The diatonic class vector is a numerical representation of the diatonic distances present in a set of notes, obtained by grouping as equivalent the intervals that are complementary within the octave. In this system, the second and the seventh belong to the same class, the third and the sixth to the same class, and the fourth and the fifth to the same class.
The vector can therefore be written as a sequence of three digits enclosed in square brackets, in the form:
[a b c]
where the three positions correspond, in order, to:
- seconds and sevenths
- thirds and sixths
- fourths and fifths
The octave is not counted separately, since it coincides with the return to the initial degree.
Example
Considering the major triad C–E–G, the intervallic pairs are:
C–E= thirdE–G= thirdC–G= fifth
In the diatonic class vector, the two thirds belong to the second class, while the fifth belongs to the third class. The resulting vector is therefore:
[0 2 1]
Considering instead the minor triad C–E♭–G, the intervallic pairs are again:
C–E♭= thirdE♭–G= thirdC–G= fifth
In this case too the diatonic class vector is:
[0 2 1]
since the vector records only the diatonic class of the interval, without distinguishing between major and minor quality.
Invariance under Inversions
The diatonic class vector is invariant under inversions. This occurs because complementary intervals within the octave are collected in the same class: second and seventh in the first, third and sixth in the second, fourth and fifth in the third.
Consequently, when a chord is inverted, some diatonic distances may transform into their complementaries, but the class of membership remains the same.
For example, the triad C–E–G in root position produces:
C–E= thirdE–G= thirdC–G= fifth
The corresponding diatonic class vector is:
[0 2 1]
The first inversion E–G–C produces instead:
E–G= thirdG–C= fourthE–C= sixth
The diatonic class vector remains nonetheless:
[0 2 1]
since the third and sixth both belong to the second class, while the fourth belongs to the third class.
The second inversion G–C–E produces:
G–C= fourthC–E= thirdG–E= sixth
and maintains the same vector:
[0 2 1]
In this sense, the diatonic class vector describes a more abstract and synthetic form of the diatonic intervallic content: it does not record the concrete disposition of the chord within the octave, but preserves the fundamental relational structure even through inversions.
Difference from the Diatonic Interval Vector
The diatonic interval vector distinguishes six categories: second, third, fourth, fifth, sixth and seventh. The diatonic class vector instead reduces these six categories to three classes alone, grouping together the intervals complementary within the octave. In this sense, it represents a more synthetic form of the diatonic content of a chord.