IFDO: Difference between revisions

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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).

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 IFDOs are subsets of [[just intonation]], and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger 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.

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 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).
+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 IFDOs are subsets of [[just intonation]], and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger 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.