Definitions

# 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.

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 (1/11 semitones)
2) major seconds/minor sevenths (2/10 semitones)
3) minor thirds/major sixths (3/9 semitones)
4) major thirds/minor sixths (4/8 semitones)
5) perfect fourths/perfect fifths (5/7 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 dasbncmdpetf 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 chord, {C E G}, 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 sets with the same interval vector are not always equivalent (See Z-relation).

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 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.

## Z-relation

In musical set theory, a Z-relation, also called isomeric relation, is a relation between two pitch-class sets in which the two sets have the same intervallic content (i.e. they have the same interval vector), but they are of different Tn-type and Tn/TnI-type. That is to say, one set cannot be derived from the other through transposition or inversion.

For example, the two sets {0,1,4,6} and {0,1,3,7} have the same interval vector (<1,1,1,1,1,1>) but they are not transpositionally or inversionally related.

The term originated with Allen Forte, but the notion seems to have first been considered by Howard Hanson. Hanson termed this the isomeric relationship, defining two such sets to be isomeric.

Though it is commonly observed that Z-related sets always occur in pairs, David Lewin noted that this relation is a result of twelve-tone equal temperament (12-ET). In 16-ET, Z-related sets are found as triplets. Lewin's student Jonathan Wild continued this work for other tuning systems, finding Z-related tuplets with up to 16 members in higher ET systems.

Some argue that the "relation" is often so remote as to be imperceptible, but certain composers have exploited the Z-relation in their work. For instance, the play between {0,1,4,6} and {0,1,3,7} is clear in Elliot Carter's second string quartet.

• Rahn, John (1980). Basic Atonal Theory. ISBN 0-02-873160-3.
• Forte, Allen (1973/1977). Structure of Atonal Music. ISBN 0-300-01610-7/ISBN 0-300-02120-8.
• Hanson, Howard (1960). The Harmonic Materials of Modern Music. Appleton-Century-Crofts.
• Straus, Joseph N. (1990/2000/2005). Introduction to Post-Tonal Theory  3rd Ed. ISBN 0-13-189890-6.

### Z-relation references

• Allen Forte, The Structure of Atonal Music, Yale University Press, 1977. ISBN 0-300-02120-8
• Howard Hanson, Harmonic Materials of Modern Music, Appleton-Century-Crofts, 1960