Dyadic chord: Difference between revisions

{{Distinguish| Dyad }}

A '''dyadic chord''', also described as a '''dyadically consonant chord''' or '''pairwise consonant chord''', is a [[chord]] each of whose [[interval]]s belongs to a specified set of intervals considered to be [[consonance and dissonance|consonant]]; it is therefore relative to the set of intervals in question.  

For example, the [[tetrad]]

* 1 – 6/5 – 7/5 – 8/5

is a dyadic chord in the [[7-odd-limit]] since every interval involved in it is an element of the 7-odd-limit [[tonality diamond]]. Now if we replace 7/5 with 10/7:

The significance of dyadic chords and of the paradigm where all interval pairs are examined in the chord has the psychoacoustic basis of timbral fusion and emergence of the [[virtual fundamental]]. In the above examples, it can be shown that the lower harmonics of each note in the first chord blends better than in the second. Meanwhile, the virtual fundamental of the first chord appears 5/1 below the bass, whereas that of the second appears much lower, at 35/1 below the bass as the denominators "fight" each other. For these reasons we tend to find the first chord more consonant than the second.

== Essentially tempered dyadic chord ==

In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in just 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. An ''essentially just'' dyadic chord is 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, [[4:5:6|1 – 5/4 – 3/2]] is a just dyadic chord when the consonance set is the [[5-odd-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.

[[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]

An '''essentially tempered dyadic chord''' is a chord defined in a [[abstract regular temperament|regular temperament]] and relative to a set of consonances such that each interval is within that 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 makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-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]]s, 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.

== Innate comma chord ==

An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#Essentially_tempered_dyadic_chord|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals. For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)

== Anomalous saturated suspension ==

An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-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]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

* [[12:14:18:21]]

 

== List of just intonation tetrads ==

* [[Seven limit tetrads]]

* [[1-3-7-11 tetrads]]

 

== List of essentially tempered dyadic chords ==

Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available. See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.

 

=== 7-odd-limit ===

{| class="wikitable"

|-

! Chords !! Associated Temperament !! Associated Commas

|-

|-

| [[Starling chords]] || [[Starling]] || [[126/125]]

 

=== 9-odd-limit ===

{| class="wikitable"

! Chords !! Associated Temperament !! Associated Commas

|-

|-

|-

 

=== 11-odd-limit ===

|-

! Chords !! Associated Temperament !! Associated Commas

|-

|-

|-

|-

| [[Valinorsmic chords]] || [[Valinorsmic]] || [[176/175]]

|-

|-

|-

|-

| [[Werckismic chords]] || [[Werckismic]] || [[441/440]]

|-

| [[Swetismic chords]] || [[Swetismic]] || [[540/539]]

| [[Pentacircle chords]] || [[Pentacircle]] || [[896/891]]

| [[Undecimal marvel chords]] || [[Marvel]] || 225/224, 385/384

| [[Prodigy chords]] || [[Prodigy]] || 225/224, 441/440

| [[Undecimal sensamagic chords]] || [[Sensamagic]] || 245/243, 385/384

|-

| [[Jove chords]] || [[Jove]] || 243/242, 441/440

|-

| [[Miracle chords]] || [[Miracle]] || 225/224, 243/242, 385/384

|-

| [[Magic chords]] || [[Magic]] || 100/99, 225/224, 245/243

| [[Supermagic chords]] || [[Supermagic]] || 100/99, 385/384

|-

| [[Orwell tetrad]] || [[Guanyin]] || 176/175, 540/539

|-

| [[Tutonic hexad|Tutonic hexads]] || [[Meantone]] || 81/80, 99/98, 126/125

| [[Baldanders chords|Baldanders hexads]] || [[Andromeda]] || 100/99, 225/224, 245/242

|-

| [[Porcupine heptad]] || [[Porkypine]] || 55/54, 100/99

 

{| class="wikitable"

| [[Buzurgismic chords]] || [[Buzurgismic]] || [[169/168]]

| [[Mynucumic chords]] || [[Mynucumic]] || [[196/195]]

|-

| [[Gassormic chords]] || [[Gassormic]] || [[275/273]]

| [[Marveltwin chords]] || [[Marveltwin]] || [[325/324]]

| [[Ratwolfsmic chords]] || [[Ratwolfsmic]] || [[351/350]]

| [[Major minthmic chords]] || [[Major minthmic]] || [[352/351]]

| [[Minor minthmic chords]] || [[Minor minthmic]] || [[364/363]]

|-

|-

|-

| [[Cuthbert chords]] || [[Cuthbert]] || [[847/845]]

| [[Sinbadmic chords]] || [[Sinbadmic]] || [[1001/1000]]

| [[Kestrel chords]] || [[Kestrel]] || [[1188/1183]]

| [[Catadictmic chords]] || [[Catadictmic]] || [[1287/1280]]

|-

| [[Lambeth chords]] || [[Lambeth]] || [[1573/1568]]

| [[Petrmic chords]] || [[Petrmic]] || [[2200/2197]]

|-

| [[Parapyth chords]] || [[Parapyth]], [[gentle]] || 352/351, 364/363

| [[Hecate hexad]] || [[Hecate]] || 225/224, 325/324, 385/384

| [[Woodpecker octad]] || [[Woodpecker]] || 66/65, 121/120, 126/125

|-

| [[Miraculous decad]] || [[Miraculous]], [[Revelation]] || 105/104, 196/195, 512/507

 

=== 15-odd-limit ===

{| class="wikitable"

! Chords !! Associated Temperament !! Associated Commas

|-

| [[Island chords]] || [[The Archipelago|Island]] || [[676/675]]

| [[Nicolic chords]] || [[Nicolic]] || [[1575/1573]]

|-

| [[Myhemiwell chords]] || [[Myhemiwell]] || [[3388/3375]]

|-

| [[Battaglia chord]] || [[Marvel]] || 225/224

| [[Tetracot chords]] || [[Tetracot]] || 100/99, 243/242

|-

 

=== 17-odd-limit ===

{| class="wikitable"

! Chords !! Associated Temperament !! Associated Commas

|-

| [[Augustmic chords]] || [[Augustmic]] || [[154/153]]

| [[Charismic chords]] || [[Charismic]] || [[256/255]]

| [[Prototannismic chords]] || [[Prototannismic]] || [[273/272]]

|-

| [[Ursulismic chords]] || [[Ursulismic]] || [[375/374]]

|-

| [[Seminaiadmic chords]] || [[Seminaiadmic]] || [[442/441]]

| [[Monardismic chords]] || [[Monardismic]] || [[561/560]]

| [[Dakotismic chords]] || [[Dakotismic]] || [[595/594]]

| [[September chords]] || [[September]] || [[715/714]]

| [[Horizmic chords]] || [[Horizmic]] || [[833/832]]

| [[Ainismic chords]] || [[Ainismic]] || [[936/935]]

| [[Twosquare chords]] || [[Twosquare]] || [[1089/1088]]

|-

| [[Cimbrismic chords]] || [[Cimbrismic]] || [[1275/1274]]

|-

| [[Fidesmic chords]] || [[Fidesmic]] || [[2025/2023]]

| [[Heptacircle chords]] || [[Heptacircle]] || [[2431/2430]]

| [[Sextantonismic chords]] || [[Sextantonismic]] || [[2601/2600]]

 

=== 19-odd-limit ===

{| class="wikitable"

| [[Photismic chords]] || [[Photismic]] || [[324/323]]

|-

| [[Nutrismic chords]] || [[Nutrismic]] || [[343/342]]

|-

| [[Dudon chords]] || [[Dudon]] || [[361/360]]

| [[Devichromic chords]] || [[Devichromic]] || [[400/399]]

| [[Abnobismic chords]] || [[Abnobismic]] || [[456/455]]

| [[Hedwigmic chords]] || [[Hedwigmic]] || [[476/475]]

| [[Eulalismic chords]] || [[Eulalismic]] || [[495/494]]

|-

| [[Boethius chords]] || [[Boethius]] || [[513/512]]

| [[Kingfisher chords]] || [[Kingfisher]] || [[969/968]]

|-

| [[Bihendrixmic chords]] || [[Bihendrixmic]] || [[1083/1078]]

|-

|-

| [[Solvejgsmic chords]] || [[Solvejgsmic]] || [[1331/1330]]

| [[Aureusmic chords]] || [[Aureusmic]] || [[1445/1444]]

| [[Pinkanberry chords]] || [[Pinkanberry]] || [[1521/1520]]

 

 

 

 

 

 

 

 

 

 

 

 

 

| <font style="font-size:0.85em">8:10:14:19 ~ 12:15:21:28</font>

 

 

 

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