IFDO

An IDO (inverse-arithmetic division of the octave) is a periodic tuning system which divides the octave according to the inverse-arithmetic mean.

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-IDO 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 IDOs are subsets of JI, and thus the more precise and appropriate equivalence of an n-IDO is to an n-UDO (utonal division of the octave).

Formula

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

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

Individual pages for IDOs

• 2ido
• 3ido

See also

• EDL – equal division of length, a similar concept
• ADO – arithmetic division of the octave
• EDO – equal division of the octave