Dyadic chord: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~Omegatron~~|~~Omegatron~~]] and made on <tt>~~2014~~-08-~~13 20~~:30:06 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-06 05:15:49 UTC</tt>.<br>

: The original revision id was <tt>~~518470238~~</tt>.<br>

: The original revision id was <tt>608163375</tt>.<br>

: The revision comment was: <tt>~~can't find any explanation for these~~</tt><br>

: The revision comment was: <tt></tt><br>

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<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">[[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, [[Ratios|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.

Kite Giedraitis has proposed the term "innate comma chord" to describe the type of chord that can't be mapped to just intonation in a given prime limit and odd limit, hence a chord that won't "ring". This is broader than the term "essentially tempered chord" because it includes the possibility that the chord is not tempered at all, and contains a wolf interval. For example, the augmented triad in 5-limit JI is an innate comma chord, because it's impossible to tune all three major 3rds to 5/4. In practice, it might be sung or played justly as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5, with a wolf interval. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. However in 19-limit JI (or in the 2.3.5.19 subgroup), the augmented chord can be tuned 12:15:19, and thus isn't an innate comma chord.

=Anomalous Saturated Suspensions=

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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"><html><head><title>Dyadic chord</title></head><body><a href="#Definitions">Definitions</a> | <a href="#Anomalous Saturated Suspensions">Anomalous Saturated Suspensions</a> | <a href="#Just intonation tetrads">Just intonation tetrads</a> | <a href="#Essentially tempered dyadic chords">Essentially tempered dyadic chords</a>

<h1 id="toc0"><a name="Definitions"></a>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, <a class="wiki_link" href="/Ratios">1-5/4-3/2</a> 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 />

Kite Giedraitis has proposed the term &quot;innate comma chord&quot; to describe the type of chord that can't be mapped to just intonation in a given prime limit and odd limit, hence a chord that won't &quot;ring&quot;. This is broader than the term &quot;essentially tempered chord&quot; because it includes the possibility that the chord is not tempered at all, and contains a wolf interval. For example, the augmented triad in 5-limit JI is an innate comma chord, because it's impossible to tune all three major 3rds to 5/4. In practice, it might be sung or played justly as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5, with a wolf interval. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. However in 19-limit JI (or in the 2.3.5.19 subgroup), the augmented chord can be tuned 12:15:19, and thus isn't an innate comma chord.<br />

<br />

<h1 id="toc1"><a name="Anomalous Saturated Suspensions"></a>Anomalous Saturated Suspensions</h1>

@@ 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:Omegatron|Omegatron]] and made on <tt>2014-08-13 20:30:06 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-06 05:15:49 UTC</tt>.<br>
-: The original revision id was <tt>518470238</tt>.<br>
+: The original revision id was <tt>608163375</tt>.<br>
-: The revision comment was: <tt>can't find any explanation for these</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">[[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, [[Ratios|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 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, [[Ratios|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.
+Kite Giedraitis has proposed the term "innate comma chord" to describe the type of chord that can't be mapped to just intonation in a given prime limit and odd limit, hence a chord that won't "ring". This is broader than the term "essentially tempered chord" because it includes the possibility that the chord is not tempered at all, and contains a wolf interval. For example, the augmented triad in 5-limit JI is an innate comma chord, because it's impossible to tune all three major 3rds to 5/4. In practice, it might be sung or played justly as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5, with a wolf interval. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. However in 19-limit JI (or in the 2.3.5.19 subgroup), the augmented chord can be tuned 12:15:19, and thus isn't an innate comma chord.
 =Anomalous Saturated Suspensions=
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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:22:&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;a href="#Definitions"&gt;Definitions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Anomalous Saturated Suspensions"&gt;Anomalous Saturated Suspensions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Just intonation tetrads"&gt;Just intonation tetrads&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Essentially tempered dyadic chords"&gt;Essentially tempered dyadic chords&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:34 --&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, &lt;a class="wiki_link" href="/Ratios"&gt;1-5/4-3/2&lt;/a&gt;  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;
+  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, &lt;a class="wiki_link" href="/Ratios"&gt;1-5/4-3/2&lt;/a&gt; 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;
 &lt;br /&gt;
 By an &lt;em&gt;essentially tempered&lt;/em&gt; dyadic chord is meant a chord defined in an &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt; 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 &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pumps&lt;/a&gt;, 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.&lt;br /&gt;
+&lt;br /&gt;
+Kite Giedraitis has proposed the term &amp;quot;innate comma chord&amp;quot; to describe the type of chord that can't be mapped to just intonation in a given prime limit and odd limit, hence a chord that won't &amp;quot;ring&amp;quot;. This is broader than the term &amp;quot;essentially tempered chord&amp;quot; because it includes the possibility that the chord is not tempered at all, and contains a wolf interval. For example, the augmented triad in 5-limit JI is an innate comma chord, because it's impossible to tune all three major 3rds to 5/4. In practice, it might be sung or played justly as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5, with a wolf interval. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. However in 19-limit JI (or in the 2.3.5.19 subgroup), the augmented chord can be tuned 12:15:19, and thus isn't an innate comma chord.&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="Anomalous Saturated Suspensions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Anomalous Saturated Suspensions&lt;/h1&gt;