Musical Cantor

Cantor’s ternary set, or more briefly, the Cantor set, is a well-known construction that illustrates some interesting topological properties of the real numbers. To describe it briefly, we define an iterative process as follows: Let \( C_0 \) be the interval from \( 0 \) to \( 1\). Then \( C_1 \) is obtained by removing the middle third from \( C_0 \), leaving the two intervals from \( 0 \) to \( \frac{1}{3} \) and \( \frac{2}{3} \) to  \( 1 \). The set \( C_2 \) is obtained by removing the two middle thirds from \( C_1 \). So \( C_2 \) will then contain four intervals, each of length \( \frac{1}{9}. \) In general, the set \( C_n \) will contain \( 2^{n} \) intervals, each of length \( \left(\frac{1}{3}\right)^{n} \). The Cantor set, denoted \( C \), is then defined as

\(C=\displaystyle\bigcap_{n=0}^{\infty} C_n. \)

The Cantor set satisfies many curious properties. For instance, in a certain sense, it has length zero. This is more or less because at stage \( n \), it has a total length of \( \left(\frac{2}{3}\right)^{n} \), which approaches \( 0 \) as \( n \) approaches infinity. However, from the perspective of cardinality, it can be shown that the Cantor set is uncountable. This is actually fairly simple to see, using the following argument. In order to be an element of the Cantor set, you must be a member of every set \( C_n \). Going from \( C_0 \) to \( C_1 \), you have to be either ‘left’ or ‘right’ of the middle third. Let’s say hypothetically that you are ‘left.’ Then going from \( C_1 \) to \( C_2 \), again, you must be either ‘left’ or ‘right’ of the middle third. It’s clear that this process repeats indefinitely. If we represent ‘left’ and ‘right’ by \( 0 \) and \( 1 \), then we see that elements of the Cantor set can be put into one-to-one correspondence with infinite binary sequences. Thus, by Cantor’s diagonal argument, the Cantor set must be uncountable. So although it has measure zero, it is an uncountable set!

One can also try to “hear” the Cantor set in the following way. Start with a particular frequency, let’s say, middle C. Define \( C_0 \) as sustaining that frequency for some duration, let’s say, one second. Then choose an interval of modulation, for instance, a tone. Then \( C_1 \) would be the triplet D-C-D, since we have modulated the outer thirds by a tone. We can proceed similarly to the general form of \( C_n \), and in principle, we could even define the full Cantor set \( C \), although it would be difficult to program. The following audio tracks illustrate a few of these ideas.

Here’s Cantor for \( n = 0, 1, 2, 3, 4: \)

Cantor with \( n = 6: \)

Inverted Cantor:

Cantor with the modulation factor as a perfect fourth:

A mash-up of many of these ideas:

I hope this post allows you to hear the Cantor set better!

(Note: Once again, I’ve coded all the musical examples in Gamma.

Rhythm and Frequency

If you happen to have a subscription to JSTOR, Karlheinz Stockhausen’s 1962 article “The Concept of Unity in Electronic Music” is a classic explanation of some of the concepts I’m about to describe. It turns out that some musical parameters, such as rhythm and frequency, are in fact manifestations of one underlying mathematical principle.

Consider a simple rhythm, for instance, a pulse that divides a second into three equal beats. We can view the speed of this rhythm as 3 Hz. If we speed this pulse up until it is several hundred Hz, we will no longer process it as a rhythm but instead as a pitch with a frequency, as the following audio example demonstrates:

More interestingly, suppose we take a polyrhythm. That is, we take a single pulse, let’s say, 1 Hz, and divide it evenly into two different parts. One of the simplest possibilities is a 3:2 pattern, which would initially start out as a 3 Hz : 2 Hz ratio. Again, if we speed this polyrhythm up until it is several hundred Hz, our brains begin to perceive the rhythm as frequency. In particular, we will hear a perfect fifth, since the ratio of the frequencies is 3:2. Try to determine in the middle of the following audio example where you stop perceiving the sounds as a rhythm and start perceiving them as frequency:

Here’s a similar example, but with a 4:3 ratio, so that the frequencies play a perfect fourth:

And the irresistible 6:5:4 ratio, which produces a major triad:

It may seem counterintuitive, but these examples demonstrate that rhythm and frequency are in a certain sense the same concept. It’s merely our perception of them that varies.

(Note: I’ve coded all the musical examples in Gamma.)