1/6-comma meantone: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
1/6 comma meantone is the tuning of [[Meantone_family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo|55edo]] in terms of its tuning.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-30 13:11:39 UTC</tt>.<br>
: The original revision id was <tt>260052498</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">1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo]] in terms of its tuning.


=Fractional projection matrix=
=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
The [[Fractional_monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 || 0 || 0 || 0&gt; ||
 
|| [2/3 || 1/3 || 1/6 || 0&gt; ||
{| class="wikitable"
|| [-4/3 || 4/3 || 2/3 || 0&gt; ||
|-
|| [-19/3 || 10/3 || 5/3 || 0&gt; ||
| | [1
| | 0
| | 0
| | 0&gt;
|-
| | [2/3
| | 1/3
| | 1/6
| | 0&gt;
|-
| | [-4/3
| | 4/3
| | 2/3
| | 0&gt;
|-
| | [-19/3
| | 10/3
| | 5/3
| | 0&gt;
|}


=Links=
=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]</pre></div>
<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;1-6 Syntonic Comma Meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;1/6 comma meantone is the tuning of &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt; in terms of its tuning.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Fractional projection matrix"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Fractional projection matrix&lt;/h1&gt;
The &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional projection map&lt;/a&gt; defining 7-limit 1/6 comma meantone is&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;[1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[2/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[-4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[-19/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe_Monzo|Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]      [[Category:1/6-comma]]
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Links&lt;/h1&gt;
[[Category:meantone]]
&lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&lt;/a&gt; by Ross W. Duffin &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&gt;permalink&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&lt;/a&gt; by &lt;a class="wiki_link" href="/Joe%20Monzo"&gt;Joe Monzo&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&gt;permalink&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

1/6 comma meantone is the tuning of meantone temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to 55edo in terms of its tuning.

Fractional projection matrix

The fractional projection map defining 7-limit 1/6 comma meantone is

[1 0 0 0>
[2/3 1/3 1/6 0>
[-4/3 4/3 2/3 0>
[-19/3 10/3 5/3 0>

Links

Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone by Ross W. Duffin permalink

Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone by Joe Monzo permalink

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