Monday, 19 November 2012

Tau of Phi

By Professor Phil Moriarty
(to accompany Numberphile's Tau of Phi video)

The song can be downloaded from Soundcloud.

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, it just 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” 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 aficionado).

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” is based around Bb minor. My first thought was to transpose ViHart’s vocals down to Bb.

But she ended up sounding not too unlike Barry White at times. 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 fads out with tau riff gently looping on guitar.

The song can be downloaded from Soundcloud.


  1. Hi, I love the way Dr Moriarty explains stuff. Can you do a video of him explaining how a boomerang works? With aerodynamics, cyclic precession and all sorts of cool physics going on, plus your Australian connection, I think it would make for an excellent topic. I've been told that a boomerang "is a boomerang shaped object that boomerangs back when you boomerang it"!

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