Conway’s Game of Life (CGoL)has always fascinated me. It is probably the most well known of all cellular automata and also probably the most intuitive. Yet even simple patterns can turn into complex sequences of shapes, patterns, and noise.
Years ago, when learning about the HTML5 WebAudio API, I came across a fun little demo called Blips of Life by Mike M. Fleming. Use your mouse to draw some dots and then click the triangle Play icon in the bottom left. Great, right? I’ll let you play around with that for a while. Leave it running while you read, perhaps?
This is in 1U Eurorack format.
When it came time to start prototyping new modules for my modular synth, I was inspired to recreate Mike’s work in hardware. I didn’t have exactly the parts to fully recreate his Blips of Life, but using the parts I had in hand I made a prototype.
My version has only an 8×8 grid and only has a major pentatonic scale. The small grid means that there are fewer possible patterns, although not so few it is monotonous. The major pentatonic scale is fine. The largest problem with the prototype is that I used CircuitPython to write it, which has no interrupt support. I love Adafruit – they’re a great company and they design terrific boards. But removing interrupts from their fork of MicroPython has cut several projects short.
The prototype works pretty well and exposed a new design challenge: how do you deal with “games” that end in loops? They’re a subset of steady state patterns in CGoL – a pattern can go “extinct”, “steady”, or loop in a finite sequence. The first case is easy to detect and deal with. If all the cells of the grid are off, repopulate the board. You can detect a steady state by comparing the next board with the previous. If they’re identical, repopulate.
But loops can be any arbitrary length, and can step through rather complex patterns. The only way I know to detect them is to have a list of boards known to be part of or lead to a loop. I’ve got some ideas how to do that either via live loop detection or with a precomputed list of boards. As yet, the performance limitations of CircuitPython really prevent tackling it. I’ll need to reimplement the code in C++ using Arduino. Hats off to Adafruit for supporting both Python and Arduino on their boards.
I’ve continued to work on the optical ray diagram tool prototype. I added a way to measure the effective focal length (EFL) of the lens system. It isn’t automatic, but by adjusting the parameters you can align an intersection at the optical axis and read off the EFL. Obviously this should be a one button click sort of thing, but it is kind of interesting to see how the various parameters affect EFL.
The UI is still very rough and the code is even worse. But I’ve actually been using it!
My main area of interest before going to automation is identifying and coding all the various measurements that you want of a lens. To identify these, I’ve been reading the excellent Applied Optics and Optical Engineering edited by Rudolf Kingslake. Chapter 3, Photographic Objectives, traces the history of the development of lenses from the development of photography to the book’s publication in the 1960s. I would recommend either starting there if you have some familiarity with optics already. If you’re new to optical design, start with Chapter 1, Lens Design. I got my copy from the public library but you can also borrow a digital copy from the Open Library on archive.org.
I realized I should change the example lens configuration in my prototype to a Cooke Triplet after reading Chapter 3. As the book points out, a lot of modern lens designs can be traced to or analyzed as variants of the Cooke Triplet. It is also unique in being only three elements but having performance that is good enough to warrant doing the work of designing one yourself. It is also non-trivial enough that you want an automated tool to design one, so it makes a good example.
The next step will be to add proper measurements of the various aberrations and distortions. I’ll be using worked examples from Applied Optics and Optical Engineering to check the calculations of my tool. The current default configuration is from this student project in MIT OpenCourseWare by Choi, Cooper, Ong, and Smith. I think I’ve already found a discrepancy in my results so my work is cut out for me.
Another source for a worked example is Dennis Taylor’s original patent from 1895. While Taylor invented it, the design is named after the company he was working for at the time – T. Cooke & Sons of York.
I have a few prototype Eurorack modular synth modules in the works. I tend to get them working well enough to be musically interesting and then move to work on the next prototype. It’s not because I don’t plan on finishing them – it’s more that all the biggest questions are answered and I want to move on to the next prototype and answer whatever questions it is trying to answer.
The prototype laid out
This module is based around the Yamaha YM3812 chip, also known as the OPL2. You might know it as part of the capabilities of the AdLib and original Sound Blaster sound cards. Think of the classic sound of the Doom soundtrack – that’s coming out of a YM3812 (emulated or otherwise).
But you can do a lot more than what you hear on the Doom soundtrack – even though I’d be fine if that was the limit of it’s sonic capabilities. FM synthesis is “weird” in the way it can produce wild sounds that are very hard to produce with subtractive synthesis. As an added bonus, the YM3812 has multiple symmetrical channels and is this capable of impressive polyphony. One module isn’t just one voice – it’s 6. Also it has a drum synth mode… It makes sense that it is so capable if you think about all the great PC game soundtracks made with one, but it isn’t what you’d expect to be coming out of a single modular synth module.
The Problem
In the modular world, you tend to break apart and de-integrate as much of the synth chain as possible. This is so you have the freedom to reconfigure the synthesis signal path in wild and fun ways. So a module with not only a complete voice but six complete voices is swimming against the current in how you typically design these things.
How the front and back circuit boards attach together in the prototype
One outcome of such a non-modular module is the matter of the number of possible parameters available. Typically a module might have 1-4 inputs and 1-2 outputs. That’s a gross simplification but gives you an idea of the majority of signal complexity involved. A single channel of a YM3182 has about 16. And then you have 6-8 copies of those – each voice can be configured more or less independently. So we’re talking hundreds of possible inputs.
It has one output.
So on the face of it, this is a bad match. And therein lies the hypothesis of this design. “How can you adapt a YM3812 to the modular synth design norms?” How do you make it understandable to someone thinking in terms of fairly straightforward signal chains? How do you present the configuration of a YM3812 so it matches the mental model of someone used to something like the Behringer Neutron?
A Lone Voice
I can’t do anything about the output space. There is literally only a single pin for the output and there isn’t any access to the individual voices. So right off, I decided that this prototype would be a single voice. That might seem wasteful, but I can use the other voices to “mirror” the main voice to fill it out by slightly detuning them or by playing notes related by harmonics such as octaves or triad chords.
That also reduces the input space. We’re down from hundreds to a dozen or so inputs if we’re only treating this as a single voice. Some existing designs stop there, but I wanted to go further.
Time Variations
There are broadly two types of inputs to a voice: time varying and time invariant. The time varying inputs configure, for example, the way the amplitude of the sound changes over time. In a modular synth, input like this are controlled by other modules. So I decided to discard all time-varying parameters. Parameters like the amplitude of the voice would be modified externally using voltage controlled amplifiers (VCA) just like you would do with a standard modular signal path.
This reduces the input space by half. We’re looking at about 6 inputs. That isn’t bad – there are definitely synth modules with 6 inputs. But I wanted to go further.
Digital Zippers
The YM3812 is a digital chip. All of the OPL series of synth chips are. This is what made them such a great product for Yamaha. It was easy to make a digital chip out of silicon so they could produce the entire sound path out of a handful of parts that would take thousands of separate analog components to replicate. And because it’s digital, it’s very easy to use in a PC sound card. The CPU just sets the input registers of the chip and away you go.
In a modular synth, all of the patch paths are analog: continuous time varying signals between about -10 to 10 volts. To adapt these kind of signals to the YM3812, I would need to digitize them using an analog to digital converter (ADC). But there’s a problem here too – what sort of digital resolution should I use? If I use too low a resolution, the continuous varying signals end up being converted to broad, stair step patterns. It means your smooth subtle varying input gets turned in to sudden chunky sound changes. People call this effect “zippering” because it can cause a sound similar sound when a parameter moves through those discrete stair step values. That isn’t intrinsically bad in the world of analog synths, but you’d like to at least have the option to avoid it.
Some of the input parameters of the YM3812 have a very limited range of possible input values. As an example, the strength of the feedback from one internal voice generator to itself is controlled by just three bits. That’s only 8 possible values! That does not map well to a 20v swing input.
So I took all the parameters that did not have enough bits of configuration available to be used with an analog input off the table. I would still have them accessible, but through manual switches and control knobs. They’d be more for setting the broad mode of the voice, not for use inside the time of a single note playing. That removes a handful more inputs from consideration. In fact, you’re down to only four. That is a completely respectable number of inputs for a synth module. But I wanted to go further.
A page from my notebook where I went through all the possible inputs to a voice
Getting Rational
One of the interesting things about FM synthesis is that a lot of the timbre results from the mathematical ratio between the different frequencies of the oscillators involved. In the YM3812, each oscillator has 12 possible frequency multipliers to aid in defining these ratios. So while there are only 12 values for a given oscillator, there are 144 combinations between the two oscillators of each voice. Twelve steps isn’t enough for an analog input but 144 is fine. So my final reduction was to combine the two ratio inputs into a single input.
I wanted to think about the ratios as actual ratios so I wrote this out by hand. This is the type of thing I think about to fall asleep sometimes.
And that leaves us with just three inputs: one that controls the frequency of the voice, one that controls the amount of mixing between the two internal oscillators, and one that controls the ratio between the oscillators. To put it another way: one controls the pitch and the other two control the timbre. That sounds like a perfectly understandable module. It is still more integrated than you would see in a traditional module where the timbre modification would occur in a separate module (or sets of modules), but it is much closer.
And there you have how I arrived at the final design of the prototype. All other design considerations stem from the decision of which inputs to use: the physical layout, the specifics of how signals map to sound changes, the size of the module, etc.
There are a lot of details I’m glossing over here, and I’ll talk about them more in future articles.
I’ve dug into this ray diagram sketch on CodePen because it’s pretty satisfying to twiddle the properties of the simulation and see how things change. I’ve added some sliders, but beware of the code – it isn’t pretty. I’d say it’s about reached the point of unmaintainability.
Screenshot as of this post
The UI is a total wreck, but you can currently alter all the major parameters. There aren’t any measurements yet, which is probably the next most important feature. It’s fine and dandy to move these virtual lenses around and see how the rays refract, but without the proper measurements it’s not actually a useful tool. Also there isn’t a way to change the lens type or order, and once you can do that, you’ll really want to be able to save and load a given configuration.
And this hasn’t even gotten into the optimization part! That’s the whole purpose!
