IFDO: Difference between revisions

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An '''IDO''' ('''inverse-arithmetic ~~division of the octave''') or '''ELDO''' ('''equal length~~ division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] [[~~equal length division~~|~~inverse-arithmetically~~]] ~~rather than logarithmically~~, which ~~means it is divided proportional~~ to the length of a string to be ~~divided~~. ~~All~~ IDOs are subsets of [[~~just intonation~~]].

An '''IDO''' ('''inverse-arithmetic division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic mean.

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]].

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 ==

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-An '''IDO''' ('''inverse-arithmetic division of the octave''') or '''ELDO''' ('''equal length division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] [[equal length division|inverse-arithmetically]] rather than logarithmically, which means it is divided proportional to the length of a string to be divided. All IDOs are subsets of [[just intonation]].
+An '''IDO''' ('''inverse-arithmetic division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic mean.
+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]].
+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 ==