IFDO: Difference between revisions

An '''IFDO''' ('''inverse-arithmetic frequency division of the octave'''), or '''UDO''' ('''utonal division of the octave''') is a [[period]]ic [[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 [[Wikipedia: Harmonic mean|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 [[Wikipedia: Arithmetic mean|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).

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

== Individual pages for IFDOs ==

* [[2ifdo]]

* [[3ifdo]]

* Through other [[Pythagorean means]]:

** [[AFDO]] – arithmetic frequency division of the octave

** [[EDO]] – equal division of the octave

[[Category:IFDO]]