IFDO
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 progression of frequency.
The inverse-arithmetic progression is known in general mathematics as the 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 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 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[1]. 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
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.