Complement (music)

In traditional music theory

Music theory

Music theory is the field of study that deals with how music works. It examines the language and notation of music. It identifies patterns that govern composers' techniques. In a grand sense, music theory distills and analyzes the parameters or elements of music – rhythm, harmony , melody,...

 a complement is the interval

Interval (music)

In music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....

 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 which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...

 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

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 which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...

 and the unison

Unison

-Interval of the unison:Unison may refer to the pseudo-interval formed by a tone and its duplication , for example c-c, as differentiated from the second, c-d, etc. In the unison the two pitches have the ratio of 1:1 or 0 half steps and zero cents...

 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

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 F is denoted −F....

 sense of the same melodic interval in the opposite direction - e.g.

In traditional music theory

Music theory

Music theory is the field of study that deals with how music works. It examines the language and notation of music. It identifies patterns that govern composers' techniques. In a grand sense, music theory distills and analyzes the parameters or elements of music – rhythm, harmony , melody,...

 a complement is the interval

Interval (music)

In music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....

 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 which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...

 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

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 which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...

 and the unison

Unison

-Interval of the unison:Unison may refer to the pseudo-interval formed by a tone and its duplication , for example c-c, as differentiated from the second, c-d, etc. In the unison the two pitches have the ratio of 1:1 or 0 half steps and zero cents...

 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

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 F is denoted −F....

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

The perfect fifth is the musical interval between a note and the note seven semitones above it on the musical scale. For example, the note G lies a perfect fifth above C; D is a perfect fifth above G, C is a perfect fifth above F...

, or 7, is the complement of the perfect fourth

Perfect fourth

The perfect fourth is a musical interval which spans four scale degrees. It consists of the note and the note five semitones above it on the musical scale. For example, the interval between a C and the next F above it is a perfect fourth; similarly the interval between a G and the next C above...

, or 5, 7+5 = 12 = 0 mod 12).

Complementation

In twelve-tone music and serialism

Serialism

In music, serialism is a technique, method , "highly specialized technique" , or "way" of composition, but also "a philosophy of life , a way of relating the human mind to the world and creating a completeness when dealing with a subject"...

 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.

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. With 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 the twelve notes of the chromatic scale. Tone rows are the basis of Arnold Schoenberg's twelve-tone technique and most types of serial music...

s are used simultaneously, thereby creating, "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."

Hexachordal complementation is the use of the potential for pairs of hexachords to each contain six different pitch classes and thereby complete an aggregate.

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 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." 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," even though not its literal pc complement.