# tag "math"

## A fractal sequencer toy

Here's a fractal chord sequencer you can play right in your browser. READ MORE

## Aconcagua

Every so often I come back to the idea of generating self-similar or "fractal" chord progressions by recursively applying grammar rules, as in my 2015 composition Dharmapala. I like the general method described in that article, but one thing I don't like so much about the finished piece is that it sounds the same all the way through. There's a little bit of "development" or shift in texture over the course of the piece, largely driven just by my performance-time changes to the synthesizer settings and the fact that I allow the notes to be chosen from a wider range toward the end. READ MORE

## Listening longer to the Mandelbrot Set

I've long been interested in ways to sonify fractals, and I wrote an article here about Listening to the Mandelbrot Set in 2017. At that time, I was thinking of it as a way to generate a waveform: basically, imagine running a point along the (infinitely wiggly!) boundary of the set, making many complete circuits per second, and use the real and imaginary coordinates of the moving point as waveforms for synthesis. Then the entire Mandelbrot Set (often abbreviated to "M") defines a timbre that may be interesting, and which can be used as a musical building block. It's a little bit like putting a phongraph needle on the edge of the set and playing it like a record. READ MORE

## What's Euclid got to do with it?

There's a lot of talk in modular synthesis circles about what people call "Euclidean" rhythms, which are claimed to exist in most kinds of traditional music around the world. Unfortunately, this concept is often oversold, and presented in a way that makes it sound much more complicated than it is. READ MORE

## Level up on circuit simplification

In my last entry I talked about the rules for simplifying series and parallel circuits. Two resistors in series can be replaced by one with a value equivalent to the pair of them; two in parallel can similarly be replaced; there are other rules in the same general form for capacitors and inductors (assuming theoretically perfect components); and by applying these rules repeatedly you can simplify complicated circuits down to much simpler equivalents. I also set up an interactive reverse calculator for finding combinations of standard-value components to make up a desired, maybe non-standard, value. READ MORE

## Walking the hypercube

Here is the skeleton of a four-dimensional hypercube, or tesseract, with the vertices labelled by musical note names. READ MORE

## Alternate harmony with additive synthesis

Much of musical harmony comes down to combining notes that share harmonics. Sounds produced by (some...) physical objects typically have consistent waveforms, where each wave is the same shape as the last. That is also typical of modular-synth oscillators; and its consequence is that the spectrum always consists of a sum of sine waves all at integer multiples called harmonics of one frequency called the fundamental. The proportions and phases of the different harmonics determine the shape of the waves, and those can vary a lot, but the general pattern of integer multiples is fixed. If you play a note like D with a fundamental frequency of 293.7 Hz, it will have its harmonics at 293.7 Hz, 587.4 Hz, 881.1 Hz, 1174.8 Hz, 1468.5 Hz, 1762.2 Hz, and so on. READ MORE

## Frequency, latency, and uncertainty

Some wigglers want "fast" envelopes that will turn a signal on and off without any delay and without any perceptible clicking sound. Some wigglers want spectral effects, like frequency shift or vocoding, without any latency between the input and output. Some wigglers just want accurate pitch tracking of external inputs. All of these wigglers are doomed! None of those things can ever work perfectly. But the funny thing is that they're all doomed for the same reason. There is a fundamental principle that limits the performance of all these seemingly different things, and I'll try to explain the connection in this posting without resorting to any particularly complicated math. READ MORE

## Live-coding a permutation-based fugue

If you've been following North Coast for a long time, you may remember the live video streams I did back in 2016. This week I spent some time getting the software and hardware set up to do that again, which pretty much meant starting over from scratch because enough things have changed that my former config was no longer appropriate. You can read about the technical side over on my personal site. I'm still not sure exactly how I'll be using this capability, but it's something I've wanted for a long time as a way to raise the profile of some of the stuff I do. In the days to come I'll probably plan out some scheduled broadcasts on music, electronics, and other topics. READ MORE

## Listening to the Mandelbrot set

Okay, it's New Year's Eve, I'm overdue to write this week's Web log entry, and the topic I wanted to write about is held up because of (among other things) a package that apparently was stolen from my front porch after it was "delivered" during my vacation and I told them not to mail it so early but blah blah blah... Instead of the exciting mystery topic, let's dig into the back catalog again and think about how to listen to the Mandelbrot set. READ MORE

## Maximizing inharmonicity

As I've said before, I'm interested in slow ambient drones - sounds that smoothly change their texture over a period of time without any sharp boundaries or beats. If you have something like this playing in the background, I want for it to never grab your attention, but when you occasionally do focus on it, you notice something different each time. In my " Totally tubular" posting last month, I talked a bit about making inharmonic bell sounds using a hardware modular synth. This week I've been playing with additive synthesis in the Csound software modular. READ MORE

## Ribbet!

The Leapfrog VCF does not actually go "ribbet," at least not without some effort. But after being told by one friend I showed it to that most people don't want to make real music with their modulars but just want something that goes "uuuuuuuuuuuLAAAAAAAAAAAAARGH" (which, let's face it, is true) I've added a demo highlighting that. Uulargh? Yeah, we've got this. READ MORE

## Preferred values for resistors and capacitors

You've probably noticed that although resistors and capacitors come in many different values, some of which seem like they could be randomly chosen, there's nonetheless some sort of logic to it. You'll see power-of-ten sizes like 1kΩ, 10kΩ, 100kΩ and it's understandable that those would be "round" numbers it might be convenient to use and manufacture... but for instance there also seems to be something special about the number 47, so you see many resistor values like 4.7kΩ, 47kΩ, 470kΩ, and capacitors like 0.47µF and 470pF. Why 47? Why not 46 or 48 - especially when 47 is a prime number and 48 has many small divisors, which would seem like it might be convenient? Here are some notes on the commonly-used numbers and where they come from. READ MORE

## Notes on notes on the plane

For this week's Web log entry, I'm digging into my back catalogue again to look at some of the geometry behind the Black Swan Suite. This is all written up in a PDF file with some musical notation and stuff that doesn't work well in HTML; read that if you want details, but I'll give a taste of the construction here. READ MORE

## Fun with fractal chord progressions

I make a lot of music that is intended to be "ambient": something you can put on in the background and have it create a mood without demanding conscious attention in the foreground all the time. I'm also interested in automatic generation: I'd like to define some kind of pattern or structure and then have machines expand that into a complete composition, and have it be interesting all the way through, without too much human supervision. In this week's entry I'm going to go through a piece called Dharmapala which I wrote a couple years ago using a fractal chord progression. I've used this same technique in several other works, with varying success; I even built a mode for it into one of my homemade modules for realtime performance. But so far I've gotten the best results applying it in offline composition with a lot of careful planning and experiment. READ MORE