Millioctave

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

1 History and use
2 See also
3 Notes
4 External links

[edit] History and use

The millioctave was introduced by the German physicist Arthur von Oettingen in his book Das duale Harmoniesystem (1913). The invention goes back to John Herschel, who in 1871 proposed a division of the octave into 1000 parts in On Sound and Atmospheric Vibrations with the Mathematical Elements of Music.

Compared to the cent, the millioctave has not been as popular. It is, however, occasionally used by authors who wish to avoid the close association between the cent and equal temperament. However, it has been criticized that it introduces a bias for the less familiar 10-tone equal temperament.^[1]