Superparticular Woman (1992)
When I came to study privately with Ben Johnston in 1984, having already completed my doctorate, I loved his music, but had no intention of getting involved with microtonality - too much work for too little effect, I thought. (Well, as it turned out, I was right about the work part.) But at my first lesson Ben, who never proselytized for just intonation, off-handedly mentioned how nice a chord I'd written would sound if tuned correctly, and he reeled off the appropriate fractions. Having been the math star of my high school, I instantly realized that I had the necessary brain talents for microtonality. Also, as devoted as I was to minimalism, I had been striving to combine it with the extreme chromaticism I also loved; Ben's tiny pitch shifts were chromaticism squared. Within that moment it was as though I felt a large iron door slam shut behind me, and I realized there was no going back.
Even so, it took years of exploring pitch structures before I felt I could hear in my head chord progressions not reducible to a 12-pitch scale. All through the late 1980s I filled notebooks with fractions and logarithms, and not until 1992 did I come up with a rather Johnstonian harmonic conception, but a simple one, that I felt I could compose in intuitively and hear what I was doing. I began (as I have often begun pieces since) by writing a melody of superparticular ratios: 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, and 5/4, seven pitches which lie within only 221 cents (a slightly large whole-step). A superparticular ratio is a ratio in the form (x+1)/x; that's the joke of the title. I looked for how many harmonic series' were needed to justify each of these superparticular ratios along with the pitch G as 1/1, and found that I needed only five different ones. (I have often ended up basing my just-intonation music on G because I start out thinking in Harry Partch's 43-tone scale, which was centered on G at 392 cps.) Five was sufficient because the series that contained G as fundamental was the same as the ones that contained it as 2nd harmonic, 4th harmonic, and so on.
Every harmony in Superparticular Woman, then, is chosen from a harmonic series, and every harmonic series contains the pitch G, the pivot point of the work. In the chart below, the fundamental of each series lies at the bottom of each column. Going upward, the 3rd harmonic is next, then the 5th, the 7th, the 9th, and in two cases the 11th as well (even-numbered harmonics are omitted because they duplicate harmonics lower in the series):
C^ A+ F7+ D B Bb7 F+ DL^- D E Db7 BL G = G = G = G = G C Bb C#L Eb7 Eb EL A AL C F
I have notated these (as closely as html will allow) in Ben Johnston's excellent microtonal notation, in which:
+ raises a pitch by 81/80 (21.5 cents)
- lowers a pitch by 80/81 (-21.5 cents)
# raises a pitch by 25/24 (71 cents)
b (flat) lowers a pitch by 24/25 (-71 cents)
7 lowers a pitch by 35/36 to get a seventh harmonic (-49 cents)
L (upside-down 7) raises a pitch by 36/35 (+49 cents)
^ (upward arrow) raises a pitch by 33/32 (53 cents) to get an eleventh harmonic
F-A-C, C-E-G, and G-B-D are all perfectly tuned 4:5:6 major triads.
(If you don't have enough experience with just intonation to make sense of these charts, try reading the step-by-step Just Intonation Explained section.)
This was the basic harmonic structure I had to play with. As it turned out, this yielded a scale of 22 pitches to the octave (though I never cared how many nor even counted them until putting this page together):
Pitch: G Ab^ A A+ AL Bb7 Bb B BL C C^ Db7 C#L D DL^- Eb7 Ratio: 1/1 11/10 10/9 9/8 8/7 7/6 6/5 5/4 9/7 4/3 11/8 7/5 10/7 3/2 11/7 14/9 Cents: 0 165 182 204 231 267 316 386 435 498 551 583 617 702 782 765
Eb E EL F7+ F F+ 8/5 5/3 12/7 7/4 16/9 9/5 814 884 933 969 996 1018
As Superparticular Woman's bass line leaps around in a kind of permutational passacaglia pattern, the melodies in the upper overtones shift ever so slightly for just the shimmering fusion of harmonic weirdness and resonant consonance I wanted.
A PDF score of the piece can be found here. The score is the piece; the recording is merely a performance, made with primitive early-1990s retunable technology. A better rendering could be made today, but there is other music to write.
Copyright 1997 Kyle Gann
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