“I wonder if Xenakis is that far away from what I'm trying to do - not of course in what he says, but in what he achieves.” - John Cage, in an interview with Daniel Charles1
The apparent anarchy of Cage's chance music seems the antithesis to the rigorously mathematical constructions of Xenakis. Why did Cage volunteer this correlation between their work?
Earlier in his interview with Charles, Cage hinted that dichotomies such as the one between disorder and exactitude were not necessarily mutually exclusive. “You can't choose, because everything comes at once - there is temporal simultaneity.”2 This essay explores the very notion of temporal simultaneity, and examines Xenakis' and Cage's respective use of chance operations in creating music in which “everything comes at once.”
This essay analyzes chance operations used in Xenakis' Pithoprakta (1956) and Cage's Fontana Mix (1958). The analysis focuses first on the stochastic calculations used in mm. 53-60 of Pithoprakta. The next section examines the aleatoric methods Cage used to determine event durations in his tape piece Fontana Mix. Both composers used chance to create sound spaces in which synchronicity was not a requirement for simultaneity. Their common interest in extending the idea of simultaneity reveals itself in the form of similar methods which in turn yield comparable results.
Chance music is naturally associated with a relinquishing of compositional control. Paradoxically, Xenakis argued that in certain cases chance operations could represent a composer's intentions more closely than any other compositional process.
In measures 53 through 60 of Pithoprakta, Xenakis coupled Bernoulli's Law of Large Numbers with Boltzmann's Kinetic Gas Theory. The Law of Large Numbers states that the aggregate of a large number of events can be perceived as an event in its own right. Kinetic Gas Theory states that the temperature of air can be understood to be the mean velocity of its composite molecules. The cloud of pizzicato glissandi in this section of Pithoprakta represents a snapshot of a large number of discrete molecular velocities in a body of air standing at a constant temperature.
While individual molecules in a body of air may range from very fast to very slow, we perceive the average effect of thousands of particles bombarding us at once. Bernoulli's Law of Large Numbers tempers the randomness of single instances, allowing heat sensors in our skin to register a gestalt interpretation of an incomprehensibly complex phenomenon.
By applying Boltzmann's formula [ex. 1 below], Xenakis was able to calculate the probability that a given molecule would have a given velocity at a given instant. He determined the probabilities for 58 distinct velocities, and used a Gaussian distribution [ex. 2] to generate 1,148 likely speeds of molecules in a gas at a constant temperature.
a = temperature of the gas
v = molecular speed
f(v) = the probability a given molecule has a speed of v
The Kinetic Gas formula allowed Xenakis to statistically construct a characteristic distribution of molecular velocities in air - a distribution that could have existed in the real world. The precise measurement of actual molecules in a real world body of air, on the other hand, would have been impossible due to indeterminacy. Far from relinquishing compositional control, Xenakis used the chance operations of stochastics to represent as closely as possible a phenomenon which defied accurate measurement.
Each of the molecular speeds was translated into a pizzicato glissando event, with the slope of the glissando proportional to the corresponding particle's speed. As a group, these sound events represent a likely distribution of molecular speeds in a gas at a given instant. This large number of glissandi, however, is not played in synchrony. Over a period of 18.5 seconds, 46 independent string parts play an average of 25 notes each. How could Xenakis have meant the 1,148 sound events to be taken as a group when only 46 could be played at a time?
Statistically speaking, forty-six is not a large number of instances. To demonstrate this, the gas analogy will be taken one step further. Assuming a fair distribution of molecular velocities and a large enough sample of molecules, the actual physical distribution of individual molecules within the gas must be more or less homogenous. Similarly, a fair distribution of glissando slopes would result in sounds evenly spaced within the pitch spectrum.
Xenakis plotted out his 1,148 velocities on graph paper [ex. 3]. The pattern of lines which represent glissandi illustrates a chaotic process that could not be sufficiently tempered by the Law of Large Numbers. The distribution of pitches varies greatly from measure to measure.
The chaotic process not only affected the patterns of distribution, it shifted the center of gravity for the entire mass of strings. Xenakis' graph was divided into 15 ranges, each representing a major third. Each tessitura was assigned a unique value, with higher pitches receiving higher values. The number of instruments in each range at the beginning of a measure were counted, then multiplied against the value of the range. The 46 products were added together and then divided by 46 for the weighted average pitch. Even with the ranges weighted logarithmically (to compensate for the logarithmic nature of pitch), the weighted averages migrate widely from one measure to another [ex. 4].
With his strong background in mathematics and science, Xenakis knew the Law of Large Numbers was meaningless unless the collective events occurred simultaneously. He also understood that 46 outcomes could not possibly have qualified as a statistically large sample. Yet somehow, the 1,148 events were meant to be taken in simultaneity, despite the 18.5 seconds necessary for execution.
In Formalized Music, Xenakis stated his Einsteinian perspective on time: “...its end and beginning coincide (negative time) disengaging itself endlessly...Time, Causality [parenthetical and italics his].”3 In this passage, he expressed the idea that time was a process of cause and effect, rather than a series of temporal frames. The contrapositive of this equation (absence of causality = absence of time) allowed for a simultaneity that depended on non-causality rather than on strict synchronicity.
The 1,148 values were calculated by a non-recursive process, meaning the outcome of one calculation was not passed back into the formula for the next calculation. The calculation was repeated 1,148 times, but was always started with the original source of numbers. While related in origin, none of the outcomes are causally related to each other. During the 18.5 seconds of Newtonian time needed to play this passage, Einsteinian time stands still.
Paradoxically, the flurry of sonic activity caused by 46 pizzicato strings helps to obscure any sense of motion through time. By superimposing factorially unrelated metric subdivisions, Xenakis created an extremely complex temporal grid. By assigning twelve players to quintuplets, nineteen to triplets, and fifteen to quarter notes [ex. 5], he defined metrical articulations with denominators of fifteen or even twenty [ex. 6]!
Pizzicato notes on non-fretted string instruments decay rapidly, due to the mute-like elasticity of the human finger. This is especially the case of plucked glissandi. While bowed glissandi have a strong linear drive, plucked glissandi barely survive to their terminal note. This lack of staying power also helps the perception that the notes occur independently, rather than in a continuum. The pointillistic character of plucked strings and complex metrical subdivisions reinforce the sense of atemporality created by Xenakis' stochastic processes.
In the composition of Fontana Mix, John Cage also utilized chance operations to suspend the flow of time. The sounds for Fontana Mix were determined by the chance intersections of lines and dots on a series of transparencies [ex. 7]. Transparencies with dots and curved lines were superimposed with a 100 x 20 matrix, also on transparency. A transparency with a single straight line was then laid down in a way that 1) two dots were connected and 2) the line intersected both of the long sides of the matrix [ex. 8]. The intersections of the straight line and the curved lines determined the taped sound sample to be used for a given sound event. The absolute horizontal distance between the points of intersection of the graph and the straight line determined the duration of a single sound event.
Like Xenakis, Cage believed that time was not fixed. “Time is inevitably beyond measure. It can't ever again be clock time.”4 Cage was very interested in the connection between imposed structures, and their effect on the perception of how time passes. By removing imposed structures from his music, he hoped to do away with clock time. The chance operations used to compose Fontana Mix were non-recursive, meaning each instance was generated in its own discrete process. The non-causality between each of the instances disallowed any semblance of internalized structure.
The irony of Cage's aesthetic of anarchy was in the number of rules required to effect it. His rules for determining tape lengths (which in turn determined the durations) were actually quite detailed [ex. 9]. The structurist conditioning of musicians was so complete, he reasoned, that a certain level of structure was necessary for doing away with structure itself.
His manner of setting out the parameters for determining the temporality of Fontana Mix showed a great deal of specificity. So specific, in fact, that the probability distribution inherent to this chance operation can be formally calculated.
The dots, the graph matrix, and the straight line play a role in the divination of event durations. The curved lines are involved only in the selection of the sounds themselves. The straight line connects two random dots. Since the dots themselves are distributed at random, the resultant orientation of the straight line is therefore random.
A random orientation means that the line's relationship to the matrix has an equal probability of being any one of the 360 degrees. Does it follow that the tape lengths defined by the intersections would have an equal distribution between 0 and 100? (The maximum horizontal length is enforced by the 100 cell width of the matrix.)
Contrary to what our intuition may tell us, the probable distribution among the 100 possible tape lengths (assuming an even distribution of angles) is not equal. The degree at which a line intersects two parallel lines is not proportional to the absolute horizontal distance defined by the points of intersection. The length is based on the tangent function. The trigonometric relationship between orientation and tape length adds a systematic bias to the selection of tape lengths.
To demonstrate this relationship, let us take an ordered sample, with all possible angles [ex. 10]. By slowly rotating the line counter-clockwise at 1 degree intervals, the full range of degree orientations and tape lengths can be represented. The lengths may be calculated using the tangent function [ex. 11 below]
Lengths were calculated for degree intervals between 0 and 79. Angles higher than 79 had to be thrown out of the sample, as they could not satisfy Cage's criteria for intersections between the straight line and the two parallel lines. The frequencies of each of the 100 possible line lengths were compiled [ex. 12].
The distribution pattern of durations generated by Cage's matrix are strikingly similar to the Gaussian curve employed by Xenakis to distribute his 58 possible glissandi [ex. 13]. The mathematics behind their chance operations are different, yet alike in their intended purpose: to create bodies of data in which the individual elements are causally unrelated.
While Xenakis and Cage held antithetically divergent ideas on what their roles were as composers, they held a common perspective regarding the relationship between causality and time. To them, time was not something that flowed independently of cause and effect. The strategy of generating random numbers non-recursively allowed them to create groups of sound events in which the individual events could not possibly have been construed as coming in any sequence. As the above distribution graphs illustrate [ex. 13], this curious convergence in their compositional strategies resulted in quantifiable similarities between the products of their chance operations.
The 'vertical' time of Pithoprakta and Fontana Mix is not based primarily in the subjective experience of the audience. Rather, the verticality rests on formal compositional grounds:
Non-recursive chance operations => non-causality => extended simultaneity
This notion of time adds a twist to the definition of simultaneity: “occurrence at the same time”5. If time is suspended as long as causality is absent, simultaneity may be translated to: occurrence within a period in which no event may be understood to be the result of another.
These composers abandoned the world of 'clock' time. In the world of 'clock' time, sequences of events were driven by the unstoppable flow of time. Xenakis' equivalence of time and causality built a temporality in which time itself was driven by the flow of events. By removing cause and effect, Cage and Xenakis removed past and future.