An Acoustic Analysis of Col Legno Articulation in Iannis Xenakis's Nomos Alpha

Iannis Xenakis's Nomos Alpha for solo cello (1966) exploits the three principal methods of sound production on a cello string—bowing, plucking, and col legno (hitting the string with the stick of the bow)—to a remarkably extensive degree. Each of these methods of sound production has been the focus of acoustic research. Conceptually, the simplest of these three sound production methods is the plucked string, the physics of which is summarized in Backus (1977) and elsewhere. The spectrum of harmonic partials activated by the action of plucking depends upon the location along the string where plucking occurs. The sinusoidal components of the resulting waveform will reflect, in their respective amplitudes, the degree to which the point of plucking approached or coincided with their own peaks and nodes. The displacement of the string caused by plucking forms a "hump" that propagates along the string; upon reaching one end of the string, the hump is reflected to form a "dip" that travels back in the reverse direction. Through Fourier analysis, this rapid circulation of humps and dips can be reduced to the harmonic partials that comprise, in varying amplitudes, the spectrum of the generated sound.

The idealized image of the waveform produced by bowing a string is that of a sawtooth shape (produced by the bow sticking to and slipping against the string in quick alternation) in which the amplitude of the nth harmonic is that of the fundamental divided by n. In order that no slipping occurs during the long period of "sticking" during a bowed stroke, a minimum bow force is required, which rises in proportion to an increasing bow speed but which tends towards a finite non-zero minimum at very slow bow speeds (Schelleng 1973). The minimum-required bow force can as much as double when the fundamental frequency coincides with the air resonance or the principal wood resonance—commonly known as "wolf tones" (Raman 1918). Using a machine described by Saunders (1937), Schelleng (1973) identified a "dropoff" from perfect inverse frequency harmonic strength between the fifth and tenth partials. Response was found to fall off sooner with less flexible strings or bow pressure closer to the minimum.

The physics of a struck string, however, has been studied more from the perspective of the mechanism of a piano than that of col legno playing on a string instrument. Studies such as Bacon and Bowsher (1978) and Suzuki (1987), for example, focus on issues not entirely relevant to the situation of a col legno attack on a cello. For example, they discuss the hardness of the hammering agent, which in the case of a piano hammer is assumed always to be less hard than a cello bow stick. Certain aspects of sound production are common to both instrumental contexts. When a string is plucked or struck, for example, a peak is created at a point in the wave-action of the string where some of the sound's harmonic components would otherwise have a node. This has the result of nullifying those components, a process known as "harmonic damping." In the case of a bowed string, however, the effects of harmonic damping are circumvented.

Parametric Extremes in Xenakis's Nomos Alpha

An acoustic description of Nomos Alpha is of interest because of the wide variety of extended techniques that the composer juxtaposes in quick succession. In this piece, Xenakis achieves extremes of affect (setting new standards of potential for what might be characterized as "eerieness," aggression, and hyperactivity), extremes of register (exploiting a range of over seven octaves, retuning the lowest string down an octave to C1 = 32.7 Hz and vaulting as high as E8 = 5274 Hz), and extremes of timbre (from the composer's dictum forbidding vibrato completely to the frequent employment of col legno, sul ponticello, and tremolo effects). Given the inclinations to disorder in the variety of effects in Nomos Alpha, it is not difficult to agree with Griffiths (1975), who questions the significance of the logical constructions governing both in-time and out-of-time pitch procedures in the piece. A...