Octave reduction

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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.

See also