Exploring the Harmonic Series

In writing about Sofia Gubaidulina's "The Light of the End," performed by the San Francisco Symphony last week, I made a comment to the effect that composers who have explored the sounds of natural harmonics often fall into the trap of what I called "aimless wandering;" and I realize that this may deserve some further explanation. The history of our Western music tradition is, among other things, a tradition of normative usage. This is already evident in the manuscripts collected by Edmond de Coussemaker, particularly those in his Scriptorum de Musica Medii aevi. One of my favorite examines is the "Ars Cantus Mensurabilis" of Magistri Franconis (Franco of Cologne), one of the first manuscripts to document the synchronization of the voices of polyphonic music through rhythmic patterns. Rather like a pioneer of knowledge engineering, Franco attempted to infer "rules of practice" from examples of how both he and those around him were "making" polyphonic music around the end of the thirteenth century. By the middle of the twentieth century we had teachers like Arnold Schoenberg departing from the traditional pedagogy of the "rules" of harmony and species counterpoint to try to identify how every note of a composition served a "structural function," based on where it was situated in the score (structure) and how it served what I have called the composition's journey through time (function).

When one departs from the twelve-tone scale (particularly under equal-tempered tuning but, to some extent, also under other tuning systems), one also departs from a massive body of normative usage. What I meant by "aimless wandering" is that many composers interested in exploring natural harmonics can easily fall into the trap of just "noodling" (as one of my own composition teachers put it) up and down the harmonic series. (I offered Glenn Branca's third symphony as an example.) At the very end of "The Light of the End," Gubaidulina has a solo cello do this sort of thing, playing a series of arpeggios that rise from a fundamental, each one ascending to a slightly higher harmonic. To my ears, however, this was less a matter of "noodling" and more an effort to summarize the vocabulary of natural harmonics that had supported so much of the harmonic and contrapuntal structures we had been hearing since the beginning of the piece.

This morning I discovered, quite by accident, that there was a precedent to this "structural function" of arpeggiating through natural harmonics. It occurs at the end of the first movement ("Dialogo") of Benjamin Britten's C major sonata for cello and piano (Opus 65). The notes (no author cited) for my Etcetera CD of this sonata describe this "dialog" as a "wistful, oddly Brahmsian discussion of a rising or falling second, a nervous rhythmic tension being imparted by the piano's scalic thirds." Now I may be reading too much into how I heard this movement, but I came away feeling that the invocation of the harmonic series in the coda of this movement served as a reflection on the origins of those thirds and particularly the seconds. When it came to natural harmonics, we know from the open horn solo that begins and ends his Opus 31 serenade that Britten was too disciplined to give in to "noodling." The few upper harmonics that deviate the most from the equal-tempered scale frame an extremely visceral climax in that solo. Britten may not have been familiar with what Schoenberg was teaching about "structural function;" but this solo offers an excellent demonstration of the principle. I am thus willing to assume, with a relatively high level of confidence, that the arpeggiation of the harmonic series in his later cello sonata is guided by that same principle.

Was Gubaidulina influenced by Britten's precedent? Whether or not she had any direct relationship with Britten, there is a good chance that she was aware of performances by Mstislav Rostropovich, for whom Britten's sonata was composed. Indeed, Rostropovich was still alive and active when Gubaidulina was working on "The Light of the End;" so there is at least a remote possibility that she would have been honoring him by assigning a solo cello passage to the coda of her work and then recognizing another coda that had been composed for him. This is a rather tenuous chain of hypotheses; but, in the context of this blog, it is close enough for rehearsal practices!