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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | =Definitions= |
| : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-06 05:46:09 UTC</tt>.<br>
| | 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. |
| : The original revision id was <tt>608164341</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
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| =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. | |
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| 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. | | By an ''essentially tempered'' dyadic chord is meant a chord defined in an [[Abstract_regular_temperament|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. |
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| 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 term 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. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. In 7-limit JI, one of the 3rds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three 3rds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord isn't an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it's debatable that this chord qualifies as an augmented triad, because the upper 3rd hardly sounds major. | | 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 term 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. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. In 7-limit JI, one of the 3rds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three 3rds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord isn't an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it's debatable that this chord qualifies as an augmented triad, because the upper 3rd hardly sounds major. |
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| =Anomalous Saturated Suspensions= | | =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. | | An ''anomalous saturated suspension'', or ASS, is a term [http://www.webcitation.org/60VBgPSUS introduced] by [[Graham_Breed|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|Paul Erlich]]. Below are listed two 9-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone. |
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| [[just added sixth chord]] | | [[just_added_sixth_chord|just added sixth chord]] |
| [[swiss tetrad]]
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| =Just intonation tetrads=
| | [[swiss_tetrad|swiss tetrad]] |
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| [[seven limit tetrads]]
| | =Just intonation tetrads= |
| [[nine limit tetrads]]
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| [[1-3-7-11 tetrads]]
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| [[thirteen limit tetrads]]
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| [[fifteen limit tetrads]]
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| [[seventeen limit tetrads]]
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| =Essentially tempered dyadic chords= | | [[seven_limit_tetrads|seven limit tetrads]] |
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| | [[nine_limit_tetrads|nine limit tetrads]] |
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| | [[1-3-7-11_tetrads|1-3-7-11 tetrads]] |
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| | [[thirteen_limit_tetrads|thirteen limit tetrads]] |
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| | [[fifteen_limit_tetrads|fifteen limit tetrads]] |
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| | [[seventeen_limit_tetrads|seventeen limit tetrads]] |
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| | =Essentially tempered dyadic chords= |
| Here are some pages on certain essentially tempered dyadic chords. | | Here are some pages on certain essentially tempered dyadic chords. |
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| ==7-limit== | | ==7-limit== |
| [[starling chords]] | | [[starling_chords|starling chords]] |
| [[hendrix chord]] | | |
| | [[hendrix_chord|hendrix chord]] |
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| | ==9-limit== |
| | [[Meantone_add6-9_pentad|meantone add6-9 pentad]] |
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| | [[marvel_chords|marvel chords]] |
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| | [[sensamagic_chords|sensamagic chords]] |
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| | ==11-limit== |
| | [[mothwellsmic_triad|mothwellsmic triad]] |
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| | [[ptolemismic_triad|ptolemismic triad]] |
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| | [[valinorsmic_chords|valinorsmic chords]] |
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| | [[rastmic_chords|rastmic chords]] |
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| | [[keenanismic_chords|keenanismic chords]] |
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| | [[pentacircle_chords|pentacircle chords]] |
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| | [[werckismic_chords|werckismic chords]] |
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| | [[swetismic_chords|swetismic chords]] |
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| | [[undecimal_marvel_chords|undecimal marvel chords]] |
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| | [[jove_chords|jove chords]] |
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| | [[prodigy_chords|prodigy chords]] |
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| | [[miracle_chords|miracle chords]] |
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| | [[magical_seventh_chord|magical seventh chord]] |
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| | [[orwell_tetrad|orwell tetrad]] |
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| | [[tutonic_sextad|tutonic sextad]] |
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| | ==13-limit== |
| | [[ratwolf_triad|ratwolf triad]] |
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| | [[gentle_chords|gentle chords]] |
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| | [[minthmic_chords|minthmic chords]] |
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| | [[huntmic_chords|huntmic chords]] |
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| | [[kestrel_chords|kestrel chords]] |
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| | [[mynucumic_chords|mynucumic chords]] |
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| | [[squbemic_chords|squbemic chords]] |
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| | [[sinbadmic_tetrad|sinbadmic tetrad]] |
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| | [[marveltwin_triad|marveltwin triad]] |
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| | [[petrmic_triad|petrmic triad]] |
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| | [[cuthbert_triad|cuthbert triad]] |
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| | [[hecate_hexad|hecate hexad]] |
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| | ==15-limit== |
| | [[orwell_tetrad|guanyin tetrad]] |
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| | [[island_tetrad|island tetrad]] |
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| ==9-limit==
| | [[nicolic_tetrad|nicolic tetrad]] |
| [[meantone add6-9 pentad]] | |
| [[marvel chords]]
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| [[sensamagic chords]]
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| ==11-limit==
| | [[battaglia_chord|battaglia chord]] |
| [[mothwellsmic triad]] | |
| [[ptolemismic triad]]
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| [[valinorsmic chords]]
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| [[rastmic chords]]
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| [[keenanismic chords]]
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| [[pentacircle chords]]
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| [[werckismic chords]]
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| [[swetismic chords]]
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| [[undecimal marvel chords]]
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| [[jove chords]]
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| [[prodigy chords]]
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| [[miracle chords]]
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| [[magical seventh chord]]
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| [[orwell tetrad]]
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| [[tutonic sextad]]
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| ==13-limit== | | ==19-limit== |
| [[ratwolf triad]] | | [[hendrix_chord|hendrix chord]] |
| [[gentle chords]]
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| [[minthmic chords]]
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| [[huntmic chords]]
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| [[kestrel chords]]
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| [[mynucumic chords]]
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| [[squbemic chords]]
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| [[sinbadmic tetrad]]
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| [[marveltwin triad]]
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| [[petrmic triad]]
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| [[cuthbert triad]]
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| [[hecate hexad]]
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| ==15-limit==
| | [[rootminor_triad|rootminor triad]] |
| [[orwell tetrad|guanyin tetrad]] | |
| [[island tetrad]]
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| [[nicolic tetrad]]
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| [[battaglia chord]]
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| ==19-limit==
| | [[rootsubminor_triad|rootsubminor triad]] |
| [[hendrix chord]]
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| [[rootminor triad]]
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| [[rootsubminor triad]] | |
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| ==21-limit== | | ==21-limit== |
| [[slendric pentad]]</pre></div> | | [[slendric_pentad|slendric pentad]] [[Category:chord]] |
| <h4>Original HTML content:</h4>
| | [[Category:dyadic]] |
| <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><!-- 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: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: -->
| | [[Category:overview]] |
| <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definitions</h1>
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| 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 />
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| <br />
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| 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 />
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| 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 term 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. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. In 7-limit JI, one of the 3rds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three 3rds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord isn't an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to &quot;ring&quot;. However, it's debatable that this chord qualifies as an augmented triad, because the upper 3rd hardly sounds major.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Anomalous Saturated Suspensions"></a><!-- ws:end:WikiTextHeadingRule:2 -->Anomalous Saturated Suspensions</h1>
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| 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 />
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| <br />
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| <a class="wiki_link" href="/just%20added%20sixth%20chord">just added sixth chord</a><br />
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| <a class="wiki_link" href="/swiss%20tetrad">swiss tetrad</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Just intonation tetrads"></a><!-- ws:end:WikiTextHeadingRule:4 -->Just intonation tetrads</h1>
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| <br />
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| <a class="wiki_link" href="/seven%20limit%20tetrads">seven limit tetrads</a><br />
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| <a class="wiki_link" href="/nine%20limit%20tetrads">nine limit tetrads</a><br />
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| <a class="wiki_link" href="/1-3-7-11%20tetrads">1-3-7-11 tetrads</a><br />
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| <a class="wiki_link" href="/thirteen%20limit%20tetrads">thirteen limit tetrads</a><br />
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| <a class="wiki_link" href="/fifteen%20limit%20tetrads">fifteen limit tetrads</a><br />
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| <a class="wiki_link" href="/seventeen%20limit%20tetrads">seventeen limit tetrads</a><br />
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| <br />
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| <!-- 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>
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| Here are some pages on certain essentially tempered dyadic chords.<br />
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| <br />
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| <!-- 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>
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| <a class="wiki_link" href="/starling%20chords">starling chords</a><br />
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| <a class="wiki_link" href="/hendrix%20chord">hendrix chord</a><br />
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| <br />
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| <!-- 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>
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| <a class="wiki_link" href="/meantone%20add6-9%20pentad">meantone add6-9 pentad</a><br />
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| <a class="wiki_link" href="/marvel%20chords">marvel chords</a><br />
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| <a class="wiki_link" href="/sensamagic%20chords">sensamagic chords</a><br />
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| <br />
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| <!-- 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>
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| <a class="wiki_link" href="/mothwellsmic%20triad">mothwellsmic triad</a><br />
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| <a class="wiki_link" href="/ptolemismic%20triad">ptolemismic triad</a><br />
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| <a class="wiki_link" href="/valinorsmic%20chords">valinorsmic chords</a><br />
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| <a class="wiki_link" href="/rastmic%20chords">rastmic chords</a><br />
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| <a class="wiki_link" href="/keenanismic%20chords">keenanismic chords</a><br />
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| <a class="wiki_link" href="/pentacircle%20chords">pentacircle chords</a><br />
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| <a class="wiki_link" href="/werckismic%20chords">werckismic chords</a><br />
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| <a class="wiki_link" href="/swetismic%20chords">swetismic chords</a><br />
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| <a class="wiki_link" href="/undecimal%20marvel%20chords">undecimal marvel chords</a><br />
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| <a class="wiki_link" href="/jove%20chords">jove chords</a><br />
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| <a class="wiki_link" href="/prodigy%20chords">prodigy chords</a><br />
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| <a class="wiki_link" href="/miracle%20chords">miracle chords</a><br />
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| <a class="wiki_link" href="/magical%20seventh%20chord">magical seventh chord</a><br />
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| <a class="wiki_link" href="/orwell%20tetrad">orwell tetrad</a><br />
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| <a class="wiki_link" href="/tutonic%20sextad">tutonic sextad</a><br />
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| <br />
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| <!-- 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>
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| <a class="wiki_link" href="/ratwolf%20triad">ratwolf triad</a><br />
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| <a class="wiki_link" href="/gentle%20chords">gentle chords</a><br />
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| <a class="wiki_link" href="/minthmic%20chords">minthmic chords</a><br />
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| <a class="wiki_link" href="/huntmic%20chords">huntmic chords</a><br />
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| <a class="wiki_link" href="/kestrel%20chords">kestrel chords</a><br />
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| <a class="wiki_link" href="/mynucumic%20chords">mynucumic chords</a><br />
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| <a class="wiki_link" href="/squbemic%20chords">squbemic chords</a><br />
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| <a class="wiki_link" href="/sinbadmic%20tetrad">sinbadmic tetrad</a><br />
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| <a class="wiki_link" href="/marveltwin%20triad">marveltwin triad</a><br />
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| <a class="wiki_link" href="/petrmic%20triad">petrmic triad</a><br />
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| <a class="wiki_link" href="/cuthbert%20triad">cuthbert triad</a><br />
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| <a class="wiki_link" href="/hecate%20hexad">hecate hexad</a><br />
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| <br />
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| <!-- 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>
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| <a class="wiki_link" href="/orwell%20tetrad">guanyin tetrad</a><br />
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| <a class="wiki_link" href="/island%20tetrad">island tetrad</a><br />
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| <a class="wiki_link" href="/nicolic%20tetrad">nicolic tetrad</a><br />
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| <a class="wiki_link" href="/battaglia%20chord">battaglia chord</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Essentially tempered dyadic chords-19-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->19-limit</h2>
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| <a class="wiki_link" href="/hendrix%20chord">hendrix chord</a><br />
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| <a class="wiki_link" href="/rootminor%20triad">rootminor triad</a><br />
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| <a class="wiki_link" href="/rootsubminor%20triad">rootsubminor triad</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Essentially tempered dyadic chords-21-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->21-limit</h2>
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| <a class="wiki_link" href="/slendric%20pentad">slendric pentad</a></body></html></pre></div>
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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 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 term 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. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. In 7-limit JI, one of the 3rds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three 3rds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord isn't an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it's debatable that this chord qualifies as an augmented triad, because the upper 3rd hardly sounds major.
Anomalous Saturated Suspensions
An anomalous saturated suspension, or ASS, is a term 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 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
seventeen limit tetrads
Essentially tempered dyadic chords
Here are some pages on certain essentially tempered dyadic chords.
7-limit
starling chords
hendrix chord
9-limit
meantone add6-9 pentad
marvel chords
sensamagic chords
11-limit
mothwellsmic triad
ptolemismic triad
valinorsmic chords
rastmic chords
keenanismic chords
pentacircle chords
werckismic chords
swetismic chords
undecimal marvel chords
jove chords
prodigy chords
miracle chords
magical seventh chord
orwell tetrad
tutonic sextad
13-limit
ratwolf triad
gentle chords
minthmic chords
huntmic chords
kestrel chords
mynucumic chords
squbemic chords
sinbadmic tetrad
marveltwin triad
petrmic triad
cuthbert triad
hecate hexad
15-limit
guanyin tetrad
island tetrad
nicolic tetrad
battaglia chord
19-limit
hendrix chord
rootminor triad
rootsubminor triad
21-limit
slendric pentad