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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<span style="display: block; text-align: right;">Other languages: [[:de:mitteltönig Deutsch]]</span>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2017-07-22 22:15:35 UTC</tt>.<br>
: The original revision id was <tt>615834047</tt>.<br>
: The revision comment was: <tt>added link (deorphaning)</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">&lt;span style="display: block; text-align: right;"&gt;Other languages: [[xenharmonie/mitteltönig|Deutsch]]
&lt;/span&gt;
Meantone is a familar historical [[temperament]] based on a chain of fifths (or fourths), which is discussed [[Meantone family|here]] in the context of the associated family of temperaments, and [[Meantone vs meanpop|here]] in terms of 11-limit extensions.


=History=
Meantone is a familar historical [[temperament|temperament]] based on a chain of fifths (or fourths), which is discussed [[Meantone_family|here]] in the context of the associated family of temperaments, and [[Meantone_vs_meanpop|here]] in terms of 11-limit extensions.
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[Well Temperament|Well Temperaments]] and eventually 12-tone [[Equal Temperament]] won in popularity.


=Theory and Classification=  
=History=
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[Well_Temperament|Well Temperaments]] and eventually 12-tone [[Equal_Temperament|Equal Temperament]] won in popularity.


[[Meantone intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val]] &lt;7 11 16|), with a modifier {..."double diminished", "diminished", "minor", "major", "augmented", "double augmented"...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.
=Theory and Classification=
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular_Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.


=Meantone Temperaments (ie, tunings)=
[[Meantone_intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val|val]] &lt;7 11 16|), with a modifier {..."double diminished", "diminished", "minor", "major", "augmented", "double augmented"...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.
* [[19edo|19-edo]]
* [[1-3 Syntonic Comma Meantone|1/3 Syntonic Comma Meantone]]
* [[Golden Meantone]]
* [[Quarter-comma meantone|1/4 Syntonic Comma Meantone]]
* [[31edo|31-edo]]
* [[1-5 Syntonic Comma Meantone|1/5 Syntonic Comma Meantone]]
* [[1-6 Syntonic Comma Meantone|1/6 Syntonic Comma Meantone]]
* [[12edo|12-edo]]
* [[Lucy tuning]]
* [[50edo|50-edo]]
* [[55edo|55-edo]]
* [[Tungsten meantone]]


=Spectrum of Meantone Tunings by Eigenmonzos=  
=Meantone Temperaments (ie, tunings)=
||~ [[Eigenmonzo]] ||~ Fifth size (usual name) ||
<ul><li>[[19edo|19-edo]]</li><li>[[1-3_Syntonic_Comma_Meantone|1/3 Syntonic Comma Meantone]]</li><li>[[Golden_Meantone|Golden Meantone]]</li><li>[[Quarter-comma_meantone|1/4 Syntonic Comma Meantone]]</li><li>[[31edo|31-edo]]</li><li>[[1-5_Syntonic_Comma_Meantone|1/5 Syntonic Comma Meantone]]</li><li>[[1-6_Syntonic_Comma_Meantone|1/6 Syntonic Comma Meantone]]</li><li>[[12edo|12-edo]]</li><li>[[Lucy_Tuning|Lucy tuning]]</li><li>[[50edo|50-edo]]</li><li>[[55edo|55-edo]]</li><li>[[Tungsten_meantone|Tungsten meantone]]</li></ul>
|| 10/9 || &lt;span class="cwcot"&gt;691.202 (1/2 comma)&lt;/span&gt; ||
|| 15\26 || 692.308 ||
|| 56/45 || 694.651 ||
|| 28/27 || 694.709 ||
|| 81/70 || 694.732 ||
|| 11\19 || 694.737 ||
|| 6/5 || 694.786 (1/3 comma) ||
|| 35/27 || 695.389 ||
|| 51\88 || 695.455 ||
|| 1\2 + 1\(4π) || 695.493 (Lucy tuning) ||
|| 9/7 || 695.614 ||
|| f^4 = 2f + 2 || 695.630 (Wilson fifth) ||
|| 40\69 || 695.652 ||
|| 25/24 || 695.810 (2/7 comma) ||
|| 13/10 || 695.838 (ratwolf fifth, meanpop eigenmonzo) ||
|| 36/35 || 695.936 ||
|| 54/49 || 695.987 ||
|| 29\50 || 696.000 ||
|| 15/14 || 696.111 ||
|| 78125/73728 || 696.165 (5-limit least squares) ||
|| (8 - φ)\11 || 696.214 (Golden meantone) ||
|| 49/45 || 696.245 ||
|| 47\81 || 696.296 ||
|| 7/6 || 696.319 ||
|| 48/35 || 696.399 ||
|| [19 9 -1 -11&gt; || 696.436 (9-limit least squares) ||
|| 5/4 || 696.578 (5- 7- and 9-limit minimax, 1/4 comma) ||
|| 49/48 || 696.616 ||
|| 60/49 || 696.626 ||
|| [-55 -11 1 25&gt; || 696.648 (7-limit least squares) ||
|| 18\31 || 696.774 ||
|| 35/32 || 696.796 ||
|| 8/7 || 696.883 ||
|| 49/40 || 696.959 ||
|| 7/5 || 697.085 ||
|| 43\74 || 697.297 ||
|| 21/16 || 697.344 ||
|| 16/15 || 697.654 (1/5 comma) ||
|| 25\43 || 697.674 ||
|| 64/63 || 697.728 ||
|| 21/20 || 697.781 ||
|| 28/25 || 698.099 ||
|| 32\55 || 698.182 ||
|| 80/63 || 698.303 ||
|| 45/32 || 698.371 (1/6 comma) ||
|| 39\67 || 698.507 ||
|| 46\79 || 698.734 ||
|| 25/21 || 699.384 ||
|| 7\12 || 700.000 ||
|| 31\53 || 701.887 ||
|| 3/2 || 701.955 ||
[5/4 7] eigenmonos: [[meanwoo12]], [[meanwoo19]]


=Links=
=Spectrum of Meantone Tunings by Eigenmonzos=
* http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by [[Kyle Gann]]</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;Meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;Other languages: &lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/mittelt%C3%B6nig"&gt;Deutsch&lt;/a&gt;&lt;br /&gt;
&lt;/span&gt;&lt;br /&gt;
Meantone is a familar historical &lt;a class="wiki_link" href="/temperament"&gt;temperament&lt;/a&gt; based on a chain of fifths (or fourths), which is discussed &lt;a class="wiki_link" href="/Meantone%20family"&gt;here&lt;/a&gt; in the context of the associated family of temperaments, and &lt;a class="wiki_link" href="/Meantone%20vs%20meanpop"&gt;here&lt;/a&gt; in terms of 11-limit extensions.&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="History"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;History&lt;/h1&gt;
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various &lt;a class="wiki_link" href="/Well%20Temperament"&gt;Well Temperaments&lt;/a&gt; and eventually 12-tone &lt;a class="wiki_link" href="/Equal%20Temperament"&gt;Equal Temperament&lt;/a&gt; won in popularity.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Theory and Classification"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Theory and Classification&lt;/h1&gt;
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;rank-2 temperament&lt;/a&gt;. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Meantone%20intervals"&gt;Intervals in meantone&lt;/a&gt; have standard names based on the number of steps of the diatonic scale they span (this corresponds to the &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt;7 11 16|), with a modifier {...&amp;quot;double diminished&amp;quot;, &amp;quot;diminished&amp;quot;, &amp;quot;minor&amp;quot;, &amp;quot;major&amp;quot;, &amp;quot;augmented&amp;quot;, &amp;quot;double augmented&amp;quot;...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Meantone Temperaments (ie, tunings)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Meantone Temperaments (ie, tunings)&lt;/h1&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/19edo"&gt;19-edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/1-3%20Syntonic%20Comma%20Meantone"&gt;1/3 Syntonic Comma Meantone&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Golden%20Meantone"&gt;Golden Meantone&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Quarter-comma%20meantone"&gt;1/4 Syntonic Comma Meantone&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/31edo"&gt;31-edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/1-5%20Syntonic%20Comma%20Meantone"&gt;1/5 Syntonic Comma Meantone&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/1-6%20Syntonic%20Comma%20Meantone"&gt;1/6 Syntonic Comma Meantone&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/12edo"&gt;12-edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Lucy%20tuning"&gt;Lucy tuning&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/50edo"&gt;50-edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/55edo"&gt;55-edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Tungsten%20meantone"&gt;Tungsten meantone&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Spectrum of Meantone Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Spectrum of Meantone Tunings by Eigenmonzos&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;th&gt;&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzo&lt;/a&gt;&lt;br /&gt;
! | [[Eigenmonzo|Eigenmonzo]]
&lt;/th&gt;
! | Fifth size (usual name)
        &lt;th&gt;Fifth size (usual name)&lt;br /&gt;
|-
&lt;/th&gt;
| | 10/9
    &lt;/tr&gt;
| | <span style="">691.202 (1/2 comma)</span>
    &lt;tr&gt;
|-
        &lt;td&gt;10/9&lt;br /&gt;
| | 15\26
&lt;/td&gt;
| | 692.308
        &lt;td&gt;&lt;span class="cwcot"&gt;691.202 (1/2 comma)&lt;/span&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| | 56/45
    &lt;/tr&gt;
| | 694.651
    &lt;tr&gt;
|-
        &lt;td&gt;15\26&lt;br /&gt;
| | 28/27
&lt;/td&gt;
| | 694.709
        &lt;td&gt;692.308&lt;br /&gt;
|-
&lt;/td&gt;
| | 81/70
    &lt;/tr&gt;
| | 694.732
    &lt;tr&gt;
|-
        &lt;td&gt;56/45&lt;br /&gt;
| | 11\19
&lt;/td&gt;
| | 694.737
        &lt;td&gt;694.651&lt;br /&gt;
|-
&lt;/td&gt;
| | 6/5
    &lt;/tr&gt;
| | 694.786 (1/3 comma)
    &lt;tr&gt;
|-
        &lt;td&gt;28/27&lt;br /&gt;
| | 35/27
&lt;/td&gt;
| | 695.389
        &lt;td&gt;694.709&lt;br /&gt;
|-
&lt;/td&gt;
| | 51\88
    &lt;/tr&gt;
| | 695.455
    &lt;tr&gt;
|-
        &lt;td&gt;81/70&lt;br /&gt;
| | 1\2 + 1\(4π)
&lt;/td&gt;
| | 695.493 (Lucy tuning)
        &lt;td&gt;694.732&lt;br /&gt;
|-
&lt;/td&gt;
| | 9/7
    &lt;/tr&gt;
| | 695.614
    &lt;tr&gt;
|-
        &lt;td&gt;11\19&lt;br /&gt;
| | f^4 = 2f + 2
&lt;/td&gt;
| | 695.630 (Wilson fifth)
        &lt;td&gt;694.737&lt;br /&gt;
|-
&lt;/td&gt;
| | 40\69
    &lt;/tr&gt;
| | 695.652
    &lt;tr&gt;
|-
        &lt;td&gt;6/5&lt;br /&gt;
| | 25/24
&lt;/td&gt;
| | 695.810 (2/7 comma)
        &lt;td&gt;694.786 (1/3 comma)&lt;br /&gt;
|-
&lt;/td&gt;
| | 13/10
    &lt;/tr&gt;
| | 695.838 (ratwolf fifth, meanpop eigenmonzo)
    &lt;tr&gt;
|-
        &lt;td&gt;35/27&lt;br /&gt;
| | 36/35
&lt;/td&gt;
| | 695.936
        &lt;td&gt;695.389&lt;br /&gt;
|-
&lt;/td&gt;
| | 54/49
    &lt;/tr&gt;
| | 695.987
    &lt;tr&gt;
|-
        &lt;td&gt;51\88&lt;br /&gt;
| | 29\50
&lt;/td&gt;
| | 696.000
        &lt;td&gt;695.455&lt;br /&gt;
|-
&lt;/td&gt;
| | 15/14
    &lt;/tr&gt;
| | 696.111
    &lt;tr&gt;
|-
        &lt;td&gt;1\2 + 1\(4π)&lt;br /&gt;
| | 78125/73728
&lt;/td&gt;
| | 696.165 (5-limit least squares)
        &lt;td&gt;695.493 (Lucy tuning)&lt;br /&gt;
|-
&lt;/td&gt;
| | (8 - φ)\11
    &lt;/tr&gt;
| | 696.214 (Golden meantone)
    &lt;tr&gt;
|-
        &lt;td&gt;9/7&lt;br /&gt;
| | 49/45
&lt;/td&gt;
| | 696.245
        &lt;td&gt;695.614&lt;br /&gt;
|-
&lt;/td&gt;
| | 47\81
    &lt;/tr&gt;
| | 696.296
    &lt;tr&gt;
|-
        &lt;td&gt;f^4 = 2f + 2&lt;br /&gt;
| | 7/6
&lt;/td&gt;
| | 696.319
        &lt;td&gt;695.630 (Wilson fifth)&lt;br /&gt;
|-
&lt;/td&gt;
| | 48/35
    &lt;/tr&gt;
| | 696.399
    &lt;tr&gt;
|-
        &lt;td&gt;40\69&lt;br /&gt;
| | [19 9 -1 -11&gt;
&lt;/td&gt;
| | 696.436 (9-limit least squares)
        &lt;td&gt;695.652&lt;br /&gt;
|-
&lt;/td&gt;
| | 5/4
    &lt;/tr&gt;
| | 696.578 (5- 7- and 9-limit minimax, 1/4 comma)
    &lt;tr&gt;
|-
        &lt;td&gt;25/24&lt;br /&gt;
| | 49/48
&lt;/td&gt;
| | 696.616
        &lt;td&gt;695.810 (2/7 comma)&lt;br /&gt;
|-
&lt;/td&gt;
| | 60/49
    &lt;/tr&gt;
| | 696.626
    &lt;tr&gt;
|-
        &lt;td&gt;13/10&lt;br /&gt;
| | [-55 -11 1 25&gt;
&lt;/td&gt;
| | 696.648 (7-limit least squares)
        &lt;td&gt;695.838 (ratwolf fifth, meanpop eigenmonzo)&lt;br /&gt;
|-
&lt;/td&gt;
| | 18\31
    &lt;/tr&gt;
| | 696.774
    &lt;tr&gt;
|-
        &lt;td&gt;36/35&lt;br /&gt;
| | 35/32
&lt;/td&gt;
| | 696.796
        &lt;td&gt;695.936&lt;br /&gt;
|-
&lt;/td&gt;
| | 8/7
    &lt;/tr&gt;
| | 696.883
    &lt;tr&gt;
|-
        &lt;td&gt;54/49&lt;br /&gt;
| | 49/40
&lt;/td&gt;
| | 696.959
        &lt;td&gt;695.987&lt;br /&gt;
|-
&lt;/td&gt;
| | 7/5
    &lt;/tr&gt;
| | 697.085
    &lt;tr&gt;
|-
        &lt;td&gt;29\50&lt;br /&gt;
| | 43\74
&lt;/td&gt;
| | 697.297
        &lt;td&gt;696.000&lt;br /&gt;
|-
&lt;/td&gt;
| | 21/16
    &lt;/tr&gt;
| | 697.344
    &lt;tr&gt;
|-
        &lt;td&gt;15/14&lt;br /&gt;
| | 16/15
&lt;/td&gt;
| | 697.654 (1/5 comma)
        &lt;td&gt;696.111&lt;br /&gt;
|-
&lt;/td&gt;
| | 25\43
    &lt;/tr&gt;
| | 697.674
    &lt;tr&gt;
|-
        &lt;td&gt;78125/73728&lt;br /&gt;
| | 64/63
&lt;/td&gt;
| | 697.728
        &lt;td&gt;696.165 (5-limit least squares)&lt;br /&gt;
|-
&lt;/td&gt;
| | 21/20
    &lt;/tr&gt;
| | 697.781
    &lt;tr&gt;
|-
        &lt;td&gt;(8 - φ)\11&lt;br /&gt;
| | 28/25
&lt;/td&gt;
| | 698.099
        &lt;td&gt;696.214 (Golden meantone)&lt;br /&gt;
|-
&lt;/td&gt;
| | 32\55
    &lt;/tr&gt;
| | 698.182
    &lt;tr&gt;
|-
        &lt;td&gt;49/45&lt;br /&gt;
| | 80/63
&lt;/td&gt;
| | 698.303
        &lt;td&gt;696.245&lt;br /&gt;
|-
&lt;/td&gt;
| | 45/32
    &lt;/tr&gt;
| | 698.371 (1/6 comma)
    &lt;tr&gt;
|-
        &lt;td&gt;47\81&lt;br /&gt;
| | 39\67
&lt;/td&gt;
| | 698.507
        &lt;td&gt;696.296&lt;br /&gt;
|-
&lt;/td&gt;
| | 46\79
    &lt;/tr&gt;
| | 698.734
    &lt;tr&gt;
|-
        &lt;td&gt;7/6&lt;br /&gt;
| | 25/21
&lt;/td&gt;
| | 699.384
        &lt;td&gt;696.319&lt;br /&gt;
|-
&lt;/td&gt;
| | 7\12
    &lt;/tr&gt;
| | 700.000
    &lt;tr&gt;
|-
        &lt;td&gt;48/35&lt;br /&gt;
| | 31\53
&lt;/td&gt;
| | 701.887
        &lt;td&gt;696.399&lt;br /&gt;
|-
&lt;/td&gt;
| | 3/2
    &lt;/tr&gt;
| | 701.955
    &lt;tr&gt;
|}
        &lt;td&gt;[19 9 -1 -11&amp;gt;&lt;br /&gt;
[5/4 7] eigenmonos: [[meanwoo12|meanwoo12]], [[meanwoo19|meanwoo19]]
&lt;/td&gt;
        &lt;td&gt;696.436 (9-limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.578 (5- 7- and 9-limit minimax, 1/4 comma)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;49/48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.616&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;60/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.626&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[-55 -11 1 25&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.648 (7-limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18\31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.774&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;35/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.796&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.883&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;49/40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.959&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.085&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;43\74&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.297&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.344&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.654 (1/5 comma)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25\43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.674&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;64/63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.728&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21/20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;697.781&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;28/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;698.099&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;32\55&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;698.182&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;80/63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;698.303&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;45/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;698.371 (1/6 comma)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;39\67&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;698.507&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;46\79&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;698.734&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;699.384&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;700.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;31\53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;701.887&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;701.955&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


[5/4 7] eigenmonos: &lt;a class="wiki_link" href="/meanwoo12"&gt;meanwoo12&lt;/a&gt;, &lt;a class="wiki_link" href="/meanwoo19"&gt;meanwoo19&lt;/a&gt;&lt;br /&gt;
=Links=
&lt;br /&gt;
<ul><li>[http://www.kylegann.com/histune.html http://www.kylegann.com/histune.html] -- An Introduction to Historical Tunings, by [[Kyle_Gann|Kyle Gann]]</li></ul>     [[Category:meantone]]
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Links&lt;/h1&gt;
[[Category:temperament]]
&lt;ul&gt;&lt;li&gt;&lt;!-- ws:start:WikiTextUrlRule:500:http://www.kylegann.com/histune.html --&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/histune.html" rel="nofollow"&gt;http://www.kylegann.com/histune.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:500 --&gt; -- An Introduction to Historical Tunings, by &lt;a class="wiki_link" href="/Kyle%20Gann"&gt;Kyle Gann&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Category:theory]]

Revision as of 00:00, 17 July 2018

Other languages: de:mitteltönig Deutsch

Meantone is a familar historical temperament based on a chain of fifths (or fourths), which is discussed here in the context of the associated family of temperaments, and here in terms of 11-limit extensions.

History

Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various Well Temperaments and eventually 12-tone Equal Temperament won in popularity.

Theory and Classification

Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a rank-2 temperament. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.

Intervals in meantone have standard names based on the number of steps of the diatonic scale they span (this corresponds to the val <7 11 16|), with a modifier {..."double diminished", "diminished", "minor", "major", "augmented", "double augmented"...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.

Meantone Temperaments (ie, tunings)

Spectrum of Meantone Tunings by Eigenmonzos

Eigenmonzo Fifth size (usual name)
10/9 691.202 (1/2 comma)
15\26 692.308
56/45 694.651
28/27 694.709
81/70 694.732
11\19 694.737
6/5 694.786 (1/3 comma)
35/27 695.389
51\88 695.455
1\2 + 1\(4π) 695.493 (Lucy tuning)
9/7 695.614
f^4 = 2f + 2 695.630 (Wilson fifth)
40\69 695.652
25/24 695.810 (2/7 comma)
13/10 695.838 (ratwolf fifth, meanpop eigenmonzo)
36/35 695.936
54/49 695.987
29\50 696.000
15/14 696.111
78125/73728 696.165 (5-limit least squares)
(8 - φ)\11 696.214 (Golden meantone)
49/45 696.245
47\81 696.296
7/6 696.319
48/35 696.399
[19 9 -1 -11> 696.436 (9-limit least squares)
5/4 696.578 (5- 7- and 9-limit minimax, 1/4 comma)
49/48 696.616
60/49 696.626
[-55 -11 1 25> 696.648 (7-limit least squares)
18\31 696.774
35/32 696.796
8/7 696.883
49/40 696.959
7/5 697.085
43\74 697.297
21/16 697.344
16/15 697.654 (1/5 comma)
25\43 697.674
64/63 697.728
21/20 697.781
28/25 698.099
32\55 698.182
80/63 698.303
45/32 698.371 (1/6 comma)
39\67 698.507
46\79 698.734
25/21 699.384
7\12 700.000
31\53 701.887
3/2 701.955

[5/4 7] eigenmonos: meanwoo12, meanwoo19

Links