BETWEEN U S: A HyperHistory of American Microtonalists
Eleven-limit tuning means that we define intervals and pitches by multiplying and dividing the numbers 2, 3, 5, 7, and 11, with no prime numbers larger than 11 used as factors. Adding 11 into the mix gives us a very interesting range of new intervals:
12/11 = 150.637 cents
11/10 = 165.004 cents
11/9 = 347.408 cents
14/11 = 417.508 cents
15/11 = 536.951 cents
11/8 = 551.318 cents
16/11 = 648.682 cents
22/15= 663.049 cents
11/7 = 782.492 cents
18/11 = 852.592 cents
20/11 = 1034.996 cents
11/6 = 1049.363 cents
Note that of these intervals, nine are within 15 cents of a quartertone (50 cents between two equal-tempered steps in 12-tone equal temperament), and six of those – 12/11, 11/9, 11/8, 16/11, 18/11, 11/6 – are within three cents. Eleven-limit tuning produces many of the pitches that we think of as quartertones. The peculiar quality of eleven-limit tuning is to smooth out the scale by giving us mediating pitches half-way in-between the pitches we’re used to.
The great champion of eleven-limit tuning, of course, is Harry Partch. His 43-tone scale, 43 non-equal steps to the octave, uses no prime factors larger than 11. I won’t give his scale here, because you can easily find it in his book Genesis of a Music and other places. There is loads of Harry Partch information on the Internet, the best sites being Corporeal Meadows and a British Harry Partch web page. In addition, all of Harry Partch’s major and minor works are being released on the innova and CRI labels.
The other person I can mention as having written a significant-sized output in eleven-limit tuning is myself, Kyle Gann. Eleven-limit tuning appeals to me for all those in-between notes, those quarter-tones that slide so easily between the pitches we’re used to. My own attraction to microtonality is the potential for extreme chromaticism and a minimalist approach to voice-leading in which lines can remain almost motionless while the harmony changes key wildly. You can hear this effect in my 1997 piece How Miraculous Things Happen, of which you can find an audio excerpt here. The piece uses 24 pitches to the octave (but very unevenly distributed, not quartertone), and capitalizes on the interval of 11/9 (347 cents) to slide smoothly between the minor (316 cents) and major (386) thirds. My largest just-intonation work so far is a 35-minute, one-man opera, Custer and Sitting Bull. The four movements of this work expand from 20 to 31 pitches, each movement based on a different tuning principle. You can also find audio samples of this piece at the same place, and program notes and tuning charts at my Custer and Sitting Bull web page.
Partch charmingly expressed his reasons for not proceeding past 11 to the 13-limit: “When a hungry man has a large table of aromatic and unusual viands spread before him he is unlikely to go tramping along the seashore and in the woods for still other exotic fare. And however skeptical he is of the many warnings regarding the unwholesomeness of his fare – like the ‘poison’ of the ‘love-apple’ tomato of a comparatively few generations ago – he has no desire to provoke further alarums.” Personally, I’ve never been able to get comfortable enough with my perception of the 13th harmonic as a consonance to pass that limit myself. But if you’re hungrier than Partch, and more attuned than myself, you’ll want to go on to the thirteen-limit and beyond.
From BETWEEN U S: A HyperHistory of American Microtonalists
by Kyle Gann
© 2001 NewMusicBox