Музыкалық жиын теориясында интервалдық векторлар

In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interval class vector), PIC vector (or pitch class interval vector) and APIC vector (or absolute pitch class interval vector, which Michiel Schuijer states is more proper.) for what would now be written The modern notation, introduced by Allen Forte, has considerable advantages and is extendable to any equal division of the octave. A scale whose interval vector has six unique digits is said to have the deep scale property. The major scale and its modes have this property. For a practical example, the interval vector for a C major triad (3 11B) in the root position, {C E G} , is 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 entire set class, meaning that 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 called Z related sets, explained below). For a set of n pitch classes, the sum of all the numbers in the set's interval vector equals the triangular number Tn−1 =
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.

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 (and thus the same interval vector) but they are not transpositionally related (are of different Tn type ) or inversionally related (are of different Tn/TnI type). See: isomer. According to Michiel Schuijer (2008), the hexachord 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 by Milton Babbitt, and, "the discovery of the relation," was, "reported," by David Lewin in 1960 as an example of the complement 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). Mathematical proofs of the hexachord theorem were published by Kassler (1961), Regener (1974), and Wilcox (1983). Straus argues, "[sets] in the Z relation will sound similar because they have the same interval content," 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 in Elliott Carter's Second String Quartet.

Some Z related chords are connected by M or IM (multiplication by 5 or multiplication by 7), due to identical entries for 1 and 5 on the interval vector.