The octave complement or inverse interval of an interval is its interval distance from the octave. The octave complement can be seen as a binary symmetric relation over intervals. The concept important in musical practice and most musical theories. Its use is typically restricted to octave-reduced intervals (including the octave).
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 (~).
- minor third vs. major sixth
- (just) unison vs. (just) octave
- just fifth vs. just fourth
Octave-complement 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).
- 5/4 vs. 2*4/5 = 8/5
- 4/3 vs. 2*3/4 = 6/4 = 3/2
Intervals represented as Monzos can be transformed into their octave complement by inverting all arguments and increasing the 2-argument.
- |-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>
Octave-complement intervals represented as s\n meaning s steps of n-EDO follow this relation
s1 + s2 = n. For given s and n, the unknown x can be calculated by the formula
x := n-s.
- 7\12 vs. (12-7)\12 = 5\12
- 1\7 vs. (7-1)\7 = 6\7
Octave-complement 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.
- 333¢ vs. (1200-333)¢ = 867¢
- 701.955¢ vs. (1200-701.955)¢ = 498.045¢