Standard Interval Vector
In twelve-tone equal temperament, the standard interval vector is a sequence of six digits, one for each interval class from 1 to 6. The six classes are the following:
| Semitones | Interval name | Interval class |
|---|---|---|
| 1 / 11 | minor second / major seventh | 1 |
| 2 / 10 | major second / minor seventh | 2 |
| 3 / 9 | minor third / major sixth | 3 |
| 4 / 8 | major third / minor sixth | 4 |
| 5 / 7 | perfect fourth / perfect fifth | 5 |
| 6 | tritone | 6 |
Class 0, corresponding to the unison and octave, is not counted.
The vector is written as a sequence of six digits enclosed in angle brackets, in the form <a b c d e f>, where each position indicates the number of times the corresponding interval class is present.
For example, considering the major triad C–E–G, the intervallic pairs are three:
| Note pair | Semitones | Interval class |
|---|---|---|
| C–E | 4 | 4 |
| E–G | 3 | 3 |
| C–G | 7 | 5 |
The corresponding vector is <0 0 1 1 1 0>, since the chord contains one interval of class 3, one of class 4, one of class 5, and no other intervals.
The standard interval vector is therefore a count of the interval classes present in a set of pitches. In the musical set theory made canonical above all by the work of Allen Forte, it does not distinguish the order of the notes nor the ascending or descending direction, but treats as equivalent the intervals and their inversions with respect to the octave. In essence, in this system the vector describes the intervallic content of a set of pitch classes — that is, pitches considered independently of the octave.
For a general treatment of the standard concept, see the Wikipedia entry: Interval vector.
Spatial Interval Vector
In the theory of Lorenzo Frizzera, the spatial interval vector is defined as a representation of the distances actually present in the sonic content of a set of notes. The adjective spatial serves to distinguish it from the standard vector: here, indeed, the intention is not an abstract set-theoretic vector, but a description of the intervallic distances as they emerge concretely in the sonic structure.
In this usage, complementary intervals within the octave are not reduced to the same class, but treated as different relationships. Consequently, intervals such as the minor second and the major seventh, or the major third and the minor sixth, are no longer considered equivalent. The vector records which specific distances appear within the octave and how many times they occur.
| Semitones | Interval name | Class in standard vector | Value in spatial vector |
|---|---|---|---|
| 1 | minor second | 1 | 1 |
| 2 | major second | 2 | 2 |
| 3 | minor third | 3 | 3 |
| 4 | major third | 4 | 4 |
| 5 | perfect fourth | 5 | 5 |
| 6 | tritone | 6 | 6 |
| 7 | perfect fifth | 5 | 7 |
| 8 | minor sixth | 4 | 8 |
| 9 | major sixth | 3 | 9 |
| 10 | minor seventh | 2 | 10 |
| 11 | major seventh | 1 | 11 |
From this it follows that, while the standard vector operates on six interval classes, the spatial vector separately distinguishes the eleven possible intervals from 1 to 11 semitones, leaving the tritone as an autonomous central value. It can be expressed as a sequence of eleven digits enclosed in square brackets.
Considering again the major triad C–E–G, the sonic content of the chord is not reduced to the classes 3–4–5, but maintained in its effective distances of 3–4–7 semitones, and the resulting spatial vector is therefore:
[0 0 1 1 0 0 1 0 0 0 0]
where the eleven positions correspond, in order, to the intervals of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 semitones.
It should be noted that, like the standard interval vector, the spatial interval vector also counts the occurrences of intervals and not merely their presence or absence. Consequently, if the same distance appears multiple times within a chord, it is recorded as many times as it actually recurs. For example, the chord C–E♭–G♭–A has the spatial vector [0 0 3 0 0 2 0 0 1 0 0].
Difference between Standard Vector and Spatial Vector
The standard interval vector is essentially a classificatory tool: it serves to compare sets that are theoretically equivalent even when the order of the notes, the vertical disposition or the register changes. For this reason, in set theory, it ignores direction and reduces intervals to their complementary classes within the octave.
The spatial interval vector, by contrast, is a more strictly musical descriptive tool: it does not aim to classify abstract sets, but to make readable the internal sonic profile of a chord, preserving differences that the reduction to interval classes tends to cancel. In this sense, while reducing intervals above the octave within the frame of a single octave, it is closer to the concrete sonic datum than to mere theoretical equivalence between sets.
Standard theory also treats intervallic content as a property of a set class, not of a single concrete disposition: the vector indeed remains invariant under transposition, inversion, permutation and vertical disposition of the set.
For example, the chords C–E–G and C–E–A have the same standard vector <0 0 1 1 1 0>, despite having different sonic contents. In the first case the effective distances are 3, 4 and 7 semitones; in the second they are 4, 5 and 9 semitones. The spatial vector arises precisely from the need not to cancel such differences, so as to preserve their musical relevance.
Relationship with Directed-Interval Vectors
The spatial interval vector used in this entry should not be confused with the directed-interval vectors present in mathematical music theory. A useful reference is the article by Robert W. Peck, All-(Generalized-)Interval(-System) Chords, where the directed-interval vector is used to count all directed relationships between the notes of a set in a cyclic chromatic space.
The difference can be clarified with a major triad, for example:
C–E–G
In the directed-interval vector, all possible internal transformations between the notes of the chord are considered. One therefore counts also the unisons — that is, the relationship of each note with itself:
C → C
E → E
G → G
and one also counts the reverse movements:
C → E and E → C
C → G and G → C
E → G and G → E
In this way the vector describes not only the distances contained in the chord, but all possible directed relationships between its elements. Hence the term directed.
In the case of the major triad C–E–G, the directed-interval vector, written from 0 to 11 semitones, is:
[3 0 0 1 1 1 0 1 1 1 0 0]
The three initial occurrences indicate the unisons:
C → C
E → E
G → G
The other occurrences indicate the directed relationships between the different notes of the chord, including the inverse ones.
The spatial interval vector, by contrast, functions more simply. It does not count all possible transformations, but only the positive intervallic spaces present between the notes of the chord within the octave.
In the case of the same major triad C–E–G, only these three distances are considered:
E–G = 3 semitones
C–E = 4 semitones
C–G = 7 semitones
The spatial interval vector is therefore:
[0 0 1 1 0 0 1 0 0 0 0]
This vector distinguishes complementary intervals, for example 3 from 9, 4 from 8, 5 from 7, but does not count unisons and does not double the relationships by also considering the reverse movement.