euler

Euler’s Identity

By | Modular, Music | No Comments

Another addition to my modular soundscape series… “Euler’s Identity” was recorded in a  single take while I was live tweaking various settings. You can download a lossless version on my Bandcamp page – name your price or enter 0 to get it for free.

The track is named after a mathematical equation that some consider “the most beautiful equation in math”.

 

Euler’s Identity: eiπ + 1 = 0.

Euler’s Identity is an Equation about constants π and e. Both are “Transcendental” quanti­ties; in decimal form, their digits unspool into Infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect Symmetry and closure of the Circle; it’s in Planetary Orbits, the endless up and down of light waves. e (2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore’s law. It’s used to model everything that grows.

What Euler showed is that π and e are deeply related, connected in a dimension perpendicular to the world of real things – a place measured in units of i, the square root of -1, which of course doesn’t … exist. Mathematicians call it an imaginary number. These diagrams are visual metaphors. Imagine a graph with real numbers on the horizontal axis and imaginary ones on the vertical. Exponential function, f(x) = ex, ordinarily it graphs as an upward swooping curve – the very paradigm of progress. But put i in there, Euler showed, and eix instead traces a circle around the origin – an endless wheel of Samsara intercepting Reality at –1 and +1. Add another axis for Time and it’s a helix winding into the Future; viewed from the side, that helix is an oscillating sine wave.The rest is easy: Take that function f(x) = eix, set x = π, and you get eiπ = -1. Rearrange terms and you have the famous identity: eiπ + 1 = 0.

That’s the essence of Euler’s alchemy: By ventur­ing off the real number line into this empyrean dimension, he showed that disruptive, exponential change (the land of e) reduces to infinite repeti­tion (π). These diagrams combine the five most fundamental numbers in math – 0, 1, e, i, and π – in a relation of irreducible simplicity. e and π are infinitely long decimals with seemingly nothing in common, et they fit together perfectly – not to a few places, or a hundred, or a million, but all the way to forever.

You can take this farther, too. If you write that function above in a more general but still simple form as f(x) = e(zx), where z = (a + bi), what you get is no longer a circle but a logarithmic spiral, combining rotation and growth – now both at the same time- These graceful spirals are also found everywhere in Nature, from the whorls in a nautilus shell to the sweep­ing arms of Galaxies. And they’re related, in turn, to the Golden Ratio (yet another infinite deci­mal, 1.61803 …) and the Fibonacci Sequence of Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …).

But the weirdest thing about Euler’s formula – given that it relies on imaginary numbers – is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems into more tractable algebra – like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digi­tize music, and it tames all manner of wavy things in quantum mechanics, electron­ics, and signal processing; without it, computers would not exist.

fe

Metanoia

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This is a little modular soundscape I made one night. Sometimes it’s nice to sit down with just the modular and see how far you can push it, I’m always amazed at the huge sound that can come out of my little setup.

If you would like to download a lossless WAV version you can do so on my Bandcamp page. Just enter 0 to download it for free, if you have some spare change lying around please feel free to buy it. All the money made from my music goes directly back into gear upgrades etc so helps me make more music.

modular1

Going Modular 2: The Plan

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In Part 1 I talked about my reasons for entering the modular world and also where I’ve been gathering my information to get started. In this article I’ll talk a little about the basic things I’ve learnt as a complete novice and my first steps towards assembling a case. There is no doubt that on the surface Modular Synthesis can look too hard to delve into, but the more you talk to experienced users and research the clearer things become. I’ll try to impart some of the basics to help get you started. I am still learning about modular synthesis so please keep that in mind if you are a more experienced user reading this article, this is entirely my perspective as someone new to the modular world.

My modular is going to be Eurorack format, I touched on why I selected this format in the previous article – the small size, affordability and large selection of modules were the primary reasons for choosing this format.

Eurorack Module Specifications

Eurorack format calls for modules of 128.5mm height. The height is referred to as “U” and one module is 3U high (3U = 128.5mm).

Horizontal width is measured in “horizontal pitch”, where 1 HP = 5mm.

3.5mm phone jacks are used for interconnection or “patching”.

Eurorack requires ±12V power, in addition to +5V required by some modules. The format uses ribbon cables for power and a two-row ribbon cable connector containing either 10, 12, or 16 pins. The 16-pin connector uses some of the extra pins to distribute control voltage and gate signals from a keyboard to the modules via a common case connection.

Eurorack Modules

If you visit some forums like the Muff Wiggler forum you will come across plenty of people asking the question “what do I need to get first?” or “what is the best basic setup to start with?” – there will no doubt be pages of debate between seasoned modular users about how you should begin. The truth is it really depends on what you want to achieve, you need a plan.

Because you are assembling your unit from scratch you can take it in any direction you want, maybe you want to replicate your favourite vintage synth or explore drones or drum machines, or perhaps you are interested in creating an effects unit for processing audio. Your goal will decide how you begin. I want to add another synthesiser to my setup as I feel my noise sources are currently a little limited. So this is the direction I’ll explore in this series of articles.

To start off, I had a look at my current hardware synthesisers and the various sections that help generate and shape the sound, this provides a direction for the type of modules you will need when assembling a modular synthesiser, things like Oscillators, Filters, Envelopes, LFO and some controls to help shape or sequence the sound. Here are some common terms you will come across as you start to shop around for modules, I’ve included the Wikipedia definitions but don’t worry if they don’t make a lot of sense at this stage! I’ve put some notes under each one about what it does in simple terms.

  • VCO – Voltage Controlled Oscillator, a continuous voltage source, which will output a signal whose frequency is a function of the settings. In its basic form these maybe simple waveforms (most usually a square wave or a sawtooth wave, but also includes pulse, triangle and sine waves), however these can be dynamically changed through such controls as sync, frequency modulation, and self-modulation.
    Translation: It is a sound source, it produces sound.
  • LFO – A Low Frequency Oscillator may or may not be voltage-controlled. It may be operated with a period anywhere from a fortieth of a second to several minutes. It is generally used as a control voltage for another module. For example, modulating a VCO will produce frequency modulation, and may create vibrato, while modulating a VCA will produce amplitude modulation, and may create tremolo, depending on the control frequency. The rectangular wave can be used as a logic / timing / trigger function.
    Translation: It makes the sound move or turn on and off.
  • VCF – Voltage Controlled Filter, which attenuates frequencies below (high-pass), above (low-pass) or both below and above (band-pass) a certain frequency. VCFs can also be configured to provide band-reject (notch), whereby the high and low frequencies remain while the middle frequencies are removed. Most VCFs have variable resonance, sometimes voltage-controlled.
    Translation: It shapes the sound by cutting or accentuating highs and lows.
  • VCA – Voltage Controlled Amplifier, is usually a unity-gain amplifier which varies the amplitude of a signal in response to an applied control voltage. The response curve may be linear or exponential. Also called a two-quadrant multiplier.
    Translation: Modular Synthesisers don\’t have a volume knob, so this is what controls how loud a signal is.
  • Mixer – a module that adds voltages.
    Translation: It takes a bunch of different signals (sounds) and mixes them together.
  • ADSREnvelope – Sound synthesis techniques often employ an envelope generator that controls a sound’s parameters at any point in its duration. Most often this is an “ADSR” (Attack Decay Sustain Release) envelope, which may be applied to overall amplitude control, filter frequency, etc.
    Translation: A VCO outputs a constant sound, so an envelope is used to shape the sound and decide how long it plays etc.

Essentially what you need to get started is a  VCO (something that makes noise) or multiple VCO\’s depending on your budget, and some modules to help shape and contour the sound into something musical. An entry level setup might look something like this:

  • Minimal setup: VCO + 1 VCF + 1 VCA + 1 ADSR + 1 LFO + MIXER
  • Better minimum setup: VCO + 1 VCF + 1 VCA + 2 ADSR + 2 LFO + MIXER
  • More advanced setup: VCO + 2 VCF + 1 VCA + 2 ADSR + 2 LFO + NOISE + RING MOD + 2 MIXER + SAMPLE AND HOLD

This will give you enough to start making sounds, from there you can add some of the wonderfully insane modules that developers are dreaming up to bend and twist the sound in seemingly unlimited ways.

Case and Power Supply

Another component you will need to consider is the kind of case/box you are going to mount your modules in, and how they will be powered. Cases can range from a simple wooden box (sometimes called a skiff or boat – a boat is slightly deeper) right through to more expensive flight cases that have built in power supplies. If you are starting out with a limited budget you will probably want an entry level kit like the TipTop Audio Happy Ending kit or you will take a DIY approach and build your own. I decided to take a DIY approach and build a wooden case to fit my specifications.

The main things you will need to build your own box are some modular rack rails (also known as Z Rails – TipTop Audio also manufacture these) and a power supply. There are some great module based power supplies available to power small cases and make setup easy for us newbies. The important thing is to ensure that it is +12V with +5V available as a lot of new modules (particularly digital) require this. Power Modules simply mount onto the front rails like a standard module and have a “bus board” that runs through the back of your case allowing around 10 modules to be connected.

I opted for the TipTop Audio uZeus as a lot of people seem to use them with very few complaints. Because the uZeus is designed to power smaller modular’s I decided to use 104HP rails (528.3mm) and make it a single row of modules (3U) – this info allows you to work out the internal measurements you will need to mount the rails correctly (where 3U = 128.5mm). How the case  looks outside of that is entirely up to you. I was fortunate enough to have some solid Rimu wood available so my case is built entirely from this.

Module Brands

There are some amazing modules out there and some really innovative companies producing them, so what modules should you get? I took some time to check out the main companies making Eurorack modules  (You can check out a short list in Part 1) and watch the demo videos and/or audio demos that are available on most of their websites. YouTube is also a great resource, if you find a module that peaks your interest then search for it on YouTube so you can see some examples of it in use.

I stumbled across Mutable Instruments when I was doing my initial research and their modules really stood out as being innovative and creative, something that would help me achieve a great sound. They also tend to be multi-functional and contain hidden features so you can do a little more with them. I took the plunge and ordered a few of their modules to get started, my first case will be primarily filled with Mutable Instruments modules but I’d like to expand it with some TipTop Audio, Harvestman and Make Noise modules in the future – when you go modular it seems to become a life long obsession/sickness so so I have no doubt I’ll be building another box sometime soon.

Control

Modular synthesisers use control voltage to trigger sound – you may also see this referred to as CV/Gate. You can use devices like the Arturia Minibrute, Microbrute or BeatStep or some of the Elektron range to send CV/gate signals or you can get a MIDI to CV converter and use your preferred Midi controller to play/sequence your modular. Hopefully my case will be ready in the next few days and I can start putting my modular together, this will be the focus of Part 3 in this series. Please feel free to ask questions or discuss anything I’ve talked about in the comments.

modular0

Going Modular 1: The Idea

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I’ve been wanting to add another sound source to my studio setup for a while, I’m really enjoying the Microbrute and how it has changed my creative process so getting another analog synth seemed like a good idea. The more I looked around at affordable mono synth’s the more I found myself questioning if I really needed another one… the only other obvious way I could improve my setup would be to add a larger polyphonic analog synth like a DSI prophet 12 or a Moog but they are at the more expensive end of the scale and I can’t afford to buy one. As I looked around I started to wonder what I could do… how could I get a monster synth without having to sell my car to finance it.

I’ve long admired Modular synthesisers but never really thought about putting one together… it always seemed like it would be far too complicated and yet another huge learning curve to tackle. If you aren’t familiar with Modular synthesisers they are basically a synthesiser that is assembled from individual modules, each module has a specific purpose, some modules generate noise while others help you craft that nose into something musical. They look like a cross between a 1940’s telecommunications patch board and a mad scientists laboratory  – A giant machine covered in cables and blinking lights that generates weird squeaking and fart noises. It appealed to me for a few reasons – for a start you can assemble something entirely custom with modules that will all work together even though they are made by different manufacturers. Secondly, you can spread the cost out and make smaller purchases more frequently rather than having to outlay a lot of cash initially (of course if you can afford this then by all means spend away!) and you can also assemble it in a way that best fits the space you have available. You aren’t restricted by the box the manufacturer gives you like a standard synthesiser – you create it and add to it over time to build the sound you want. Obviously if you are like me and working to a budget it is going to take some time to create your dream setup but if you get a few key pieces at the beginning you will be able to start making some noise.

There are a few different formats available but the Eurorack format appealed to me the most. The modules are small, widely available and for someone working to a budget this format works well. There also seems to be a lot of innovation happening in the Eurorack format at the moment with some really inspiring modules coming onto the market so it is a great time to go modular. I have a basic understanding of synthesis and how the various sections on a synthesiser fit together to shape a sound, but I felt like I needed to learn a lot more before I got serious about all this modular stuff. There is a great series of articles at Sound on Sound that I would recommend, I am still working through them at the moment (it’s split into about 30 parts) and for the majority it confuses more than enlightens me but that seems to come with the territory.

I also found a great community of modular heads at muffwiggler.com who are really friendly and helpful. I posted about what I wanted to achieve (it is important to set some goals right away) with a modular and how I was thinking of going about it and received a lot of advice about what I should focus on to get the basic building blocks in place. Another great resource is Modular Grid – This site gives you the ability to sketch out your potential setup and create a virtual modular, it features different formats and modules from most of the major manufacturers. A few minutes on this site and you will start to see how your modular can get out of control really quickly! 17 Modules later I realised it might be time to cut back.

If videos are more your thing then there are some introductory series on youtube, Flux has just started an intro to Eurorack series that seems like a good place to start. You are going to have to spend a lot of time reading and learning and asking stupid newbie questions but the modular community is cool with that so don’t be afraid – everyone needs to start somewhere. It’s also a good idea to look at some manufacturers websites so you can get a feel for the different flavours that are out there, here are a few that I have found so far:

Pittsburgh Modular
Mutable Instruments
Harvestman
Doepfer
Analogue Solutions
Tip Top Audio
Intellijel
Make Noise

So after a fairly intensive round of research and advice from some Modular masters I feel like I am starting to get my head around the world of modular synths, some things are still a complete mystery to me but hopefully I’ll be able to write more in-depth about this stuff soon and use a language that anyone can understand when talking about Oscillators, VCA’s, VCFs and all the crazy terminology you need to become fluent in. Here is a taste of the creative potential that these instruments have…

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