Musical Pi, Part 1

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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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10 Responses to “Musical Pi, Part 1”

  1. Maria H. Andersen Says:

    Trackback: http://tcmtechnologyblog.blogspot.com/2008/01/from-360-musical-pi.html

  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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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 […]

  10. jimmyz Says:

    Have you heard this interpretation of pi base 12?

    As a waltz, the digits can be timed to create order and incorporate common chord progressions.

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