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Recently, and I need not cite specific instances
since a segment of the recent literature abounds with them, the scope of
music apparently has been extended to include the interpretation in musical
notation of arithmetical expressions. On this scale, at least, this is
a new role for mathematics in the history of Western music: as a compositional
prescriptive. Although the word “mathematics” has figured
prominently in
this history as both epithet and encomium, beyond the incidental notational use
of number symbols, mathematics and music have been conjoined on the
non-compositional
level: in a Pythagorean, numerological, Quadrivium-oriented association,
or quasi-analytically—as in such a familiar, characteristic case as the
Esthétique
musicale (1876) of Wronski and Durutte. The compositional invocations
of “mathematics” which are current pose so many fundamental
questions that
only one of them will be even stated at this time. This centers around
the apparent relationship of a mathematical expression and its musical
representation inherent in these procedures, which parallels the relation
between a formal theory and an interpretation of it. What, then, determines
the choice of the mathematical expression? Usually, a formal theory is
chosen or constructed in the light of an intended interpretation; what
properties are possessed by an arithmetical progression, for example, which
make it appropriate for interpretation as a metrical or ordinal pitch,
or durational, or dynamic, or timbral, or density determinant? What data
in the field of musical perception determine the rules of correlation?
And what of the possible overpredictability and the assumed separability
of the components of even the atomic musical event? All of these questions
and others must be probed thoroughly before a decision can be even tentatively
reached as to the extent to which such procedures enlarge the domain of
music.
Milton Babbitt
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