I do not think it would be an exaggeration to say that much of the history of Western music has been involved with the evolution of tuning systems to deal with the tension between the "naturally occurring" intervals of the harmonic series and the need for a more manageable collection of pitches and intervals (which is not necessarily a subset of the "natural" set). Joseph Yasser's Theory of Evolving Tonality begins his own model with the pentatonic scale (as in the pitches of the black keys on a piano keyboard), demonstrating how its pitches emerge from the first five perfect fifths in the harmonic series (if we begin on C, that would be C-G-D-A-E). In other words our first effort at a "scale" emerged from a sequence of 2:3 ratios.
This is fine as far as it goes, but it does not go very far. Most importantly, it overlooks the emergence of the tonic-dominant relation that has dominated almost all of Western harmonic theory. That relation is the inverse of the 2:3 ratio, which, mathematically, would be a 3:4 ratio; but that ratio never emerges as you keep piling on more perfect fifths. From a mathematical point of view, the problem is that there is no pair of non-zero integers, n and m, such that 3n = 2m. In slightly more mystical language, you can always "depart" from your fundamental pitch to a fifth; but, if you keep going on that path, you can never return to that fundamental. To the extent that much of musical structure can be approached through the metaphor of a journey, the physical fact that you can never return is problematic.
By the eighteenth century those who made instruments, particularly keyboard instruments, came to deal with this problem through equal-tempered tuning. The octave would be divided into equal intervals, each of which was based on the same ratio of 1:2(1/12) (the twelfth root of two). Mathematically, this meant that a "journey" of twelve of these intervals would take you back to the simplest harmonic ratio of a pitch to its octave, 1:2. It also meant that all of the natural intervals of the harmonic series based on integer ratios (i.e. rational numbers) could only be approximated by the irrational ratios of 1:2(1/12), 1:2(2/12), 1:2(3/12), 1:2(4/12), 1:2(5/12), 1:2(6/12), 1:2(7/12), 1:2(8/12), 1:2(9/12), 1:2(10/12), and 1:2(11/12). Since the adoption of equal-tempered tuning, music history has seen the efforts of several composers to return to the natural harmonic series for both harmony and counterpoint.
This week Kurt Masur introduced to the San Francisco Symphony and its audiences one of the most recent of these efforts, "The Light of the End," which Sofia Gubaidulina completed in 2003. Masur conducted the premier of this work with the Boston Symphony Orchestra; so he was, in may ways, the best person to bring it now to San Francisco. In his program notes for this work, Thomas May spoke of the intervals of the harmonic series as "deviations from conventional tuning," meaning that we perceive them as "out of tune." This is, to a great extent, true and is even reinforced by the psychological theory of categorical perception. However, since the rise of electronic and other experimental musics following the Second World War, we have been exposed to a wide variety of synthesized sounds; and many of those sounds have returned to their roots (pun somewhat intended) in the harmonic series. We are thus less susceptible to those out-of-tune judgments induced by categorical perception than we were around fifty years ago.
From this new vantage point, we can listen to Gubaidulina's score for what it is, an exploration of the dialectic between the traditional sonorities of equal-tempered tuning and those of musical instruments' natural harmonics. In Gubaidulina's personal aesthetic there is pain in this dialectical opposition; and there is no doubt that "The Light of the End" reveals that pain through considerable tension in both harmonic and contrapuntal constructs. However, listening to it is also a marvelous exploratory experience. It invites the same sense of wonder that John Cage had summoned in his most famous experiment based on the hypothesis that one could make a composition of those sounds that occurred naturally while one sat in silence for four minutes and thirty-three seconds. However, while Cage's innovation reflected a let-it-be philosophy, Gubaidulina's is one of meticulous construction with an extremely broad palette of sonorities. It is also well informed by what Arnold Schoenberg called the "structural functions" of harmony and counterpoint. This is important because much of the pioneering work with such "deviant" intervals, whether in La Monte Young's "Well-Tuned Piano" performances or Glenn Branca's third ("Gloria") symphony (subtitled "Music for the first 127 intervals of the harmonic series"), has been little more than the sort of "aimless wandering" I recently attributed to Giuseppe Verdi. "The Light of the End" is very much a journey along which both harmony and counterpoint are steadily at work to move the mind behind the ear in a forward direction.
Needless to say, this is a process that is as complex as it is tense. So I can account for little more than the superficial impressions of a first immediate exposure. I suspect that I would have done well to attend all three performances this week, but I hope that it will not be a long wait before I have another opportunity to hear this piece. It has certainly benefitted from having Masur as a champion, but I hope that other conductors will soon follow Masur's lead. (I could make a pun about picking up his baton, but Masur now conducts without one.)
One might think that Anton Bruckner's fourth symphony ("Romantic" in E-flat major) would be an odd work to pair with "The Light of the End." However, in its own way this particular symphony is its own "Gloria" (as Branca had put it) to the 2:3 and 3:4 harmonic ratios. Horns figure significantly throughout the entire symphony; and much of their language is kept at a relatively fundamental intervallic level, at least until the full brass section erupts in the harmonic richness of the third movement's "hunting scene." This is not a particularly sophisticated piece of music; but, as I have written about Olivier Messiaen, the music is a product of a highly devout sense of faith. Thus, matters such as those "structural functions" of harmony and counterpoint (not to mention overall form) are secondary to Bruckner's invocation of a devotional spirit. Successful performance thus depends on being able to summon that sense of devotion without succumbing to self-indulgence. Masur achieved this through a sure sense of pace. Bruckner may have preferred meditation to journey; but Masur knew how to bring out the journey in this symphony and advance us through its four movements, even when the discourse of those movements was at its most repetitive.
As a result the entire evening amounted to a sustained meditation on natural phenomena. This made for a rare experience of aesthetic explorations grounded in philosophical assurances. This is hardly the stuff for a steady diet of concert-going; but, when it comes around, it becomes far more memorable than we can anticipate.