Inmusical set theory, aninterval vector is an array ofnatural numbers which summarize theintervals present in aset ofpitch classes. (That is, a set ofpitches whereoctaves are disregarded.) Other names include:ic vector (or interval-class vector),PIC vector (or pitch-class interval vector) andAPIC vector (or absolute pitch-class interval vector, which Michiel Schuijer states is more proper.)^[1]: 48

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 class. That is, sets with high concentrations of conventionally dissonant intervals (i.e., seconds and sevenths) sound more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e., thirds and sixths) sound moreconsonant. While the actual perception of consonance and dissonance involves many contextual factors, such asregister, an interval vector can nevertheless be a helpful tool.

Intwelve-tone equal temperament, an interval vector has six digits, with each digit representing the number of times aninterval class appears in the set. Because interval classes are used, the interval vector for a given set remains the same, regardless of the set'spermutation or vertical arrangement. The interval classes designated 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 isinversionally equivalent to itself.)

Interval class 0, representing unisons and octaves, is omitted.

In his 1960 book,The Harmonic Materials of Modern Music,Howard Hanson introduced amonomial method of notation for this concept, which he termedintervallic content: p^em^dn^c.s^bd^at^f for what would now be written⟨abcdef⟩.^{[2][note 1]} The modern notation, introduced by Donald Martino in 1961, has considerable advantages^[specify] and is extendable to anyequal division of the octave.^[3] Allen Forte in his 1973 workThe Structure of Atonal Music notated the interval vector using square brackets, citing Martino;^[4]: 15 subsequent authors, e.g.John Rahn, use angled brackets.^[5]: 100

A scale whose interval vector has six unique digits is said to have thedeep scale property. The major scale and its modes have this property.

For a practical example, the interval vector for a Cmajor triad (3-11B) 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 does not change with transposition or inversion, it belongs to the entireset class, meaning that⟨001110⟩ is the vector of all major (and minor) triads. Some interval vectors correspond to more than one sets that cannot be transposed or inverted to produce the other. (These are calledZ-related sets, explained below).

For a set ofn pitch classes, the sum of all the numbers in the set's interval vector equals thebinomial coefficient${\displaystyle (\genfrac{}{}{0ex}{}{n}{2})=\frac{n(n-1)}{2}}$, since the interval vector elements are computed comparing each pair of pitch classes from the set consisting of n elements. This corresponds also to thetriangular number${\displaystyle T_{n-1}}$.

An expanded form of the interval vector is also used intransformation theory, as set out inDavid Lewin'sGeneralized Musical Intervals and Transformations.^{[full citation needed]}

In musical set theory, aZ-relation, also calledisomeric relation, is a relation between two pitch class sets in which the two sets have the same intervallic content (and thus the same interval vector) but they are nottranspositionally related (are of different T_n-type ) orinversionally related (are of different T_n/T_nI-type).^[1]: 99 For example, the two sets 4-z15A {0,1,4,6} and 4-z29A {0,1,3,7} have the same interval vector⟨111111⟩ but one can not transpose and/or invert the one set onto the other.

In the case ofhexachords each may be referred to as aZ-hexachord. Any hexachord not of the "Z" type is its owncomplement while the complement of a Z-hexachord is its Z-correspondent, for example 6-Z3 and 6-Z36.^[4]: 79 See:6-Z44,6-Z17,6-Z11, andForte number.

The symbol "Z", standing for "zygotic" (from the Greek, meaning paired oryoked, such as the fusion of two reproductive cells),^[1]: 98 originated with Allen Forte in 1964, but the notion seems to have first been considered by Howard Hanson. Hanson called this theisomeric relationship, and defined two such sets asisomeric.^[2]: 22 See:isomer.

According to Michiel Schuijer (2008), thehexachord theorem, that any two pitch-class complementary hexachords have the same interval vector, even if they are not equivalent under transposition and inversion, was first proposed byMilton Babbitt, and, "the discovery of the relation," was, "reported," byDavid Lewin in 1960 as an example of thecomplement theorem: that the difference between pitch-class intervals in two complementary pitch-class sets is equal to the difference between the cardinal number of the sets (given two hexachords, this difference is 0).^{[1]: 96–7 [6]} Mathematical proofs of the hexachord theorem were published by Kassler (1961), Regener (1974), and Wilcox (1983).^{[1]: 96–7}

Though it is commonly observed that Z-related sets always occur in pairs,David Lewin noted that this is a result of twelve-toneequal temperament (12-ET).^{[citation needed]} 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.^{[citation needed]}

The equivalence relationship of `having the same interval content', allowing the trivial isometric case, was initially studied in crystallography and is known asHomometry. For instance the complement theorem is known to physicists asBabinet's principle. For a recent survey see.^[7]

Straus argues, "[sets] in the Z-relation will sound similar because they have the same interval content,"^{[8][1]: 125} which has led certain composers to exploit the Z-relation in their work. For instance, the play between {0,1,4,6} and {0,1,3,7} is clear inElliott Carter'sSecond String Quartet.^{[citation needed]}

SomeZ-related chords are connected byM orIM (multiplication by 5 or multiplication by 7), due to identical entries for 1 and 5 on the interval vector.^{[1]: 83, 110}

• Interval cycle
• Pitch interval

• Set classes and interval-class content
• Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology, by Robert T. Kelley
• Twentieth Century Pitch Theory: Some Useful Terms and Techniques

• Musical set theory

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