Class notes are reminders of some of the main points discussed in class. They are not always presented in a rigorous format.
John Cage uses his chart technique: there is a chart or matrix of small groupings of pitches, one of durations, and one of dynamics. The I-Ching, The Book of Changes , contains 64 hexagrams made of continuous or broken lines and is used to select (through some chance operations) a location from each chart and then the result is assigned to a layer. A density chart controls the number of simultaneous layers, also determined with the help of I-Ching (
).
I-Ching operations that Cage used from there on.
It is interesting to note that the procedure resembles to some extent (pre-existent pool of pitches, matrix-like setup, a well organized “system”) the domain technique of Boulez. In spite of the difference between deterministic (Boulez) and chance
(Cage) procedures the two them sustained a lively and instructive correspondence between 1949 and 1954.
Before determining through chance the pitches, durations, and dynamics of a group of sounds, Cage decided if that event will be a sound or a silence, giving them a 50-50 probability of occurrence. We have here an example of conditional probability . The probability that and event A occurs once E has occurred or the conditional probability of A given E is defined as:
P(A | E) = P(A ∩ E) / P(E)
(see also:
)
Conditional probability can also be represented in the form of a tree structure:
.60
●
.40
●
●
.55
.45
.20
.80
● .33
● .27
● .08
● .32
0.00
0.33
0.60
0.68
1.00
There are many possible applications of conditional probability in music eg. assigning probabilities to problems of orchestration: a piano (● of origin) can not sustain a very long duration (● at the end of .40 branch) in the very high register
(● at the end of .20 branch) and this is given a relatively small chance to happen (.40 * .20 = .08).
Hiller's program and many other computer applications use random numbers to select an element from a list if its probability is given. In the example above for instance, if the random number happens to be 0.62354, the element whose probability falls within the range 0.60 – 0.68 (right column) will be selected. Random numbers are produced through methods that differ in their complexity but they all are only pseudo random, since they need to be “seeded” (i.e. a first number needs to be fed to the computer) and the same seed will always produce the same sequence of random numbers.
John Cage used other chance procedures besides the I-Ching such as enhancing the impurities of cheap manuscript paper and considering them sounds in a composition or using a star atlas and marking sounds in a score according to star locations
( Atlas Eclipticalis ). These were decisions deriving from his interest in Far-Eastern philosophy and religion, especially Zen
Buddhism and aimed at removing as much as possible the composer's subjectivity.