The diatonic interval vector is a numerical representation of the diatonic distances present in a concrete arrangement of notes, considered without specific quality. In this sense, it does not distinguish between major, minor, augmented or diminished intervals, but takes into account only their generic diatonic number — that is, second, third, fourth, fifth, sixth and seventh.
This approach is close to the notion of generic interval already used in music theory. For a general treatment, see the Wikipedia entries Generic and specific intervals and Interval (music).
In the seven-note diatonic system, considering the internal distances within an arrangement within the octave, the vector can be written as a sequence of six digits enclosed in square brackets:
[a b c d e f]
where the six positions correspond, in order, to the intervals of second, third, fourth, fifth, sixth and seventh. The octave is not counted separately, since it coincides with the return to the initial degree in a heptatonic system and does not constitute a new generic interval class. The same theory of generic intervals indeed considers as the maximum value, in the diatonic collection, six generic steps — that is, the seventh.
Example
Considering the major triad C–E–G, the intervallic pairs are:
C–E= thirdE–G= thirdC–G= fifth
The corresponding diatonic interval vector is therefore:
[0 2 0 1 0 0]
since the chord contains two thirds and one fifth.
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 interval vector is:
[0 2 0 1 0 0]
since the diatonic vector does not distinguish between a major third and a minor third, but records only the diatonic number of the interval.
Variation under Inversions
Inversions modify the diatonic interval vector when the chord is considered in its concrete disposition within the octave. Indeed, while preserving the same notes, the inversion changes the generic distances between the notes arranged from bottom to top.
For example, the triad C–E–G in root position produces two thirds and one fifth:
[0 2 0 1 0 0]
The first inversion E–G–C produces instead:
E–G= thirdG–C= fourthE–C= sixth
The corresponding vector is therefore:
[0 1 1 0 1 0]
The second inversion G–C–E produces:
G–C= fourthC–E= thirdG–E= sixth
In this case too the vector is:
[0 1 1 0 1 0]
In this sense, the diatonic interval vector describes not only the abstract identity of the chord, but also its concrete disposition. The root position and the inversions can therefore produce different vectors.
Difference from the Standard Interval Vector
Unlike the standard interval vector of set theory, which operates on chromatic interval classes and reduces intervals to their octave equivalences, the diatonic interval vector describes only the generic intervallic content of a set of notes. It is therefore better suited to representing structures built on diatonic relationships, while it is not designed for general chromatic analysis.