Dyadic chord: Difference between revisions

A '''dyadic 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. Such a chord may also be described as ''dyadically'' or ''pairwise consonant''.  

In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. 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, [[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.

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 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.  

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, hence a chord that will not "ring". 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.)

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.  

! Chords !! Associated Temperament !! Associated Commas

| [[Archytas chords]] || [[Archytas clan|Archytas]] || [[64/63]]

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

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

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

! Chords !! Associated Temperament !! Associated Commas

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

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

| [[Slendric pentad]] || [[Gamelismic clan|Gamelismic]] || [[1029/1024]]

| [[Schisminic chords]] || [[Schisminic]] || [[4096/4095]]

! Chords !! Associated Temperament !! Associated Commas

| [[Rhodesismic chords]] || [[Rhodesismic]] || [[1197/1196]]

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

| [[Ragismic chords]] || [[Ragismic microtemperaments|Ragismic]] || [[4375/4374]]

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

| [[Kibismic chords]] || [[Kibismic]] || [[Kilobyte comma|1024/1023]]

| [[Tricetertismic chords]] || [[Tricetertismic]] || [[Tricetertisma|29792/29791]]

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas

! Chords !! Associated Temperament !! Associated Commas