Like just about everyone of my generation in the United States and Europe, I grew up thinking that music was made from the twelve pitches in the chromatic scale best known through the white and black keys on a conventional piano keyboard. Thus, when, for my senior thesis, I implemented software that could play music in six independent voices, tuning was simply a matter of representing the frequency of the lowest possible note and then doing successive multiplications by the twelfth root of two for all of the other notes. (To be fair, this was a time when representation of irrational numbers required so-called “floating point” representation, which was not particularly precise. Thus, what I really did was compute the twelve pitches of a single octave and then multiply those numbers by two over and over to account for the higher octaves.)
My advisor, Marvin Minsky, then pointed out that I could augment this representation with enough of the constructs of a programming language to achieve Turing completeness. In other words I would have a programming language as powerful as any other programming language that had these extra “instructions” that sounded notes at given times for given durations. The question of how expressive that language could be from a musical point of view, rather than a computational one, became the basis for my doctoral research, conducted, again, under Minsky’s supervision. (In some ways this could be seen as a predecessor of MIDI, but Minsky was never one to get into battles over prior art!)
However, if I was going to study musical expressiveness, I knew that I could not base my research on my own feeble efforts as a composer. Fortunately, I came to know the composer Ezra Sims, who initially did not want to have anything to do with my ideas. This was because he had been composing with both quarter tones and third tones and basically wanted nothing to do with the traditional chromatic scale.
My response came immediately: What if I divided the semitone into six equal subintervals? Three of them would make a quarter tone and two of them would make a third tone. This was no big deal for my software. 72 successive multiplications of the 72nd root of two was no more difficult than twelve multiplications by the twelfth root of two. Furthermore, in the programming language itself, the “letters” of the pitches translated into integers; and the accidentals involved addition or subtraction. All that was necessary was an additional set of symbols to handle different sizes of additions and subtractions.
I quickly discovered that Sims’ interest in these microtones arose from his desire to achieve closer approximations to the pitches in the natural overtone series. With that objective in mind, I started doing some calculations of my own. If the octave was divided into 72 equal parts, how close could one get to the integer ratios of those overtones? Where the perfect fifth was concerned (the 3:2 ratio), the answer was, “Very close indeed” (within two cents). (A cent is the ratio of the 1200th root of two, meaning that 100 cents is the interval of a semitone, hence the name.)
Furthermore, the actual note was the same as the one on the conventional piano keyboard dividing the octave into only twelve equal parts. Where the major third (the 5:4 ratio) is concerned, however, the difference is 13.7 cents, meaning that you get a better approximation by knocking down the third step of the “white key” major scale by just one of those intervals defined by the 72nd root of two. As you then ascend to the higher overtones, things just get worse.
There have been, of course, composers that deliberately exploited how worse they got. My favorite example is in the horn solo that begins and concludes Benjamin Britten’s Opus 31 serenade, which was written to be played on a “natural” instrument (i.e. without valves). There are a lot of perfect fifths from the tonic to the dominant (as is so often the case in horn music); but, near the end of the solo, the fifth is approached from above by a note that is disturbingly different from either a semitone or a whole tone. That is because it is the thirteenth harmonic.
(The last time the San Francisco Symphony played this piece, one of my colleagues wanted to know why Robert Ward was having trouble with his solo. I replied, “He wasn’t having trouble. He was playing it the way Britten wanted!”)
Sims’ influence on my interest in the more remote overtones came back to me recently while I have been reading the new biography of Lou Harrison, Lou Harrison: American Musical Maverick by Bill Alves and Brett Campbell. I became more aware of Harrison’s own interest in just intonation tuning systems, in which all intervals are defined by integer ratios. I had known about this work ever since I had added the Etcetera Records release of Harrison’s music for guitar and percussion, which came out in 1990. (This album was a valuable resource for keeping me connected to my personal tastes in music during my years doing multimedia research in Singapore.)
The album indicated that guitarist John Schneider was playing an instrument called the “Well-Tempered Guitar.” It offered the advantage of interchangeable fingerboards invented by Tom Stone. One could thus design and use fret systems for different approaches to tuning. The booklet illustrated two of these fret systems, one for an approach to just intonation that Harrison had been exploring and another for “Pythagorean” tuning (just intonation in which all ratios involve only two and three).
Long after my return from Singapore, in 2001, the Other Minds festival commissioned a new guitar piece from Harrison. The composer was looking for an instrument that would have stronger sonorities than the guitars for which he had previously written pieces. The result was a new approach to the design of a metal-body tricone resonator guitar that become known as the National Steel guitar. The work that Harrison wrote was “Scenes from Nek Chand,” which would turn out to be his final composition. When Schneider released a recording on Mode collecting Harrison’s five suites for guitar, he was using both the well-tempered guitar of his Etcetera recording and the more recent National Steel instrument.
Since that time I have discovered that I have been putting a generous amount of time into listening to these alternative tunings, not to mention exploring the mathematics behind the interval ratios. I was reminded of both scientific and anecdotal evidence about listening. On the scientific side there is a phenomenon known as “categorical perception.” This is the tendency of mind to classify a pitch according to a familiar frame of reference (such as an equal-tempered piano), even if the pitch differs by some significant amount. (How much is “significant” is argued among those interested in this theory with great vigor.) I tend to believe that this is a phenomenon that does, indeed, work; but it only works until it doesn’t! For example, when Britten gradually brings in higher harmonics in his Opus 31, we tend to hear them as “piano” pitches until the thirteenth harmonic confronts us as something that “just does not fit.”
In contrast when I went to Israel for my first teaching job in 1971, I had with me the Columbia recording of the music of Harry Partch. Like Harrison, Partch had a great interest in just intonation; and he had designed his own scale system to accommodate it. I remember playing that recording for the wife of one of my colleagues, who had extensive training as a musician; and she just couldn’t take it. It was as if Partch’s music was an affront to all of the ear training skills that she had honed so well!
Where Harrison’s music is concerned, I find that my own listening comes down closer to the categorical perception side of these two extreme approaches. One reason for this may be that, at least where his guitar music is concerned, Harrison is more interested in melody than in harmony. (If you want to experience the impact of his tuning systems on harmony, his works for American Gamelan make stronger cases. In those performances one can experience the vibrancy of intervals that reveal their “ancestry” in the harmonic series.)
Nevertheless, I would not dismiss all this attention to alternative guitar designs as having been more trouble than they were worth. Harrison was as interested in playing music as he was in writing it. I do not know to what extent Harrison used alternative instrument designs to allow him to experiment with what could be expressed through just intonation. However, I know from the Alves-Campbell book that he retuned at least one of his own keyboard instruments. It would thus appear that, in his own work, there was a strong link between thinking about what he wanted and experiencing what he could do. In retrospect, I would say that Sims used my software with that same link in mind.
Perhaps this tells us something about how things tended to go off the rails in those early “experimental” days following the Second World War. This was a time when getting the technology to do anything constituted a major achievement, particularly when that “anything” was a product of “batch processing,” rather than real-time interaction. Those who became too wrapped up in technology lost touch with the premise that musicians needed to “play around” with their materials, just as their predecessors had done for millennia. These days, when I encounter musicians improvising with software, I figure that, even if I am not that convinced by the results, I am still glad that they have that capacity to “play around” in search of what they will do next.