Meantone intervals: Difference between revisions

Revision as of 20:02, 31 October 2011

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Original Wikitext content:

This table shows all the simple intervals of [[Meantone family#Septimal%20meantone|septimal meantone]], which includes the entire 7-limit tonality diamond.

In [[12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.

In [[19edo]], in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.

See also [[http://en.wikipedia.org/wiki/List_of_meantone_intervals|Wikipedia's list of meantone intervals]]
||~ Name ||~ Size* ||~ Ratios ||
||||||~ Unisons ||
|| Perfect unison (P1) || 0 || 1/1 ||
|| Augmented unison (A1) || 75.5 || 28/27~25/24~21/20 ||
||||||~ Seconds ||
|| Diminished second (d2) || 42.0 || 64/63~50/49~36/35 ||
|| Minor second (m2) || 117.5 || 16/15~15/14 ||
|| Major second (M2) || 193.0 || 10/9~9/8 ||
|| Augmented second (A2) || 268.5 || 7/6 ||
||||||~ Thirds ||
|| Diminished third (d3) || 235.0 || 8/7 ||
|| Minor third (m3) || 310.5 || 6/5 ||
|| Major third (M3) || 386.0 || 5/4 ||
|| Augmented third (A3) || 461.5 || 21/16 ||
||||||~ Fourths ||
|| Diminished fourth (d4) || 428.0 || 9/7 ||
|| Perfect fourth (P4) || 503.5 || 4/3 ||
|| Augmented fourth (A4) || 579.0 || 7/6 ||
|| Double augmented fourth (AA4) || 654.5 || Close to 16/11 ||
||||||~ Fifths ||
|| Double diminished fifth (dd5) || 545.5 || Close to 11/8 ||
|| Diminished fifth (d5) || 621.0 || 10/7 ||
|| Perfect fifth (P5) || 696.5 || 3/2 ||
|| Augmented fifth (A5) || 772.0 || 14/9 ||
||||||~ Sixths ||
|| Diminished sixth (d6) || 738.5 || 32/21 ||
|| Minor sixth (m6) || 814.0 || 8/5 ||
|| Major sixth (M6) || 889.5 || 5/3 ||
|| Augmented sixth (A6) || 965.0 || 7/4 ||
||||||~ Sevenths ||
|| Diminished seventh (d7) || 931.5 || 12/7 ||
|| Minor seventh (m7) || 1007.0 || 16/9~9/5 ||
|| Major seventh (M7) || 1082.5 || 15/8 ||
|| Augmented seventh (A7) || 1158.0 || 35/18~49/25~63/32 ||
||||||~ Octaves ||
|| Diminished octave (d8) || 1124.5 || 27/14 ||
|| Perfect octave (P8) || 1200.0 || 2/1 ||
|| Augmented octave (A8) || 1275.5 || 25/12~21/20 ||
``*`` In POTE septimal meantone

Original HTML content:

<html><head><title>Meantone intervals</title></head><body>This table shows all the simple intervals of <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone">septimal meantone</a>, which includes the entire 7-limit tonality diamond.<br />
<br />
In <a class="wiki_link" href="/12edo">12edo</a> the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.<br />
<br />
In <a class="wiki_link" href="/19edo">19edo</a>, in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.<br />
<br />
See also <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/List_of_meantone_intervals" rel="nofollow">Wikipedia's list of meantone intervals</a><br />


<table class="wiki_table">
    <tr>
        <th>Name<br />
</th>
        <th>Size*<br />
</th>
        <th>Ratios<br />
</th>
    </tr>
    <tr>
        <th colspan="3">Unisons<br />
</th>
    </tr>
    <tr>
        <td>Perfect unison (P1)<br />
</td>
        <td>0<br />
</td>
        <td>1/1<br />
</td>
    </tr>
    <tr>
        <td>Augmented unison (A1)<br />
</td>
        <td>75.5<br />
</td>
        <td>28/27~25/24~21/20<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Seconds<br />
</th>
    </tr>
    <tr>
        <td>Diminished second (d2)<br />
</td>
        <td>42.0<br />
</td>
        <td>64/63~50/49~36/35<br />
</td>
    </tr>
    <tr>
        <td>Minor second (m2)<br />
</td>
        <td>117.5<br />
</td>
        <td>16/15~15/14<br />
</td>
    </tr>
    <tr>
        <td>Major second (M2)<br />
</td>
        <td>193.0<br />
</td>
        <td>10/9~9/8<br />
</td>
    </tr>
    <tr>
        <td>Augmented second (A2)<br />
</td>
        <td>268.5<br />
</td>
        <td>7/6<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Thirds<br />
</th>
    </tr>
    <tr>
        <td>Diminished third (d3)<br />
</td>
        <td>235.0<br />
</td>
        <td>8/7<br />
</td>
    </tr>
    <tr>
        <td>Minor third (m3)<br />
</td>
        <td>310.5<br />
</td>
        <td>6/5<br />
</td>
    </tr>
    <tr>
        <td>Major third (M3)<br />
</td>
        <td>386.0<br />
</td>
        <td>5/4<br />
</td>
    </tr>
    <tr>
        <td>Augmented third (A3)<br />
</td>
        <td>461.5<br />
</td>
        <td>21/16<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Fourths<br />
</th>
    </tr>
    <tr>
        <td>Diminished fourth (d4)<br />
</td>
        <td>428.0<br />
</td>
        <td>9/7<br />
</td>
    </tr>
    <tr>
        <td>Perfect fourth (P4)<br />
</td>
        <td>503.5<br />
</td>
        <td>4/3<br />
</td>
    </tr>
    <tr>
        <td>Augmented fourth (A4)<br />
</td>
        <td>579.0<br />
</td>
        <td>7/6<br />
</td>
    </tr>
    <tr>
        <td>Double augmented fourth (AA4)<br />
</td>
        <td>654.5<br />
</td>
        <td>Close to 16/11<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Fifths<br />
</th>
    </tr>
    <tr>
        <td>Double diminished fifth (dd5)<br />
</td>
        <td>545.5<br />
</td>
        <td>Close to 11/8<br />
</td>
    </tr>
    <tr>
        <td>Diminished fifth (d5)<br />
</td>
        <td>621.0<br />
</td>
        <td>10/7<br />
</td>
    </tr>
    <tr>
        <td>Perfect fifth (P5)<br />
</td>
        <td>696.5<br />
</td>
        <td>3/2<br />
</td>
    </tr>
    <tr>
        <td>Augmented fifth (A5)<br />
</td>
        <td>772.0<br />
</td>
        <td>14/9<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Sixths<br />
</th>
    </tr>
    <tr>
        <td>Diminished sixth (d6)<br />
</td>
        <td>738.5<br />
</td>
        <td>32/21<br />
</td>
    </tr>
    <tr>
        <td>Minor sixth (m6)<br />
</td>
        <td>814.0<br />
</td>
        <td>8/5<br />
</td>
    </tr>
    <tr>
        <td>Major sixth (M6)<br />
</td>
        <td>889.5<br />
</td>
        <td>5/3<br />
</td>
    </tr>
    <tr>
        <td>Augmented sixth (A6)<br />
</td>
        <td>965.0<br />
</td>
        <td>7/4<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Sevenths<br />
</th>
    </tr>
    <tr>
        <td>Diminished seventh (d7)<br />
</td>
        <td>931.5<br />
</td>
        <td>12/7<br />
</td>
    </tr>
    <tr>
        <td>Minor seventh (m7)<br />
</td>
        <td>1007.0<br />
</td>
        <td>16/9~9/5<br />
</td>
    </tr>
    <tr>
        <td>Major seventh (M7)<br />
</td>
        <td>1082.5<br />
</td>
        <td>15/8<br />
</td>
    </tr>
    <tr>
        <td>Augmented seventh (A7)<br />
</td>
        <td>1158.0<br />
</td>
        <td>35/18~49/25~63/32<br />
</td>
    </tr>
    <tr>
        <th colspan="3">Octaves<br />
</th>
    </tr>
    <tr>
        <td>Diminished octave (d8)<br />
</td>
        <td>1124.5<br />
</td>
        <td>27/14<br />
</td>
    </tr>
    <tr>
        <td>Perfect octave (P8)<br />
</td>
        <td>1200.0<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>Augmented octave (A8)<br />
</td>
        <td>1275.5<br />
</td>
        <td>25/12~21/20<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextRawRule:00:``*`` -->*<!-- ws:end:WikiTextRawRule:00 --> In POTE septimal meantone</body></html>

@@ Line 1: / Line 1: @@
 <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:keenanpepper|keenanpepper]] and made on <tt>2011-10-31 19:51:10 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-31 20:02:32 UTC</tt>.<br>
-: The original revision id was <tt>270458208</tt>.<br>
+: The original revision id was <tt>270460686</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">||~ Name ||~ Size* ||~ Ratios ||
+<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">This table shows all the simple intervals of [[Meantone family#Septimal%20meantone|septimal meantone]], which includes the entire 7-limit tonality diamond.
+In [[12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.
+In [[19edo]], in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.
+See also [[http://en.wikipedia.org/wiki/List_of_meantone_intervals|Wikipedia's list of meantone intervals]]
+||~ Name ||~ Size* ||~ Ratios ||
 ||||||~ Unisons ||
 || Perfect unison (P1) || 0 || 1/1 ||
@@ Line 44: / Line 51: @@
 || Perfect octave (P8) || 1200.0 || 2/1 ||
 || Augmented octave (A8) || 1275.5 || 25/12~21/20 ||
-``*`` In POTE septimal meantone
+``*`` In POTE septimal meantone</pre></div>
-</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 intervals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;
+<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 intervals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This table shows all the simple intervals of &lt;a class="wiki_link" href="/Meantone%20family#Septimal%20meantone"&gt;septimal meantone&lt;/a&gt;, which includes the entire 7-limit tonality diamond.&lt;br /&gt;
+&lt;br /&gt;
+In &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.&lt;br /&gt;
+&lt;br /&gt;
+In &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.&lt;br /&gt;
+&lt;br /&gt;
+See also &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/List_of_meantone_intervals" rel="nofollow"&gt;Wikipedia's list of meantone intervals&lt;/a&gt;&lt;br /&gt;
 &lt;table class="wiki_table"&gt;

Meantone intervals: Difference between revisions

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