20ed5: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~Kosmorsky~~|~~Kosmorsky~~]] and made on <tt>2011-09-~~02 20~~:48:11 UTC</tt>.

: This revision was by author [[User:guest|guest]] and made on <tt>2011-09-04 15:13:03 UTC</tt>.

: The original revision id was <tt>~~250428170~~</tt>.

: The original revision id was <tt>250647256</tt>.

: The revision comment was: <tt></tt>

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<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 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).

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.

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<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"><html><head><title>20ed5</title></head><body>20th root of 5 &quot;Hieronymus' Tuning&quot;

An <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> 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 ~~only two or three~~ pentaves within human hearing range, imagine if that were the case with octaves (yes that is a helpful analogy).

A <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> 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 (<a class="wiki_link" href="/meantone">meantone</a> 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 &quot;pentave&quot;, 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.

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-09-02 20:48:11 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-09-04 15:13:03 UTC</tt>.<br>
-: The original revision id was <tt>250428170</tt>.<br>
+: The original revision id was <tt>250647256</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>
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 <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 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 (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&gt;
 &lt;span class="commentBody"&gt;One way of looking at it comes by constructing it via four tempered 3/2&lt;/span&gt; ([[meantone]] without octaves) &lt;span class="commentBody"&gt;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, 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.
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 <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;
 &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 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;
+A &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 (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&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;span class="commentBody"&gt;One way of looking at it comes by constructing it via four tempered 3/2&lt;/span&gt; (&lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; without octaves) &lt;span class="commentBody"&gt;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, 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;