Octave reduction
Octave reduction is the process of transposing an interval by octaves so that the resulted size falls between the unison and the octave. Formally, this means multiplying the frequency ratio with a whole-number power of 2 until it has a real-number value greater or equal than 1 ("1/1", the unison) and less than 2 ("2/1", the octave):
1 <= r < 2
If r does not satisfy this inequality, it has to be
- multiplied by 2 while less than 1 or
- divided by 2 while greater than or equal to 2
Examples
- 3/4 is less than 1, so multiply by 2 to get 3/2
- 7/2 is greater than 2, so divide by 2 to get 7/4
- 4/2 is greater than 2, so divide by 2 to get 2, which is equal to 2, so divide by 2 to get 1
- Adding 4 fifths corresponds to calculating the product of 4 time (3/2 the interval ratio) leading to 81/16. This interval (5.0625 in decimal representation) is greater than 2 octaves
(2*2 = 2^2 = 4), but less than 3 octaves(2*2*2 = 2^3 = 8). So it gets divided by 2 (or multiplied by 1/2) two times:(81/16)*(1/2)*(1/2) = 81 / (16*2*2) = 81/64 - Subtracting a fourth (4/3) from minor third 6/5 corresponds to dividing 6/5 by 4/3 which is the same as
(6/5)*(3/4) = 18/20 = 9/10. The result (0.9 in decimal representation) is less than 1 but greater than 1/2 (which mean one octave down). So it gets multiplied by 2 once:9/10*2 = 18/10 = 9/5.