## Musical Pi, Part 1

by

Jon Turner, a professor of musical composition here at Nazareth, has composed a suite of music based on π! (see also part 2 and part 3) As he says:

The basic idea is to use the decimal expansion of pi to give an unendingly varying [but] related series of notes.

The first step was to convert π to base 12 (to match the chromatic musical scale), so

$\pi = 3.1415926535\ldots_{10} = 3.184809493B\ldots_{12}$

(where B represents decimal 11 in base 12). Starting with C at 0, he gets

$\pi = E\flat D\flat A\flat E A\flat C\ldots$

Here’s Jon explaining the piece “Quest 4π” (click to listen):

The drum part is a sequence of around 1400 digits (0-11 each), mapped to 12 various midi drum sounds in eighth-notes, with zeros marked by both a bass drum and a crash cymbal, also plus bass and chord: the mixolydian chord progression follows the digits as chromatic pitch-classes (mapped c=0, c#=1, d=2, etc), with a new chord articulated every time a zero arrives in the continuous 8th-note drum part. This amounts to a sequence of about 85 chord changes in an overall time of 2:27.

There are a few places where an accented rhythm repeats the same chord. If zeros immediately repeat, there is not enough time to project the chordal harmony within a single eighth-note, so that it is taken to be a special articulation of the chord. Also, harmony of the “leading zero” (C) is accented by staccato breaths or resoundings, as at the midpoint. But when a π digit immediately repeats in the sequence, the bass shifts octave up or down, depending. So, if two 6′s (F#) occur in π, then the next bass is shift an octave up or down, and back for three in a row, and so forth. Otherwise, octave placement in the bass was by my best guess.

Once this rhythmic roux was prepared, a shredding 8th-note guitar solo was applied to the drum pulse by inclination, will, and taste.

The steps of the recipe:

1. Start with a supply of 10000 digits of π in text form (see listing below courtesy of math prof Dr. Larry Turner at swau.edu). These digits can also be generated with the built-in terminal calculator “bc” in Mac OSx, whose precision can be set to any number of digits and also any numeric base.
2. The text is massaged and reformatted in BBEdit so that, upon input to HMSL, a 10000-cell table is built containing the sequence of digits as numbers from 0 to 11 (duodecimal digits include 10=”A”, 11=”B”).
3. HMSL, or Hierarchical Music Specification Language, a MIDI dialect of FORTH for classic Macs, is programmed to generate a midifile containing an arbitrarily large number of π digits as MIDI notes, across the chromatic scale within a single octave, in a standard MIDI file.
4. Using the MIDI editing capabilities of Opcode’s Musicshop 2.0 midi sequencer, the converted digits are assigned drum notes, in a sequencer track. Zero is associated with a bass drum and crash cymbal combination, which sets metric downbeat within the continuous pulse of percussion.
5. All bass drum notes are copied to a fretless bass part, then are transposed according to π’s digital sequence, giving the chord progression 03.18480… or C, D#, C#, G#, E, G#, C, etc, a progression of about 85 chords total across the 1400 eighth notes in the drum part. The harmonic and rhythmic progressions both follow the same sequence, but at different speeds, averaging 1 bass/chord advance per dozen drum 8th-notes. Although π begins on 3 (Eb), the piece starts (and ends) on C because I considered it the reference “leading zero” to measure the digits of π.
6. The resulting bass part exactly follows the bass drum-cymbal rhythm. This line is copied and arranged into a keyboard track, edited to form a mild mixolydian (7sus4) chord, which slaves to the bass in harmonic parallel.
7. The resulting jam inspires a freely-composed guitar part, a continuous swath of shredded eighth notes, custom-fitted to this particular harmonic/rhythmic form of the digits.

The full 10000 duodecimal π digits are listed below, using “A” for 10 and “B” for 11 (as in hexadecimal).

Due to the mathematical nature of π, there is absolutely no regular pattern in this sequence, an arbitrarily random shape at any detectable level. Moreover, this is only one way of many: the piece would be completely different if a different irrational number were used (2π, 1/π, e, golden mean, arctangent, etc), if a different duodecimal digit were assigned the bass drum’s organizing function, if the progression were inverted, or if it were based on the interval between digits rather than the digits themselves, to name a few yet untried options.

One of the ways that truer randomness shows itself is the way different “clumpings” or repetitions can and will occur. This effect can easily be seen by scanning any part of the sequence. The digits are not uniformly scattered, they are randomly scattered.

The accents can be thought to illustrate the distribution of zero digits, for example, and this drives the irregular accent pattern of the chord progression, as it progressively resets the meter. Other drums repeat and form audible groups of several eighth-notes. The net philosophical result is that, without any regularity, the unending chain of forever novel percussion combinations seems to be filtered by the ear and brain to perceive a willed human expression. The virtuosity seems willed, rather than mechanical, as if the series was in a dialog with itself, approaching the “truth” about π, through ever higher steps of precision.

But to make it truly human-sounding, the mix needed some human error, so I overlayed a guitar part which would “rule” it and make continuous sense in a way the other parts could not possibly intend, but assent to nonetheless, a little gangsta guitar graffiti for Xmas.

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### 9 Responses to “Musical Pi, Part 1”

1. Maria H. Andersen Says:
2. TwoPi Says:

For a completely different juxtaposition between music and the ratio of circumference to diameter, Lawrence Mark Lesser’s modified lyrics to Don McLean’s “American Pie” might be of interest…. see Prof. Lesser’s website for details….

3. Walking Randomly » Music From Mathematical Constants Says:

[...] From Mathematical Constants In a recent post over at 360 the author was discussing how Joe Turner (a prof of musical composition) has formed a [...]

4. The One-Year Anniversary Carnival of Mathematics « 360 Says:

[...] over at Walking Randomly, writes about Music From Mathematical Constants. Following up on a post here at 360, he uses Mathematica’s SoundNote[] function, along with some clever list [...]

5. Musical Pi, Part 2 « 360 Says:

[...] Pi, Part 2 Following up (at long last) on Musical Pi, Part 1, we present the remaining nine pieces (plus bonus tracks!) in the suite of music based on π, [...]

6. Musical Pi, Part 3 « 360 Says:

[...] Finally, the last four tracks in the suite of pi-based music, composed by Jon Turner. (See also part 1 and part [...]

7. Wu, Ming Says:

I’ve calculated the first 88888 digits of π in base 12 as shown below:

3.184809493B918664573A6211BB151551A05729290A7809A492742140A60A55256A\
0661A03753A3AA54805646880181A3683083272BBBA0A370B12265529A828903B4B2\
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09A8906B61B8B48A62253A88A50A43BA0944572315933664476B3AABB77583975120\
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9804922175A068A46739461990A2065BB0A30BBAB7024A585B1A84428195489784A0\
7A331A7B0A1574565B373B05B03A5A80A13AB87857734679985558A5373178A7B282\
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8. TwoPi Says:

I wonder if there’s a published record for memorizing digits of pi in base 12. Would they need to be recited, or could they be performed?

9. From 360… Musical Pi Says:

[...] computer games like Guitar Hero and Rock Band. So they might really appreciate this post about Musical Pi from the 360 [...]