Octave complement

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The octave complement or inverse interval of an interval is its interval distance from 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 octave-reduced intervals (including the octave).

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 (~).

Interval name part
base value complement
Name # # Name
unison 1 8 octave
second 2 7 seventh
third 3 6 sixth
fourth 4 5 fifth
fifth 5 4 fourth
sixth 6 3 third
seventh 7 2 second
octave 8 1 unison
Attribute parts
base value ~ ~ complement
diminished -2 +2 augmented
minor -1 +1 major
just 0 0 just
major +1 -1 minor
augmented +2 -2 diminished
Examples
minor third vs. major sixth
(just) unison vs. (just) octave
just fifth vs. just fourth

Ratio

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).

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 2-argument.

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

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`.

Examples
7\12 vs. (12-7)\12 = 5\12
1\7 vs. (7-1)\7 = 6\7

Cents

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`.

Examples
333¢ vs. (1200-333)¢ = 867¢
701.955¢ vs. (1200-701.955)¢ = 498.045¢