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Abstract
Arbor Vitae (2006) for string quartet is James Tenney's last work. The title presents the image of a tree as the central metaphor for the work's harmonic structure, which is similar to the way tree branches emanate from other branches. The harmonic form of Arbor Vitae is a series of related tonalities modulating through a richly populated, extended just intonation pitch space. The piece explores the progression of single tonalities expanding into multiple tonalities. This article examines the inner workings of Arbor Vitae and the musical result. It documents the algorithm Tenney defined to generate the piece and provides a history of the piece.
Acknowledgements
It was an honor to work with Jim Tenney during his completion of Arbor Vitae. While working, Jim, with his characteristic clarity and elegance, made all aspects of this difficult piece understandable. Jim was a vital and nurturing part of the ‘tree of life’. He championed the works of many composers, both predecessors and contemporaries. His contribution to the modern musical repertoire is prodigious. Most personally, he taught and inspired countless younger composers. It is these generous contributions for which I and I know many others thank him deeply.
A special thanks to Lauren Pratt and James Tenney's family, Larry Polansky, Michael Pisaro, Michael Byron, Nick Didkovsky and the Bozzini Quartet, all of whom contributed to the current edition of the score. And thanks to Lauren Pratt, Mark So, Ted Coffey and especially Larry Polansky, all of whom helped with this article.
Notes
[1] A subscript attached to a note name indicates octave placement. B-flat1 denotes 2 octaves and a major second below middle C, which would be denoted as C4.
[2] The pitches of Arbor Vitae are in an 11-limit just-intonation (see Partch (Citation1974) on intonation limits).
[3] Deviation from the nearest pitch in 12-tone equal temperament is expressed in cents (one hundredth of a tempered semitone) using a minus or plus sign and the cents symbol, ¢.
[4] All score examples and manuscripts of Arbor Vitae reprinted by permission.
[5] Note that the 15th partial is approximately 11.73¢ flat from the nearest equal tempered pitch, and the 5th partial is approximately 13.69¢ flat. In Arbor Vitae, cents deviations are only rounded from the harmonic of B-flat from which the pitch is derived.
[6] Throughout the article, variables are named similarly to the actual names of variables that Tenney used.
[7] All time variables are in seconds.
[8] Before I worked with Tenney on Arbor Vitae, he had already defined an algorithm to generate the piece. However, during the time we worked together, Tenney made a few changes to his original algorithm. A change in exrmax was one of these. Originally, the variable spanned from 4 to 6 instead of 4 to 5. This change was made so that roots were chosen more frequently throughout much of the piece.
[9] A close examination of the time-variant probability schemes implemented in Arbor Vitae shows that the possibility of a given branch depends on rdiag, nmult, canReqB, the rtprobs and the multprobs, which are ultimately related to the size and number of prime factors of that branch – that is, a branch's probability is based on the harmonic distance with respect to the given root and the fundamental. Also, the probabilities are continuously being recalculated so that the possibility of roots and branches that have not been chosen for an extended period of time increases. An in-depth analysis of this complex, time-variant probabilistic system and its musical result exceeds the scope of this article.
[10] The division by 2 for bdur shortens the average sounding tone durations throughout the piece. It was another one of the few changes Tenney made from the original algorithm in which bdur = 2 rand .
[11] These piecewise equations were derived from a graph that I transcribed with Tenney after he decided to alter his original one. In Tenney's original graph, which uses a linear pitch scale, some of the breakpoints determining the limits of the pitch range are drawn midway between the B-flats at the vertical position of a tritone, but the vertical axis is labeled with F +2¢ at those vertical positions suggesting that the octaves be split harmonically into a just fifth and a just fourth: breakpoints at B-flat and F +2¢. In the transcribed graph, these breakpoints are drawn at the same vertical position as in the original graph, but are labeled according to their vertical position on a linear pitch scale thus splitting the octave equally into two tritones: breakpoints at B-flat and E. Neither Tenney nor I noticed the discrepancy between the two versions while studying the data outputs or listening to synthesized realizations of the piece. The change affects only two pitch classes in the entire set of available pitch classes of the piece: the E +5¢ (derived from the 363rd partial of B-flat) and the F +2¢ (derived from the 3rd partial of B-flat) may have been assigned to different registers.
[12] For the first 40 seconds, imultset may be empty since the pitch range is less than an octave. In this case, the branch is discarded. This accounts for a silence at the beginning of the piece.