Dyadic chord: Difference between revisions

Revision as of 18:23, 26 July 2011

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[[toc|flat]]

=Definitions=
By a //dyadic chord// is meant a chord each of whose intervals belongs to a specified set of intervals considered to be consonant; it is therefore relative to the set of intervals in question. By a //just// dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an //essentially just// dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, 1-5/4-3/2 is a just dyadic chord when the consonance set is the 5-limit diamond with octave equivalence, and 0-10-18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1-5/4-3/2.

By an //essentially tempered// dyadic chord is meant a chord defined in an [[abstract regular temperament]] such that each interval belongs to a consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1-6/5-10/7, when mapped by starling temperament, which tempers out 126/125, has each of its intervals in the set of 7-limit consonances which is the tempering of the 7-limit diamond by 126/125. However, (10/7)/(6/5) = 25/21 is 25-limit, and there is no other 7-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord.

=Essentially tempered dyadic chords=
Here are some pages on certain essentially tempered dyadic chords.

==7-limit==
[[starling tetrad|septimal diminished triad]]
[[starling tetrad]]

==9-limit==
[[marvel triad]]
[[sensamagic triad]]

==15-limit==
[[orwell tetrad|guanyin tetrad]]

==21-limit==
[[gamelismic pentad]]

Original HTML content:

<html><head><title>Dyadic chord</title></head><body><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Definitions">Definitions</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Essentially tempered dyadic chords">Essentially tempered dyadic chords</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: -->
<!-- ws:end:WikiTextTocRule:19 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definitions</h1>
By a <em>dyadic chord</em> is meant a chord each of whose intervals belongs to a specified set of intervals considered to be consonant; it is therefore relative to the set of intervals in question. By a <em>just</em> dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an <em>essentially just</em> dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, 1-5/4-3/2 is a just dyadic chord when the consonance set is the 5-limit diamond with octave equivalence, and 0-10-18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1-5/4-3/2.<br />
<br />
By an <em>essentially tempered</em> dyadic chord is meant a chord defined in an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> such that each interval belongs to a consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1-6/5-10/7, when mapped by starling temperament, which tempers out 126/125, has each of its intervals in the set of 7-limit consonances which is the tempering of the 7-limit diamond by 126/125. However, (10/7)/(6/5) = 25/21 is 25-limit, and there is no other 7-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Essentially tempered dyadic chords"></a><!-- ws:end:WikiTextHeadingRule:2 -->Essentially tempered dyadic chords</h1>
Here are some pages on certain essentially tempered dyadic chords.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Essentially tempered dyadic chords-7-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit</h2>
<a class="wiki_link" href="/starling%20tetrad">septimal diminished triad</a><br />
<a class="wiki_link" href="/starling%20tetrad">starling tetrad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Essentially tempered dyadic chords-9-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->9-limit</h2>
<a class="wiki_link" href="/marvel%20triad">marvel triad</a><br />
<a class="wiki_link" href="/sensamagic%20triad">sensamagic triad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Essentially tempered dyadic chords-15-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->15-limit</h2>
<a class="wiki_link" href="/orwell%20tetrad">guanyin tetrad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Essentially tempered dyadic chords-21-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->21-limit</h2>
<a class="wiki_link" href="/gamelismic%20pentad">gamelismic pentad</a></body></html>

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 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>2011-07-26 18:20:56 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-26 18:23:35 UTC</tt>.<br>
-: The original revision id was <tt>242961665</tt>.<br>
+: The original revision id was <tt>242961955</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>
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 [[orwell tetrad|guanyin tetrad]]
+==21-limit==
+[[gamelismic pentad]]
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 </pre></div>
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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;Dyadic chord&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definitions"&gt;Definitions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Essentially tempered dyadic chords"&gt;Essentially tempered dyadic chords&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&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;Dyadic chord&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Definitions"&gt;Definitions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Essentially tempered dyadic chords"&gt;Essentially tempered dyadic chords&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definitions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definitions&lt;/h1&gt;
 By a &lt;em&gt;dyadic chord&lt;/em&gt; is meant a chord each of whose intervals belongs to a specified set of intervals considered to be consonant; it is therefore relative to the set of intervals in question. By a &lt;em&gt;just&lt;/em&gt; dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an &lt;em&gt;essentially just&lt;/em&gt; dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, 1-5/4-3/2 is a just dyadic chord when the consonance set is the 5-limit diamond with octave equivalence, and 0-10-18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1-5/4-3/2.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Essentially tempered dyadic chords-15-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;15-limit&lt;/h2&gt;
-&lt;a class="wiki_link" href="/orwell%20tetrad"&gt;guanyin tetrad&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;a class="wiki_link" href="/orwell%20tetrad"&gt;guanyin tetrad&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Essentially tempered dyadic chords-21-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;21-limit&lt;/h2&gt;
+&lt;a class="wiki_link" href="/gamelismic%20pentad"&gt;gamelismic pentad&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Dyadic chord: Difference between revisions

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