July 26, 2016
Having moved to Germany and now feeling almost free of jet-lag, I'd like to continue the story of the Tonal Plexus today by sharing some music, along with some details about the theory from which it was born.
As I mentioned earlier, my first compositions in Just Intonation were for piano, according to an idea I had developed in 1999, about which I wrote a paper, presented at Music Theory Midwest and in a different form to students and faculty at the University of Minnesota. In a nutshell, the theory is this:
Taking the above points together, the basic idea of the theory is this:
Intervals may be represented by approximate measurements with varying degrees of accuracy, within reasonable thresholds of human perception. Different measurements of intervals as ratios place the tones in different harmonic series, where all the numbers are harmonics relating to a given root tone of 1. Intervals having a power of 2 in the denominator always contain an octave duplication of the root tone of a harmonic series, which makes them the easiest to understand, especially when considering combination tones. Representing all intervals in a basic form with a power of 2 in the denominator results in the ability to see all harmonic relationships of all intervals with equal ease.
The above idea contains a kernel which later developed into the H-System. Measurements are represented ratios in Just Intonation, and so every interval can be represented by more than one ratio, with the JND margin of error. The numbers chosen to represent an interval become very important, because the Combination Tones produce a whole set of pitches revealing more than just one interval, but rather a revealing whole structure of tones within a harmonic series. The idea is to make that structure equally easy to understand for all intervals.
At the time I was starting this research, I didn't have very firm numbers on the JND, but I assumed it was a few cents (later I learned that my guess was confirmed by research). A threshold of a few cents allows a wide range of ratios to represent essentially the same interval. Obviously, the closer an approximation, the more accurate the results.
An example will help to make this clearer. With a wide range of error allowed, the equal tempered minor second of exactly 100 cents can be represented by 17/16 (16:17) at about 105 cents. The number 16 in the denominator satisfies a requirement of the theory, and this gives us a difference tone of 1, which is five octaves below the lower tone of the interval. A problem with this representation is that it conflicts with our perception. You can easily prove this to yourself. Go to a piano, and play a minor second very loudly in the middle-upper register, so that you hear a low tone buzzing in your ears. Listen to the pitch of the low tone, and you'll notice it is not a low octave duplication of the lower key on the piano that you are playing. The difference tone is flatter than that. 5 cents error doesn't correspond close enough to our perception. A closer approximation is 271/256 at about 99 cents. In this case we have a difference tone of 271-256 = 15. If you know your JI intervals, 15/8 or 8:15 is a Major Seventh, a halfstep below the root. This is pretty much where that difference tone actually is. And so, you see that representing 100 cents by 271/256 gives you a clear idea of the harmonic root of the interval, and where the combination tones are within the whole harmonic structure produced by the interval.
271/256 is four octaves higher up in a harmonic series than 17/16. These octaves can be thought of as "Levels" within a harmonic series. Within each level, there are twice as many unique tones as the level below it. For example, in the first level 2:4, there is one tone, 3. In the second level 4:8 there are two unique tones, 5 and 7, while 6 is a duplicate of 3 which belongs to the level below. From this I came up with the idea that each tone within a level has a "Place". 3 has the first place in the first level, and 5 has the first place in the second level. Every harmonic can thus be notated with two numbers which define its level and place within that level, in set notation. 3 is represented as (1,1) = level 1, place 1. 5 is (2,1) = level 2, place 1. And so on.
If you program computers, you can see how it would be very easy to write a program which selects levels and places to produce music in Just Intonation based on given fundamental frequencies. That's what I did in my first exploratory music employing this theory. The program selects levels and places while a fundamental frequency very slowly climbs from 0 Hz up to the highest frequency the machine would allow, and back down (three times faster). I had plans to compose a middle section with more deliberate, less algorithmic structure, but I never did it. The result is a composition I call Jacob's Ladder, written on the Commodore 64 and recorded with direct output into a digital reverb unit. It's about 40 minutes long, and takes some patience to listen to. To those of you who take the time to listen to it without skipping ahead, thank you in advance for your patience, and I hope you enjoy it.
Best Regards,
Aaron
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