# Octave reduction

**Octave reduction** is the process of multiplying an interval 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`

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