Operations on intervals

The following are common arithmetic operations on musical intervals.

Stacking and unstacking

Stacking two intervals feels perceptually like we are adding two distances, though it corresponds to multiplying two frequency ratios. The logarithm function is the bridge between frequency space and pitch space:

${\log }_2(ab)={\log }_{2}a+{\log }_{2}b$

for any two frequency ratios a and b. The above equation tells us that the sum of the size (in octaves) of a and the size (in octaves) of b is equal to the size in octaves of the product of two frequency ratios ab. Hence, stacking corresponds to multiplying frequency ratios in linear (frequency) space, and adding cent values in logarithmic (pitch) space.

To convert octaves to cents, we simply multiply both sides by 1200:

$1200{\log }_2(ab)=1200{\log }_{2}a+1200{\log }_{2}b.$

Also, dividing frequency ratios corresponds to subtracting their perceptual sizes:

$1200{\log }_2(a/b)=1200{\log }_{2}a-1200{\log }_{2}b.$