Dyadic chord

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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 are a related notion to [[comma pump|comma pumps]], and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.

=Anomalous Saturated Suspensions= 
An //anomalous saturated suspension//, or ASS, is a term [[http://www.webcitation.org/60VBgPSUS|introduced]] by [[Graham Breed]] for a q-limit just dyadic chord to which no pitch q-limit pitch class can be added while keeping it in the q-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5: ... :q chord or the 1:1/3:1/5: ... :1/q chord. The existence of such chords was [[http://www.webcitation.org/60VCUHe6d|discovered]] by [[Paul Erlich]]. Below are listed two 9-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

[[just added sixth chord]]
[[swiss tetrad]]

=Just intonation tetrads= 

[[seven limit tetrads]]
[[nine limit tetrads]]
[[1-3-7-11 tetrads]]
[[thirteen limit tetrads]]
[[fifteen limit tetrads]]

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

==7-limit== 
[[starling tetrad]]
[[hendrix chord]]

==9-limit== 
[[meantone add6-9 quintad]]
[[marvel triad]]
[[sensamagic triad]]

==11-limit== 
[[neutral tetrad]]
[[keenanismic tetrads]]
[[jove tetrads]]
[[werckismic chords]]
[[swetismic chords]]
[[magical seventh chord]]
[[orwell tetrad]]
[[tutonic sextad]]

==13-limit== 
[[gentle chords]]
[[minthmic chords]]
[[huntmic chords]]
[[kestrel chords]]
[[mynucumic chords]]
[[marveltwin triad]]
[[petrmic triad]]
[[cuthbert triad]]

==15-limit== 
[[orwell tetrad|guanyin tetrad]]
[[island tetrad]]
[[nicolic tetrad]]
[[battaglia chord]]

=19-limit= 
[[hendrix chord]]
[[rootminor triad]]
[[rootsubminor triad]]

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

Original HTML content:

<html><head><title>Dyadic chord</title></head><body><!-- ws:start:WikiTextTocRule:22:&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:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#Definitions">Definitions</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Anomalous Saturated Suspensions">Anomalous Saturated Suspensions</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Just intonation tetrads">Just intonation tetrads</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Essentially tempered dyadic chords">Essentially tempered dyadic chords</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#x19-limit">19-limit</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: -->
<!-- ws:end:WikiTextTocRule:34 --><!-- 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. Essentially tempered dyadic chords are a related notion to <a class="wiki_link" href="/comma%20pump">comma pumps</a>, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Anomalous Saturated Suspensions"></a><!-- ws:end:WikiTextHeadingRule:2 -->Anomalous Saturated Suspensions</h1>
 An <em>anomalous saturated suspension</em>, or ASS, is a term <a class="wiki_link_ext" href="http://www.webcitation.org/60VBgPSUS" rel="nofollow">introduced</a> by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a> for a q-limit just dyadic chord to which no pitch q-limit pitch class can be added while keeping it in the q-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5: ... :q chord or the 1:1/3:1/5: ... :1/q chord. The existence of such chords was <a class="wiki_link_ext" href="http://www.webcitation.org/60VCUHe6d" rel="nofollow">discovered</a> by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>. Below are listed two 9-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.<br />
<br />
<a class="wiki_link" href="/just%20added%20sixth%20chord">just added sixth chord</a><br />
<a class="wiki_link" href="/swiss%20tetrad">swiss tetrad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Just intonation tetrads"></a><!-- ws:end:WikiTextHeadingRule:4 -->Just intonation tetrads</h1>
 <br />
<a class="wiki_link" href="/seven%20limit%20tetrads">seven limit tetrads</a><br />
<a class="wiki_link" href="/nine%20limit%20tetrads">nine limit tetrads</a><br />
<a class="wiki_link" href="/1-3-7-11%20tetrads">1-3-7-11 tetrads</a><br />
<a class="wiki_link" href="/thirteen%20limit%20tetrads">thirteen limit tetrads</a><br />
<a class="wiki_link" href="/fifteen%20limit%20tetrads">fifteen limit tetrads</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Essentially tempered dyadic chords"></a><!-- ws:end:WikiTextHeadingRule:6 -->Essentially tempered dyadic chords</h1>
 Here are some pages on certain essentially tempered dyadic chords.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Essentially tempered dyadic chords-7-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->7-limit</h2>
 <a class="wiki_link" href="/starling%20tetrad">starling tetrad</a><br />
<a class="wiki_link" href="/hendrix%20chord">hendrix chord</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Essentially tempered dyadic chords-9-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->9-limit</h2>
 <a class="wiki_link" href="/meantone%20add6-9%20quintad">meantone add6-9 quintad</a><br />
<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:12:&lt;h2&gt; --><h2 id="toc6"><a name="Essentially tempered dyadic chords-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2>
 <a class="wiki_link" href="/neutral%20tetrad">neutral tetrad</a><br />
<a class="wiki_link" href="/keenanismic%20tetrads">keenanismic tetrads</a><br />
<a class="wiki_link" href="/jove%20tetrads">jove tetrads</a><br />
<a class="wiki_link" href="/werckismic%20chords">werckismic chords</a><br />
<a class="wiki_link" href="/swetismic%20chords">swetismic chords</a><br />
<a class="wiki_link" href="/magical%20seventh%20chord">magical seventh chord</a><br />
<a class="wiki_link" href="/orwell%20tetrad">orwell tetrad</a><br />
<a class="wiki_link" href="/tutonic%20sextad">tutonic sextad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Essentially tempered dyadic chords-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2>
 <a class="wiki_link" href="/gentle%20chords">gentle chords</a><br />
<a class="wiki_link" href="/minthmic%20chords">minthmic chords</a><br />
<a class="wiki_link" href="/huntmic%20chords">huntmic chords</a><br />
<a class="wiki_link" href="/kestrel%20chords">kestrel chords</a><br />
<a class="wiki_link" href="/mynucumic%20chords">mynucumic chords</a><br />
<a class="wiki_link" href="/marveltwin%20triad">marveltwin triad</a><br />
<a class="wiki_link" href="/petrmic%20triad">petrmic triad</a><br />
<a class="wiki_link" href="/cuthbert%20triad">cuthbert triad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Essentially tempered dyadic chords-15-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->15-limit</h2>
 <a class="wiki_link" href="/orwell%20tetrad">guanyin tetrad</a><br />
<a class="wiki_link" href="/island%20tetrad">island tetrad</a><br />
<a class="wiki_link" href="/nicolic%20tetrad">nicolic tetrad</a><br />
<a class="wiki_link" href="/battaglia%20chord">battaglia chord</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="x19-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->19-limit</h1>
 <a class="wiki_link" href="/hendrix%20chord">hendrix chord</a><br />
<a class="wiki_link" href="/rootminor%20triad">rootminor triad</a><br />
<a class="wiki_link" href="/rootsubminor%20triad">rootsubminor triad</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x19-limit-21-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->21-limit</h2>
 <a class="wiki_link" href="/gamelismic%20pentad">gamelismic pentad</a></body></html>