Complement (music)

In music the term complement refers to two distinct concepts.
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 also known as its inverse or inversion. Note that the octave and the unison are each other's complements and that the tritone is its own complement (though the latter is "respelt" as either an augmented fourth or a diminished fifth, depending on the context).
In twelvetone music and serialism the complement of one set of notes from the chromatic scale contains all the other notes of the scale. For example, ABCDEFG is complemented by B♭C♯E♭F♯A♭.
Note that musical set theory broadens the definition of both senses somewhat.
Contents
Interval complementation
Rule of nine
The rule of nine is a simple way to work out which intervals complement each other.^{[1]} Taking the names of the intervals as cardinal numbers (fourth etc becomes four), we have for example 4+5=9. Hence the fourth and the fifth complement each other. Where we are using more generic names (such as semitone and tritone) this rule cannot be applied. However, octave and unison are not generic but specifically refer to notes with the same name, hence 8+1=9.
Perfect intervals complement (different) perfect intervals, major intervals complement minor intervals, augmented intervals complement diminished intervals, and double diminished intervals complement double augmented intervals.
Rule of twelve
Using integer notation and modulo 12 (in which the numbers "wrap around" at 12, 12 and its multiples therefore being defined as 0), 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).
Thus the #Sum of complementation is 12 (= 0 mod 12).
Set theory
In musical set theory or atonal theory, complement is used in both the sense above (in which the perfect fourth is the complement of the perfect fifth, 5+7=12), and in the additive inverse sense of the same melodic interval in the opposite direction  e.g. a falling 5th is the complement of a rising 5th.^{[citation needed]}
Aggregate complementation
In twelvetone music and serialism complementation (in full, literal pitch class complementation) is the separation of pitchclass collections into complementary sets, each containing pitch classes absent from the other^{[2]} or rather, "the relation by which the union of one set with another exhausts the aggregate".^{[3]} To provide, "a simple explanation...: the complement of a pitchclass set consists, in the literal sense, of all the notes remaining in the twelvenote chromatic that are not in that set."^{[4]}
In the twelvetone technique this is often the separation of the total chromatic of twelve pitch classes into two hexachords of six pitch classes each. In rows with the property of combinatoriality, two twelvenote tone rows (or two permutations of one tone row) are used simultaneously, thereby creating, "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."^{[2]} In other words, the first and second hexachord of each series will always combine to include all twelve notes of the chromatic scale, known as an aggregate, as will the first two hexachords of the appropriately selected permutations and the second two hexachords.
Hexachordal complementation is the use of the potential for pairs of hexachords to each contain six different pitch classes and thereby complete an aggregate.^{[5]}
Sum of complementation
For example, given the transpositionally related sets:
0 1 2 3 4 5 6 7 8 9 10 11  1 2 3 4 5 6 7 8 9 10 11 0 ____________________________________ 11 11 11 11 11 11 11 11 11 11 11 11
The difference is always 11. The first set may be called P0 (see tone row), in which case the second set would be P1.
In contrast, "where transpositionally related sets show the same difference for every pair of corresponding pitch classes, inversionally related sets show the same sum."^{[7]} For example, given the inversionally related sets (P0 and I11):
0 1 2 3 4 5 6 7 8 9 10 11 +11 10 9 8 7 6 5 4 3 2 1 0 ____________________________________ 11 11 11 11 11 11 11 11 11 11 11 11
The sum is always 11. Thus for P0 and I11 the sum of complementation is 11.
Abstract complement
^{[clarification needed]}In set theory the traditional concept of complementation may be distinguished as literal pitch class complement, "where the relation obtains between specific pitchclass sets",^{[3]} 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 invertedandtranposed form of the literal complement,"^{[8]} which may be described as abstract complement,^{[9]} "where the relation obtains between set classes".^{[3]} This is because 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,"^{[10]} even though not its literal pc complement. Originator Allen Forte describes this as, "significant extension of the complement relation," though George Perle describes this as, "an egregious understatement".^{[11]}
As a further example take the chromatic sets 71 and 51. If the pitchclasses of 71 span CF♯ and those of 51 span GB then they are literal complements. However, if 51 spans CE, C♯F, or DF♯, then it is an abstract complement of 71.^{[9]} As these examples make clear, once sets or pitchclass sets are labeled, "the complement relation is easily recognized by the identical ordinal number in pairs of sets of complementary cardinalities".^{[3]}
See also: SideslippingSee also
 Twelvetone technique#Invariance
 Inclusion (music)
Sources
 ^ Blood, Brian (2008 rev 2009). "Inversion of Intervals". Music Theory Online. Dolmetsch Musical Instruments. http://www.dolmetsch.com/musictheory13.htm#intervals. Retrieved 25 December 2009.
 ^ ^{a} ^{b} Whittall, Arnold. 2008. The Cambridge Introduction to Serialism, p.272. New York: Cambridge University Press. ISBN 9780521682008 (pbk).
 ^ ^{a} ^{b} ^{c} ^{d} Nolan, Catherine (2002). The Cambridge history of Western music theory, p.292. Thomas Street Christensen, editor. ISBN 0521623715.
 ^ Pasler, Jann (1986). Confronting Stravinsky: Man, Musician, and Modernist, p.97. ISBN 0520054032.
 ^ Whittall 2008, p.273.
 ^ Whittall, 103
 ^ Perle, George (1996). TwelveTone Tonality, p.4. ISBN 0520201426.
 ^ Schmalfeldt, Janet (1983). Berg's Wozzeck: Harmonic Language and Dramatic Design, p.64 and 70. ISBN 0300027109.
 ^ ^{a} ^{b} Berger, Cayer, Morgenstern, and Porter (1991). Annual Review of Jazz Studies, Volume 5, p.25051. ISBN 0810824787.
 ^ Schmalfeldt, p.70
 ^ ^{a} ^{b} Perle, George. "PitchClass Set Analysis: An Evaluation", p.16971, The Journal of Musicology, Vol. 8, No. 2 (Spring, 1990), pp. 151172. http://www.jstor.org/stable/763567 Accessed: 24/12/2009 15:07.
Allinterval hexachord · Allinterval twelvetone row · Atonality · Chromatic scale · Combinatoriality · Complementation · Derived row · Hexachord · List of pieces that use serialism and twelvetone · Semitone · Serialism · Tone row (Aggregate) · Trope
Prime row · Retrograde · Inversion · Retrograde inversion · Second Viennese SchoolCategories: Intervals
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