12-10-2012, 03:32 PM
Mathematics and the Twelve-Tone System: Past, Present, and Future
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The first major encounter of non-trivial mathematics and non-trivial music was in the
conception and development of the twelve-tone system from the 1920s to the present.
Although the twelve-tone system was formulated by Arnold Schoenberg, it was Milton
Babbitt whose ample but non-professional background in mathematics made it possible
for him to identify the links between the music of the second-Viennese school and a
formal treatment of the system. This paper sketches a rational reconstruction of the
twelve-tone system as composers and researchers applied mathematical terms, concepts,
and tools to the composition and analysis of serial music. It also identifies some of the
major trends in twelve-tone topics that have led up to the present, provides a brief
account of our present mathematical knowledge of the system, and suggests some future
directions for research, including open questions and unproven conjectures.
What is the twelve-tone system? For this purposes of this paper it is described as the
musical use of ordered sets of pitch-classes in the context of the twelve-pitch-class
universe (or aggregate) under specified transformations that preserve intervals or other
features of ordered-sets or partitions of the aggregate. Thus the objects treated by the
twelve-tone system are not only rows, but any series or cycle of any number of pitchclasses,
with or without repetition or duplication, or multi-dimensional constructs such as
arrays and networks.
The earliest research into the nature of the system is captured by Schoenberg’s phrase,
“The unity of musical space,” Undoubtedly, Schoenberg understood this space as the
combination of pitch and time. But aside from the basic transformations of the row, R
and I, plus RI for closure (and P as the identity), the details of this space are quite vague.
Similar lapses of clarity fostered misconceptions about the aural reality of the system on
one hand, and the justification of its application to structuring other so-called parameters
of music on the other. Future research would correct this ambiguity, differentiating it into
different musical spaces and entities. First, the system itself was shown to preserve
musical properties such as interval and interval-class (Babbitt, 1960); Second, Babbitt
(1962) showed that rows were not chosen by Schoenberg and his students capriciously,
but would depend on features such as shared ordered and unordered sets.
Early pre-mathematical research also concerned itself with the relations of the system to
tonality. But once again a lack of clarity that conflated reference, quotation, suggestion,
analogy, and instantiation made the question impossible to define, much less answer.
This obsession with tonality retarded work on the vertical or harmonic combination of
rows in counterpoint. It took until the 1970s for research on the nature of musical systems
and their models helped make the tonality issue manageable. Understanding tonality as
recursive but level invariant made it possible to conceive of the multiple order number
function rows (Batstone, 1972).
Other developments included ways to extend the relationships among pitch-classes to
time and other musical dimensions (Stockhausen, 1959 and Babbitt, 1962) and the
construction of networks of pitches or other musical entities connected by succession,
intersection or transformation. two-partition graphs (Morris, 1987), transformation
networks (Lewin, 1987) and some types of compositional spaces (Morris, 1995a).
Today, the field is supported by an application of mathematical group theory, where
various kinds of groups act on pcs, sets, arrays, etc. The most important group is the
affine group including the Tn, and Mm operations either in Z12 or simply Z. Other
subgroups of the background group S12 have been used to relate musical entities; these
fall into two categories; the so-called context sensitive groups some of which are simplytransitive,
and groups that are normalized by operations in the affine group.