There is nothing that musicians take more for granted than the fact that there are twelve pitches to an octave, and that these pitches divide the octave into twelve equal steps. Apparently few musicians question this arrangement, and only a tiny minority can explain whence it arose, why, and from what principles its authority derives. This 12-pitch assumption, however, is far from innocent. Twelve-tone equal temperament, as this common tuning is called, is a 20th-century phenomenon, a blandly homogenous tuning increasingly imposed on all the world’s musics in the name of scientific progress. In short, twelve-tone equal temperament is to tuning what the McDonald’s hamburger is to food.
How can this be so? What is so unnatural about twelve-tone equal temperament?
The basis of any natural system of tuning is that two pitches sound consonant (that is to say, sweet, or intelligible to the ear) when their sound waves vibrate at ratios of relatively small whole numbers. In an octave, for example, two pitches vibrate at a ratio of 2 to 1, one pitch vibrating twice as fast as the other. In a perfect fifth, such as C up to G, the ratio is 3 to 2. In a major third C to E, the ratio is 5 to 4.
The great problem that nature bequeaths to us in the mathematics of tuning – not an obstacle, but a wonderful challenge when viewed the right way – is that these simple intervals aren’t divisible by each other. To illustrate, we need a perceptual measure of interval size. The one invented by the great acoustician Alexander Ellis in the late 19th century is called a cent, and is equal, by definition to one 1200th of an octave, or 1/100th of a half-step.
An octave: (ratio 2:1) = 1200 cents
A perfect fifth: (ratio 3:2) = 701.955 cents
A major third: (ratio 5:4) = 386.3 cents
In the equal temperament we’re used to, three major thirds – C to E, E to G#, G# to C – equal one octave. But as you can see, three pure major thirds of 386.3 cents do not equal one octave, because 3 x 386.3 does not equal 1200. So equal temperament, our McDonald’s hamburger tuning, stretches every major third out to an arbitrarily out-of-tune 400 cents, somewhat the way McDonalds standardizes every patty to a flat quarter-pound of dubious relation to beef. These means that every major third on the piano is out of tune by 13.7 cents, creating busy little beat patterns between the overtones of every major third we hear. Unless you’ve had some exposure to Indian or Indonesian or some other non-Western musical tradition (or authentic barbershop quartet music, the last pure-tuned tradition in America), it’s quite likely that you’ve never heard a true major third in your life, nor a true major or minor triad.
Music schools teach that this Big Mac tuning has been around for centuries and represents an immutable endpoint of progress. It’s a lie. History, even in Europe, has provided many alternatives, Arabic and Asian cultures have provided rich tuning resources unknown to us, and many recent American composers have explored alternative tuning possibilities.
There are many reasons to write in other tunings, seemingly as many as there are composers who do it. La Monte Young seeks absolute purity of pitch so he can explore complex combinations of distant overtones never heard before. Harry Partch wanted to imitate in melody the subtle contours of the human voice, without compromise. Lou Harrison wants to recapture the sensuous presence that true intervals had before the 20th century. Ben Johnston wants his music perfectly in tune so it will have a healthful psychological effect on the listener. Myself, I enjoy the expanded composing resources of 30 or so pitches to the octave, and the option of creating amazing chromatic effects through minimal voice-leading. Some composers are seeking a magical harmonic alchemy written about in ancient treatises. Others just enjoy exotic out-of-tuneness. One of the exciting things about the microtonal field is that, despite its grounding in natural laws of acoustics, its diverse practitioners hardly agree on anything.
For those intrigued but unfamiliar with the wide range of microtonal strategies, this quick survey in four sections will explore several options for escaping equal temperament. We’ll look at forms of historical tunings, take a regrettably brief glimpse at other tunings of the world, and examine tunings devised by several American composers, both in the areas of just intonation and of equal temperaments based on divisions other than 12.
And for those who want more information, there are a lot of Web sites. Book publishers and academic musicians are absolutely convinced that alternate tuning is a strange, esoteric subject that no one except a few weirdos is interested in. If they’d look on the Web, they’d find thousands of tuning aficionados. You can learn everything you wanted to know about meantone at meantone.com, and Terry Blackburn, Zeke Hoscan, and Stephen Malinowski have excellent pages on the mathematics of different
European tunings. There’s a Pythagorean Web page. And one of the most forward-looking theoretical thinkers, with a lot of new tuning conceptions for new composition, is Joe Monzo. The tuning of our music evolves historically more rapidly than people realize, and it’s on the move again.