# Interval Vector

In musical set theory, an interval vector (also called an interval-class vector or ic vector) is an array that expresses the intervallic content of a pitch-class set. Often referred to as a PIC vector (or pitch-class interval vector), Schuijer suggests that APIC vector (or absolute pitch-class interval vector) is more accurate.

In 12 equal temperament it has six digits, with each digit standing for the number of times an interval class appears in the set. (Interval classes, not regular intervals, must be used, in order that the interval vector remains the same, regardless of the set's permutation or vertical arrangement.) The interval classes represented by each digit ascend from left to right. That is:

1) minor seconds/major sevenths (1 or 11 semitones)
2) major seconds/minor sevenths (2 or 10 semitones)
3) minor thirds/major sixths (3 or 9 semitones)
4) major thirds/minor sixths (4 or 8 semitones)
5) perfect fourths/perfect fifths (5 or 7 semitones)
6) tritones (6 semitones) (The tritone is inversionally related to itself.)

Interval class 0 (representing unisons and octaves) is omitted.

The concept was named intervalic content by Howard Hanson in his The Harmonic Materials of Modern Music, where he introduced the monomial notation pemdnc.sbdatf for what would now be written . The modern notation, which has considerable advantages and is extendable to any equal division of the octave was introduced by Allen Forte.

A scale whose interval vector contains six different numbers is said to have the deep scale property. Major, natural minor and modal scales have this property.

For a practical example, the interval vector for a C major triad in the root position, {C E G} ( Play), is <001110>. This means that the set has one major third or minor sixth (i.e. from C to E, or E to C), one minor third or major sixth (i.e. from E to G, or G to E), and one perfect fifth or perfect fourth (i.e. from C to G, or G to C). As the interval vector will not change with transposition or inversion, it belongs to the entire set class, and <001110> is the vector of all major (and minor) triads. It should, however, be noted that some interval vectors correspond to more than one sets that cannot be transposed or inverted to produce the other. (These are called Z-related sets, explained below).

For a set of x elements, the sum of all the numbers in the set's interval vector equals (x*(x-1))/2.

While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the sound qualities that are created by different collections of pitch classes. That is, sets with high concentrations of conventionally dissonant intervals (i.e. seconds and sevenths) will generally be heard as more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e. thirds and sixths) will be heard as more consonant. (While the actual perception of consonance and dissonance involves many contextual factors, such as register, an interval vector, nevertheless, can be a helpful tool.)

An expanded form of the interval vector is also used in transformation theory, as set out in David Lewin's Generalized Musical Intervals and Transformations.

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