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Millioctave

The millioctave (mO) is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave. From this it follows that one millioctave is equal to the ratio 2, the 1000th root of 2, or approximately 1.0006934 .

Given two frequencies a and b, the measurement of the interval between them in millioctaves can be calculated by


$$n = 1000 \log_2 \left( \frac{a}{b} \right) \approx 3322 \log_{10} \left( \frac{a}{b} \right)$$

Likewise, if you know a note b and the number n of millioctaves in the interval, then the other note a may be calculated by:


$$a = b \times 2 ^ \frac{n}{1000}$$

Like the more common cent, the millioctave is a linear measure of intervals, and thus the size of intervals can be calculated by adding their millioctave values, instead of multiplication, which is necessary for calculations of frequencies.

A millioctave is exactly 1.2 cents.