IFDO: Difference between revisions

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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 progression of frequency.

The inverse-arithmetic progression is known in general mathematics as the {{W|Harmonic progression (mathematics)|harmonic progression]], 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 progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W|arithmetic progression}}.

The inverse-arithmetic progression is known in general mathematics as the {{W|Harmonic progression (mathematics)|harmonic progression}}, 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 progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W|arithmetic progression}}.

For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All integer IFDOs are subsets of [[just intonation]], and up to transposition, any integer IFDO is a superset of a smaller integer IFDO and a subset of a larger integer IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).

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 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 progression of frequency.
-The inverse-arithmetic progression is known in general mathematics as the {{W|Harmonic progression (mathematics)|harmonic progression]], 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 progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W|arithmetic progression}}.
+The inverse-arithmetic progression is known in general mathematics as the {{W|Harmonic progression (mathematics)|harmonic progression}}, 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 progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W|arithmetic progression}}.
 For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All integer IFDOs are subsets of [[just intonation]], and up to transposition, any integer IFDO is a superset of a smaller integer IFDO and a subset of a larger integer IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1).