Interval vector
)[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 more consonant.
While the actual perception of consonance and dissonance involves many contextual factors, such as register, an interval vector can nevertheless be a helpful tool.
In his 1960 book, The Harmonic Materials of Modern Music, Howard Hanson introduced a monomial method of notation for this concept, which he termed intervallic content: pemdnc.sbdatf 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 any equal division of the octave.
The symbol "Z", standing for "zygotic" (from the Greek, meaning paired or yoked, 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.
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).
[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-tone equal temperament (12-ET).
[citation needed] The equivalence relationship of `having the same interval content', allowing the trivial isometric case, was initially studied in crystallography and is known as Homometry.
