TITLE: String Quartet #1Edgerton_StringQuartetTitlePage
CAT# (YEAR COMPOSED): 64 (2002)
INSTRUMENTATION: string quartet
PAGES: 141

PREMIERE BY: first movement by Kairos Quartet (Germany) on 4 April 2004, 18. Tage für Neue Musik Rottenburg (am Neckar) A Quattro! – Festival für Streichquartett Kulturverein Zehntscheuer E.V. Rottenburg am Neckar.
NOTE: The influence of nonlinear phenomena and the scaling of multidimensional phase space served as generating principles for musical composition in this First String Quartet.

movement one

movement two

movement three

movement four

movement five

1st Movement

„Korpereinsatz mit Sogwirkung“ (body movement with staying power)
“… Edgerton looked after performance practice and tradition of the string quartet, with only a few pauses were intricate figurations with new means of articulation reorganized – such as finely differentiated rules for bow angle and the intensity of the finger pressure. After the first movement I was curious to see how Edgerton applies the findings of his research when referencing Giacinto Scelsi (third movement) and Helmut Lachenmann (fifth Movement). “

Michael Eidenbenz, RADIOMAGAZIN 25/2004 (SWIZERLAND, JUNE 2004)
“… Edgerton, in his 1 Stringquartet writes a complex music that one can either simply intuitively understand, or can parse out a deeper understanding from the composers’ instructions. … is a recent heir to a venerable tradition, in which the last word has not yet been spoken.”
(Michael Eidenbenz, translated by M. Edgerton)

“… Edgerton’s Quartet #1 … virtuosic special effects and calculated dramaturgy.”
Werner M. Grimmel (translated by M. Edgerton)

Michael Edgerton: Streichquartett Nr. 1,1. Satz (2002)
” … The many-layered nuanced colors of the string sound may possibly be associated with the diffusion of light through a prism, to awaken the impression of the existence of a larger string ensemble.”
Christoph Sramek (translated by M. Edgerton)

The following is from the article:
Edgerton, M.E., Neubauer, J., Herzel, H. (2003). Nonlinear Phenomena in Contemporary Musical Composition and Performance. Perspectives of New Music, 41:2, 30-65.

Example 14 shows an excerpt from movement one (Edgerton, 2002b), in which multiple variables are placed within a scalable framework and simultaneously shifted. These variables include bow rotation, bow angle, bow portion, bow length, effleure (left hand pressure), bow speed, pitch, rhythm, intensity, placement, bow attack, and release.

example 14

Formally, movement one references sonata form (without its tonal implications) and pre-tonal contrapuntal complexes. However, polyphony is not only an organizing principle between instruments, but within each instrument as well. This internal polyphony occurs through the selection of the prominent variables of sound production that may be shifted within a scalable environment and that carry a robustness to affect the acoustic output. Particular gestural quantities or compositional procedures, such as transposition, retrograde, and inversion, that act upon a temporal field may be applied to other parameters, such as the retrograde form of bow speed applied in stretto to bow placement. The result of such maneuvers is to increase the weighting of redundancy across an increasingly complex and potentially disorganized sound mass. To be clear, this approach extends beyond traditional notions of timbre, as it suggests that multiple components of sound production may shift differently over time. This activity will produce not only bifurcations to nonlinear states, but will lend itself to an enlarged perceptual bandwidth of the vertical grouping characteristics within extra-complex and temporally shifting sonorities of the formerly monophonic output (Edgerton and Auhagen, 2002). Further, tempi relations are formed from the fractal dimension 1.2618 of the Koch curve, in which an infinite length fits within a finite area (such as may be found in a coastal outline).

Movement two is influenced by the musical riddle of the composition UT, RE, MI, FA, SOL, LA by the renaissance composer John Bull, in which the procedure of transcribing letters to integers serves as a formal, generative process (Verkade, 2002). Referencing the typical slow second movement of the European classic/romantic era, this movement uses a special tuning designed to capitalize on closeness of frequency correspondence during the production of higher integer natural harmonics. As before, multiple variables outside of rhythm/pitch are shifted in nonidiomatic ways.

Movement three suggests the concept of divergence through the decoupling of bow speed and portion with tempi. In practice this is a nontrivial situation, as string players are taught to correspond bow speed with tempo. Even slight deviations from this norm often result in “nonmusical” performance, and in this string quartet, present significant psycho-physiological hurdles to navigate. In order to heighten this decoupling explicitly, easily identifiable gestures were chosen that are to be performed at extremely quick speed.

The fourth movement is inspired metaphorically by the phase space portrait of a strange attractor (a behavior that is stable and non-periodic, that stays within a definable phase space, infinitely deep, yet never quite repeating). The process involved abstracting and subsequently unfolding a few geometrical shapes featuring the characteristics of a strange attractor within a two-dimensional graph. Then, the data from these unfolded shapes were applied to many of the multidimensional parameters.

The fifth movement reinterprets the first four movements and introduces new material dealing primarily with low-amplitude inharmonic sonorities. Globally, the fifth movement is conceptually governed by the rough structure of the first crude computer printouts of the Mandelbrot set, in which a central active region is joined by thin, filament-like strands to far-reaching islands as resolution is increased (see Example 15).

example 15

The islands of visual activity seemed to provide an appropriate landscape in which to reassemble the sonically active material from the first four movements. As Mandelbrot and other mathematicians found, when the quality of computation improved, each level of magnification roughly followed the principle of self-similarity at different scales, but showed that none of the successive molecules exactly matched one another— there were always new species appearing (Gleick, 1988). Musically, this rough resemblance, implicating a dynamic process of change and a broadening of the fractal concept, served as a guide to the reinterpretation of the materials from the first four movements. Specifically, the elements reappeared along a continuum from roughly similar to dramatically altered in gesture, technique, and expression. Secondly, the islands of visual activity are not spatially isolated, as the rough visualization (Example 15) suggests, but rather are connected by a delicate web connected to the main body. The suggestions of a fine, delicate and unseen web served the intention of exploring primarily soft, inharmonic sections of activity. Therefore, the previous multidimensional conception was applied to sound production extending beyond the normal nut-tobridge bowed/plucked string sound. These inharmonic sonorities were somewhat strictly governed by numbers from the logistic equation in the following way.

The logistic equation, a model of population change, was applied. This recursive quadratic function generates sequences of numbers xn by the relation:


When the control parameter r is less than 1, all iterated values decay to zero (meaning that a particular population becomes extinct), which is not much use for the musical intentions of this work. Then as the control parameter increases between r = 1 and r = 3, iterations converge to a single value, which is still not much use. Between r = 3.0 and r∞  = 3.5699…, a series of bifurcations occur that are highly sensitive to the value of the control parameter, forming a period-doubling cascade. These results are somewhat predictable and offer little to stimulate perceptually interesting artistic sequences, and are also not much use creatively. However, between r∞ ≈ 3.57 and r = 4.0 the iterated values produce a complex behavior, nearly, though not entirely, chaotic for all values of r, as windows of periodicity occur (Williams, 1997). It was from these regions that the numbers from a single iterated r-value were chosen to strictly determine the temporal and production characteristics of the inharmonic sonorities.

example 16

The resultant values were assigned to classes of sound production that identified degree of pitchedness (pitched, pitched/nonpitched, and nonpitched), placement of sound generator (string above nut, on scroll, etc.), type of sound generator (bow, hand, etc.), manner of sound generation (rub, pluck, damp, etc.), and an additional class of production variable (angle, rotation, etc.). Additionally, each value was assigned a temporal characteristic (duration, placement within a time series, etc.). To each temporal unit was added a second numeric sequence in order to offer a greater choice when selecting the internal variables within each sound production class identified above. In this way, each temporal unit featured fifteen separate control variables over the last five sound production classes. The final choice of which variables were selected from the available choices was accomplished by intuition. All of the elements were scaled within minimal to maximal values. The intention of utilizing the logistic equation was to focus on the deeply-hidden windows of periodicity within chaos. Therefore it was pleasing to see how ratios of redundancy versus novelty were compositionally well balanced. This suggests that further examination of the hidden structures within complexity is warranted (see Examples 16 and 17).

example 17

In conclusion, nonlinear phenomena have been reported in many diverse disciplines, including physics, health sciences, engineering, literature, neurology, geology and music. As this paper has shown, nonlinear phenomena have robust potential for both performance and composition. It has been shown that nonlinear phenomena occur in the sound production of extra-normal extended techniques for both voices and musical instruments. Further, it is clear that many of the tools used to describe nonlinear phenomena may offer appropriate philosophical and/ or constructional processes in the service of contemporary musical composition.


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