Maths In Action

Just back from London where I had a fun — and ever-so-slightly daunting — time talking about the beauty of maths in music and physics for an audience of 700 GCSE students at the “Maths In Action” conference. The venue was stunning…


Just about visible at the front of the cavernous auditorium is the speaker before me, Hugh Hunt (Engineering, Cambridge), doing a remarkable job of entertaining and engaging the audience with his demonstration-packed talk on angular momentum, gyroscopic precession, and all things spin-y. Hugh’s talk was an impossible act to follow — he set the bar exceptionally high indeed. I did my usual spin on the quantum-metal interface but tilted it towards a discussion of the role of mathematics in physics. (I had to come clean right at the start and confess to the students that I am most definitely not a mathematician.)

The students were great throughout — they certainly were not shy to shout out answers to the questions I asked (and to sing musical notes back to me, occasionally even in tune). An extra big thank you to Korbyn — and my sincere apologies if I’ve got the spelling wrong — for coming up on stage to play the opening riff to Black Sabbath and help me introduce the concept of the diabolical flattened fifth.

And, of course, I have to say a huge thank you to David Matthews, coordinator of the event (and Maths Programme Manager for The Training Partnership, who run a very broad series of GCSE events of this type)  for both the invitation to speak and for such an impressively organised day. As someone who too often struggles to manage just two teenagers*, attempting to coordinate 700 would bring me out in a cold sweat…

* I’m joking, Niamh and Saoirse. You’re great.

Denim and leather brought us all together

After the thrash metal overkill of last weekend, this week it was the turn of some heavy duty NWOBHM to crank the volume to the point of pain. I’m still recovering from the early eighties flashbacks induced by Saxon’s set at the Royal Concert Hall on Saturday night so please forgive this relatively short review. (Fortunately, my voice is now rather less croaky — it spent most of Sunday recuperating from my hollering along to “Wheels Of Steel”, “Dallas 1 pm”, “Strangers In The Night” and a bevy of other (head)banging metal classics…)

Up before Biff and the Barnsley boys, we had the classic rock of Wayward Sons, fronted by Toby Jepson, the erstwhile Little Angels frontman. (Bonus review points before they played a note for wandering on stage to the strains of Johnny Cash’s “I Shot A Man In Reno”). As a thrash metal fan in the late eighties I had to keep my penchant for Little Angels material well under wraps — “Too posh to mosh” and all that — but I always had a soft spot for Jepson’s voice. (After all, he comes close to Geddy Lee’s stratospheric vocal register at times. Even now in the throes of middle age. Bastard.) She’s A Little Angel is still very welcome indeed when it appears on shuffle on my phone.

But Wayward Sons aren’t about capturing Mr. Jepson’s past glories — they’re a much more seventies-style hard riffing, hard rocking band with none of the hair metal pretensions of Little Angels. Their down to earth rock ‘n’ roll is imbued with a wonderfully infectious sense of enthusiasm, some beefy Bonham-esque backbeats, and even a very welcome Thin Lizzy feel at times. Last year’s “Until The End” is as good an introduction as you’ll get…

I’ll definitely be looking out for Wayward Sons in the future — high octane rock with a sound that may well not be fashionable ever again. But that’s just the way we like it.

Doro Pesch, the long-crowned Queen of Metal, was next to tread the Royal Concert Hall boards. I’ve never been a fan, I’ve got to admit, but Doro deserves quite some respect for flying the metal flag so loyally for quite this long. And she and her band certainly can command a crowd…

Doro’s set took the Spinal Tap influence and, it’s got to be said, cranked it far beyond 11. Here are the lyrics for the chorus of the closing song of the set, “All For Metal”…

Oh-oh-oh, oh-oh, oh-oh
Oh-oh-oh, oh-oh, oh-oh
Oh-oh-oh, oh-oh
All for metal (Metal)
Oh-oh-oh, oh-oh, oh-oh
Oh-oh-oh, oh-oh, oh-oh
Oh-oh-oh, oh-oh
Gods of metal (Metal)

How much more metal could this be? None. None more metal. (Metal)

And then it was time for the main event. Saxon were not only one of the first metal bands I got into but were a formative part of my musical education back when I were a lad. I distinctly remember feeling rather pleased with myself when I nailed the riff for “Strong Arm Of The Law” as a fledgling guitarist — one of the first “proper” guitar parts I learned — after rewinding and replaying the cassette of the album for what seemed like weeks (but was probably months). Here’s that riff reverberating across the Concert Hall on Saturday night…

Classic old school metal tunes were mixed with Saxon’s newer (and rather heavier) material. I’ll be brutally honest, however — I was there for the nostalgia trip. Kudos to Biff and the boys for not resting on their laurels and continuing to write and perform new material, but give me “And The Bands Played On” over “Sacrifice” any time. Fortunately, they did. Better still, the Barnsley ‘bangers closed their fan-pleasing set by belting out this anthem for the NWOBHM generation…

I can’t hear you…

Where were you in ’79 when the dam began to burst?
Did you check us out down at the local show?
Were you wearing denim, wearing leather?
Did you run down to the front?

Did you listen to the radio every Friday night?

Decades of Aggression



Ahhh, the comforting sound of the hirsute and happily inebriated Slayer fan at full bellow in their natural environment. Walking to the SSE Arena from Wembley Park tube station on Saturday evening, the streets were alive (undead) with the sounds of the Slayer swarm. I was in London, with my friend (and fellow physicist) James Theobald, his 11 year-old son, Jonathan, and twelve thousand or so other thrash metal fans for this:


Billed as The Farewell Tour, Slayer are retiring their plectrums, profanity, and pentagrams for a life of quiet reflection. Maybe. Given that Sabbath completed what seemed to be their fifth or sixth farewell tour a couple of years back, and that Ozzy has recently said that his “No More Tours” tour doesn’t actually mean that there will be no more tours (naturally), you’ll excuse my lack of surprise if Slayer give in to temptation and are reborn a few years hence.

Regardless of whether or not the bell truly has tolled for Slayer, it’s been a couple of years since I last did a review of a gig and there was no way I was letting this one go without putting pixels to screen.  Dom Lawson, one of the very best metal writers there is [1], reviewed Slayer’s performance in The Guardian [2] the day after the gig. He nailed it:

But tonight is all about the band that created one of metal’s most unarguably flawless albums, Reign in Blood (1986). From the discordant malevolence of their trademark riffs to the reliably unapologetic grimness of the lyrics, Slayer’s sound is seminal and ageless.

And here’s one of those seminal tracks [3]… (Some would say the seminal track.)

But Saturday night wasn’t all about Slayer. We also had three other major metal bands to whet our palette — sorry, assault our eardrums — before Araya, King, Holt [4], and Bostaph unleashed their demons. First up was the old school death metal of Obituary. Unfortunately, I’ve always found the Floridian deathsters to be rather one dimensional. (Perhaps it’s got something to do with the Tampa climate not exactly being conducive to metal. It’s not the sodden, grey landscape of Birmingham, after all.) “Slowly We Rot” hit the spot, however.

Up before Slayer were Lamb Of God. I’m again not a massive fan but they’re certainly an impressive metal machine; their drummer, Chris Adler, is worth the price of admission alone…

There are very, very few modern metal drummers who can match Adler’s proficiency, power, and prowess on double bass drums. And his band certainly can incite some fascinating non-equilibrium crowd dynamics (of a type discussed in a classic Physical Review Letter a few years back)…

I was, however, especially keen to experience Anthrax bringing the noise before they had to cede the stage to Lamb Of God. One of the Big Four of thrash metal, the New Yorkers were my “gateway” in the eighties from classic metal and NWOBHM (like Priest, Maiden, and Dio) to the heavier, darker, faster side. I didn’t “get” Metallica on the first, second, or even third listen — they were an acquired taste — but Anthrax I loved from the start, not least because of Joey Belladonna’s melodic and powerful vocals. (Scott Ian’s fandom for 2000 AD, a comic I devoured throughout the eighties and much of the nineties, also didn’t exactly put me off…) So it was wonderful to see Anthrax firing on all cylinders last Saturday, despite (a) them not playing I Am The Law (sob), and (b) Scott Ian berating those of us in the nosebleed seats for a certain lack of allegiance to the metal cause…

I’ll transcribe the key sentence from Mr. Ian’s rebuke, for those of you who may not have quite discerned what he said in the video above: “For those of you in the seats, stand the f**k up, you’re embarrassing yourselves“.

I dutifully got to my feet, suitably shame-faced.


Thanks to Taj Panesor for the wonderful shot of Anthrax in full flow above.

I’m writing this on the day of the US midterms. In an era when prejudice and hatred are stoked up on a daily basis in an increasingly polarised sociopolitical landscape, the lyrics of Anthrax’s Indians ringing out through the arena on Saturday night really paid testament to the cathartic power of music (be it metal or any other lesser variant)…

Territory, it’s just the body of the nation
The people that inhabit it make its configuration
Prejudice, something we all can do without
Flag of many colors is what this land’s all about

Or, as the Lemmy-endorsed — and what greater accolade could a metal/rock band have? – Skunk Anansie put it, Yes, It’s Fucking Political


[1] …although it’s a close run thing between Lawson and Andrew O’Neill, author of the fabulous “A History of Heavy Metal“. Even if you pull out the old air guitar only very occasionally, you owe it to yourself to get hold of O’Neill’s hilarious overview of the evolution of metal. Sample quote:

There are two types of people in the world: people who like heavy metal, and dicks. (Don’t worry if you fall into the latter category; I’m very persuasive.)

[2] Slayer in The Guardian.

Slayer. In. The Guardian.

Let that sink in, old school metal fans…

[3] It is verboten to write a review of any metal concert/album/song that does not include the word “seminal”. (Similarly, every physicist is contractually obliged to mention Feynman in every lecture they deliver.)

[4] Gary Holt is also with Exodus. I owe Gary and his bandmates a huge debt of gratitude because they were one of three bands who allowed the use of their lyrics in “When The Uncertainty Principle Goes To 11” free of charge. Other bands — well, more correctly, other publishers — charged up to $750 for the privilege of using a single line of lyrics as an epigraph at the start of a chapter.

Science Friction Braves The Pit Again

A very short post to say a massive thank you to Natasha Mitchell and Raphael Dixon (of the Australian Broadcasting Corporation) for putting together the video trailer below. As Natasha put it, “Watch until the very end if you’re the real deal .”

A physicist eulogises…

If you’re gonna die, die with your boots on
If you’re gonna try, well, stick around
Gonna cry, just move along
If you’re gonna die, you’re gonna die

from Iron Maiden’s Die With Your Boots On. Track 4 (Side 1) of Piece of Mind (EMI, 1983). Songwriters: Smith, Harris, Dickinson

On Friday I had the great pleasure of chatting with the dynamic duo of Olivier Larvor and Keith Clarke for their Death Hangout podcast. Notwithstanding the podcast moniker, it was a fun, upbeat, and rather uplifting conversation. (I kid you not. Cross my heart and hope to die..)  Here are Olivier (L) and Keith (R), with their friend, the not-so-Grim Reaper (who disappointingly didn’t put in an appearance on Friday).


Find out more about their intriguing podcast and the associated upcoming book here.

Olivier, Keith and I chatted about the physics of death (with some departures here and there to muse over such themes as rules for life (apparently there’s about a dozen of them), quantum woo, and the broader connotations of the second law of thermodynamics). It’ll take a while for the podcast to be uploaded as Olivier and Keith will have to summon their editing and production demons. In the meantime I wanted to post the eulogy at the foot of this post, which I stumbled across while I was doing some homework for the podcast. As a humanist and a physicist I find it incredibly moving.

I was also invited to a second “Death Hangout” podcast recorded on Friday. The guest this time was Jon Wiederhorn (pictured below), a highly respected figure in the heavy metal community and the author of a number of hugely entertaining books about the genre and a number of its larger-than-life musicians: Louder than Hell, I’m The Man (the biography of Anthrax’s Scott Ian), and Ministry: The Lost Gospels According To Al Jourgensen.

Embed from Getty Images

Death, of course, is at the very core of so much metal music. (The “die, die, die” refrain in Metallica’s “Creeping Death” immediately creeps to mind but there are so very many examples. A drinking game based on mentions of death in metal songs would have a fairly short half-life due to incipient alcohol poisoning…) When the episode featuring Jon goes live, I’ll link to it here at “Symptoms…”. Jon gave a wonderfully engaging, and indeed some might say life-affirming, overview of the catharsis and excitement that metal can generate, gory and/or morbid lyrics notwithstanding

Anyway, I’ll fade to black for now with that beautiful eulogy from a physicist I mentioned. It’s from an NPR broadcast back in 2005 given by the Chicago-based writer and performer Aaron Freeman

You want a physicist to speak at your funeral. You want the physicist to talk to your grieving family about the conservation of energy, so they will understand that your energy has not died. You want the physicist to remind your sobbing mother about the first law of thermodynamics; that no energy gets created in the universe, and none is destroyed. You want your mother to know that all your energy, every vibration, every Btu of heat, every wave of every particle that was her beloved child remains with her in this world. You want the physicist to tell your weeping father that amid energies of the cosmos, you gave as good as you got.

And at one point you’d hope that the physicist would step down from the pulpit and walk to your brokenhearted spouse there in the pew and tell him that all the photons that ever bounced off your face, all the particles whose paths were interrupted by your smile, by the touch of your hair, hundreds of trillions of particles, have raced off like children, their ways forever changed by you. And as your widow rocks in the arms of a loving family, may the physicist let her know that all the photons that bounced from you were gathered in the particle detectors that are her eyes, that those photons created within her constellations of electromagnetically charged neurons whose energy will go on forever.

And the physicist will remind the congregation of how much of all our energy is given off as heat. There may be a few fanning themselves with their programs as he says it. And he will tell them that the warmth that flowed through you in life is still here, still part of all that we are, even as we who mourn continue the heat of our own lives.

And you’ll want the physicist to explain to those who loved you that they need not have faith; indeed, they should not have faith. Let them know that they can measure, that scientists have measured precisely the conservation of energy and found it accurate, verifiable and consistent across space and time. You can hope your family will examine the evidence and satisfy themselves that the science is sound and that they’ll be comforted to know your energy’s still around. According to the law of the conservation of energy, not a bit of you is gone; you’re just less orderly. Amen.

Metallizing and Melodifying Phi

Apologies for the radio silence on the blog of late. I’ll be posting more regularly in the coming days/weeks. For now, this one is a bit of a blast from the past. Over six years ago (gulp), Brady Haran and I collaborated with the talented and prolific Dave Brown (boyinaband) on a suitably metallized rendition of a fundamental constant — the golden ratio, otherwise known as φ

MoriartyAs I’ve said during various talks about the metal-maths-physics interface (including this) over the years since that video was uploaded , some people buy a Porsche for their mid-life crisis. Mine involved attempting to reconnect with my — substantially less follicularly challenged, see image to right for pictorial evidence — halcyon heavy metal days…

At the time of uploading the video, I wrote the first blog post below. It’s been loitering at Brady’s, now discontinued, original blog for quite some time. I’m reblogging it here, along with another post from many moons ago on a more sedate rendering of φ (and its cousin τ).

Metallizing Phi

13 July 2012

Here are all the gory details for the musicians amongst you…

Guitar tuning: Bb F Bb Eb G C

(This is traditional “drop D” tuning, i.e. D A E D G B E, dropped two tones in order to approximate the math metal/Djent sound without a seven string guitar.).

We stick almost exclusively to riffs derived from the Bb harmonic minor scale (although the chorus is based around the natural Bb minor scale)

I used Guitar Rig to record the riffs (both clean and effected) which I then sent to Dave who used his studio wizardry and musical acumen to arrange and structure the song. This involved quite a number of e-mail exchanges to hone the structure of the song during which Dave had to rein in my old school metal tendencies on more than one occasion…


0:00. We kick off with a clean picked piece which looks like this:


The digits of phi are “embedded” in the notes played on the 4th string. I make use of octaves and finger picking to embellish the riff.

0:08 Dave comes in with sixteenth note “chugs” (on Bb) which are timed to match the digits of phi (as explained in the video).

00:16 All hell breaks loose. Same idea as for ‘chug’ pattern starting at 0:08, except this time  matched by kick drums. (All drum programming by Dave – visit his website for tutorials on how he lays down those impressive drum tracks).

00:40 The riff for verse 1 is basically power chords given by the digits 161803398, as follows (where ^ represents a higher octave):

1 – Bb

6 – Gb

1 – Bb

8 – ^Bb

0 – ^Db

3 – Db

3 – Db

9 – ^C

8 – ^Bb

8 – ^Bb

00:55 The chorus is a similar idea but this time in Bb minor (not harmonic minor).

1 – Bb

6 – Gb

1 – Bb

8 – ^Bb

0 – [rest]

3 – Db

3 – Db

9 – ^C

8 – ^Bb

8 – ^Bb

7 – Ab

1:11 Here we switch to ‘encoding’ the [1 + sqrt (5)]/2 representation of phi in the riff. It’s a much more old school metal riff  and involves lots of use of the open sixth string (first note of the Bb harmonic minor scale) to incorporate ‘1’.

The digits of sqrt (5) are then encoded as shown in the tablature below.

I wanted to get a somewhat Mastodon-esque feel here so used lots of octaves (and slides into octaves).

I tried to down-pick as much as possible to ‘drive’ the riff . The ½ of (1+sqrt(5))/2 is built in as half-time on the drums.


1:27. I very much wanted to have a heavily Tool-influenced riff in the song. Tool are math metal  pioneers and, as many of those who have watched the “Golden Ratio – Making a Math Metal Anthem” video have pointed out, their song Lateralus has lyrics which are based around the Fibonacci sequence. So, the following is my ‘homage’ to Tool…


The digits of phi are encoded in the notes on the sixth and fifth strings and I ‘pedal’ around Bb notes on the third, fourth, and fifth strings.

2:15 As explained by Dave in the video, his riff here is also derived from (1 + root 5 )/ 2.

Sqrt (5) is embedded in the number of chugs again and the drums are half time. The “1” is a sustained and ringing Bb note.



Real but uncountable


At the root of the problem

Patterns will!

Emerge from the equation

Golden Angle!

Sprials out of control


Chorus: The proportion is divine, you’ll find your way

To Phi (to Phi) (to Phi)

The ratio defined, you can’t deny

It’s Phi


The five-fold way

Forbidden symmetry

Crossing points define

Demonic  geometry

[Verse 1 is fairly self-explanatory.Verse 2  above is a little more obscure. It refers to the pentagram which, of course, is a key piece of metal ‘iconography’. The verse refers to five-fold symmetry which is directly linked to phi.].

Phi = root(1 + Phi = root(1 +Phi = root(1 +Phi = root(1 +…

[This stems from the equation φ = sqrt (1 + φ) which, of course, is recursive – hence the looping lyric). 




The Tau of Phi

To accompany Numberphile’s Tau of Phi video…

The music is here:

For some unfathomable reason, not everyone is a fan of heavy metal so I thought it might be helpful to compose a piece of ‘mathemusic’ which didn’t involve growling, screaming, and/or distorted, detuned guitars. If nothing else, I thought it might win Brady back a few  of those subscribers who unsubscribed from Numberphile in protest when our Golden Ratio Song was uploaded.

There are, of course, a number of great pieces of music out there whose composers have used fundamental mathematical constants as their basis (long before we decided to ‘metallize’ phi in the way we did). ViHart’s “A Song About A Circle Constant” and Michael Blake’s “What tau sounds like” are great examples and highly recommended. And both Tool (with ‘Lateralus’) and After The Burial (with “Pi”) have written songs directly inspired by constants in Nature. (More on Tool below).

But what do we get if we mix melodies and riffs based around a number of different constants? This was one of the motivations for the “Tau of Phi(bonacci)” piece. I was intrigued as to how a piece inspired by the digits of both tau and phi would sound.

Here’s how the piece of music works. (I used Audacity for all of the recording, effects, and mixing).

0:00 – 0:17. Opens with a gently looping piano melody derived from the first eight digits of tau mapped onto a Bb harmonic minor scale. (The same scale as we used for the math metal song). The sound in the background is a combination of strings and a crescendo involving Bb octaves which I then time-reversed. The strings throughout the piece are based on the digits of tau.

0:18 – 0:43. The tau riff continues to play. The chords underlying this are an interpretation on piano of the opening of the math metal Golden Ratio song. I take some ‘liberties’ here, however, and first play the sequence: “1…6…1” three times in a row, (starting at 0:18, 0:27, and at 0:36). That is, I repeat the first three digits of phi three times. This adds to the overall ‘atmosphere’ of the piece. (What’s important, I feel, is to use the constants to inspire the composition, rather than to slavishly reproduce the sequence of digits. Music and maths (and physics!) are all about creativity.)

0:45 – 0:51. Chords represent the “8” and “0” of phi.

0:52 – 1:00. …and then the “3..3..9..8” of phi.

1:02 seconds (and ~ 0.8 of a second!) – “Reprise” of opening tau riff on guitar and piano..

1:09 Tool’s “Lateralus” riff (downtuned to Bb and played on electric piano, rather than guitar). There were very many comments about “Lateralus”, and its relationship to the Fibonacci series, under the video for our golden ratio song. I felt it only right to ‘allude’ to Lateralus here. Timing of riff not coincidental (for Tool aficionados…).

1:20 ViHart, in her wonderfully crystal-clear vocal tones, sings 6..2..8..3..1..8..5..3. [Lots of delay and reverb courtesy of Audacity’s standard effects base].

I sampled the numbers from Vi’s “Oh No,  Pi Politics Again” video.

…except for the “6”. Unfortunately, she didn’t sing the digit “6” in that video so I add to resort to sampling her rendition of “6” from her tau song. But in her tau song, she’s singing along with a guitar. This meant quite a bit of manipulation of the frequencies of the sample to attempt to isolate the vocal.

[Warning – ‘tech-y’ musical bit:

ViHart sings the notes in her songs/melodies in the key of C major. But the music in the “Tau of Phi(bonacci)” is based around Bb minor. My first thought was to transpose ViHart’s vocals down two tones (i.e. from C to Bb major). But she ended up sounding not too unlike Barry White.

Not good.

So I instead transposed her vocals up a semitone to C#. C# major is the tonic major key of Bb minor so shifting Vi’s vocals up a semitone (a) doesn’t modify her overall vocal tone too much, and (b) works harmonically (in principle!).]

1:28 – 1:37. Piece fades out with tau riff gently looping on guitar.


A few years later I collaborated with another exceptionally talented musician (and physics teacher), Alan Stewart, on this piece of maths-influenced instrumental prog rock. (I learnt so very much from Alan about how melody and harmony work.)

Alan’s original version without my everything-one-louder-than-everything-else guitar on top (and with a full explanation of the links to the maths) is here: