Musical Symmetry Revisited

Symmetric 8-note scale

In a blog post last year I talked about symmetric musical sets – such as the tritone, the augmented triad, the diminished 7th, the whole-tone scale and the chromatic scale – which divide the octave into 2, 3, 4, 6 and 12 equal parts respectively. For various reasons musicians dislike these groupings, so they’re used very sparingly in classical music and virtually never in pop music. But as someone who’s always been more into maths than music, I’m fascinated by any kind of symmetry.

Traditionally the octave is divided into 12 semitones, so the symmetric sets I just mentioned are the only possible ones. But what if you wanted to divide the octave into 8 equal parts? That seems an obvious choice, because it’s what the word octave implies. But to do it we need to invoke quarter tones. There are 24 of these in an octave, and 24 divided by 8 is 3, so we’re looking for notes 3 quarter tones (or one and a half semitones) apart.

Writing music in quarter tones isn’t easy, because the MIDI format defines pitch as an integer number of semitones. But it does allow something called “pitch bending” (presumably to simulate bending the string of a guitar), and with a bit of patience you can use that feature to raise the necessary notes by a quarter tone.

Here’s a short (1 minute) piece I wrote to see what it would sound like. It’s basically a random composition using the 8 equally spaced notes shown in the diagram above.

Fake Physics: Spoofs, Hoaxes and Fictitious Science

Fake Physics Springer Science and Fiction

Out now – my third contribution to Springer’s “Science and Fiction” series, after Pseudoscience and Science Fiction (2017) and Rockets and Ray Guns (2018). Those two were based around pet subjects of mine – the two-way interaction between SF and pseudoscience in the first case, and genuine Cold War science in the second. In contrast, this new book came out of a suggestion by the series editor, who drew my attention to a large number of spoofs – often vey funny – produced by professional scientists and presented in the form of serious academic papers.

These papers tend to fall into two categories – spoofs written purely for entertainment, such as April Fool jokes, and hoaxes designed to make a serious point – of which Alan Sokal’s nonsensical paper on quantum gravity, which was accepted by the editors of a professional journal simply because it pandered to their preconceptions, is the best-known example. I’ve combined these with more familiar examples of science-fictional “fake physics”, particularly when perpetrated by writers who were also professional scientists – such as Isaac Asimov, whose “Endochronic Properties of Resublimated Thiotimoline” (1948) bridges both genres: t’s a spoof science paper as well as an SF story.

The result is Fake Physics: Spoofs, Hoaxes and Fictitious Science – on sale now from all the usual places, including Amazon.com and Amazon UK. Here are some words from the back cover:

People are used to seeing “fake physics” in science fiction – concepts like faster-than-light travel, antigravity and time travel to name a few. The fiction label ought to be a giveaway, but some SF writers – especially those with a background in professional science – are so adept at “technobabble” that it can be difficult to work out what is fake and what is real. The boundaries between fact and fiction can also be blurred by physicists themselves … examples range from hoaxes aimed at exposing poor editorial standards in academic publications, through “thought experiments” that sound like the plot of a sci-fi movie to April Fools’ jokes. This entertaining book is a joyous romp exploring the whole spectrum of fake physics – from science to fiction and back again.

Apollo Nostalgia

Apollo 11 souvenirs

This time 50 years ago I was getting very excited about the forthcoming Moon landing. As I mentioned in a previous post, my serious interest in space travel started with the Apollo 8 mission, which took place soon after my 11th birthday. So with the 50th anniversary of the Apollo 11 landing fast approaching (I’m posting this 50 years to the day after the launch of the previous mission, Apollo 10), I thought it would be fun to look back through some of the souvenirs I collected at the time.

This is my first attempt at a video of this type, and I know it isn’t very professional-looking – but here it is anyway:

Good reviews for Cosmic Impact

Sky at Night Cosmic Impact review

As you can see from the above picture, the current issue of the BBC’s Sky at Night magazine includes a review of my book Cosmic Impact. It’s a really nice review, too, by Katrin Raynor-Evans – who says, among other positive things: “The text is superb … It is informative and clear, and May manages to encapsulate everything you need to know about the potential risk to our planet and species.” She gives it four stars out of five.

Cosmic Impact also gets four stars from Brian Clegg, at his Popular Science  book review site. Again the review has lots of positive comments, including the following:

This short book is ideal to get a good overview of the subject without having to delve into too much technical detail – and May makes it approachable by giving the subject context from the many science fiction and popular culture scenarios … where something hits the Earth from outer space.

Finally, although I haven’t seen it myself, I’m told I got a very brief but favourable mention in New Scientist, in the issue dated 2 February 2019. It’s in the “Don’t Miss” column, under the subheading “Read”. After recommending the Penguin Book of Outer Space Exploration to “armchair adventurers” they go on to say:

But if hiding under the sofa is more your style, try Cosmic Impact: Understanding the Threat to Earth from Asteroids and Comets by Andrew May.

If you want to do just that, you can find it in any good bookstore or via the following Amazon links:

Postscript 10 May 2019

Another excellent review in the May issue of How It Works magazine – this time with a five-star rating:

How It Works Cosmic Impact review

Cosmic Impact – out now!

Cosmic Impact

Following in the footsteps of Destination Mars, my second contribution to the Hot Science series has just been published by Icon Books. It’s called Cosmic Impact: Understanding the Threat to Earth from Asteroids and Comets, and here is what the publishers say about it:

As end-of-the-world scenarios go, an apocalyptic collision with an asteroid or comet is the new kid on the block, gaining respectability only in the last decade of the 20th century with the realisation that the dinosaurs had been wiped out by just such an impact.

Now the science community is making up for lost time, with worldwide efforts to track the thousands of potentially hazardous near-Earth objects, and plans for high-tech hardware that could deflect an incoming object from a collision course – a procedure depicted, with little regard for scientific accuracy, in several Hollywood movies.

Astrophysicist and science writer Andrew May disentangles fact from fiction in this fast-moving and entertaining account, covering the nature and history of comets and asteroids, the reason why some orbits are more hazardous than others, the devastating local and global effects that an impact event would produce, and – more optimistically – the way future space missions could avert a catastrophe.

Cosmic Impact is available either as a paperback or an ebook from all the usual places. If you’re an Amazon customer, here are some quick links for you:

Another Book in Progress

Science and fiction books

I’m part way through writing my third contribution to the Springer “Science and Fiction” series, after Pseudoscience and Science Fiction last year, and Rockets and Ray Guns a few months ago.

As I’ve done in the past, I thought I’d share a photograph of some of the books I’ve been using as “research material” – one of the most enjoyable things about writing for this series! Unlike the previous two books, the unifying theme of this one has more to do with style than subject-matter, so it may not be obvious from the photo. So you’ll have to wait until Springer formally announce the new book – hopefully some time in the first half of next year.

Random Encounters on the London Tourist Trail

Random Encounters on the London tourist trail

Just out – my second book in collaboration with Paul Jackson, following on from Weird Wessex a few years ago. Like its predecessor the new book is “a tourist guide to strange and unusual sights” – but this time they’re right in the centre of London.

There’s an Egyptian Goddess in Mayfair and Karl Marx in Soho, a tiny police station in Trafalgar Square and an 18-inch-wide alley in Covent Garden (careful you don’t get stuck!). Alongside the iconic landmarks that the regular guidebooks tell you about, central London has an impressive assortment of quirky and unusual sights, from art installations in the form of human body parts to hundred-year-old advertising signs and a forgotten tube station. This book gives you a guided tour of all these sights and more – without straying far from the places you were going to see anyway, like Big Ben and Buckingham Palace, the Tower of London and St Paul’s Cathedral, the museums of South Kensington and the entertainment hotspots of the West End.

Like Weird Wessex, Random Encounters on the London Tourist Trail is packed with full-colour photographs. With improved paper and print quality, the colours really stand out in this one too. You can get it from Amazon UK or any other Amazon store. There’s also a Kindle version (although that won’t give you colour pictures if you use an ordinary monochrome Kindle reader).

Algorithmic Beatles

Markov music matrix

When Eric Morecambe mangled Grieg’s Piano Concerto on a TV special in 1971, he insisted he was “playing all the right notes, but not necessarily in the right order”. That’s a valid point, because there aren’t that many different notes on a piano and the only thing that distinguishes one tune from another is the order in which you play them.

To a mathematician or computer programmer the situation is crying out for quantitative analysis. The diagram above shows the “transition matrix” for one specific Beatles tune (using the MIDI standard where middle C is C5). It’s clear there’s a lot of order here. One thing that jumps out is that there’s only one “black” note, G#5, and it’s always followed by A5. In fact A5 is a very popular note, cropping up after no fewer than 8 different pitches. On the other hand, G#5 itself is very rare, only ever coming after D6, and then only 6% of the time.

As well as analysing the original tune, this allows us to write a new tune of our own using the same transition matrix. The result (as the aforementioned mathematicians and computer programmers will recognize) is a first-order Markov chain. Producing an algorithm of this type from scratch would be rather tedious (as indeed the initial analysis would be), but fortunately there’s some free software called OpenMusic which includes built-in Markov functions that make the process much simpler.

Of course, there’s more to a tune than the pitch of the notes – there’s the duration of a note too. But that can be analysed and reproduced by exactly the same method. I experimented with an algorithmic composition of my own, based on the Beatles song analysed above. As a first step, I used the OpenMusic Markov functions to generate a series of tune-fragments for both the “right hand” and “left hand” of the piano. Then, to give the composition some structure, I arranged the fragments in a rough approximation to classical sonata form.

I won’t say what the original song was, because I want to see if anyone can guess it. As a hint, I’ve inserted a brief quotation from the original at the mid-point of the piece. Here it is on YouTube:

Rockets and Ray Guns

My second contribution to Springer’s Science and Fiction series is out now – Rockets and Ray Guns: The Sci-Fi Science of the Cold War. It’s in a similar style to Pseudoscience and Science Fiction, and even includes some pseudoscience of its own (e.g. UFOs, ESP and mind control) as well as the more obvious topics such as nuclear weapons, guided missiles and space warfare.

The Cold War saw scientists in East and West racing to create amazing new technologies, the like of which the world had never seen. Yet not everyone was taken by surprise. From super-powerful atomic weapons to rockets and space travel, readers of science fiction had seen it all before.

Sometimes reality lived up to the SF vision, at other times it didn’t. The hydrogen bomb was as terrifyingly destructive as anything in fiction, while real-world lasers didn’t come close to the promise of the classic SF ray gun. Nevertheless, when the scientific Cold War culminated in the Strategic Defence Initiative of the 1980s, it was so science-fictional in its aspirations that the media dubbed it “Star Wars”.

This entertaining account, offering a plethora of little known facts and insights from previously classified military projects, shows how the real-world science of the Cold War followed in the footsteps of SF – and how the two together changed our perception of both science and scientists, and paved the way to the world we live in today.

The book has already received a couple of nice reviews:

  • By Tom Reale (“a work that will delight science, history, and SF buffs alike”) on the AIPT website
  • By Brian Clegg (“a solid contribution to the history of science fiction and its relation to the real world”) on his Popular Science blog

Needless to say, Rockets and Ray Guns is available from all good booksellers, including Amazon.com and Amazon UK.

The joy of (musical) sets

Music set-theory

I mentioned musical set theory in a previous post, and now that I understand it better I’m getting very enthusiastic about it. It’s a really powerful technique for analysing and composing music. The mathematical connection may give the impression that it “dehumanizes” music by imposing mechanistic constraints and artificial rules – but the exact opposite is true. It’s traditional music theory that forces arbitrary rules and constraints on you – set theory liberates you from them. It’s a framework for organizing your own creativity – with no rules whatsoever.

I’ll explain how it works in a moment, but first a few words about my sources. The bible of the subject is Allen Forte’s The Structure of Atonal Music, which is divided into two roughly equal parts. The first is packed with useful stuff, although the second part was much too advanced for me. But Forte’s book is really about musical analysis, and what I was interested in was composition. On that front, I found a great little book by Stanley Funicelli called Basic Atonal Counterpoint (which is a CreateSpace book, but very professionally done). I also found a lot of practical tips on Frans Absil’s YouTube channel – he also produced the Pitch-Class Set Graphical Toolkit you can see on my iPad in the photograph above.

Musical set theory starts from a few basic observations:

  • The notes of the chromatic scale can be represented by integer “pitch-classes”: C = 0, C# = 1, D = 2 etc. After B = 11 you get back to C = 0, so additions and subtractions have to be done with mod-12 arithmetic.
  • Intervals between pitch-classes are much more important than absolute pitches. So C major [0, 4, 7] and E flat major [3, 7, 10] are just different transpositions of the same set (it’s called 3-11).
  • Inverting an interval (i.e. subtracting it from 12) doesn’t change its basic nature. So interval 7 (perfect fifth) can be grouped with 5 (perfect fourth), interval 8 (minor sixth) with 4 (major third) etc. This leaves us with just six “interval classes”: 1, 2, 3, 4, 5, 6.
  • The characteristic sound of a set is mainly determined by its interval vector. For example, the major chord 3-11 = [0, 4, 7] has an interval vector 001110 (one minor third, one major third, one perfect fifth and nothing else).

Traditional Western music depends heavily on set 7-35 [0, 2, 4, 5, 7, 9, 11] – the white notes on a piano, aka the major or minor scale (remember you can transpose these notes up by any integer between 1 and 11 to get all the other major and minor scales). Within that 7-element set, there are a number of strongly favoured subsets – most notably the aforementioned 3-11 (the major triad and its inversion, the minor triad).

The purpose of set theory should be obvious now. It gives you access to dozens of other sets, all with their own unique sound. You might think “but they’re going to sound terrible”, and in some cases they do. Set theory helps you to avoid the terrible-sounding ones! But there are some great-sounding sets that simply don’t exist in traditional music theory, such as 4z-29 = [0, 1, 3, 7], with an eyecatching interval vector of 111111.

To teach myself how the system works, I wrote a short “symphony” using the above ideas. It’s my first ever musical composition, and the result sounds a lot more interesting than if I’d struggled with all that traditional stuff about sharps and flats, majors and minors, dominants and subdominants etc. That wouldn’t have told me how to get close to the kind of spooky, spacey, quirky music I wanted to write.

Here is a link to the YouTube video: