Meantone
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<span style="display: block; text-align: right;">Other languages: [[xenharmonie/mitteltönig|Deutsch]]
</span>
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 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=
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 intervals|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)=
* [[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=
||~ [[Eigenmonzo]] ||~ Fifth size (usual name) ||
|| 10/9 || <span class="cwcot">691.202 (1/2 comma)</span> ||
|| 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=
* http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by [[Kyle Gann]]Original HTML content:
<html><head><title>Meantone</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/mittelt%C3%B6nig">Deutsch</a><br />
</span><br />
Meantone is a familar historical <a class="wiki_link" href="/temperament">temperament</a> based on a chain of fifths (or fourths), which is discussed <a class="wiki_link" href="/Meantone%20family">here</a> in the context of the associated family of temperaments, and <a class="wiki_link" href="/Meantone%20vs%20meanpop">here</a> in terms of 11-limit extensions.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="History"></a><!-- ws:end:WikiTextHeadingRule:0 -->History</h1>
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various <a class="wiki_link" href="/Well%20Temperament">Well Temperaments</a> and eventually 12-tone <a class="wiki_link" href="/Equal%20Temperament">Equal Temperament</a> won in popularity.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Theory and Classification"></a><!-- ws:end:WikiTextHeadingRule:2 -->Theory and Classification</h1>
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a <a class="wiki_link" href="/Regular%20Temperaments">rank-2 temperament</a>. 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.<br />
<br />
<a class="wiki_link" href="/Meantone%20intervals">Intervals in meantone</a> have standard names based on the number of steps of the diatonic scale they span (this corresponds to the <a class="wiki_link" href="/val">val</a> <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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Meantone Temperaments (ie, tunings)"></a><!-- ws:end:WikiTextHeadingRule:4 -->Meantone Temperaments (ie, tunings)</h1>
<ul><li><a class="wiki_link" href="/19edo">19-edo</a></li><li><a class="wiki_link" href="/1-3%20Syntonic%20Comma%20Meantone">1/3 Syntonic Comma Meantone</a></li><li><a class="wiki_link" href="/Golden%20Meantone">Golden Meantone</a></li><li><a class="wiki_link" href="/Quarter-comma%20meantone">1/4 Syntonic Comma Meantone</a></li><li><a class="wiki_link" href="/31edo">31-edo</a></li><li><a class="wiki_link" href="/1-5%20Syntonic%20Comma%20Meantone">1/5 Syntonic Comma Meantone</a></li><li><a class="wiki_link" href="/1-6%20Syntonic%20Comma%20Meantone">1/6 Syntonic Comma Meantone</a></li><li><a class="wiki_link" href="/12edo">12-edo</a></li><li><a class="wiki_link" href="/Lucy%20tuning">Lucy tuning</a></li><li><a class="wiki_link" href="/50edo">50-edo</a></li><li><a class="wiki_link" href="/55edo">55-edo</a></li><li><a class="wiki_link" href="/Tungsten%20meantone">Tungsten meantone</a></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Spectrum of Meantone Tunings by Eigenmonzos"></a><!-- ws:end:WikiTextHeadingRule:6 -->Spectrum of Meantone Tunings by Eigenmonzos</h1>
<table class="wiki_table">
<tr>
<th><a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a><br />
</th>
<th>Fifth size (usual name)<br />
</th>
</tr>
<tr>
<td>10/9<br />
</td>
<td><span class="cwcot">691.202 (1/2 comma)</span><br />
</td>
</tr>
<tr>
<td>15\26<br />
</td>
<td>692.308<br />
</td>
</tr>
<tr>
<td>56/45<br />
</td>
<td>694.651<br />
</td>
</tr>
<tr>
<td>28/27<br />
</td>
<td>694.709<br />
</td>
</tr>
<tr>
<td>81/70<br />
</td>
<td>694.732<br />
</td>
</tr>
<tr>
<td>11\19<br />
</td>
<td>694.737<br />
</td>
</tr>
<tr>
<td>6/5<br />
</td>
<td>694.786 (1/3 comma)<br />
</td>
</tr>
<tr>
<td>35/27<br />
</td>
<td>695.389<br />
</td>
</tr>
<tr>
<td>51\88<br />
</td>
<td>695.455<br />
</td>
</tr>
<tr>
<td>1\2 + 1\(4π)<br />
</td>
<td>695.493 (Lucy tuning)<br />
</td>
</tr>
<tr>
<td>9/7<br />
</td>
<td>695.614<br />
</td>
</tr>
<tr>
<td>f^4 = 2f + 2<br />
</td>
<td>695.630 (Wilson fifth)<br />
</td>
</tr>
<tr>
<td>40\69<br />
</td>
<td>695.652<br />
</td>
</tr>
<tr>
<td>25/24<br />
</td>
<td>695.810 (2/7 comma)<br />
</td>
</tr>
<tr>
<td>13/10<br />
</td>
<td>695.838 (ratwolf fifth, meanpop eigenmonzo)<br />
</td>
</tr>
<tr>
<td>36/35<br />
</td>
<td>695.936<br />
</td>
</tr>
<tr>
<td>54/49<br />
</td>
<td>695.987<br />
</td>
</tr>
<tr>
<td>29\50<br />
</td>
<td>696.000<br />
</td>
</tr>
<tr>
<td>15/14<br />
</td>
<td>696.111<br />
</td>
</tr>
<tr>
<td>78125/73728<br />
</td>
<td>696.165 (5-limit least squares)<br />
</td>
</tr>
<tr>
<td>(8 - φ)\11<br />
</td>
<td>696.214 (Golden meantone)<br />
</td>
</tr>
<tr>
<td>49/45<br />
</td>
<td>696.245<br />
</td>
</tr>
<tr>
<td>47\81<br />
</td>
<td>696.296<br />
</td>
</tr>
<tr>
<td>7/6<br />
</td>
<td>696.319<br />
</td>
</tr>
<tr>
<td>48/35<br />
</td>
<td>696.399<br />
</td>
</tr>
<tr>
<td>[19 9 -1 -11><br />
</td>
<td>696.436 (9-limit least squares)<br />
</td>
</tr>
<tr>
<td>5/4<br />
</td>
<td>696.578 (5- 7- and 9-limit minimax, 1/4 comma)<br />
</td>
</tr>
<tr>
<td>49/48<br />
</td>
<td>696.616<br />
</td>
</tr>
<tr>
<td>60/49<br />
</td>
<td>696.626<br />
</td>
</tr>
<tr>
<td>[-55 -11 1 25><br />
</td>
<td>696.648 (7-limit least squares)<br />
</td>
</tr>
<tr>
<td>18\31<br />
</td>
<td>696.774<br />
</td>
</tr>
<tr>
<td>35/32<br />
</td>
<td>696.796<br />
</td>
</tr>
<tr>
<td>8/7<br />
</td>
<td>696.883<br />
</td>
</tr>
<tr>
<td>49/40<br />
</td>
<td>696.959<br />
</td>
</tr>
<tr>
<td>7/5<br />
</td>
<td>697.085<br />
</td>
</tr>
<tr>
<td>43\74<br />
</td>
<td>697.297<br />
</td>
</tr>
<tr>
<td>21/16<br />
</td>
<td>697.344<br />
</td>
</tr>
<tr>
<td>16/15<br />
</td>
<td>697.654 (1/5 comma)<br />
</td>
</tr>
<tr>
<td>25\43<br />
</td>
<td>697.674<br />
</td>
</tr>
<tr>
<td>64/63<br />
</td>
<td>697.728<br />
</td>
</tr>
<tr>
<td>21/20<br />
</td>
<td>697.781<br />
</td>
</tr>
<tr>
<td>28/25<br />
</td>
<td>698.099<br />
</td>
</tr>
<tr>
<td>32\55<br />
</td>
<td>698.182<br />
</td>
</tr>
<tr>
<td>80/63<br />
</td>
<td>698.303<br />
</td>
</tr>
<tr>
<td>45/32<br />
</td>
<td>698.371 (1/6 comma)<br />
</td>
</tr>
<tr>
<td>39\67<br />
</td>
<td>698.507<br />
</td>
</tr>
<tr>
<td>46\79<br />
</td>
<td>698.734<br />
</td>
</tr>
<tr>
<td>25/21<br />
</td>
<td>699.384<br />
</td>
</tr>
<tr>
<td>7\12<br />
</td>
<td>700.000<br />
</td>
</tr>
<tr>
<td>31\53<br />
</td>
<td>701.887<br />
</td>
</tr>
<tr>
<td>3/2<br />
</td>
<td>701.955<br />
</td>
</tr>
</table>
[5/4 7] eigenmonos: <a class="wiki_link" href="/meanwoo12">meanwoo12</a>, <a class="wiki_link" href="/meanwoo19">meanwoo19</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:8 -->Links</h1>
<ul><li><!-- ws:start:WikiTextUrlRule:500:http://www.kylegann.com/histune.html --><a class="wiki_link_ext" href="http://www.kylegann.com/histune.html" rel="nofollow">http://www.kylegann.com/histune.html</a><!-- ws:end:WikiTextUrlRule:500 --> -- An Introduction to Historical Tunings, by <a class="wiki_link" href="/Kyle%20Gann">Kyle Gann</a></li></ul></body></html>