Improvisations Continued

I recently started fooling around with GarageBand for the first time, and even though it’s presumably not a product aimed at very serious musicians, it’s quick and easy to make some nice sounds. Here are a couple of ambient pseudo-improvisations I made (they’re more or less improvised, although since they are each several tracks, they weren’t made 100% in real time).

The first one is made from a couple of GarageBand’s built-in keyboards and synths, and has a “theme song to a sci-fi show” vibe to it:

The second one is built in a few layers of repetition, and has a very earnest vibe to it. It’s just a recording or two of me playing the piano, so I was only using GarageBand for mixing:

Piano improvisation

I’ve always had trouble producing reasonable-quality recordings of piano playing, but I managed to dig up an old microphone that works okay. While I’ve been doodling around with improvisations for years now, I only recently came across the improvisations of Keith Jarrett, which are highly original and inspirational. His Cologne Concert performance is probably the most famous, but I’m partial to his Paris Concert, perhaps because of its classical strains.

Here’s a clip of one of my own unpremeditated improvisations. I still find myself resorting to little crutches here and there, but if I deliberately try to push the envelope, the results are usually surprisingly–occasionally they are jarring, but you learn to roll with it! The rhythmic staccato portion around 2:45, for instance, definitely represents a departure from my comfort zone, but in a positive way.

‘Cantor’ at the 2014 MAT End of Year Show

For the 2014 MAT End of Year Show, my piece “Cantor” was performed in the audiovisual concert. Here are the program notes:

“In ‘Cantor’, the mathematical process of removing middle thirds to generate the Cantor ternary set (introduced by mathematician Georg Cantor in the 1870s) inspires a musical process across multiple timescales. Different levels and mappings are layered together to create a a sort of Cantor mosaic.”

And the piece itself:

You can read about some of my earlier projects and explorations with the Cantor set here and here.

Music Information Retrieval tools for Algorithmic Composition

The field of music information retrieval, while relatively young, still abounds with a variety of interesting mathematical techniques. Many of these techniques take as a point of departure a collection of ‘features’ or ‘feature vectors’; e.g., spectral flux, windowed root-mean-square energy, short-time Fourier transform vectors, etc. The hope is to glean some sort of recognition of genre or mood based on certain properties of these feature vectors.

One concept that is occasionally used is the self-similarity matrix (SSM). Assume that we begin with a set of \( n \) feature vectors for which we would like to construct the corresponding SSM. By definition, this matrix is the \( n \times n \) matrix whose entry at position \( (i, j) \) is give by a ‘similarity’ (e.g., Euclidean distance) between vectors \( v_i \) and \( v_j \). Here is a graphical example of an SSM of approximately \( 100 \) windows of a spectrogram:



This matrix may remind you of a related concept from probability. Given a set of states, we can define the stochastic matrix, which determines with what probability we will transition from one state to another. In other words, the entry at position \( (i, j) \) of the stochastic matrix is given by the probability of transitioning from state \( i \) to state \( j \). Here is an example of a stochastic matrix (note how the rows sum to \( 1 \)):

0 & 0 & 1/2 & 0 & 1/2\\
0 & 0 & 1 & 0 & 0 \\
1/4 & 1/4 & 0 & 1/4 & 1/4\\
0 & 0 & 1/2 & 0 & 1/2\\
0 & 0 & 0 & 0 & 1

Given an SSM, we can obtain a stochastic matrix by normalizing each row to sum to \( 1 \). A stochastic matrix can then drive an algorithmic process such as a Markov chain. The main question is determining how to map the different ‘states’ into audio. One approach, assuming the SSM has been computed from a spectrogram, is to map each window to a stochastic frequency cloud, where the various frequencies occur in proportion to the energy in the corresponding bin. Here’s a basic audio example:

The previous example focused mostly on frequencies, but we can use other typical MIR features to map to other musical parameters, such as amplitude envelopes. For instance, we can compute a windowed RMS of an audio sample to get an idea of where energy peaks are located in time. We may then use the RMS signal itself as an envelope, creating a sound whose contours follow the energy of the original sound through time. Here’s an example:


Another musical parameter we can control is rhythm. By locating relative extrema (using a simple first-difference approach), we can also approximate onsets. The spacing between those onsets can then be mapped into rhythms.In my example, I randomly choose a spacing with weight corresponding to the intensity of the onset.

Finally, there is the parameter of form. Since music typically evolves through time, it is aesthetically important to devise some system whereby the music will be changing. A purely algorithmic approach, based on self-similarity, is intriguing: starting with an initial SSM, we generate a few seconds of audio using the previously described techniques. That audio is then analyzed to generate a new SSM, which in turn drives new audio. Other data such as RMS can also be sequentially updated. The piece essentially composes itself forever. Of theoretical interest might be the question of whether it converges to some steady state (or perhaps ‘converges’ into a steady loop).

Here is an étude based on combining several of the aforementioned ideas:



The Devil’s Staircase

The Cantor function, mapping the unit interval to itself, is a relative to the Cantor set, which was described previously on this website. Being uniformly continuous, but not absolutely continuous, it is a classic mathematical counterexample. Although its derivative is zero almost everywhere, its range of outputs still encompasses the full unit interval. Wikipedia has a nice pictorial representation of the iterative process that defines it.


Cantor iteration, courtesy of Wikipedia


The concept starts from considering the Cantor set probabilistically. The Cantor function is in fact the cumulative distribution function for a random variable which is uniformly distributed on the Cantor set. But we can also define the Cantor function directly. Consider the Cantor set as the set of all ternary expansions whose entires are all either \( 0 \) or \(2\). As mentioned previously, there is then a bijection between the Cantor set and the full unit interval by ‘pretending’ the \(2\)s are \(1\)s and interpreting the numbers as binary expansions. Hence, for an input \(x\) which belongs to the Cantor set, we define \(c(x)\) as this correspondence. We then further define the Cantor function on inputs \(x\) that do not belong to the Cantor set as follows: since \(x\) does not belong to the Cantor set, it must contain a \(1\) in its ternary expansion. By truncating \(x\) immediately after the first \(1\) in its expansion, ‘pretending’ any previous \(2\)s are \(1\)s, and then interpreting the result as a binary expression as before, we obtain the appropriate output \(c(x)\).

I’ve recently written the code for a Cantor function UGen in SuperCollider for use in digital audio. The UGen uses an approximation to the Cantor function based on a recursive sequence of functions which converge pointwise to the Cantor function. The user can control the depth of the recursion for more or less precision. Here are a few very primitive demonstrations. To illustrate the recursion of the function sequence that converges to the Cantor function, I’ve taken a sine oscillator and modulated its frequency from \(200\) to \(600\) Hz using different convergents of the Cantor function. Here they are, for \( n = 0, 1, 2, \text{and } 3\):

\( n = 0\): (equivalent to one straight line between \(200\) and \(600\); i.e., no plateaus)

\(n = 1\): (comprises three straight lines, the middle one being completely flat; i.e., one plateau)

\(n = 2\): (three plateaus connected by straight lines)

\(n = 3\): (seven plateaus connected by straight lines)

(In general, there will be \( 2^n – 1\) plateaus.)

Of course, those examples aren’t especially interesting from a musical perspective. However, I found that the UGen was quite expressive when used at control rate to manipulate such parameters as envelopes and rhythms. Here is a recording  of an improvisation I generated in SuperCollider, using almost exclusively the Cantor UGen to shape the sounds. (The actual audio rate UGen is a pulse oscillator, and the end result is put through a reverberator.)

If others are interested in trying out the Cantor function UGen, I will post the code shortly!



Granular experiments

Recently, I’ve been exploring new ideas based on basic granular synthesis. As part of my work for a digital audio programming course, I composed a short three-part work, ‘Sweep’, which is largely experimental but based on some of these ideas. The audio is generated using a basic sequencer that we designed in C using our granular synthesis module from an earlier course taught by Andrés Cabrera. The code reads in a file in a Csound score format (in this case, one that is generated from a Python script) and sequences it in real time. Each movement is actually generated from the same Python script, but modulating a few of the basic parameters. The idea is to be able to create lots of different pieces in a modular fashion, so this is just one particular possible realization.

Movement I:

Movement II:

Movement III:

Barkley and Doppler

Recently we’ve learned a bit about physical modeling of PDEs in my pattern formation class. Here is a video depicting Barkley turbulence, which is a type of reaction-diffusion system.

On an unrelated note, here are a few of my results from implementing a simulation of the Doppler effect. In our first example, a source is traveling at about Mach \( \frac{1}{10} \) horizontally from right to left across a distance of \(200\) meters. When it is directly in front of us, the distance between us and the source is \(10\) meters.

In the second example, the source travels twice as fast, twice as far, and we are twice as close. The shift is correspondingly more dramatic.

One of the more counterintuitive aspects of the Doppler effect is the situation in which the source is coming directly at us. We never really experience this in real life (unless we are actually getting run over by the source). In this situation, the source does not in fact slide continuously from a higher frequency to a lower frequency but rather it does a discrete jump. (This is because the Doppler effect depends on the projections of the velocity vectors of the source and the receiver onto the line connecting the two, so unless the source and receiver are traveling directly toward each other, these projections vary continuously. However, when the source and receiver travel directly toward each other, the projections are constant until they meet, at which point they discretely jump to a new value.) In the particular example I have implemented, the source is traveling at Mach \( \frac{1}{4} \). (Warning: this example might be somewhat uncomfortable to listen to!)

Notice that we heard what sounded like a major sixth! This is actually a pretty straightforward consequence of the mathematics behind the Doppler shift. The emitted frequency, \(f\), is distorted by a frequency factor of \( \frac{c}{c + v_s} \), where \(c\) is the speed of sound in the relevant medium and \(v_s\) is the velocity of the source approaching you, with the convention that it is negative as it approaches you and positive as it recedes. Hence, the frequency ratio between the approaching sound and the receding sound is given by \( \frac{c + |v_s|}{c – |v_s|} \). In particular, at Mach \( \frac{1}{4} \), this ratio is \(\frac{\frac{5}{4}}{\frac{3}{4}} = \frac{5}{3} \), which is a just major sixth. Other musical intervals are, of course, also easily obtainable — Mach \( \frac{1}{3} \) yields an octave, for instance. In general, the reader can verify with some basic algebra that in order to get the frequency ratio \(P:Q\), the source must travel at Mach \( \frac{P – Q}{P + Q} \).

You might have noticed the expression \( c – v_s \) in the denominator above and wondered about the situation in which we are traveling Mach \(1\) or faster, but that’s another story for a future post.




Media mashup

I’ve been working on a bunch of small projects recently that I wanted to share. The first two are related to visualizing Newton fractals in the complex plane. One is a video showing what happens when you move around the three zeros of a cubic polynomial while coloring its corresponding Newton fractal, while the other shows the Newton fractal generated from the function \( f(z) = \sin z \).

The next image is a basic implementation of diffusion-limited aggregation.

I also experimented a bit with an iterative edge detection procedure on MAT’s canonical bunny image, which yielded the following video.

In the audio world, I’m currently experimenting with ideas based on granular synthesis and the Doppler effect, so stay tuned for future content!



Sine Wall

It’s been a while since I’ve updated, but I’ve still been trying to come up with new creative content whenever possible. I recently wrote a brief electronic piece as a project for one of my MAT courses. Although it is generated entirely with sine oscillations, it still packs a bit of a punch. Entitled “Wall,” it is more playful and less serious than my previous works. Hope you enjoy!


In my MAT 201B course, “Programming with Media Data,” I’ve developed an audio/visual representation of different molecular random walks, including a loose model of Brownian motion. The fluctuations in the spatial displacements are mapped to color (in terms of HSV) and pitch (in terms of frequency). The user can interact with the simulation, discretely or continuously varying the temperature to affect the magnitude of the displacements. I’ve created two other stochastic processes in addition to the Brownian motion and included a mode which cycles rapidly between the three random walks, essentially producing a ‘random walk between random walks.’ I also programmed a short piece which illustrates the various aspects of my project in action. You can see a video screen capture of it below.