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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).


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:
    • AFDO – arithmetic frequency division of the octave
    • EDO – equal division of the octave


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