The absolute spatial interval vector is an extension of the spatial interval vector in which distances are no longer considered solely within the span of a single octave, but can be recorded across an arbitrary extension of the sonic material.
While the ordinary spatial interval vector reports the relationships present within the octave, the absolute spatial vector considers the distance in semitones of each note relative to the lowest note of the set, without reducing it within the twelve-semitone limit. In this sense, it no longer measures only the local intervallic content of a structure, but its overall development in the sonic space.
The vector can therefore extend as far as necessary, depending on the breadth of the chord, scale or musical configuration under consideration. If a structure develops over two, three or more octaves, the vector will have a correspondingly greater number of positions. Each position represents a specific distance in semitones from the lowest note, and the digit placed at that position indicates how many times that distance appears in the set.
In this perspective, the absolute spatial interval vector does not merely describe which intervals exist in the abstract, but how the entire sonic material is distributed in the vertical space starting from its point of origin.
Form of the Vector
The vector can be written as a sequence of digits enclosed in square brackets, in the form:
[a₁ a₂ a₃ a₄ ... aₙ]
where each position corresponds, in order, to an increasing distance in semitones from the lowest note of the set.
For example:
- the first position corresponds to 1 semitone above the lowest note
- the second to 2 semitones
- the third to 3 semitones
- and so on, up to the maximum extension considered
Difference from the Ordinary Spatial Vector
The ordinary spatial interval vector is confined within the octave and describes distances from 1 to 11 semitones. The absolute spatial interval vector, by contrast, imposes no such limit and can therefore represent structures distributed across multiple octaves.
Example
Considering the set:
C–G–D'
that is, a structure that develops over more than one octave, the distances relative to the lowest note C are:
G= 7 semitonesD'= 14 semitones
The corresponding absolute spatial interval vector will therefore be:
[0 0 0 0 0 0 1 0 0 0 0 0 0 1]
since the distance of 7 semitones appears once and the distance of 14 semitones appears once.
Usage
The absolute spatial interval vector is useful for describing scales, chords and melodic-harmonic structures that extend beyond the octave, while keeping their real spatial disposition visible. In this sense, it represents a possible theoretical extension of the ordinary spatial vector.