In music the term complement refers to two distinct concepts.

In traditional music theory

Music theory

Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

 a complement is the interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 which, when added to the original interval, spans an octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 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

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 and the unison

Unison

In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

 are each other's complements and that the tritone

Tritone

In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

 is its own complement (though the latter is "re-spelt" as either an augmented fourth or a diminished fifth, depending on the context).

In twelve-tone music and serialism

Serialism

In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

 the complement of one set of notes from the chromatic scale

Chromatic scale

The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...

 contains all the other notes of the scale. For example, A-B-C-D-E-F-G is complemented by B-C-E-F-A.

Note that musical set theory

Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

broadens the definition of both senses somewhat.

Rule of nine

The rule of nine is a simple way to work out which intervals complement each other. 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

Semitone

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

and tritone

Tritone

In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

) this rule cannot be applied. However, octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

and unison

Unison

In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

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

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

 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

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

, or 7, is the complement of the perfect fourth

Perfect fourth

In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

, 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

Additive inverse

In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....

 sense of the same melodic interval in the opposite direction - e.g. a falling 5th is the complement of a rising 5th.

Aggregate complementation

In twelve-tone music and serialism complementation (in full, literal pitch class complementation) is the separation of pitch-class

Pitch class

In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

 collections into complementary sets, each containing pitch classes absent from the other or rather, "the relation by which the union of one set with another exhausts the aggregate". To provide, "a simple explanation...: the complement of a pitch-class set consists, in the literal sense, of all the notes remaining in the twelve-note chromatic that are not in that set."

In the twelve-tone technique this is often the separation of the total chromatic of twelve pitch classes into two hexachord

Hexachord

In music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...

s of six pitch classes each. In rows with the property of combinatoriality

Combinatoriality

In music using the twelve tone technique combinatoriality is a quality shared by some twelve-tone tone rows whereby the row and one of its transformations combine to form a pair of aggregates...

, two twelve-note tone row

Tone row

In music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...

s (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." 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

Permutation (music)

In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...

 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.

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

Tone row

In music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...

), in which case the second set would be P1.

In contrast, "where transposition

Transposition

Transposition may refer to:Mathematics* Transposition , a permutation which exchanges two elements and keeps all others fixed* Transposition, producing the transpose of a matrix AT, which is computed by swapping columns for rows in the matrix AGames* Transposition , different moves or a different...

ally related sets show the same difference for every pair of corresponding pitch classes, inversionally related sets show the same sum." 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

In set theory

Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

 the traditional concept of complementation

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

 may be distinguished as literal pitch class complement, "where the relation obtains between specific pitch-class sets", 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," which may be described as abstract complement, "where the relation obtains between set classes". This is because since P

Variable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 is equivalent to M, and M is the complement of M, P is also the complement of M, "from a logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

al and musical point of view," even though not its literal pc complement. Originator Allen Forte

Allen Forte

Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...

 describes this as, "significant extension of the complement relation," though George Perle

George Perle

George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...

describes this as, "an egregious understatement".
As a further example take the chromatic sets 7-1 and 5-1. If the pitch-classes of 7-1 span C-F and those of 5-1 span G-B then they are literal complements. However, if 5-1 spans C-E, C-F, or D-F, then it is an abstract complement of 7-1. As these examples make clear, once sets or pitch-class sets are labeled, "the complement relation is easily recognized by the identical ordinal number in pairs of sets of complementary cardinalities".