Euler’s Number and the price of fish

Sonify thisA short listening exercise for you:  This morning, I turned two streams of data into music. One of these ditties expresses the first 48 digits of the transcendental number e through the medium of handbells. The other ‘plays’ the metric tonnes of salmon sold, day by day, in the second quarter of 2010 on the London Stock Exchange.



Any idea which is which?

The popular science press is awash with music projects of this ilk – a genre of sound art known as ‘data sonification’. If you’re looking for a sonification project that’s guaranteed to go viral, first identify a datastream with sufficient exotic charm to lure the discerning journalist. Anything far too big or small to imagine is particularly fitting – data from quark collisions, junk DNA and coronal mass ejections are good – numbers of double-decker buses won’t cut it.  Spit out this datastream, map it number by number onto notes, chords or rhythms et voila! You have a texbook sciart project: a string of inscrutable music with science running all the way through it like a stick of Blackpool rock.

I have to be honest and say I’m unconvinced by some of the data sonification oeuvre. Arguably, music from data can be an entertaining way to get some more arcane research projects into the public consciousness. These catchy numbers, for instance, recently helped to make the ATLAS detector talk of the town – and that’s no bad thing. Of course sonification is nothing new. The Geiger Counter sonifies the presence of ionising radiation. And musical instruments can be thought of as machines for sonifying human gestures (especially the theremin). Admittedly, I occasionally dabble with sonification. I’ve recently been grapping with some optical flow algorithms, for instance, which I’m using to make music from the flight of butterflies in London Zoo. I think sonification projects should point up the salient attributes of a body of data, revealing barely discernable patterns and relationships, for instance. And – more importantly – should somehow express the purpose of the datastream (tricky to do – but therein lies the art). Check out these sonified sorting algorithms for example (by @andrut).

Sonified sorting algorithms (@andrut)

They’re let down by some cheesy general midi sounds but nevertheless let you hear a jumble of data gradually being sorted from big to small.  Fascinating – and very different to the species of sonification that I personally find irksome. I have a problem with sonifications that express very little about the data stream under scrutiny, apart from the fact it’s made up of numbers.

Take my hastily prepared bell tune, for example, a digit-by-digit sonification of Euler’s Number (e). In case you’re wondering, it’s song number 1 on the list. There’s enough to say on e, its cosmic significance and its applications, to fill a library. Just for starters, this beautiful number can be used to express the decay of isotopes or capacitive charge or the growth of populations. Embedded in integrals, it enables us to analyse the world using new geometries (for instance with Fourier and Laplace transforms – indispensible tricks that make me feel like I’m putting on my psychonautic maths glasses). But when I hear tune 1, all I can sense is that e is made of a seemingly random stream of digits.* My ‘Euler music’ might as well be about the price of fish.

This technique I’ve used isn’t dissimilar to one which has been heard quite a lot this week, as musicians have been sonifying pi, digit by digit, for ‘Pi Day’. You can hear various entries on YouTube (Ryan has written about a rather fine example on this blog). Yes, many of them sound beguiling – but do they express anything beyond the fact that pi is irrational, like e? If you close your eyes, do you sense anything, erm, circly? Listening to these paens to pi, I feel there’s something slightly disappointing about the mappings many of the composers have used to turn numbers to notes. Somehow, their digit-by-digit sonifications reduce pi, a number of universal significance, into something quotidian. Sadly, many express the number in decimal then map it onto an eight-note, diatonic, tempered Western scale. About as cosmic as representing the surface of Jupiter with Lego bricks.

Maybe my analysis is unfair, too prescriptive or simply oversensitive. Perhaps there’s nothing wrong than music whose sole aim is to make the observation that pi is made of a never-ending, random* stream of digits. That, in itself, may be a sufficiently interesting music project. But here’s the rub – these numbers are random and random music is boring. Don’t take my word for it. There’s a raft of psychology (to be filed under The Bleedin’ Obvious) that relates arousal to the complexity and novelty of art and music. This work is summed up neatly by the Wundt Curve (1974). Put simply, we find music boring if it’s too predictable or too random. We seem to love the stuff in between – the music that presents us with that glorious, sonic game, where musical patterns sometimes meet, sometimes defy, our expectations.

So, that sums up my gripe about certain kinds of data sonification. Of course, regardless of its origins, responses to music are largely down to personal taste. With this in mind, I felt I should seek a second opinion on this thorny topic. So I went to the experts: MC Zirconium and DJ Bongoboy, aka Project Moonbase, a duo of self-styled ‘retro futurists’ with an encyclopaedic knowledge of scientifically-inspired music. Unsurprisingly, they’re no strangers to the world of data sonfication, a genre Zirconium has described as ‘the extreme sport of music’. Over a salmon tikka, Bongoboy summed up their approach: ‘“More important than “have we discovered the Higgs boson?” or “how did the universe ultimately begin?”, the key questions for us are 1. Does it have a beat? and 2. Is there a corresponding ukulele tuning?”‘

‘I like my sonification to fill me with terror’, Zirconium added. ‘If I am listening to radiation emanating from distant parts of the universe, I want to feel horrified by the scientific majesty of it all.’

Well said Sirs! With that in mind, I’ll sign off with this little treat: my Euler’s Number music mixed with my sonatina on salmon sales. Put on your unitards, tune your ukeleles into the cosmic background radiation and imagine a mighty salmon dancing in spectral space. Dance my pretties, dance!

Fish Euler (music for dancing)


* They are random – but I’m lacking the space and the maths chops to describe the species of randomness we are dealing with here.

My next article may be of special interest to Steampunk friends. It features a short chat with a man in a tall hat that made me change my opnions on the way the artists cut up and resample past technologies.



A musician, writer and roboticist, Sarah Angliss (Spacedog) is known for her dreamlike performances, incorporating Edison phonographs, theremins, animated vent dolls and other curious machines. Sarah regularly performs live and is particularly known for her skills on the theremin and the musical automata which she makes to accompany her on stage. Sarah takes a keen interest in acoustics, the psychology of listening and in unusual physical interfaces. Her solo and collaborative work has explored musicians’ attitudes to the first drum machines and samplers, the uncanny valley, caged birds as primordial sound recorders, the stranger obsessions of early adopters of phonography and the reputed psychological effects of infrasound.

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  1. Math nerd checking in:

    Pi is believed to be random in that it is believed to be a normal, transcendental number.

    Normal, in this case, means that the digits 0 through 9 appear in equal proportion to each other (1/10 of the time)

    Trancendental means that it never ends and never settles into any kind of pattern.

    Being normal and transcendental together make it effectively random although to my knowledge this has not been proven. It has just been demonstrated up to several trillion decimal places.

    Euler’s number (e) is also believed to be transcendental and normal and hence, random in the same sense as pi.

    For those that are curious e = the limit of (1+ 1/n)^n as n approaches infinity. e^x is the only function that is its own derivative.

    I love math… it’s so elegant.

  2. This was hilarious, and spot-on. Cheers.

    Let it be known there is always a beat and a corresponding ukulele tuning. Always.

  3. Being normal also means that every possible numerical subsequence appears somewhere in the complete sequence, an infinite number of times. (And not just in base 10, but in every base.)

    No, it has not been proven for either pi or e, though nobody really doubts it. It’s easy to prove that the vast majority of numbers are normal, but it’s extremely difficult to prove that a particular number is normal. To my knowledge it’s only been done for a few special cases.

  4. Thanks Ryan and breadbox for the fascinating mathematical add-ons.

    Yes, the digits of e and pi both offer fascinating example osf randomness (but that doesn’t make their digits sound interesting!) 😉

  5. Nice article. The interpretation is all I guess – I don’t know if data sonification should automatically be called music, that’s a very media friendly thing to do, and it works. Journo’s are less interested in writing about a new sound or noise.

    How can anyone define what music is for anyone else? I have spent today listening to one of my fluorescent bulbs dying (complete with flashing). It took me a while to realise I hadn’t put any music on in my studio and was tuning in to it. It seems random but now I’m listening closely it resembles the pattern that Rudd make when it rains on a lake*… 😉 My personal listening preferences are for noisier sounds rather than musical sound, but not everyone feels the same (although I didn’t dance to your bells, I’m sorry).

    Perhaps sonification should be seen as the inspiration, the first sketches or base layer of something rather than the end result. Or simply as a way of attracting attention to an otherwise (media-perceived) dull scientific study. Because you’re quite right they can sound shite.

    Saying that data visualisations used to be pretty darn sketchy and now there’s a whole raft of really very beautiful imagery being generated from information… there is hope for sonification yet!


  6. I was made aware of this yesterday at an art & sci conference (nasn): the vOICe – video image sonification to enable partially sighted and blind to ‘see’. The presenter stated that he thought “it wasn’t a very nice sound” but I liked it.

    I wonder if those using it hear it as music, apparently some users cease to hear it after a while as it becomes their sight.

  7. Yes – sensory substitution is a fascinating area of study – and one where the sounds *have* to bear some meaningful relationship to the sense data they’re representing.
    BTW The work of Bach-y-Rita is also very interesting. He made one of the earliest sensory substitution devices – one which represented images as pins on the chest:

    ..I love this photo of his early prototype. This technology has evolved into the ‘brainport’ which represents sense data using an array of tiny electrodes on the tongue.

  8. I just stumbled across this while looking up “normal transcendental number” because of today’s comic. In happy coincidence, I’m interested in math and in aleatoric “music”.

    I’d like to expand a little on Ryan’s math comment: “trancendental” is more specific than “never ends and never settles into any kind of pattern”.

    For example, the decimal expansion of the square root of 2 never ends and doesn’t have an infinitely repeating block of digits (so is irrational), and may be normal, but it is not transcendental because it is the solution to a polynomial expression with integer coefficients (x^2 = 2).

    On the other hand, the “Champernowne constant” (0.123456789101112…) is strongly patterned, but is transcendental, and if I understand “normal” correctly, is normal as well.

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