About This Issue

As we are reminded by the contents of Gareth Loy's two-volume Musimathics (MIT Press, 2006–2007), computer music can be viewed as one of the latest developments in the centuries—indeed, millennia—of intellectual endeavor connecting mathematics and music. In particular, the ancient expression of musical intervals as integer ratios has, throughout history and to this day, fascinated musically inclined scientists and scientifically inclined musicians. Towering over most of these was the nineteenth-century physicist Hermann Helmholtz, who finally found, in the interference patterns of overtones in the cochlea, a scientifically satisfactory explanation for consonance and tonal harmony, connected to the integer ratios of just intonation. Twentieth-century modernism, however, largely viewed tuning theory, and particularly the concerns of just intonation, as anachronistic: a movement proclaiming the liberation of dissonance had no need for purer consonances. It remained for digital sound synthesis to give composers a means of transferring Helmholtz's insights to thoroughly nontraditional pitch worlds. At Bell Laboratories, John Pierce and Max Mathews experimented with synthetic spectra having corresponding, nonstandard tuning systems—a line of thinking that led to John Chowning's 1977 composition Stria (documented comprehensively in the preceding issue of this journal and on the present issue's DVD). William Sethares's book Tuning, Timbre, Spectrum, Scale (Springer, 1999) has further explored this terrain. Besides composition with synthetic timbres, though, there are other areas of computer music where tuning theory can crop up—which brings us to the first two articles in the present issue of Computer Music Journal. Each article treats a visible manifestation of tuning: microtonal controllers in the first case, and microtonal notation in the second.

The article by Andrew Milne, William Sethares, and James Plamondon explains, in considerable depth, the mathematics underlying certain principles of microtonal keyboard layout. One of these principles is transpositional invariance. On instruments having this property, geometrical shapes (e.g., chords) can be transposed intact, much like sliding a barre chord along a guitar fretboard. Such keyboards are termed isomorphic. (Non-microtonal examples include the chromatic button accordion and the Janko piano; the most influential microtonal example was the groundbreaking "generalized keyboard" of R. H. M. Bosanquet, whose design is described in appendices to Helmholtz's On the Sensations of Tone as a Physiological Basis for the Theory of Music.) Another, related principle is tuning invariance. This term, introduced here by Milne et al., describes the ability, on an isomorphic keyboard, to preserve geometrical shapes across a continuum of related tuning systems. The authors explain what it means for an interval be "the same" as its tuning changes, and they provide examples of related tuning systems across which musical patterns can be tuning-invariant. Many historically (and even ethnomusicologically) relevant tuning systems can be generated from a single interval (plus the octave), as in the cycle of fifths. As the interval's tuning changes, so does the tuning of all the pitch classes in the tuning system. This means that an isomorphic instrument can feature a continuous controller for dynamically retuning all the instrument's pitches along the continuum that includes these tuning systems, without requiring the performer to learn new fingerings for different systems (or different keys). Although their article is chiefly a mathematical exposition, the authors briefly consider the educational and creative ramifications of such an instrument. Mr. Plamondon's forthcoming controller, the "Thummer," embodies these design principles.

Microtonality presents interesting challenges, not only for the design of new instruments (and, of course, for performers of traditional instruments), but also for notation. Composers have adopted different, often idiosyncratic, conventions for naming the pitches of nonstandard tuning systems and displaying them on the page. For example, Ben Johnston's music, like that of his mentor Harry Partch, employs a microtonal just intonation for which the composer developed his own notation system, which in Mr. Johnston's case is common Western music notation with added accidentals. Not only the unusual accidentals, but also the nonstandard intonation of normally notated notes, pose difficulties for performers. It comes as no surprise that computer technology can help. In "An Automatic Translator for Semantically Encoded Musical Languages," Andreas Stefik, Melissa Stefik, and Mark Curtiss...