# Octave reduction

(Redirected from Octave-reduced)

**Octave reduction** is the process of multiplying an interval with a whole-number power of 2 (2/1 = octave) until it has a real-number value greater or equal than 1 and less than 2.

In other words, an **octave-reduced** interval *r* satisfies the equation 1 <= r < 2.

## Examples

- 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`

.