Complement (music)

In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism. In interval complementation 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 "re-spelt" as either an augmented fourth or a diminished fifth, depending on the context). In the aggregate complementation of twelve-tone music and serialism the complement of one set of notes from the chromatic scale 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 broadens the definition of both senses somewhat. 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 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. 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).