Tungsten meantone: 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:MasonGreen1|MasonGreen1]] and made on <tt>2016-10-24 00:42:36 UTC</tt>.

: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-10-24 00:47:14 UTC</tt>.

: The original revision id was <tt>~~596666782~~</tt>.

: The original revision id was <tt>596666880</tt>.

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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">**Tungsten meantone** is the name given by Mason Green in 2016 to the variant of meantone in which the size of the tempered whole tone is equal to 1/S6 of an octave, where S6 is the sixth [[https://en.wikipedia.org/wiki/Metallic_mean|metallic mean]], equal to (6 + sqrt(10))/2 or about 6.162. This gives a whole tone of ~194.733 cents, and a generator width of ~697.367 cents.

Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to [[golden meantone]], since in each case the same ratios ~~appear~~ between interval sizes appear repeatedly and the scale is thus self-similar.

Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to [[golden meantone]], since in each case the same ratios between interval sizes appear repeatedly and the scale is thus self-similar.

Successively better approximations of tungsten meantone are given by the following edos: 2, 12, 74, 456... These form a [[https://en.wikipedia.org/wiki/Lucas_sequence|Lucas sequence]], 2 * //Un//(6,−1).

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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>Tungsten meantone</title></head><body>Tungsten meantone is the name given by Mason Green in 2016 to the variant of meantone in which the size of the tempered whole tone is equal to 1/S6 of an octave, where S6 is the sixth <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metallic_mean" rel="nofollow">metallic mean</a>, equal to (6 + sqrt(10))/2 or about 6.162. This gives a whole tone of ~194.733 cents, and a generator width of ~697.367 cents.

Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to <a class="wiki_link" href="/golden%20meantone">golden meantone</a>, since in each case the same ratios ~~appear~~ between interval sizes appear repeatedly and the scale is thus self-similar.

Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to <a class="wiki_link" href="/golden%20meantone">golden meantone</a>, since in each case the same ratios between interval sizes appear repeatedly and the scale is thus self-similar.

Successively better approximations of tungsten meantone are given by the following edos: 2, 12, 74, 456... These form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lucas_sequence" rel="nofollow">Lucas sequence</a>, 2 * Un(6,−1).

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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:MasonGreen1|MasonGreen1]] and made on <tt>2016-10-24 00:42:36 UTC</tt>.<br>
+: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-10-24 00:47:14 UTC</tt>.<br>
-: The original revision id was <tt>596666782</tt>.<br>
+: The original revision id was <tt>596666880</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>
@@ Line 8: / Line 8: @@
 <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">**Tungsten meantone** is the name given by Mason Green in 2016 to the variant of meantone in which the size of the tempered whole tone is equal to 1/S&lt;span style="vertical-align: sub;"&gt;6 &lt;/span&gt;of an octave, where S&lt;span style="vertical-align: sub;"&gt;6 &lt;/span&gt;is the sixth [[https://en.wikipedia.org/wiki/Metallic_mean|metallic mean]], equal to (6 + sqrt(10))/2 or about 6.162. This gives a whole tone of ~194.733 cents, and a generator width of ~697.367 cents.
-Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to [[golden meantone]], since in each case the same ratios appear between interval sizes appear repeatedly and the scale is thus self-similar.
+Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to [[golden meantone]], since in each case the same ratios between interval sizes appear repeatedly and the scale is thus self-similar.
 Successively better approximations of tungsten meantone are given by the following edos: 2, 12, 74, 456... These form a [[https://en.wikipedia.org/wiki/Lucas_sequence|Lucas sequence]], 2 * //&lt;span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"&gt;U&lt;span style="font-size: 11.2px;"&gt;n&lt;/span&gt;&lt;/span&gt;//&lt;span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"&gt;(6,−1).&lt;/span&gt;
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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;Tungsten meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Tungsten meantone&lt;/strong&gt; is the name given by Mason Green in 2016 to the variant of meantone in which the size of the tempered whole tone is equal to 1/S&lt;span style="vertical-align: sub;"&gt;6 &lt;/span&gt;of an octave, where S&lt;span style="vertical-align: sub;"&gt;6 &lt;/span&gt;is the sixth &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metallic_mean" rel="nofollow"&gt;metallic mean&lt;/a&gt;, equal to (6 + sqrt(10))/2 or about 6.162. This gives a whole tone of ~194.733 cents, and a generator width of ~697.367 cents.&lt;br /&gt;
 &lt;br /&gt;
-Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to &lt;a class="wiki_link" href="/golden%20meantone"&gt;golden meantone&lt;/a&gt;, since in each case the same ratios appear between interval sizes appear repeatedly and the scale is thus self-similar.&lt;br /&gt;
+Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to &lt;a class="wiki_link" href="/golden%20meantone"&gt;golden meantone&lt;/a&gt;, since in each case the same ratios between interval sizes appear repeatedly and the scale is thus self-similar.&lt;br /&gt;
 &lt;br /&gt;
 Successively better approximations of tungsten meantone are given by the following edos: 2, 12, 74, 456... These form a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lucas_sequence" rel="nofollow"&gt;Lucas sequence&lt;/a&gt;, 2 * &lt;em&gt;&lt;span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"&gt;U&lt;span style="font-size: 11.2px;"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/em&gt;&lt;span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"&gt;(6,−1).&lt;/span&gt;&lt;br /&gt;