“The drum beats out of time…”

Far back in the mists of time, in those halcyon days when the Brexit referendum was still but a comfortably distant blot on the horizon and Trump’s lie tally was a measly sub-five-figures, I had the immense fun of working with Brady Haran and Sean Riley on this…

As that video describes, we tried an experiment in crowd-sourcing data via YouTube for an analysis of the extent to which fluctuations in timing might be a signature characteristic of a particular drummer (or drumming style). Those Sixty Symbols viewers who very kindly sent us samples of their drumming — all 78 of you [1] — have been waiting a very, very long time for this update. My sincere thanks for contributing and my profuse apologies for the exceptionally long delay in letting you know just what happened to the data you sent us. The good news is that a paper, Rushing or Dragging? An Analysis of the “Universality” of Correlated Fluctuations in Hi-hat Timing and Dynamics (which was uploaded to the arXiv last week), has resulted from the drumming fluctuations project. The abstract reads as follows.

A previous analysis of fluctuations in a virtuoso (Jeff Porcaro) drum performance [Räsänen et al., PLoS ONE 10(6): e0127902 (2015)] demonstrated that the rhythmic signal comprised both long range correlations and short range anti-correlations, with a characteristic timescale distinguishing the two regimes. We have extended Räsänen et al.’s approach to a much larger number of drum samples (N=132, provided by a total of 58 participants) and to a different performance (viz., Rush’s Tom Sawyer). A key focus of our study was to test whether the fluctuation dynamics discovered by Räsänen et al. are “universal” in the following sense: is the crossover from short-range to long-range correlated fluctuations a general phenomenon or is it restricted to particular drum patterns and/or specific drummers? We find no compelling evidence to suggest that the short-range to long-range correlation crossover that is characteristic of Porcaro’s performance is a common feature of temporal fluctuations in drum patterns. Moreover, level of experience and/or playing technique surprisingly do not play a role in influencing a short-range to long-range correlation cross-over. Our study also highlights that a great deal of caution needs to be taken when using the detrended fluctuation analysis technique, particularly with regard to anti-correlated signals.

There’s also some bad news. We’ll get to that. First, a few words on the background to the project.

Inspired by a fascinating paper published by Esa Rasanen (of Tampere University) and colleagues back in 2015, a few months before the Sixty Symbols video was uploaded, we were keen to determine whether the correlations observed by Esa et al. in the fluctuations in an iconic drummer’s performance — the late, great Jeff Porcaro — were a common feature of drumming.

Why do we care — and why should you care — about fluctuations in drumming? Surely we physicists should be doing something much more important with our time, like, um, curing cancer…

OK, maybe not.

More seriously, there are very many good reasons why we should study fluctuations (aka noise) in quite some detail. Often, noise is the bane of an experimental physicist’s life. We spend inordinate amounts of time chasing down and attempting to eliminate sources of noise, be they at a specific frequency (e.g. mains “hum” at 50 Hz or 60 Hz [2]) or, sometimes more frustratingly, when the signal contamination is spread across the frequency spectrum, forming what’s known as white noise. (Noise can be of many colours other than white — just as with a spectrum of light it all depends on which frequencies are present.)

But noise is most definitely not always just a nuisance to be avoided/eliminated at all costs; there can be a wealth of information embedded in the apparent messiness. Pink noise, for example, crops up in many weird and wonderful — and, indeed, many not-so-weird-and-not-so-wonderful — places, from climate change, to fluctuations in our heartbeats, to variations in the stock exchange, to current flow in electronic devices, and, indeed, to mutations occurring during the expansion of a cancerous tumour.  An analysis of the character and colour of noise can provide compelling insights into the physics and maths underpinning the behaviour of everything from molecular self-assembly to the influence and impact of social media.

The Porcaro performance that Esa and colleagues analysed for their paper is the impressive single-handed 16th note groove that drives Michael McDonald’s “I Keep Forgettin’…” I wanted to analyse a similar single-handed 16th note pattern, but in a rock rather than pop context, to ascertain whether Procaro’s pattern of fluctuations in interbeat timing were characteristic only of his virtuoso style or if they were a general feature of drumming. I’m also, coincidentally, a massive Rush fan. An iconic and influential track from the Canadian trio with the right type of drum pattern immediately sprang to mind: Tom Sawyer.

So we asked Sixty Symbols viewers to send in audio samples of their drumming along to Tom Sawyer, which we subsequently attempted to evaluate using a technique called detrended fluctuation analysis. When I say “we”, I mean a number of undergraduate students here at the University of Nottingham (who were aided, but more generally abetted, by myself in the analysis.) I’ve set a 3rd year undergraduate project on fluctuations in drumming for the last three years; the first six authors on the arXiv paper were (or are) all undergraduate students.

Unfortunately, the sound quality (and/or the duration) of many of the samples submitted in response to the Sixty Symbols video was just not sufficient for the task. That’s not a criticism, in any way, of the drummers who submitted audio files; it’s entirely my fault for not being more specific in the video. We worked with what we could, but in the end, the lead authors on the arXiv paper, Oli(ver) Gordon and Dom(inic) Coy, adopted a different and much more productive strategy for their version of the project: they invited a number of drummers (twenty-two in total) to play along with Tom Sawyer using only a hi-hat (so as to ensure that each and every beat could be isolated and tracked) and under exactly the same recording conditions.

You can read all of the details of the data acquisition and analysis in the arXiv paper. It also features the lengthiest acknowledgements section I’ve ever had to write. I think I’ve thanked everyone who provided data in there but if you sent me an MP3 or a .wav file (or some other audio format) and you don’t see your name in there, please let me know by leaving a comment below this post. (Assuming, of course, that you’d like to be acknowledged!)

We submitted the paper to the J. New Music Research last year and received some very helpful referees’ comments. I am waiting to get permission from the editor of the journal to make those (anonymous) comments public. If that permission is given, I’ll post the referees’ reports here.

In hindsight, Tom Sawyer was not the best choice of track to analyse. It’s a difficult groove to get right and even Neil Peart himself has said that it’s the song he finds most challenging to play live. In our analysis, we found very little evidence of the type of characteristic “crossover” in the correlations of the drumming fluctuations that emerged from Esa and colleagues’ study of Porcaro’s drumming. Our results are also at odds with the more recent work by Mathias Sogorski, Theo Geisel, Viola Priesemann (of the Max Planck Institute for Dynamics and Self-Organization, and the Bernstein Center for Computational Neuroscience, Göttingen, Germany) — a comprehensive and systematic analysis of microtiming variations in jazz and rock recordings spanning a total of over 100 recordings.

The likelihood is that the conditions under which we recorded the tracks — in particular, the rather “unnatural” hi-hat-only performance — may well have washed out the type of correlations observed by others. Nonetheless, this arguably negative result is a useful insight into the extent to which correlated fluctuations are robust (or not) with respect to performance environment and style. It was clear from our results, in line with previous work by Holger Hennig, Theo Geisel and colleagues, that the fluctuations are not so much characteristic of an individual drummer but of a performance; the same drummer could produce different fluctuation distributions and spectra under different performing conditions.

So where do we go from here? What’s the next stage of this research? I’m delighted to say that the Sixty Symbols video was directly responsible for kicking off an exciting collaboration with Esa and colleagues at Tampere that involves a number of students and researchers here at Nottingham. In particular, two final year project students, Ellie Hill and Lucy Edwards, have just returned from a week-long visit to Esa’s group at Tampere University. Their project, which is jointly supervised by my colleague Matt Brookes, Esa, and myself, focuses on going that one step further in the analysis of drumming fluctuations to incorporate brain imaging. Using this wonderful device.

I’m also rather chuffed that another nascent collaboration has stemmed from the Sixty Symbols video (and the subsequent data analysis) — this time from the music side of the so-called “two cultures” divide. The obscenely talented David Domminney Fowler, of Australian Pink Floyd fame, has kindly provided exceptionally high quality mixing desk recordings of “Another Brick In The Wall (Part 2)” from concert performances. (Thanks, Dave. [3]) Given the sensitivity of drumming fluctuations to the precise performance environment, the analysis of the same drummer (in this case, Paul Bonney) over multiple performances could prove very informative. We’re also hoping that Bonney will be able to make it to the Sir Peter Mansfield Imaging Centre here in the not-too-distant future so that Matt and colleagues can image his brain as he drums. (Knock yourself out with drummer jokes at this point. Dave certainly has.) I’m also particularly keen to compare results from my instrument of choice at the moment, Aerodrums, with those from a traditional kit.

And finally, the Sixty Symbols video also prompted George Datseris, professional drummer and PhD student  researcher, also at the Max Planck Institute for Dynamics & Self-Organisation, to get in touch to let us know about his intriguing work with the Giesel group: Does it Swing? Microtiming Deviations and Swing Feeling in Jazz. Esa and George will both be visiting Nottingham later this year and I am very enthusiastic indeed about the prospects for a European network on drum/rhythm research.

What’s remarkable is that all of this collaborative effort stemmed from Sixty Symbols. Public engagement is very often thought of exclusively in terms of scientists doing the research and then presenting the work as a fait accompli. What I’ve always loved about working with Brady on Sixty Symbols, and with Sean on Computerphile, is that they want to make the communication of science a great deal more open and engaging than that; they want to involve viewers (who are often the taxpayers who fund the work) in the trials and tribulations of the day-to-day research process itself. Brady and I have our spats on occasion, but on this point I am in complete and absolute agreement with him. Here he is, hitting the back of the net in describing the benefits of a warts-and-all approach to science communication…

They don’t engage with one paper every year or two, and a press release. I think if people knew what went into that paper and that press release…and they see the ups and the downs… even when it’s boring… And they see the emotion of it, and the humanity of it…people will become more engaged and more interested…

With the drumming project, Sixty Symbols went one step further and brought the viewers in so they were part of the story — they drove the direction of the science. While YouTube has its many failings, Sixty Symbols and channels like it enable connections with the world outside the lab that were simply unimaginable when I started my PhD back in (gulp…) 1990. And in these days of narrow-minded, naive nationalism, we need all the international connections we can get. Marching to the beat of your own drum ain’t all it’s cracked up to be…

Source of cartoon: https://xkcd.com/1736/

[1] 78. “Seven eight”.

[2] 50 Hz or 60 Hz depending on which side of the pond you fall. Any experimental physicist or electrical/electronic engineer who might be reading will also know full well that mains noise is generally not only present at 50 (or 60) Hz — there are all those wonderful harmonics to consider. (And the strongest peak may well not even be at 50 (60) Hz, but at one of those harmonics. And not all harmonics will contribute equally.  Experimental physics is such a joy at times…)

[3] In the interests of full disclosure I should note that Dave is a friend, a fan of Sixty Symbols, Numberphile, etc.., and an occasional contributor to Computerphile. He and I have spent quite a few tea-fuelled hours setting the world to rights



A Patter of Podcasts*

* Following extensive research — the best part of three whole minutes on Google — there shockingly appears not to be a collective noun for podcasts. Henceforth, I’m using “patter”. Given at least one OED definition of the word, I think it fits.

I’ve been very fortunate — if I were a religious man, I’d say “blessed” — to have had the support of not only the fantastic marketing team at Ben Bella (including, in particular, Lindsay Marshall) but a number of colleagues and friends when it comes to “plugging” that book I’ve recently written.

I don’t want to turn Symptoms… into a series of adverts for ‘Uncertainty to 11′ — and I won’t. Promise. I’ve got a stack of non-book-related posts coming up if I can ever find time — but I’ve done a series of podcasts and interviews recently that I’ve enjoyed so much I wanted to say a public thank you to all those involved (including Lindsay for setting up and coordinating the majority.) I’ve already blogged about The Aussie Pink Floyd pinkcast and The Death Hangout, and there are a few other podcasts to be uploaded/broadcast in future that I’ll blog about (briefly) in due course, but for now…

The Unmade Podcast

Unmade…” is the brainchild of Brady Haran, with whom I’ve worked just a little over the last decade or so, and Tim Hein. The premise is that they chat about ideas for podcasts that might get made, but probably never will. Occasionally, they invite a guest or two on to join in the conversation and come up with their own ideas for podcast themes. Not only did Brady and Tim let me do that — although, as I noted in the podcast, I can’t claim credit for all of the suggestions I made — but they very kindly let me waffle on at length about that bloody book…


Although the Ikonokast podcast with Greg Laden and his co-host Mike Haubrich started off on that Spinal Tap-inspired theme, we diverged from there quite quickly and chatted about a much broader variety of academic (and non-academic) concerns than just the metal-quantum interface…

Coincidentally, that piece of metal that opens up the Ikonokast conversation (and closes the Unmade podcast) is something called The Root Of All Things that I recorded a while ago as background music for a video. I’m hoping to find time to expand this short piece, with the help of a few musician (and scientist-cum-musician) friends, to a full-blown nano-themed sci-fi metal track over the course of the next year or so. (After all, there’s EPSRC funding to do so.) For now, however, that piece has found its place as backing music for some of Pete McPartlan’s wonderfully quirky animations and art…

The Quantum Podcast

height_90_width_90_15817639_1066803776798865_1271 The Uncertainty Principle and Metal

Maria, the host, is a second year undergraduate physics student who explains a variety of topics covered in her degree via her podcast. We had a fun time discussing everything from Devin Townsend to string theory and the state-of-the-art in theoretical physics. The latter is a theme I’m going to return to very soon here at Symptoms… (and elsewhere) in the context of Sabine Hossenfelder‘s impassioned, sharp, and brilliant critique of the state of 21st century physics, “Lost In Math“. If you have any interest at all in physics, you owe it to yourself to go get Hossenfelder’s book.


I spent most of this podcast trying to stop laughing. Byrne and Wade, your genial hosts, are both very funny guys. Unfortunately, when tasked, I failed spectacularly to come up with a musician joke on the spot. Usually I fall back on one of the drummer classics — “How can you tell a drummer’s at the door? The knocking speeds up” — but it was clearly too early in the morning and/or insufficient caffeine had been imbibed.

A big thank-you to Brady, Tim, Greg, Mike, Maria, Byrne, and Wade for the invitation to join them for a natter.

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:


Brady Haran, Doctor of Letters

A short blog post to say just how delighted I am that Brady Haran was awarded an honorary degree by the University of Nottingham earlier this week. It’s been my great pleasure to work with Brady on Sixty Symbols (and a number of his other channels) over the past seven years. Despite — no, make that because ofour occasional tête-à-tête on just how to put across a piece of physics for a broad audience, I always look forward immensely to Brady (+ camera + bag of accessories) appearing at my door.

Brady’s work, and his remarkable work ethic, have put Nottingham on the map — and then some — when it comes to public engagement and communicating science. As Mike Merrifield describes in the video below, Brady’s ever-expanding portfolio of videos has topped 400 million views./ That’s nearly 2 billion minutes’ worth of viewing worldwide. All of us at the University of Nottingham owe Brady a huge debt of gratitude and it’s wonderful that this has been formally recognised by the award of Doctor of Letters.

Brady is his usual modest self in his acceptance speech (starting at around the 6 minute mark below), but it’s no exaggeration to say that he has fundamentally and radically changed my approach to explaining science and, by extension, my teaching.

I have learnt so much from him over the years.

Thank you, Brady, and congratulations.