Dyadic chord: Difference between revisions

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: ''Not to be confused with [[dyad]].''

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

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:

* 1-6/5-10/7-8/5

is ''not'' a dyadic chord in the 7-odd-limit. Although each note is 7-odd-limit over the bass, the interval between 10/7 and 6/5 is 25/21, and that between 10/7 and 8/5 is 28/25 – these are not 7-odd-limit.

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 ==

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-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.

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, [[Ratios|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]]

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[[Category:Dyadic chords| ]]

[[Category:Lists of chords]]

~~{{Todo| add introduction }}~~

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 : ''Not to be confused with [[dyad]].''
+A '''dyadic chord''' is a [[chord]] each of whose [[interval]]s belongs to a specified set of intervals considered to be [[consonant]]; it is therefore relative to the set of intervals in question. Such a chord may also be described as ''dyadically'' or ''pairwise consonant''.
+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:
+* 1-6/5-10/7-8/5
+is ''not'' a dyadic chord in the 7-odd-limit. Although each note is 7-odd-limit over the bass, the interval between 10/7 and 6/5 is 25/21, and that between 10/7 and 8/5 is 28/25 – these are not 7-odd-limit.
+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 ==
-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-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.
+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, [[Ratios|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]]
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 [[Category:Dyadic chords| ]] <!-- main article -->
 [[Category:Lists of chords]]
-{{Todo| add introduction }}