More musique concrète

I recently finished composing a slightly longer piece, “Crackle,” based on similar themes to “Snap.” In “Crackle,” I’ve used only one sample of myself snapping my fingers in an homage/satire of Steve Reich’s “Clapping Music.” Using Pro Tools again, the sample is developed in a variety of ways, but principally with time-distortion, reverb, and panning.

Aliasing

Aliasing is a common problem in signal processing, both visually and aurally. If you’ve ever watched the spokes on a wheel turn in a movie, you may have wondered why they sometimes appear to go backward or even stand still. The reason for this has to do with how film works. Typically, the illusion of continuity is created by rapidly juxtaposing many still photographs, for instance, $$24$$ per second. (This number is usually called the frame rate.) In other words, one photograph takes place in every $$\frac{1}{24}$$ seconds. Let’s assume that we are photographing a car whose wheels turn at a rate of $$\frac{1}{4}$$ revolution every $$\frac{1}{24}$$ seconds. Then on film, the car will appear to be moving forward as normal. But suppose instead that the car is traveling three times faster — that is, its wheels are turning at a rate of $$\frac{3}{4}$$ revolution every $$\frac{1}{24}$$ seconds. When we observe the film, without sufficient visual context, it will actually appear as though the wheels are turning backward $$\frac{1}{4}$$ revolution every $$\frac{1}{24}$$ seconds. In other words, the car was moving so fast that the frame rate couldn’t keep up. In fact, even if the car had only been moving twice as fast — that is, at a rate of $$\frac{1}{2}$$ revolution every $$\frac{1}{24}$$ seconds — we wouldn’t be able to distinguish whether its motion was forward or backward on film.

Thus, if we want to accurately capture the motion of a wheel on film, we actually need to choose a frame rate at least twice the rate of the wheel. (This is the essence of the Nyquist-Shannon sampling theorem.) So a rate of $$24$$ frames per second can accommodate any wheel spinning at a rate slower than $$12$$ revolutions per second. If we take the average circumference of a tire to be about $$67$$ inches, then $$12$$ revolutions per second corresponds to $$67$$ feet per second, or approximately $$45$$ mph. Hence, any motion in excess of $$45$$ mph may appear distorted by aliasing without a faster frame rate.

All these arguments apply to audio processing as well! In digital signal processing, a continuous sound is sampled by gathering rapid data about discrete instants of the sound, similar to taking rapid photographs of a visual event. The sampling rate (analogous to the frame rate) indicates just how much data are gathered in one second. For example, one common choice of sampling rate is $$44100$$ samples per second. But if the sampling rate does not exceed twice the frequency of a signal, then the signal may be subject to aliasing. This results in low-frequency signals which were not present in the analog sound manifesting themselves in the digital version. For example, a tone with a frequency of $$43660$$ Hz could be heard instead as a tone with frequency $$44100 – 43660 = 440$$ Hz, a concert A.

While aliasing is usually a drawback of digital processing that engineers try to avoid, it sometimes has an interesting sound. Here’s a quick little track that demonstrates the concept:

The program is written so that the frequency slides perpetually higher; however, you will eventually start hearing a quasi-random scattering of different pitches due to aliasing.

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.)

Everything you ever wanted to know about temperaments (but were afraid to ask)

Here’s a historical paper I wrote on different musical temperaments from Antiquity until the late Renaissance.

A History of Temperaments

Recommended reading for anyone who is confused about the differences between just, mean, and equal temperament.

Here are some supplementary audio samples:

Just major third:

Pythagorean major third (ditone):

Syntonic comma:

Just major third cycle:

The resulting comma (“diesis”):

Percussion quartet

Here’s my senior composition project from Brown, written in 2008–2009. Titled “Percussion Quartet,” it’s scored for two pianists and two percussionists, much in the spirit of Bartók’s well-known piece for the same instruments. Lasting approximately 16 minutes, the piece is divided into four movements, the first two being played attacca.

Movement III Score

Movement IV Score

An analytic commentary is also available.