I've been working in just intonation since my freshman year in high school, in 1975 or 76, when the music of Lou Harrison and Harry Partch first registered. The beauty of the pure intervals was the first attraction, but soon, the whole business of organizing tones into scales and systems became an attraction of its own. But a scale or a system is not yet music, but rather material with potential to be used musically. The diagram (or lattice — German Tonnetz — or manifold) describing a tuning system is a static entity, rather like a map from which a useful route has not yet been discovered.
My first lattices were all at right angles, following the models of Martin Vogel and Ben Johnston, with chains of fifths (ratios of 3:2) running horizontally and major thirds (ratios of 5:4) vertically. Adding ratios of higher primes, especially 7, 11, and 13, was an on-going problem, solved with transparencies or other kludges. I literally stumbled into a way of animating my lattices, turning the static information on the lattice into information about successive events in a score, when I came across an article by Shohei Tanaka (1862-1945), a Japanese scientist who wrote a dissertation in Berlin in the late 19th century on Just Intonation, in which he advocated a 53-tone equal temperament for the approximation of a just intonation based on pure thirds and fifths. Tanaka used a lattice in which the lines of fifths and thirds were at a 60-degree angle to one another, and added lines indicating the minor third relationships as well, thus presenting major triads as upward pointing triangles and minor triads as downward. This started to look more like music to me, and quickly, all of the moves that characterize smooth voice leading in tonal music started to appear as simple moves on this hexagonal lattice. The tones connected directly to a single tone were the fifth above and below, the major thirds above and below and the minor thirds above and below. The vertical mirror of a major triad was the minor on the same root, horizontal neighbors sharing the central tone were in a dominant-tonic relationship, and triad described by triangles sharing one face had mediant relationships. At the same time, I started to recognized that the most compact voice leadings I was learning simultaneously in my study of figured bass realization were compact moves on the lattice, while progressions that were more exotic tended to be represented by greater distances on the lattice.
In the spring of '78, I went to San Diego to visit the Partch instrument collection then housed at SDSU under the watch of Danlee Mitchell. I showed Mitchell my attempts at latticing Partch's scales and Mitchell said immediately that I had to visit Erv Wilson, a name I had recognized from the second edition of Partch's Genesis of a Music (Wilson, a professional draughtsman, had done the diagrams for Partch and had also suggested the layout for one of Partch's instruments, the Quadrangularis Reversum). Wilson lived in East LA, and I soon arranged to visit his house, one of the oldest in the city, located at the edge of the arroyo into which the oldest — and now, almostly quaintly small — freeway in the city, the Pasadena, had been dug. Wilson was in his front garden when I arrived, sorting bags of corn hybrids collected from his ranch in the mountains of Chihuahua, Mexico. He looked through my collection of lattices and scores and at an adapted hawai'ian guitar I had brought along and immediately took me in to his dining room, where he sat me down at the table and started to teach some more economical, elegant, and tonally suggestive ways of mapping tones on paper, or in the three dimensional models of dowels and spheres suspended throughout his house. It was a total revelation to see how Wilson could accomodate far more than the lousy two or three dimensions I could capture on paper, as he casually drew examples with four, five, even eight dimensions represented. But these examples were not Augenmusik, but acoustically immediate and vivid, as Wilson's house was filled with re-fretted guitars, a collection of flutes, and a number of keyboard percussion instruments, of metals, wood, and bamboo, each apparently to a different tuning system, with alternative keyboard layouts of Wilson's own exquisite and ergonomic designs. (Wilson's ear is amazing: once, while using his scalatron out in his garage to tune up some aluminum tubing, I was periodically interrupted by his shouting out the ratios of each new tube "256/243! 16/15!, 13/12!").
My use of collections of pitches related by just intonation remained rather unadventurous for some time, as I was essentially replicating the most familar moves in tonal music. This was hard to square with my attraction to more experimental music. Working with La Monte Young provided one decisive step in this direction. The harmonic motion in La Monte's pieces is very much understood as motion around a lattice, with formal sections of works restricted to subsets of the total collection of pitches (incidentally Xenakis is up to something similar in Herma) , but La Monte's avoidance of ratios involving the number five whether as tones used directly or as combination tones creates very different tonal environments, and a number of works allowed performers to improvise within the constraints of rules which effectively constitute the voice leading rules in these new environments. These rules are related, also, to the "cuing" pieces of Christian Wolff, which have also been very important to me. I made a number of pieces in which performers moved by a small set of rules through pitch materials, including a set of just intonation Mazes and equal tempered pieces, the best and most-played of which was Multnomah Riffs of 1981. In Multnomah Riffs, the players worked through a score of four repeated measures or frames, from which pitches could be selected, and played at any time within the frame, omitting or repeating any tone so long as the sequence of tones was preserved. My friend Jonathan Segel, known perhaps best as a member of the band Camper Van Beethoven, contributed an ostinato for celesta to keep track of the frames.
A second decisive step was provided by a small piece by the mathematician and composer David Feldman called Going Places, for two violins. Going Places is essentially a canon for the two fiddles, chasing each other in a random walk (run, actually) across an open-ended lattice of major and minor thirds. Encouraged by David's example, I soon wrote a number of pieces involving random walks across both open-ended and closed lattices of various dimensions, and eventually started trying different sets of constraints and changing the lattice, sometimes quite dramatically, in the course of a piece. These lattice moves were also related to Lou Harrison's "interval controls" (Elliot Carter, famously, uses a related technique).
I would later encounter works by Yuji Takahashi and James Tenney that use similar techniques; I am particularly fond of Tenney's principle, in Changes for six harps, of establishing a tonic, jumping to someplace tonally distant from the tonic, and returning by the simplest possible root motion, often a descending series of fifths, thus recapitulating very familar harmonic territory. As my own strategies for working with tonal motion had generally been symmetrical, even dualistic, Tenney's example was an important one, in that it better resembled the assymetry of real tonal musics, for example in the German tradition in which I can go to IV or V and IV to I or V but V can only go to I. (For some great counter examples, with their own assymetries, from non-German traditions, consider the V-IV of the blues, or the extended subdominant chains in Berlioz's Marche Troyenne.)
In Charles Seeger's breathtakingly prescient little treatise on Dissonant Counterpoint, he introduces the principle that the structure of an existing tonal system can be maintained, but the precise content be changed, in his case changing the hierarchy of consonant and dissonant intervals in a contrapuntal texture. Following Seeger's example, in these unfamilar tonal environments, whether in just intonation or mapped to an equal temperament, there is plenty of opportunity to experiment with both the rules and material hierarchies, while still preserve qualities that are clearly tonal. (The musical and mental agility on display in this and other examples of Seeger's writings has provided a decisive spark for experimental music for several generations past and will continue to provoke for several generations to come. If anyone asks, I'm a card-carrying Seegerite.)
(I haven't let go altogether of my youthful dualism: when I come across the so-called half-diminished seventh chord in classical works (Bach and Mozart, especially), it still makes a lot of sense to think of it as a subharmonic chord, the exact inversion of a dominant seventh chord, and I think that it is sufficient that this chord is arrived at melodically, by voice leading and counterpoint (the Tristan "chord" is another example), without having to argue about the physical and psychophysical status of the subharmonic.)
At the moment, I am mostly interested in the relationship between the group of tones in play locally and globally, and especially our sense that a tone belongs or doesn't belong to a moment in a piece or a whole piece. Erv Wilson's ideas about moments of symmetry come into play here, in that subsets of a tune system can exhibit similar properties of coherence. Classical diatonic tonality, with its matruschka-doll construction of chords, diatonic scales, and chromatic scales, is one example. Recently David Feldman demonstrated that there are only a small number of distinct and connected graphs of a diatonic scale onto a 3,5 lattice. (I'll let David publish his own result, but the number should be easy enough for you to figure out on your own. He has also figured out the distinct graphs for sets of 12 pitches on this and on other lattices, a calculation which is not so easy). This was very interesting to me because I have made a number of pieces that could be considered "pan-diatonic" and Feldman's lattices suggested something more to the point, in that each graph, while preserving the letter names, had an alternative harmonic profile, much like individual moments in real tonal music. Sometimes an A is the fifth above a D, sometimes it's the third above F, but not necessarily both at the same time. The graphs could be realized in just intonation, or mapped to an equal temperament, or to an unequal temperament designed to best represent a particular graph or set of graphs. Moreover, these graphs suggest particular melodic paths, rows, if you will, with built-in harmonic properties. This line of working strikes me as very rich and, possibly, very musical.
(from a talk in California, January, 2009).