with generator a meantone fifth divided in five, and Hieronymus is the generators of that. Jerome/Hieronymus only really comes into its own as a higher limit temperament, as a 13, 17, 19 or even 23 limit system.</pre></div>
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.
<h4>Original HTML 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;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;
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.
&lt;br /&gt;
 
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 high-limit harmony it also approximates the harmonics and their pentave reductions:&lt;/span&gt;&lt;br /&gt;
== Harmonics ==
&lt;span class="commentBody"&gt;‎8, 12 (or 61), 23, 27, 32, 44, 48, 52, 56, 66, 71, 77, etc. within 20 cents.&lt;/span&gt;&lt;br /&gt;
{{Harmonics in equal
&lt;br /&gt;
| steps = 20
&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, which in turn approximate 11/8 13/8 7/6 etc., and themselves end up on the &amp;quot;pentave&amp;quot;, 5/1. Adding octaves makes it &lt;a class="wiki_link" href="/Meantone%20family#Jerome"&gt;Meantone family&lt;/a&gt; temperament, with generator a meantone fifth divided in five, and Hieronymus is the generators of that. Jerome/Hieronymus only really comes into its own as a higher limit temperament, as a 13, 17, 19 or even 23 limit system.&lt;/body&gt;&lt;/html&gt;</pre></div>
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 20
| num = 5
| denom = 1
| 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}}
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