IFDO: Difference between revisions
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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 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). | 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). | ||
== Formula == | == Formula == | ||
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* [[2ifdo]] | * [[2ifdo]] | ||
* [[3ifdo]] | * [[3ifdo]] | ||
== See also == | == See also == | ||
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** [[AFDO]] – arithmetic frequency division of the octave | ** [[AFDO]] – arithmetic frequency division of the octave | ||
** [[EDO]] – equal division of the octave | ** [[EDO]] – equal division of the octave | ||
== Notes == | |||
[[Category:IFDO| ]] <!-- main article --> | [[Category:IFDO| ]] <!-- main article --> | ||
Revision as of 08:16, 26 April 2023
An IFDO (inverse-arithmetic frequency division of the octave), or UDO (utonal division of the octave) is a periodic 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 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 arithmetic mean (see Pythagorean means).
When treated as a scale, the IFDO is equivalent to the undertone scale, also known as an aliquot scale[1]. 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).
Formula
Within each period of n-ifdo, the frequency ratio c of the k-th step is
[math]\displaystyle{ \displaystyle c = (2n)/(2n - k) }[/math]
Individual pages for IFDOs
See also
- Through other Pythagorean means:
Notes
- ↑ 1/1, The Journal of the Just Intonation Network, Volume 4, Number 1, Winter 1988, p.6, Michael Sloper.