If 12-tone equal temperament is the Big Mac of tunings, then just intonation is the health food. Just intonation means that the pitches have been defined in terms of whole-number ratios between frequencies. For example, if we’re in the key of C and I refer to a 6/5 E flat, that means an E flat that vibrates at a frequency 6/5 as fast as C; in other words, if C above middle C vibrates at 500 cycles per second (cps), 6/5 E flat will vibrate at 600 cps. The number of potential pitches in a just intonation system is equal to the number of possible fractions: namely, infinite. Naturally, composers cannot deal creatively with a disordered infinity of pitches. We need schemes to limit and justify and order the world of potential pitches. In fact, I believe that good music can only issue from an elegant tuning, and the more elegant the tuning, the more fertile it will be as a generator of musics.

In just intonation, we use fractions to define pitches. To know what pitch a fraction represents, we need to know what key we’re in. If we’re in the key of C, then we define C as 1/1, and D is 9/8. That means that D is defined as the pitch that vibrates 9/8 as fast as C. 9/8 is also the name of an interval – in this case, a whole step. Normally, in talking about justly tuned pitches, we express fractions in terms within a single octave, or between 1 and 2. If 9/8 is D, then 9/2 and 9/16 are also D, but we tend to only use 9/8 because it’s in the octave between 1 and 2. We’re used to calling pitches in different octaves all Cs, or all B flats, but it can be difficult for people to get used to the notion that 7/8 = 7/4 = 7/2 = 14/1. The pitches denoted by those fractions are all octaves of each other, because multiplication or division by 2 only changes octaves.

One of the ways we differentiate between different just intonation systems is by what prime numbers are employed factors in the tuning’s fractions. For instance, five-limit tuning is tuning in which all fractions can be expressed as powers or multiples of the numbers 2, 3, and 5 (not 1 because 1 is merely identity, and not 4 because 4 is merely an octave of 2). In seven-limit tuning, the list is expanded to 2, 3, 5, and 7. Eleven-limit tuning goes up to 2, 3, 5, 7, and 11, and so on. Finally we’ll address the possibility of 13-limit and higher tunings.

If the arithmetic here confuses you, you’ll find a fuller, more gradual explanation on my tuning page. If this sparks your interest, you’ll find all sorts of just-intonation resources at the Just Intonation Network Web page, which will lead you to an encyclopedic array of tuning sites. Or if this just scares you, you can go back to the tuning page.

From BETWEEN U S: A HyperHistory of American Microtonalists

by Kyle Gann

© 2001 NewMusicBox