Complement (music)
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In traditional music theory a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is its inverse (or inversion), except for the octave and the unison which are each other's complements.
In musical set theory or atonal theory, complement is used in both the sense above, and in the additive inverse sense of the same melodic interval in the opposite direction - e.g. a falling 5th is the inverse of a rising 5th.[citation needed]
Using integer notation and modulo 12, any two intervals which add up to 0 (mod 12) are complements (mod 12). In this case the unison, 0, is its own complement, while for other intervals the complements are the same as above (for instance a perfect fifth, or 7, is the complement of the perfect fourth, or 5, 7+5 = 12 = 0 mod 12).
[edit] Complementation
In twelve-tone music and serialism complementation is the separation of pitch-class collections into complementary sets, each containing pitch classes absent from the other[1].
In the twelve-tone technique this is often the separation of the total chromatic of twelve pitch classes into two hexachords of six pitch classes each. With combinatoriality, two twelve-note tone rows are used simultaneously, thereby creating, "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."[1]
Hexachordal complementation is the use of the potential for pairs of hexachords to each contain six different pitch classes and thereby complete an aggregate[2].
In set theory the traditional concept of complementation may be distinguished as literal pitch class complement while, due to the definition of equivalent sets, the concept may be broadened to include "not only the literal pc complement of that set but also any transposed or inverted-and-tranposed form of the literal complement."[4] This is due to the fact that since P is equivalent to M, and M is the complement of M, P is also the complement of M, "from a logical and musical point of view,"[5] even though not its literal pc complement.
[edit] Source
- ^ a b Whittall, Arnold. 2008. The Cambridge Introduction to Serialism, p.272. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).
- ^ Whittall 2008, p.273.
- ^ Whittall, 103
- ^ Schmalfeldt, Janet (1983). Berg's Wozzeck: Harmonic Language and Dramatic Design, p.64 and 70. ISBN 0-300-02710-9.
- ^ Schmalfeldt, p.70
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