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 as follows:
| 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 the 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 occurs. |
For example, considering the major triad C–E–G, there are three interval pairs:
| 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 a single 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 musical set theory, made canonical above all by the work of Allen Forte, it does not distinguish the order of the notes or their ascending or descending direction, but treats intervals and their inversions with respect to the octave as equivalent. In essence, in this system the vector describes the interval content of a set of pitch classes, that is, of pitches considered independently of octave.
For a general treatment of the standard concept, see the Wikipedia entry: Interval vector.
Spatial interval vector
In Lorenzo Frizzera’s theory, 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 it is not an abstract set-theoretic vector that is meant, but a description of the interval distances as they concretely emerge in the sonic structure.
In this use, intervals complementary to the octave are not reduced to the same class, but treated as different relations. 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 the standard vector | Value in the 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 kept in its actual 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 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 occurs more than once within a chord, it is recorded as many times as it actually occurs. For example, the chord C–E♭–G♭–A has spatial vector [0 0 3 0 0 2 0 0 1 0 0].
Difference between the standard vector and the spatial vector
The standard interval vector is essentially a classificatory tool: that is, it serves to compare theoretically equivalent sets even when the order of the notes, the vertical arrangement, or the register changes. For this reason, in set theory, it ignores direction and reduces intervals to their classes complementary to the octave.
The spatial interval vector, on the other hand, is a more strictly musical descriptive tool: it does not aim to classify abstract sets, but to make the internal sonic profile of a chord readable, preserving differences that reduction to interval classes tends to erase. In this sense, although it reduces intervals larger than an octave within the framework of a single octave, it is closer to the concrete sonic data than to mere theoretical equivalence between sets.
The standard theory, moreover, treats interval content as a property of a set class and not of a single concrete arrangement: the vector remains in fact unchanged under transposition, inversion, permutation, and vertical arrangement 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 content. In the first case the actual 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 erase 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 found 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 direct relations 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. Unisons are therefore also counted, that is, the relation of each note with itself:
C → C
E → E
G → G
and the inverse movements are also counted:
C → E and E → C
C → G and G → C
E → G and G → E
In this way the vector does not describe only the distances contained in the chord, but all the possible direct relations 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 direct relations between the different notes of the chord, including the inverse ones.
The spatial interval vector, on the other hand, works more simply. It does not count all possible transformations, but only the positive interval 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 relations by also considering the inverse movement.