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I’d like to post about something different today, namely about a corpus of works of John Cage called the “Number Pieces“. One of the reasons is that I’ve always been fascinated by these pieces. Another reason is that they were the subject of my very first paper to be published in a music journal. (The last reason is that I should  post about neo-Riemannian theory and groups of transformation which act on chords, but that would need a very long post…).

The Number Pieces are a body of works by John Cage, written between 1987 and 1992, the year Cage died. In total, 52 Number Pieces were written during that time, not mentionning the other works Cage had been working on. The title of a Number Piece is always of the same form: a number written in full letters, which indicates the number of performers, followed by a superscript which indicates the rank of the work among other works for the same number of performers. Hence \text{Five}^3 is the third piece for five performers.

Cage had become extremely popular towards the end of his life, and was therefore facing a great number of commissions. The Number Pieces arose partly to keep up with such demands: Cage managed to automatize the generation of musical works, with the help of Andrew Culver who developed a computer program to randomly choose time-brackets (more on that soon) and their contents (see here for more information about the computer programs Andrew Culver has developed for Cage).

Another reason is that Cage seemed to have gradually adopted a new vision of harmony, and the Number Pieces were concrete expressions of such a vision. Cage was often cited for having said : “I certainly had no feeling for harmony, and Schoenberg thought that that would make it impossible for me to write music. He said, “You’ll come to a wall you won’t be able to get through.” I said, “Well then, I’ll beat my head against that wall.”  By the 70s and 80s, Cage advocated a new notion of harmony, in which harmony “means that there are several sounds . . . being noticed at the same time. It’s quite impossible not to have harmony [, hmm ?]” (quote from John Cage and Joan Retallack, Musicage: Cage Muses on Words, Art, Music.  John Cage in Conversation with Joan Retallack, ed. Joan Retallack (Middletown: Wesleyan University Press, 1996), 108).

This is of course a very simplified introduction, and if you’d like to know more, I suggest reading the following great references:

  • An anarchic society of sounds: the Number Pieces of John Cage, R. Haskins, Ph.D. Dissertation, University of Rochester, New York, 2004
  • Notational practice in contemporary music: a critique of three compositional models (Berio/Cage/Ferneyhough), B. Weisser, Ph.D. Dissertation, University of New York, 1998
  • John Cage «…the whole paper would potentially be sound »: time-brackets and the Number Pieces, B. Weisser, Perspectives of New Music, 41 (2), 2003, pp. 176-226

In view of such a conception of harmony, Cage came up with the idea of the time-bracket, which is central to the Number Pieces. Time-brackets are present in 50 of the 52 Number Pieces, though they had been used before, for example in his composition “Music for ____” (1984-87). What is a time-bracket ? On a score, it will show up as such : TimeBracket A fragment of staff (or staves) contains one or multiple pitches or sounds, in some cases with additional indications such as loudness, microtones, etc. On both sides lie two intervals expressed in real time. The performer chooses to start playing the indicated material whenever he wants inside the limits of the interval on the left, and chooses to stop playing whenever he wants inside the limits of the interval on the right (Cage suggests to use a stopwatch or anything equivalent to do so). Multiple time-brackets are played successively in a composition. Some of them have just a time indication instead of a time interval (for example, 2’15”), indicating that the material has to be started (or stopped) at the exact time given. The intervals may overlap inside a time-bracket (what we will refer to as the “internal overlap”) as well as between successive time-brackets (“external overlap”). The picture belows is an extract from \text{Four}^6, in which two time-brackets almost completely overlap, yet have to be played successively : TimeBracket2 I began working on the Number Pieces in 2007, with the idea that the time-bracket system is not really a score, but a tool to create a score. I’m actually borrowing this idea from Thomas DeLio and James Pritchett, and you can find more in the following references :

  • John Cage’s Variations II: the Morphology of a Global Structure, T. DeLio, Perspectives of New Music, 19 (1/2), 1980-81, pp. 351-371, in which DeLio analyzes Variations II with a similar notion.
  • The Music of John Cage, J. Pritchett, Cambridge University Press, 1993

Indeed, the time-bracket system can be seen as a framework for all possible temporal location of a sound (for the sake of simplicity, I’m considering here a time-bracket with a single sound or pitch in it), and a realization of this time-bracket is an actual choice of a starting time and an ending time. Then, how could we analyze the global structure of a time-bracket ?

I’ve taken the approach of a statistical analysis, in which all choices of starting times and ending times are possible (given we follow the instructions of the time-bracket system), and wherein we can study the probability distributions of some variables. In other words, the starting and ending times are random variables, and therefore so are the length of the sound or its temporal location. You can find the details about this approach in the papers I’ve published :

  • John Cage’s Number Pieces: The Meta-Structure of Time-Brackets and the Notion of Time, A. Popoff, Perspectives of New Music, pp. 65–84, 48/1 (2010)
  • Indeterminate music and probability spaces: The case of John Cage’s number pieces, A. Popoff, Lecture Notes in Computer Science, Volume 6726 LNAI, pp. 220-229 (2011), available here

Let’s formalize this mathematically. A time-bracket is a set of two closed intervals \{ST,ET\} over the reals, referred to as the starting time (ST) interval and ending time (ET) interval, with ST=[0,T_2], ET=[T_1,T_3] and 0 \leq T_1 \leq T_2 \leq T_3. A realization of a time-bracket is a set \{t_s,t_e\}, t_s \in ST, t_e \in ET, such that t_s < t_e. TimeBracket-Math The length of the sound is defined as L=t_e-t_s and, as we said, if we consider T_s or T_e as random variables, then so is L. The time-bracket is also a stochastic process, for we also obtain a collection of random variables S_t index by the time t, with values in 0 or 1 depending on whether one hears a sound at time t or not. In the following, I will take T_1=30, T_2=45 and T_3=75, as in the case of the time-bracket pictured above.

How can we choose the starting and ending times ? In other terms, what are the probability distributions of T_s and T_e ? This is a really tough question, as it basically asks for a model of a performer playing the Number Pieces. I won’t enter into too much details here and, for lack of a real model of human behavior, I decide to choose the simplest possible model by assuming that the distributions are uniform. However, since we must choose the times successively (as would happen in a real performance) this means that we have

dP(t<t_s<t+dt) = \displaystyle\frac{dt}{T_2}

for the starting time, and that the distribution for the ending time is conditional upon t_s, i.e we have

dP(t<t_e<t+dt | t_s) = \begin{array}{ll} \displaystyle\frac{dt}{T_3-T_1} \text{ if } t_s < T_1 \\ \\ \displaystyle\frac{dt}{T_3-t_s} \text{ if } t_s > T_1 \end{array}

Again, this is most probably not a good model at all of performers (in particular because humans are known to be poor random generators), but it has the advantage of making the calculations analytically computable. I won’t detail these calculations here, but it is relatively easy to calculate the distribution for L and the probability P(S_t = 1). Instead, I’m just going to show the corresponding graphs. With the above model for the starting and ending times, the distribution of lengths looks like this LengthDistrib (Remember that D(L_0) is such that the probability of finding a sound of length comprised between L_0 and L_0+dL is equal to D(L_0)dL. The graph is noisy as it is the result of a simulation using 10^5 random realizations of the time-bracket, rather than the graph of the analytic function).

We see that the distribution has a maximum for 30-seconds sounds, i.e the time-bracket favors sounds of medium length. We can also see that very short sounds have a non-zero probability of existence, whereas very long sounds have a much lower probability, the extreme case being a 75s-long sound with a zero probability. This may seem strange as the time-bracket system seems to allow the possibility of a such a sound. However, mathematically, this means that we have to pick up t_s exactly at 0s and t_e exactly at 75s, an event which has a null probability. The apparent paradox for long sounds therefore stems from the selection model we are using, and indirectly points to a particular conception of musical time. Indeed, if we say that the time-bracket system allows maximum-length sounds, we unconsciously think of time in terms of durations, as we would do for any ordinary piece of classical music. In our model, we think of time as a continuous, infinite flow in which events can take place: durations are not a priori, but are observed a posteriori after the starting time and ending time are selected. On the other hand, very short sounds are possible, as you can pack infinitely many of them in the internal overlap of the time-bracket.

If we now want to know where the sound is distributed inside the time-bracket, we can study P(S_t=1), the probability of hearing the sound at time t. The graph below plots this probability for all values of t PDistribWe see here that the sound is not uniformly distributed inside the time-bracket, but has a maximum chance of being heard in the middle of the interval [0,T_3]. One could wonder if the external overlaps between successive time-brackets could change this pattern. In fact, unless the overlap is really large, there will still be a clear distinction in terms of temporal location between sounds coming from different time-brackets. As a consequence, even though the characteristics of a sound belonging to a time-bracket cannot be fully determined in advance due to their random nature, a Number Piece still exhibits a large-scale morphology. As Cage himself commented: “It is not entirely structural, but it is at the same time not entirely free of parts.

We therefore see that the time-bracket system is a simple framework (its instructions being easily implementable by any performer) with rich consequences in terms of sound characteristics, structure, etc. It also opens many perspectives :

  • How could we model the performers’ behavior regarding the choice of starting and ending times ? Can we extract data from actual performances ?
  • How could we model the listener’s behavior during an interpretation of a Number Piece ? Is he puzzled / startled / bored by the long sounds / the global structure of the piece / etc. ?
  • Can we use the same methodology as above on all time-brackets in order to analyze a Number Piece as a whole ? I’m actually working on such an analysis for the Number Pieces Five and Four and I hope I’ll be able to post about it soon.
  • Could we use these results to generate multiple instances of a given Number Piece ? It is certainly possible, and you can check the following video

    for a computer-generated interpretation of Four (Section B). You can also check the following work by B. Sluchin and M. Malt :
    « A computer aided interpretation interface for John Cage’s number piece Two5 », B. Sluchin, M. Malt, Actes des Journées d’Informatique Musicale (JIM 2012), Mons, Belgique, 9-11 mai 2012

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