Sloane’s gap: cultural influences in mathematics

The Online Encyclopedia of Integer Sequences (OEIS) is a useful resource for my Tintinnabuli Mathematica music project. For each sequence, the OEIS lists the first few terms and describes the algorithm that produces it. I use these algorithms to generate number sequences of various types (regular, fractal or chaotic), which I then translate into musical notes and output as MIDI files using a program written in Mathematica.

The OEIS was created by Neil J.A. Sloane, first in the form of a handbook, now a collaborative online project which currently lists over 200,000 sequences. Each entry is assigned a number – for example, A000001 is “number of groups of order n” and the sequence of prime numbers is A000040. Each sequence can also be seen as a graph and heard as a sonification. A thousand of these graphs have been put together as a video showing the first thousand terms of each sequence, with a sonification of Recaman’s sequence (A005132) as the soundtrack:

The sequences in the video were chosen because they were interesting or mathematically important in some way. This interestingness could be intrinsic (e.g., it relates to a known sequence, or it makes a nice pattern) or relative to the other sequences chosen (e.g., it is different to the others). It turns out that this interestingness also affects the OEIS itself as a whole. Using mathematical tools to investigate the OEIS reveals that this ostensibly objective and mathematical project is also shaped by cultural factors.

Philippe Guglielmetti (2009) investigated the frequency N(n) with which each integer n appears in the OEIS. He plotted this as a graph, showing on a logarithmic scale how many times each number up to 10,000 occurs amongst all the sequences in the OEIS. As we would expect, smaller numbers occur more frequently than larger numbers, creating a logarithmic downward curve.

Guglielmetti observed that the integers are divided pretty clearly into 2 groups, noting that there are fewer “interesting” numbers in the upper band, and more common “uninteresting” numbers below. The interesting numbers include the prime numbers (red in the image below), numbers of the form a^n (green), and highly composite numbers (yellow). There are relatively few “slightly interesting” numbers in the middle, and this area is known as “Sloane’s gap” after the creator of the OEIS.

A study by Gauvrit, Delahaye and Zenil (2011) investigated this, noting that “This gap is unexpected and requires an explanation.” The study takes an algorithmic information theory approach, which uses the frequency of occurrences N(n) of each integer n in the OEIS as a measure of a number’s algorithmic complexity. This approach accounts for the overall distribution of the numbers, but not the gap:

The theory of algorithmic information thus provides a good description of what is observable from the curve N(n). That justifies an a posteriori recourse to the theoretical concepts of algorithmic complexity in order to understand the form of the curve N(n). By contrast, nothing in the theory leads one to expect a gap like the one actually observed. (p.12)

Gauvrit et al. hypothesize that the gap is the product of the social aspect of the OEIS, caused by mathematicians’ greater interest in numbers of high and low complexity, and lesser interest in those of medium complexity.

This study also considers which is the smallest number that doesn’t appear in the OEIS – that is, the smallest integer n for which N(n) = 0, or the first number that is mathematically un-interesting. Paradoxically, this makes it quite interesting. Appropriately, the idea of the smallest uninteresting number was discussed in an episode of QI (broadcast November 2011), which pointed out that such a number is culturally, rather than mathematically, interesting. In August 2008, the smallest number not to appear in the OEIS was 8,795, and in February 2009 it was 11,630, changing as new sequences are added.

I first heard of Sloane’s gap in this list of examples of unexpected mathematical images:

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Tintinnabuli in ChucK

With a little input from my own work in programming tintinnabuli music, Arve Knudsen  has created a real-time generative piece using the ChucK programming language. Although it’s based on a short and simple ascending sequence of notes, the algorithm generates enough variation in timbre and register to keep it interesting:

Arve has written about a talk he gave on the ChucKJS project here:

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A retrospective look at some of the good things I’ve listened to during 2014, in no order.

Powell – Club Music (Diagonal Records)
A belter this one. Includes a collaborative track with Russell Haswell, who also released Foxy / One Take No Dub Edit on Powell’s Diagonal label.

TCF – 415C47197F78E811FEEB7862288306EC4137FD4EC3DED8B (Liberation Technologies)
Brilliant album from TCF, which contains a hidden spectrogram image encoded in the last track (see this post for images).

Lorenzo Senni – Superimpositions (Boomkat Editions)
Pointillist trance’.

EVOL – Wormhole Shubz (Entr’acte)
Comes with a booklet on the ‘What the…?’ preset (more commonly known as the ‘Hoover’ sound) for Roland synths developed by Eric Persing. Also makes for nice spectrogram images – see here.

Paul Jebanasam – Rites (Subtext Recordings) / Live at St John Sessions 220514

FIS – Speech Spirits (Loopy)
Odd, sparse deconstructed dance music, kind of like Lee Gamble, who also had some good stuff out this year. Oren Ambarchi remix, “The Nagger”, is great – muted guitar plucks (‘Edge of Seventeen’ / ‘Eye of the Tiger’ style) with off-kilter timing. Quixotism by Oren Ambarchi is another good 2014 album, a bit like Music for 18 Musicians.

Marc Weidenbaum – Selected Ambient Works Volume II ( 33 1/3) /
Aphex Twin – Syro (Warp) /
Dave Noyze – Syrobonkers! interview with Richard D. James
Turned out to be a bumper year for Aphex Twin stuff. First Marc Weidenbaum’s excellent 33 1/3 book about SAW II, then Syro, then a two-part rambling interview by Dave Noyze with an accompanying set of 30 or so previously unheard tracks, both of which were soon taken down. Syro is a grower. Its complex arrangement of constantly morphing patterns and multiple layers of evolving sounds means that there’s no obvious catchy hooks, but is also what makes it good for repeated listens.

Ø – Konstellaatio (Sahko Recordings)

Sculpture – Plastic Infinite (Tapebox)

 Goto8o – Custom8 (self-released)
A custom-order album in the style of your choice. I asked for an album based on complexity, where the tracks go from the simplest to the most complex. Apparently that was the most difficult order to fulfil.

Karl Fousek – Relative Position of Figures (Phinery)

An album that I discovered from other people’s best-of-2014 lists is Andøya by Eric Holm (Subtext Recordings). Made from contact mic recordings of a telegraph pole in the Arctic, like Aino Tytti’s field recording of a radio mast (Touch radio 96), but these haunting sounds are processed and arranged into musical compositions similar to Chris Watson’s later work such as El Tren Fanstasma: The Signalman’s Mix.

Finally, I’m very pleased that my 2014 releases have been included in other people’s end-of-year lists:

Tintinnabuli Mathematica Vol. I (Runningonair Music) in lists by Stationary Travels and Phirnis.

Bits EP (basic_sounds) – in Vuzh Music’s list of netlabel releases and in the accompanying 2014 Netlabel Mix.

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Exploring the Adjacent Possible in Music

A few things seemed to have aligned themselves recently, so here’s an attempt to cobble them together into a meaningful form. This is an edited version, with more references, but without a proper conclusion.

Richard D James has answered 25 questions from musicians, DJs and producers in an article for the German magazine Groove. In response to a question from Mate Galic, a founder of Native Instruments, about the use of hardware and software, RDJ reveals his involvement in a new form of evolutionary software for making music:

But I’ve actually recently hired a Chinese programmer to make a music software for me. It’s taking the concept of mutation into music software. You give the program some sounds you made and then it gives you six variations of it and then you choose the one you like most and then it makes another six and it kind of keeps trying to choosing the variations by itself. It’s a bit like that, but more advanced, but basically it starts with a sound, analyzes it, then does different versions of variations. It randomizes, it compares all of them to the original and then it picks the best one. It sounds totally awesome, but it needs to be tweeked a little bit. I will continue with this. I have a whole book full of ideas for software and instruments.

It seems to me that this is an example of an approach to music making that I’ve written about previously in the post Computer Fatigue and the Rise of Sonic Complexity, which discussed a trend of moving away from digital music making as a way of seeking richer and more complex sounds. These musicians (mainly those involved in electroacoustic music) are turning away from the computer-generated sounds because they are too clean, precise and uniform. RDJ expresses the same attitude when he says that he favours analogue synths because “everything on the computer just sounds perfect”. But some musicians have decided to stick with the computer and face the challenge of creating more complex sounds. Computer musicians have at their disposal a method described by Manuel De Landa as ‘The Virtual Breeding of Sound’ (2008, in: Miller (ed.) Sound Unbound, pp.219-226). De Landa describes “topological” transformations of sound as a means of exploring the search space of musical possibilities, in a process analogous to evolutionary variation and selection.

Clearly, current uses of genetic algorithms display only the tip of an iceberg, the exploration of which will perhaps take decades. This is a sobering thought, preventing us from being overenthusiastic about our current capabilities of breeding sound, but simultaneously it is a source of excitement at all the unknown domains waiting to be discovered.

Genetic algorithms were popularised by John Holland’s (1975) book Adaptation in Natural and Artificial Systems. A friend of mine, Honar Issa, gained a PhD by developing genetic algorithms to find the optimum construction of steel portal frames for industrial buildings. Richard Dawkins also wrote about this type of evolutionary mechanism in The Blind Watchmaker (1986) with the idea of “Biomorphs” – simple virtual creatures where one organism gives rise to 8 offspring with slight variations. Selecting one of these offspring as the new parent then generates 8 new variations on that theme, and so on. There are a few examples of interactive Biomorph applications, such as this one at By picturing a (real) organism, you can fairly quickly arrive at something that resembles and ant or a crab, for example (these virtual creatures have bilateral symmetry, as do most higher lifeforms). Thus the system offers a means to explore genetic space through a process of artificial selection.

It is important to note that evolutionary search space is not pre-defined. In the book Investigations (2001), Stuart Kauffman explored how this space of possibilities expands and develops as its constituent organisms themselves evolve and multiply with variations. Kauffman called this co-evolving space of emerging possibilities the “adjacent possible”. This is the space of possibilities that opens up with each new genetic variation. For example, ornamental feathers that covered the skin of prehistoric reptiles opened up the possibility of flight. As the function of the feather changed, so did the potential for new behaviour. Kauffman also suggests that this describes the development of human economies. I think it also applies to the development of art and music. More recently, the mathematician Steven Strogatz and others published a paper called The dynamics of correlated novelties that proposes a mathematical model for the way we explore the adjacent possible to find new things. Strogatz et al. suggest that this models the way in which we find new music, for example, where we find new things that are similar to or related to the music we already know of. In this way, our cultural capital (Bourdieu) continually evolves and expands. The path taken by evolutions from one form to another

There is a subtle but significant difference between finding new music and making new music, however. In the first case, it is a matter of exploring a space that already exists; in the other, new possibilities are created with each new development. The difference is similar to the distinction that Margaret Boden draws between two types of creativity in The Creative Mind: Myths and Mechanisms (1990). In a summary of that book (PDF), Boden says:

What you might do, and what I think you should do in this situation, is to make a distinction between “psychological” creativity and “historical” creativity. (P-creativity and H-creativity, for short.) P-creativity involves coming up with a surprising, valuable idea that’s new to the person who comes up with it. It doesn’t matter how many people have had that idea before. But if a new idea is H-creative, that means that (so far as we know) no-one else has had it before: it has arisen for the Œfirst time in human history.

P-creativity is similar to finding new music because it involves personal, but not historical, novelty. In contrast, H-creativity describes the creation of new music. Algorithms for the P-creative discovery of new music are now quite widespread, such as those offered by Last FM, Spotify, and iTunes Genius. But algorithms for the H-creative creation of new music are less common. Aphex Twin’s music mutation software is one example, which realizes De Landa’s idea of genetic algorithms for topologically transforming and breeding sounds. Autechre and Rashad Becker also appear to use something like this kind of approach to creating new music by exploring the adjacent possible through the evolution of sound. Maybe there’s potential for a crossover between the creative use of these algorithms by musicians and the analytical use of statistics and information theory (e.g. by Strogatz and others) to describe the creation of new music.

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Lab Doodle

A friend who I went to art college with found some of my old artwork recently, which I didn’t even know she had. It’s no work of art, just a photocopy of some doodles that I did during quiet periods when I was working in the laboratory at Mansfield Brewery. They’re drawn in the back pages of the lab notebook that was used to record the CO2 content of samples of beer, and they probably date from around 1999 to 2001 when the brewery closed. It’s funny to see these old images again. There are fractals, references to music, literature, philosophy and religion, and lots of biological things: insects, a cross-section of a duck, yeast cells, snake skull, locust, crab, mice, thylacine, dandelion seeds. There’s also an ‘alphabetti spaghetti’ game – every letter is hidden somewhere amongst the doodles.

lab_notebook_crop2We can look at this in terms of visual complexity. This image has a roughly uniform distribution of ink to page. It is quite detailed, with a variety of marks and shapes, but with little depth. The small images occupy this space more or less evenly but they bear little meaningful relation to each other, except for some alignment of images on a slight diagonal on the left-hand page. Unlike a well-formed sentence where a set of words is arranged into meaningful information, these small pictorial units are not grouped into higher organisational units, neither as aesthetic information nor as semantic information.

In terms of complex systems theory, the properties of this image are related to chaotic systems and the things that chaotic systems produce. The pictorial arrangement of the doodles is like the distribution of pebbles on a beach, which is a result of a chaotic system in the form of the non-linear fluid dynamics of the ocean. Different sizes, shades and shapes are distributed without any order – a non-repeating pattern. These kind of patterns can provide restful background material that isn’t too distracting or too dull, but which ultimately fails to retain aesthetic interest. The image has little artistic value because of this lack of meaningful organisation (not to mention lack of skill), but it is quite interesting as a sedimentation of passing thoughts and deeper subconscious currents.

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Encoded Message

This week I’ve been listening to the latest release by TCF (Lars Holdhus) on Liberation Technologies, entitled 415C47197F78E811FEEB7862288306EC4137FD4EC3DED8B.   Like his previous releases, each track title looks like an MD5 checksum – a string of letters and numbers that encodes the digital fingerprint of a file. A description of the release on suggests that “there are hidden keys in the track’s titles that unlock the full meaning of the work”. I haven’t been able to decode those titles, but I did notice something unusual in the visualizations of the last track – ’97 EF 9C 12 87 06 57 D8 B3 2F 0B 11 21 C7 B2 97 77 91 26 48 27 0E 5D 74′. Playing in foobar2000 and using its spectrogram visualizer, some unusual shapes are visible in the mid to upper frequencies that look almost as if they are cut-out. Here’s a foobar2000 spectrogram of the whole of track 7, which is just 29 seconds long (click to enlarge):

TCF-track7-fThe five vertical red/yellow features are a slowed-down voice saying the word ‘slow’, but what caught my attention was the pattern in the background – the dark blue shapes with sharp edges. I wondered whether TCF had used some kind of spectral manipulation tool that allowed for the frequencies to be cut up in this way – like SPEAR or FreqTweak, for example. Then it occurred to me to that the shapes are squished at the top because this is a spectrogram plotted on a logarithmic frequency scale. So I made another spectrogram on a linear frequency scale, using the free software Sonic Visualizer, and after a bit of tweaking this was the result (click to enlarge):

TCF-track7-3It’s rare to see an image so clearly in a spectrogram. It’s clear enough to be able to search for visually similar images online, which reveals that it’s a photograph of violence during political protests in Greece, 2011:

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Wormhole Spectrogramz

EVOL’s music, more often than not, looks great when visualized. The re-release of Wormhhole Shubz on Entr’acte is no exception. Below is a set of spectrograms* showing the first 35 seconds of each track. Click to enlarge.

EVOL-WS-1 EVOL-WS-2 EVOL-WS-3 EVOL-WS-4 EVOL-WS-5 EVOL-WS-6 EVOL-WS-7 EVOL-WS-8 EVOL-WS-9* Spectrograms represent the time-frequency domain of sound; they show time running from left to right, and frequency from low to high in the vertical direction. Amplitude (volume) is the ‘third dimension’, usually represented by brightness or tone, e.g. black for silence and white for the loudest sound, with greys in between. In these images, amplitude is represented with a colour palette, where red is the loudest, fading through yellow, green and blue to black. Created with foobar2000 audio player.

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