BETWEEN U S: A HyperHistory of American Microtonalists

Back in the 16th century, lots of forward-looking musicians wanted to expand the range of European tuning to include intervals based not only on 3 (perfect fifths) and 5 (major and minor thirds), but also 7. After all, Indian and Arabic musicians had gaily been using 7-based intervals for centuries. But they were infidels, and one way the Christians wanted to distance them from the heathens was by insisting that no one could really perceive such tiny pitch differences. So the Italian academics won just after 1600 (over the continuing objections of the seminal mathematician Marin Mersenne), and closed off the wonderful number 7 to the Western world for more than 300 years, until Harry Partch rediscovered it and began using it in the 1920s.

Adding the number 7 and its octaves (14, 28) to our original stable of numbers (2, 3, 5) gives us a very interesting array of new intervals:

15/14 = 119.443 cents
8/7 = 231.174 cents
7/6 = 266.871 cents
9/7 = 435.084 cents
7/5 = 582.512 cents
10/7 = 617.488 cents
14/9 = 764.916 cents
12/7 = 933.129 cents
7/4 = 968.826 cents
28/15 = 1080.557 cents

Now keep in mind that the equal-tempered intervals we’re used to are all sizes divisible by 100. An equal tempered whole step is 200 cents, a perfect fourth is 500 cents, and so on. Seven-limit intervals often create intervals a third of a half-step away from their equal temperament semi-equivalents. The 8/7 “major second” is 31 cents “sharp.” The 7/6 “minor third” is 23 cents “flat.” This specific difference creates a certain flavor for seven-limit tuning, oddly off from the tuning we’re used to and sometimes bitterly flat, yet strangely consonant. In addition, the 7/5 tritone offers a much more consonant tritone than anything we’re used to in European tuning.

Add these 10 simple seven-based intervals to the basic five-limit intervals, and you get a scale that some would consider unwieldy. The most perplexing compositional problem of working in just intonation is, once you open up the field to seven, how do you choose which pitches to use?

One of the most brilliant solutions is the one La Monte Young adopted in his six-hour piano masterpiece The Well-Tuned Piano. He eliminated all factors of the number 5, so that he was only working with multiples of 2, 3, and 7. And he arrived at the eccentric yet very beautiful 12-pitch scale, suitable for piano tuning:

Eb 1/1 = 0 cents
E 567/512 = 177 cents
F 9/8 = 204 cents
F# 147/128 = 240 cents
G 21/16 = 471 cents
G# 1323/1024 = 444 cents
A 189/128 = 675 cents
Bb 3/2 = 702 cents
B 49/32 = 738 cents
C 7/4 = 969 cents
C# 441/256 = 942 cents
D 63/32 = 1173 cents

Note that the scale doesn’t uniformly ascend: G# is lower than G, and C# is lower than C. The scale is basically a five-pitch pentatonic scale around 0, 200, 450, 700, and 950 cents, with slightly different versions available for each pitch. Young kept the tuning secret for 27 years until I tuned my synthesizer to it and published it in an article (with his permission) in Perspectives of New Music, Winter 1993, Volume 31 Number 1. There’s a lot more to say about this scale, and I say some of it on my La Monte Young web page. Unfortunately, the Gramavision recording of The Well- Tuned Piano is out of print and nearly impossible to obtain. Please don’t ask me how to get a copy, because I can’t tell you. You can find out more about La Monte, though, at the Mela Foundation Web Page.

Michael Harrison, Young’s protege and piano tuner, also writes piano music in seven-limit just intonation. He has a CD available on New Albion records.

Another important masterpiece in seven-limit tuning is Ben Johnston‘s String Quartet No. 4, “Amazing Grace.” This lushly emotive 1973 work, a series of variations on the old hymn “Amazing Grace,” begins in a simple pentatonic scale and keeps adding new pitches with each variation until it runs through a glorious 22-pitch, seven-limit scale in the final variation. It’s Ben Johnston’s most popular work, and an instant favorite for everyone who hears it. The best recording is an old one by the Fine Arts Quartet on Gasparo records – unfortunately still only on vinyl. The Kronos Quartet has made a perfectly acceptable recording on Nonesuch, but their attention to tuning isn’t as meticulous.

From seven-limit tuning, the next logical step is eleven-limit tuning.

From BETWEEN U S: A HyperHistory of American Microtonalists
by Kyle Gann
© 2001 NewMusicBox