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:

Next book research

Asteroids, comets and impactsThese days I always seem to be working on a lot of things at once, so “next book” has multiple meanings. There’s the next one to be published, which I finished writing several months ago and is now making its way through the publisher’s production process. There’s the one I’ve been asked to write and given a title for, but I’ve barely started to think about it yet. And then there’s the one I’m actually writing at the moment. That’s the one I’m talking about here. There’s a clue to its subject matter in the research material pictured above!

Symmetry in Music

Symmetric and asymmetric music chordsI recently came across the idea of applying set theory to musical analysis (which apparently has been around for some time, although I’d never heard of it before). For most people, who have a stronger intuitive grasp of music than mathematics, this must seem a pointless exercise, but for anyone like me who’s the other way around it’s really very illuminating.

Take symmetry, for example. In most areas of the arts and sciences, symmetry is seen as a good thing – but in music, that’s not the case. All the most popular chords are asymmetric in terms of interval content. You can see that in the left-hand image above, which shows the three notes of the C major chord on the chromatic circle. They’re separated by intervals of 3, 4, and 5 semitones.

In contrast, an augmented C chord, shown on the right, is perfectly symmetric, with all three intervals equal to 4 semitones. The problem (as far as musicians are concerned) is that it’s not very firmly tied to C major. It could equally well be A flat or E major. In the same way, the four-note symmetric chord C – E♭ – F♯ – A can be interpreted in four different ways: as Cdim7, E♭dim7, F♯dim7 or Adim7.

There’s even a completely symmetric two-note interval, in the form of the tritone, consisting of two notes 6 semitones apart (or 3 whole tones, which is how it gets its name). That’s exactly half an octave, for example from C to F sharp. But it’s also the distance from F sharp to C, so you really don’t know which key you’re in. That’s why composers spent centuries trying to avoid it. They called it diabolus in musica, or “the devil in music”.

Being a symmetry-loving scientist rather than a musician, I decided to try writing something that consisted only of symmetric chords. It’s a sort of canon, in the key of everything.

Here’s a link to a YouTube video, with added graphics depicting the various chords on the chromatic circle. Hopefully you’ll enjoy the graphics even if you don’t like the music!

Telescopic Tourist video

I’ve just belatedly produced a promotional video for my book The Telescopic Tourist’s Guide to the Moon, which came out last summer. Here it is:

The background “music” (actually just a sequence of spacey sounding chords) is my own composition!

Needless to say, The Telescopic Tourist’s Guide to the Moon is available from all good bookshops, as well as online retailers such as Amazon.com and Amazon UK.

The Science behind Jules Verne’s Moon Novels

Science behind Jules Verne

When I wrote The Telescopic Tourist’s Guide to the Moon last year, I wanted to refer, amongst other things, to descriptions of real lunar features in works of science fiction. Surprisingly, I found that many of the most famous Moon stories don’t actually refer to specific locations. Even more surprisingly, one of the few novels that does contain realistic descriptions of lunar geography is one of the earliest – Jules Verne’s Around the Moon, dating from 1870.

The surprise comes because Around the Moon – and its predecessor, From the Earth to the Moon (1865) – are probably best known for the completely unrealistic mode of travel, i.e. by means of a projectile launched from a giant cannon. But when I reread the novels, I was struck by just how scientifically knowledgeable they were – by the standards of their time, at any rate. As well as the physical descriptions of the Moon, Verne gets other subtleties right, too – such as the way things move once they get outside the Earth’s atmosphere (something Hollywood barely understands to this day).

So I thought I’d write another little book describing all the science Verne got right – and of course the science he got wrong, too. Here’s the blurb:

The idea of using a large gun to send humans into space is as impossible today as it was a century and a half ago, when Jules Verne wrote From the Earth to the Moon and Around the Moon. Yet he went to great lengths to persuade readers it wasn’t impossible – not through arm-waving and made-up technobabble, but using real physics and astronomy. No one had done anything like that in fiction before – and even today it’s unusual to see so much “real science” discussed in a work of science fiction. But just how much did Verne get right, and what did he get wrong? This book takes a closer look at the science content of his two great Moon novels – from Newton’s laws of motion and the conservation of energy to CO2 scrubbing, retro-rockets and the lifeless grey landscape of the Moon.

The Science behind Jules Verne’s Moon Novels is available as a paperback or Kindle ebook from Amazon.com, Amazon UK and all other Amazon sites.

Dirac on Einstein

I was going through some old audio cassettes I recorded from the radio when I was a student, and came across a really interesting little snippet. It’s the physicist Paul Dirac reminiscing about Einstein on a BBC programme, though I’m afraid I’ve no idea which one. The note I made at the time says “recorded in March 1979″ – when Dirac would have been 76 (he lived to 82).

Although the quote is very short, it’s really fascinating – and a Google search didn’t turn up any other references to it. So I made a little YouTube video of it, which hopefully the following link will take you to:

Here is my transcript of what Dirac has to say about Einstein:

He wasn’t merely trying to construct theories to agree with observation. So many people do that; Einstein worked quite differently. He tried to imagine “If I were God, would I have made the world like this?” – and according to the answer to that question, he would decide on whether he liked a particular theory or not.

And I can’t resist adding a couple of Amazon links for my own book about Einstein:

Einstein book covers

Coming soon: 30-second Energy

30-second Energy

The popular series of “30-second” books from Ivy Press has been running for several years now. Next month sees the latest in the sub-series of physics-related titles edited by Brian Clegg. This is 30-second Energy, following on from 30-second Quantum Theory, 30-second Physics, 30-second Newton and 30-second Einstein. As with those earlier books, 30-second Energy includes contributions from myself as well as from Brian and several other authors.

30-second Energy is a slight departure from the previous titles in that it’s more about practical applications of physics than theory or academic research. That means that (as with virtually anything “useful” that comes out of physics), many people won’t even realize that it is physics! That’s not necessarily a bad thing, since readers who wouldn’t dream of buying 30-second Quantum Theory or 30-second Newton might still be attracted by 30-second Energy.

This book differs from the previous four in another way too. It’s the first to include a Foreword by a “big name” – in this case, Jim Al-Khalili. Apart from that, it’s very much the same style as the others – about 60 double-page spreads of get-to-the-point-quickly text and lavish full-colour illustrations. As I’ve said before though, “30 seconds” is a slight exaggeration – it will probably take you at least 90 seconds to read each of the entries properly!

In all, 30-second Energy contains 8 contributions by me, on “Kinetic Energy”, “Potential Energy”, External Combustion”, “Internal Combustion”, “Turbines”, “Fission”, “Fusion” and “Batteries”. The book is published on 1 March 2018, and you can see its Amazon UK listing by clicking on the following link:

30-Second Energy: The 50 most fundamental concepts in energy, each explained in half a minute

Under the slightly different title of Know-It-All Energy, the book is already on sale in North America. Here is a link to its Amazon.com listing:

Know It All Energy: The 50 Most Elemental Concepts in Energy, Each Explained in Under a Minute

Cold War Update

Cold War Sci-Fi ScienceI just realized that it’s been four months since I last posted an update on this blog. I’ve been too busy! Anyway, my Cold War book now has a working title – Rockets and Ray Guns: The Sci-Fi Science of the Cold War. I’m about two-thirds of the way through writing it – hopefully it should be out some time around the middle of 2018.

Basically the book is a follow-on to Pseudoscience and Science Fiction, which was published last year. While the first book looked at the way SF anticipated and cross-fertilized with various well-known tropes of the pseudoscience industry, the new one will do the same for the real (or in some cases, allegedly “real”) science of the Cold War.

The picture above gives a quick taster of the kind of thing I mean. The illustration on the left comes from a short story by John W. Campbell called “When the Atoms Failed”, from the January 1930 issue of Amazing Stories. The picture on the right is an artist’s conception of a space-based electromagnetic railgun, dating from July 1984. This was a real-world proposal for an anti-ballistic–missile defence system, using technology that had already been demonstrated in the laboratory.

British lunar lander, 1954

BIS lunar lander

The picture above shows a comparison between an Apollo-style lunar lander, on the right, and the more traditional idea of a “spaceship” on the left. More technically, the comparison is between the Lunar Orbit Rendezvous approach used by Apollo and the competing methods of Direct Ascent (going all the way from the Earth to the Moon with a single vehicle) and Earth Orbit Rendezvous (ditto, but with the vehicle first being constructed or refuelled in Earth orbit). You might guess the picture dates from circa 1962, when NASA surprised the world by selecting LOR over the (previously much more likely) other two options. Actually it comes from a book printed in 1954.

That’s the date on my copy of the book, which is the second edition of one originally published in 1952 (I don’t know if the same picture was in the first edition). It’s called Development of the Guided Missile, by Kenneth W. Gatland – a member of the British Interplanetary Society, which was the source of the lunar landing concept depicted here. In the text the lander is designated “Type B”, while the counterpart of the Apollo CSM is Type A: “The Types A and B operate together as a composite vehicle; the former acts as the propulsion component for the Type B and remains in the terminal orbit of the destination planet whilst the smaller rocket descends to the surface.”

The most famous member of the British Interplanetary Society was Arthur C. Clarke, and he touched on the same subject in his “science-fictional autobiography” Astounding Days:

We discussed many types of rendezvous and space-refuelling techniques, to break down the journey into manageable stages. One of those involved the use of a specialized “ferry” craft to make the actual lunar landing, while the main vehicle remained in orbit. This, of course, is the approach in the Apollo project – and I am a little tired of hearing it described as a new discovery. For that matter, I doubt if we thought of it first; it is more likely that the German or Russian theoreticians had worked it out years before.

Actually the concept in Gatland’s book is a mixture of Earth Orbit Rendezvous and Lunar Orbit Rendezvous, since the Type A spacecraft (which is powered by a nuclear-thermal rocket) is first constructed in Earth orbit, using smaller unmanned rockets (Type C) and a winged shuttle (Type D) to ferry the astronauts to and from Earth orbit. The whole ambitious concept is illustrated in the picture below – it strikes me as extraordinarily sophisticated for a book published in 1954!

British Interplanetary Spacecraft

Cold War research

Cold War books

Having thoroughly enjoyed doing the research for my book on Pseudoscience and Science Fiction last year (see a selection of research materials here and here), and then The Telescopic Tourist’s Guide to the Moon earlier this year (see my Lunar Research blog post) I’ve been wondering what to do next. One idea that occurred to me is something about the Cold War … so I’ve been dutifully immersing myself in research on the subject, as you can see from the picture above.