Some time ago, I promised I would talk about the MCM 2015 conference, and the contributions I had. I will start today with my poster contribution on the statistical analysis of John Cage’s Number Pieces, which I presented also shortly as a ‘poster craze’ session.


This contribution is accessible here:

  • “A Statistical Approach to the Global Structure of John Cage’s Number Piece \text{Five}^5“, Popoff, A., 5th International Conference, MCM 2015, London, UK, June 22-25, 2015, Proceedings, Springer LNCS 9110, Eds. T. Collins, D. Meredith, A. Volk, pp. 231 236, http://dx.doi.org/10.1007/978-3-319-20603-5_24

From previous posts on John Cage’s Number Pieces, you may remember that Cage used a particular system, which he called ‘time-brackets’, to determine the temporal location of sounds in the piece. Here is a time-bracket:


The musician can choose to start playing the indicated F whenever he wants between 0s and 45s, and has to stop whenever he wants between 30s and 75s. Though all pitches (or sounds) are fixed, the time-bracket system still introduce a high level of indeterminacy. I proposed previously that a statistical approach would be more appropriate to study a single time-bracket, in which the characteristics of a sound (its length, its temporal location inside a given time-bracket, etc.) would be analyzed through their probability distribution (or probability density functions, for continuous cases).

A Number Piece contains many parts, each part containing multiple time-brackets. The next logical step is therefore to analyze the whole Number Piece from a statistical point of view, which was the subject of my poster contribution. As an application, I chose to analyze the Number Piece \text{Five}^5.


The Number Piece Five5 was written by Cage during 1991 and is scored for flute, two clarinets, bass clarinet, and percussion for a total duration of five minutes. All the time-brackets contain only a single sound. The percussion part does not specify any particular instruments: since the analysis deals with pitch class sets, I therefore discarded it. If I adopt the following graphical representation for a time-bracket,


then the temporal structure of this Number Piece can be represented as such (the pitch classes are represented with the usual semitone encoding with C=0).


As one can immediately see, the diversity of the pitch classes, the numerous internal and external overlaps between time-brackets, and their diversity (some of them being very short, some quite long) give this work a very high level of indeterminacy. The idea is to approach the description of this piece from a statistical angle, by examining all the possibilities at each time t and their associated probability. More precisely, I am interested in the possible harmonies which can arise at each time; and I take the word “harmonies” in a very broad sense, by including any combination of sounds, following in this way a remark of Cage about his late works:

“I now think the simple togetherness of art – I mean of sounds – produces harmony. That harmony means that there are several sounds…being noticed at the same time, hmm?”
(John Cage and Joan Retallack, 1996. Musicage: Cage Muses on Words, Art, Music.)

Atonal music analysis has introduced the notion of set classes, i.e. classes of pitch class sets which are transpositionally related. These set classes are usually given using Forte’s notation, and you can find a complete list of set classes here. For example, set class 3-11A includes all minor chords, while set class 3-11B includes all major chord. If one consider inversionally related pitch class sets to be equivalent, then one ends up for example with the single set class 3-11 which includes minor and major chords. Set classes are convenient because, whatever the transposition, an element of, say, set class 3-11A, will always sound as a minor chord.

To analyze the possible harmonies in \text{Five}^5, I decided to use set classes at the transpositional and inversional level. Retrospectively, the analysis would perhaps benefit more from doing it only at the transpositional level, but only form is usually present in this work (though it may be different in other Number Pieces). Since, we have only four parts, the maximum cardinality of the pitch class sets is 4, which leaves at each time t 49 possibilities: silence (set class 0-1), single pitches (set class 1-1), dyads (set classes 2-1 to 2-6), triads (set classes 3-1 to 3-12), and tetrachords (set classes 4-1 to 4z-29). In other words, one has for each time t a random variable X_t taking its values in the set of the 49 possible set classes.

The procedure is as follows: each part is treated independently from the other (no interaction between performers, since in a real performance they would decide of their own actions), and in each part time-brackets are processed successively by selecting the starting and ending times of the sounds at random, following a given probability distribution. Here, I chose a uniform distribution (as exposed here), but other choices are possible. Time is discretized with a resolution of 0.1s, which is sufficient to approximate continuous time. The temporal location of all pitches is thus known, and for each time t, one can determine the set class i being played. By repeating this a certain number of time, one can determine by a frequentist approach the probability P(X_t=i) of a set class being played at a time t. This is represented in the heatmap below, with a log scale on the probabilities.


From this map, five or six different sections of varying complexity appears, which are not readily apparent from the score. While various set classes can be heard, they do not occur with the same probability. For example, the probability of hearing tetrachords in the beginning of the piece is much lower than in the end. The advantage of this representation is that one can also follow the progression of the various set classes throughout the piece. In other words, it allows to grasp at once the structure and the possibilities offered by this Number Piece, which would be otherwise quite difficult to understand given its degree of indeterminacy.

As the probabilities of hearing a given set class are known at each time t, one can use the framework of information theory to derive interesting measures regarding the structure of this piece. For example, one can calculate the entropy H(X_t) of the random variable X_t at each time t, which describes the ‘complexity’ (in terms of the various set classes which can be heard) of the piece at t. More interestingly, given a delay \tau, one can calculate the conditional entropy H(X_{t+\tau} \mid X_t=i), which describes the uncertainty about the set class in the future, knowing that the set class i is being played right now. We can plot this conditional entropy in a heatmap, as represented below (the entropy is calculated in bits, i.e. with a base 2 logarithm) for a delay \tau of one second.


Notice that, globally, the conditional entropy increases for a given set class in the time period where it is being played. This is normal, since the time-brackets have a finite length, and as we approach their end, the harmony is likely to change soon. However one can notice that the degree of uncertainty that this conditional entropy captures can vary depending on the set class being played. For example, for set class 3-7 at t=50s the conditional entropy H(X_{t+\tau} \mid X_t=\text{'3-7'}) is low, meaning that this set class has a rather predictable evolution. In comparison, the conditional entropy H(X_{t+\tau} \mid X_t=\text{'3-4'}) at the same time is higher and increases monotonously, which means that this set class is rather unstable. I refer you to the paper for more details on these two cases.

In conclusion, this statistical approach is rather convenient as it allows to analyze at once all the possibilities offered by a Number Piece. Some points can be debated, for example the selection procedure for the time-brackets, or the choice of set classes to analyze the possible harmonies. In the first case, I am currently working on an approach to compare different selection procedures and their impact on the sonic characteristics of the pice. In the second case, the same framework can be reused as is to perform the same analysis at the transpositional level only; it could even be performed at the pitch class set level, though the number of possibilities would be much greater.

 And finally, I wish you a very happy new year !

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