octave-equivalence

based on the unique property of intervals that 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 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:

the Indonesian pelog and slendro scales, which seem to be based on the ratios of the "inharmonic" timbres of the gamelan instruments

the "non-octave non-equal" scales used by Brian McLaren, Gary Morrison's 88CET scale, Wendy Carlos's alpha, beta, and gamma scales, and others.

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, 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).

[from Joe Monzo, JustMusic: A New Harmony]

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

. . . . . . . . . . . . . . . . . . . . . . . . . .

octave-reduction

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 2^x : 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 Excel^TM 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.

[from Joe Monzo, JustMusic: A New Harmony]

---

updates:

2002.9.4 - added the section on "octave reduction"
2000.1.13

---

(to download a zip file of the entire Dictionary, click here)

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to the Microtonal Dictionary index return to my home page return to the Sonic Arts home page