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).

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

Individual pages for IFDOs

• *1ifdo – identical to 1edo
• 2ifdo
• 3ifdo

Alternate names

In 1/1, The Journal of the Just Intonation Network, Volume 4, Number 1, Winter 1988, p.6, Michael Sloper refers to this type of scale as an "aliquot scale".

An EDL is the same as an IFDO. For example, 42-EDL is the same thing as 42ifdo.

See also

• Through other Pythagorean means:
  • AFDO – arithmetic frequency division of the octave
  • EDO – equal division of the octave