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 ~~mean~~ of frequency.

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 ~~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]])~~.

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}}.

When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. 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).

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).

When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

== Formula ==

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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 mean of frequency.
+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 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]]).
+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}}.
-When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. 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).
+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).
+When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.
 == Formula ==