Octave complement
The octave complement or inverse interval of an interval is its difference with the octave. It can be seen as a binary symmetric relation over intervals. The concept is important in musical practice and most musical theories. Its use is typically restricted to octavereduced intervals (including the octave).
The octave complement is often simply called complement in the context of octaveequivalent scales, but the broader concept of complement can also apply to other intervals, such as the fifth complement, the fourth complement and the tritave complement.
Calculation
Depending on the interval representation (name, ratio, monzo, edo steps, cents), it's more or less easy to retrieve the complementary interval from a given interval.
Classical interval names
The intervals in western music have names derived from numerals, starting at 1 (unison or prime, second, etc.). These names are prefixed with further size attributes (just, minor, major, etc.) which express relative size relations (the attribute just is often omitted for unison and octave). Complementary intervals are calculated by complementing both, name and attribute parts. For the name part the complement is calculated by subtracting the ordinal number from 9, the attribute part is negated, just is the negation of just . The following tables show names with ordinal numbers (#) and attributes with size hints (~).


 Examples
 minor third vs. major sixth
 (just) unison vs. (just) octave
 just fifth vs. just fourth
Ratio
Octavecomplement intervals represented as ratios r follow the relation r1*r2 = 2
. For given r the unknown x can be calculated by the formula x := 2/r
or (for the ratio representation r = a/b
) into x := 2*b/a
(the result sometimes has to be reduced by the factor 2).
 Examples
 5/4 vs. 2*4/5 = 8/5
 4/3 vs. 2*3/4 = 6/4 = 3/2
Monzo
Intervals represented as monzos can be transformed into their octave complement by inverting all arguments and increasing the 2argument.
 Examples
 [1 1⟩ vs. [(1)+1 (1)⟩ = [+2 1⟩
 [3 3 1⟩ vs. [(3)+1 (3) (1)⟩ = [2 3 1⟩
 [2 2 1 0 1⟩ vs. [(2)+1 (2) (1) (0) (1)⟩ = [+3 2 1 0 +1⟩
Edo steps
Octavecomplement intervals represented as s\n meaning s steps of nEDO follow this relation s1 + s2 = n
. For given s and n, the unknown x can be calculated by the formula x := n s
.
 Examples
 7\12 vs. (127)\12 = 5\12
 1\7 vs. (71)\7 = 6\7
Cents
Octavecomplement intervals represented as s¢ follow this relation s1 + s2 = 1200
. For given s, the unknown x can be calculated by the formula x := 1200  s
.
 Examples
 333¢ vs. (1200333)¢ = 867¢
 701.955¢ vs. (1200701.955)¢ = 498.045¢