Following on from last week and my excursion into mathematics, this week I take things further – and perhaps you could call this post…further mathematics. I never quite got as far as further mathematics, choosing “just” to do Pure & Applied Mathematics alongside Physics at A-Level, which was not among my better academic choices. Maths was fine, it was the Physics that was the problem…

Anyway, it turns out that 345 has a couple of significances in Mathematics. It is both a Sphenic number and a Self-number, apparently, neither concept I recall coming across before (so I’ve even learned something new by writing this post).

This week takes on a few more complex mathematical concepts in music – both as subjects and as a method of structuring songs. There is even a bit of history in here, too, and a few artists I’ve not featured before, too.

A quick explanation for new readers (hi there!): my Tuesday Ten series has been running since March 2007, and each month features at least ten new songs you should hear – and in between those monthly posts, I feature songs on a variety of subjects, with some of the songs featured coming from suggestion threads on Facebook.

Feel free to get involved with these – the more the merrier, and the breadth of suggestions that I get continues to astound. Otherwise, as usual, if you’ve got something you want me to hear, something I should be writing about, or even a gig I should be attending, e-mail me, or drop me a line on Facebook (details below).

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Bizarrely, I’ve now obliquely referenced dEUS in both of these posts about Mathematics, without actually featuring them. Last week the title was one of their (greatest) songs, while this week sees another mention of Buckminster Fuller – who was also featured in 268: Notable People in Song, in the form of the dEUS track The Architect.

Add N to (X), though, were rather more…different. Early exponents of “retro”, analogue synths, long before using them became a cool thing to do, their music was deliberately clever and difficult – and their name was a mathematical formula! – a broadly instrumental sound that sign-posted the way toward what became noise-rock and arguably helped popularise outré electronic rock, too. The best-known song from this, their breakthrough album, was, of course, the robot-sex-funk of Metal Fingers In My Body, but one of the better songs on this excellent album is the galloping rhythms and squalling synths of Buckminster Fuller, apparently celebrating the American visionary by simply repeated mentions of his name.

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This was unexpectedly one of three or four tracks from this much-loved electronic act suggested, perhaps suggesting an interest in mathematics from this mysterious, reclusive group – one who make delicate, elegant electronics that is often about as far from the dancefloor as it is possible to get.

Among the maths and numerology references on Geogaddi is this track. The smallest weird number – “the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself” – is apparently 70. Like many apparently obtuse and difficult number concepts, I’m sure there are reasons for knowing this, and if you know, I’d love to hear from you.

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Darren Aronofsky’s first film, and remarkably also the soundtrack debut of Clint Mansell, a few years after he left Pop Will Eat Itself for good – and a film that I’ve still never seen (we’ve had the DVD on the shelf for some years now but never got ’round to watching it).

The lead song for the film has an intense, enigmatic quality, which matches neatly with the video visuals and also the titular mathematical constant, one that is fundamental as “the ratio of a circle’s circumference to its diameter” (and has a great many other uses) – and in some respects is still a mystery, as additional digits keep being found, as the two quadrillionth was found a few years ago (although Peter Trueb holds the record for finding the most consecutive digits, at 22 trillion…

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One of the more remarkable, unexpected successes came from Laurie Anderson, whose album Big Science was a surprise hit in 1982 (mainly thanks to the song O Superman, of course) – and an album I’m more familiar with thanks to the mighty Spiritualized covering Born Never Asked on Pure Phase (and yes, I have the luminous case version of that, thanks to my wife).

This song, though. Huh, it’s odd. Let x=x effectively results in a formula that simplifies to nothing (Or more likely, x?), and perhaps that’s not the point, as this post explains, this is a song suggesting that meaningless statements often have a deeper meaning, even if they appear not to in the first place.

This post is making my head hurt.

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That was fairly simple, then, but this is rather more complex. Man Or Astro-Man? were long one of those bands I’d heard about a great many times before I finally heard them, likely on the John Peel show, and I’ve never been entirely sure I understand them. Spectacularly odd space-surf-rock, really, and this song has a title that is a spin on the formula to determine the mass of the earth (but apparently transposed to the moon). The formula in question uses Newton’s Law of Universal Gravitation, one of the more fundamental leaps forward in mathematics and physics. Published in 1686, it was only proven by Henry Cavendish in 1798. It is also not even the most complex formula featured this week.

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This song – one of Radiohead’s better songs in their more recent, post-Kid A phase – while an incorrect mathematical formula (2+2 correctly equals 4, obviously), is also a political nod to totalitarian regimes, in the first instance to The State in Nineteen Eighty-Four – as an example of how control can be used to alter perception and control the narrative.

This was based upon Hermann Göring’s rather chilling statement that “If the Führer wants it, two and two makes five!“, but in much more recent times, a similarly brazen statement was made by Giuliani this summer: “Truth isn’t Truth“.

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One of the highlights of their recent set at Infest, the EP that this track comes from rather marked a turning point in This Morn’ Omina’s sound. This was where live percussion began to be used in the recording process, alongside the electronic beats and myriad of samples, but like all of the rest of their work, it also has a strong thematic link across all of the tracks.

Rather than the often-common emphasis on ritual and religious devotion, though, here the subject is far more otherworldly – that of the Drake Equation, which is broadly a way of trying to estimate how many active, communicative extraterrestrial civilizations there are in the Milky Way, and if I understand it correctly, was the starting point for the ongoing SETI work. Unusually, the track “titles” here are effectively elements of the whole equation (which for those of us trying to write about TMO’s live performance, can be a bit of a pain – this was the track that I’d forgotten the name of in my Infest review this year!), and (f.l) is the “fraction of the average number of planets that might go on to develop life”.

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Not only a great musician and singer-songwriter, I did not know that Lehrer retired from music to teach mathematics at the University of California, but he will likely be forever remembered for his many hilarious songs.

One of these covers both maths, humour and music. I’m not entirely sure on the provenance of what is suggested in the song, but the mathematician named here – Nikolai Lobachevsky – is certainly an important name in geometry for the work that he did (Hyperbolic or Lobachevskian geometry). It’s just that here, Lehrer sings in the character of one of Lobachevsky’s students, from what I can tell, and rather suggests that most of his work was plagiarised from others (which apparently was just a device to make the song work!). I wonder,

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Tool is probably the most cerebral of the popular metal bands, so let’s talk some simple numbers first. Promises of a new album seem to be coming more often now, but it is now twelve years since their last (10,000 Days in 2006), and seventeen since Lateralus, which came out in 2001. I’ve long loved Tool, since the mid-90s at least, and perhaps their limited output – just four full-length albums since 1992 – has only created all the more demand.

The spectacular title track from Lateralus took the already complex nature of their music to mind-bending extremes. The song is a nod to humanity expanding it’s horizons and developing further, while the syllables in the lyrics are built to match the Fibonacci sequence – the chord sequences also nod to it, while the length of the introduction also reputedly references the Golden Ratio. It blows my mind, frankly, that this band can write such complex material and make it accessible like this, as few, if any, other bands can.

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See, I could have used Aphex Twin’s ∆Mᵢ⁻¹=−α ∑ Dᵢ[η][ ∑ Fjᵢ[η−1]+Fextᵢ [η⁻¹]] (Equation – a track that showed Richard D. James’ face if you ran the track through a spectrogram!), but this one is better, as the album title too uses a mathematical function to explain that 5F_55 have become 5F-X (after one half of the group left) – and serves me to remind the brilliance of one of the more, er, weird electronic acts on Hands Productions. One that created space and mathematical-themed electronics (that occasionally strayed into noisier territories), and live provided a spectacular, men-in-furry-alien-suits show (as happened at Infest to widespread acclaim a good few years ago).

I’m still fucked if I know what that formula in the title means, or does, mind. Answers on a postcard, please…