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
As 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).
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
Emerge from the equation
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
The five-fold way
Crossing points define
[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.
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: