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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

Based on the unique property of any interval which can be most easily interpreted as the 2:1 ratio, commonly called the "octave", that: although it is a different pitch from the origin 1:1, it seems to have the same aesthetic affect or properties as 1:1.

Traditional music theory assumes octave-equivalence, thus the nominals (letter-names) of the notes repeat in the different "octaves".

Many tuning systems follow this approach, but not all. Examples of tunings which do not exhibit octave equivalence are:

Modern acoustical research yields evidence that most individuals' perception of what is consonant is more complex than the long-held belief by many music-theorists and scientists that consonance is directly related to the size of the integer terms in the ratios and/or the size of the prime- or odd-number factors. [McLaren's website will have much information on and quotations from this research - one citation refers to an interval of 12.15 Semitones as that most commonly perceived as a consonant "octave".]

Johnny Reinhard wrote an interesting paper on a study he did of a song by two Sapmi (also known as Lapp) singers of northern Scandinavia. There were very minute but deliberate interval dissonances between them, and tiny changes in these intervals in each of the 9 repeating verses. One of the most prominent was a frequently-used mistuned harmonic "octave" which ranged from about 11.90 to 12.04 Semitones.

Interestingly, Schoenberg's method of 12-tone serialism, and all the theory derived from it, assumes 'octave'-equivalence as the basic relationship between pitches, in that all pitches are considered as pitch-classes irrespective of their register (relative height or depth in pitch), while at the same time the music composed in these systems studiously avoids the use of the 'octave' in the musical gestures, in contrast to the abundant use of the 'octave' in 'tonal' music (see Browne 1974).


Browne, Richmond. 1974.
Review of Allen Forte's The Structure of Atonal Music,
in Journal of Music Theory, 18.2 [Fall 1974].

. . . . . . . . .

The operation of octave-reduction follows from the assumption of octave-equivalence described above. It is a useful procedure primarily because octave-equivalent scales are nearly always specified within a range of only one particular octave, with the frequencies of pitches in higher and lower registers assumed to be in ratios of 2x : 1 (where x is any negative or postive integer) to that reference octave.

If the ratio of a pitch or interval is <1 or >2, it may be octave-reduced by this formula: 10[log10(r) mod log10(2)], where r is the ratio which is <1 or >2.

This formula can be represented by the following code which may be pasted into a Microsoft ExcelTM spreadsheet: =10^(MOD(LOG([cell]),LOG(2))), where [cell] is the address of the cell containing the ratio which is <1 or >2.

This is particularly useful for those who, like myself, prefer to use prime-factor notation instead of ratios, without bothering to specify powers of 2 when it's not necessary.

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