Tritave complement

The tritave complement or inverse interval of an interval is its interval distance from the tritave. The tritave complement can be seen as a ternary symmetric relation over intervals. The concept important in musical practice and most musical theories. Its use is typically restricted to tritave-reduced intervals (including the tritave).

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 tritave). 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 13, 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 12 tritave second 2 11 eleventh third 3 10 tenth fourth 4 9 ninth fifth 5 8 octave sixth 6 7 seventh seventh 7 6 sixth octave 8 5 fifth ninth 9 4 fourth tenth 10 3 third eleventh 11 2 second tritave 12 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 tenth

(just) unison vs. (just) tritave

just fifth vs. just octave

Ratio

Tritave-complement intervals represented as ratios r follow the relation r1*r2 = 3. For given r the unknown x can be calculated by the formula x := 3/r or (for the ratio representation r = a/b) into x := 3*b/a (the result sometimes has to be reduced by the factor 3).

Examples

5/4 vs. 3*4/5 = 12/5

3/2 vs. 3*2/3 = 6/3 = 2/1

Monzo

Intervals represented as Monzos can be transformed into their tritave complement by inverting all arguments and increasing the 3-argument.

Examples

|-1 1> vs. |-(-1) -(1)+1> = |+1 +0>

|3 -3 1> vs. |-(3) -(-3)+1 -(1)> = |-3 4 -1>

|-2 2 1 0 -1> vs. |-(-2) -(2)+1 -(1) -(0) -(-1)> = |+2 -1 -1 0 +1>

Edt steps

Tritave-complement intervals represented as s\n meaning s steps of n-EDT 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\19 vs. (19-7)\19 = 12\19

1\11 vs. (11-1)\11 = 10\11

Cents

Tritave-complement intervals represented as s¢ follow this relation s1 + s2 = 1901.9550008654. For given s, the unknown x can be calculated by the formula x := 1901.9550008654-s.

Examples

333¢ vs. (1901.9550008654-333)¢ = 1568.9550008654¢

701.955¢ vs. (1901.9550008654-701.955)¢ = 1200.0000008654¢

See also

• tritave reduction
• Inversion (music) - Wikipedia #Intervals
• Octave complement -- the analogue for octave