Crossing Fields

Since 1977, when I first took a lesson with the tuning theorist Ervin Wilson, I've found it useful to think about pitch relationships in terms of a lattice (or, more precisely, a manifold, but this is music theory, so we needn't be too fussy) in which classes of intervals -- octaves, fifths, major third, etc. -- are each assigned their own axis and pitches are located at discreet points in the space described. (The tradition of thinking about pitches in this way originates with the German theorist Hugo Riemann, with an important precedent in the Dutch scientist Huygens; it was further developed by Shohe Tanaka, Adrian Fokker, Walter O'Connell, Wilson, James Tenney and more recently, in the tuning community and, to some extent, among, "neo-Riemannian" music theorists).

The whole point of such spatial representations of pitches is that spatial proximity represents one form of audible proximity, or harmonic distance, and this is important in real music. With pitches, there is an additional form of proximity, and that is in terms of absolute distance in frequency. The two forms of proximity interact in subtle and musically important ways. Two tones an octave apart will be usually (= played on instruments with harmonic spectra in a middle register) heard as more similar to one another than two tones a major seventh apart. Although the major seventh is a smaller interval, the octave is a less complex interval described by a very simple harmonic relationship, thus trumping the major seventh.

While one can devise metrics for these two ways of measuring intervals and from the metrics derive some formulae for comparing intervals, in real music this is an extremely delicate business, subject to a large number of variables, and ultimately, the way in which a composer decides to wander about a pitch lattice may mostly be a function of personal taste, habit, or even chance. Nevertheless, I find it an enormously useful way of managing pitches and intervals.

The same structure may also be usefully applied to dimensions other than pitch. It is possible, for example, to view musical tempi as described by the duration of a basic pulse. Such pulses can be assigned places on an n-dimensional lattice in which each dimension describes a distinct form of rational relationship between tempi. An X lattice corresponds to the powers-of-two axis in harmonic space -- each discreet step on the lattice is a doubling or halving in duration. Thus, one moves from eighths to quarters to halves to wholes. Perpendicular to this is an axis with powers of three -- moving in one direction triples, in the other divides in thirds. If the two axes meet at a quarter note, then this second axis continues in one direction with a dotted half, and in the other with triplet eighths. In the space describe by the two axes, one can soon locate all the pulses described by augmentation dots or as parts of triplets. In this way, the simplest rational relationships between tempi are represented graphically, and one can move among them via a handful of common counting units. Thus you can get a sweet little field of tempi through which one may securely modulate -- i.e. mm 60:80:90:120 -- by notating only with quarters, dotted eighths, triplet quarters, and eighth notes.* If one adds to this "harmonic" set of rational tempo relationships the possibility of speeding up or slowing down the basic pulse in absolute terms, this is equivalent to the second type of pitch relationship described above and, as in the pitch dimension, the range of possibilities for subtle yet recognizable relationships expands tremendously.
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* If you're interested in pursuing this further, the two obvious English-language sources to turn to are Morley's Plain & Easy Introduction to Practical Music and Cowell's New Musical Resources, two books no composer should be without.