# IFDO

An **IFDO** (**inverse-arithmetic frequency division of the octave**), or **UDO** (**utonal division of the octave**) is a periodic tuning system which divides the octave according to the inverse-arithmetic mean of frequency.

The inverse-arithmetic mean is known in general mathematics as the harmonic mean, but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to *subharmonic* sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic mean" was coined to avoid this conflict, as well as to point to its relationship with the arithmetic mean (see Pythagorean means).

When treated as a scale, the IFDO is equivalent to the undertone scale, also known as an aliquot scale^{[1]}. An *n*-IFDO includes the pitches found by dividing the length of a string or resonating chamber into *n* equal parts, and thus may also be called an *n*-ELDO (equal length division of the octave); however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation. As divisions of the octave, which is a rational interval, all IFDOs are subsets of JI, and thus the more precise and appropriate equivalence of an *n*-IFDO is to an *n*-UDO (utonal division of the octave).

## Formula

Within each period of *n*-ifdo, the frequency ratio *c* of the *k*-th step is

[math]\displaystyle c = (2n)/(2n - k)[/math]

## Individual pages for IFDOs

## See also

- Through other Pythagorean means:

## Notes

- ↑
*1/1, The Journal of the Just Intonation Network*, Volume 4, Number 1, Winter 1988, p.6, Michael Sloper.