Transcendental Generative Music

This is a departure from my normal open content talk, so feel free to pass this post by if you’re not interested in music.

The twelve tone technique is a music composition method generally used for creating atonal music (and is a special, and perhaps the earliest, type of serialism). Using the twelve tone technique, a composer creates a sequencing of the twelve tones of the chromatic scale without repetition, which tends to diffuse any sense of a tonal center in the overall composition. Once the composer arrives at a sequencing of the twelve chromatic tones, these may be manipulated to create a 12 x 12 matrix which can be read left to right, right to left (the retrograde), top to bottom (the inversion), or bottom to top (the retrograde inversion). See this interactive matrix for examples.

While there is much art to everything that comes afterward, arriving at the original sequencing of pitches plays an important role in this type of composition.

Over a decade ago, I was sitting in my room with nothing to do when the thought occurred to me “I wonder if you could use transcendental numbers to create tone rows…” I started with pi. Staring at the number spelled out to several digits, I wondered “how would I even go about this?” The idea occurred to me to choose an arbitrary starting pitch, and then use each digit in pi as a number of half steps to move up from the current pitch. So, taking C as a starting pitch, 3.14 would give a sequence of C, D# (3 half steps up), E (1 half step up), G# (4 half steps up), etc. However, with apologies to my friend Eve, pi just wasn’t cut out for the job – the 10th pitch generated by the digits of pi is a repeat of the first note in the sequence. Oh well, I thought.

In my disappointment, and realizing how amazing it would be if a number could walk the minefield of 12 pitches without landing on a repeat somewhere, I was very pleased to discover that e in fact fits the bill! If we pick an arbitrary starting pitch (E seems like a natural choice), the digits of e give us the next ten pitches without repetition. The 11th digit produces a repetition in the final pitch of the row, but technically we don’t need it – since we already have the first 11 pitches we can determine the final pitch in the row by process of elimination.

So e, it turns out, generates a perfect tone row. Using a value for e of 2.718281828, the matrix based on e looks like this:

E	F#	C#	D	A#	C	G#	A	F	G	D#	B
D	E	B	C	G#	A#	F#	G	D#	F	C#	A
G	A	E	F	C#	D#	B	C	G#	A#	F#	D
F#	G#	D#	E	C	D	A#	B	G	A	F	C#
A#	C	G	G#	E	F#	D	D#	B	C#	A	F
G#	A#	F	F#	D	E	C	C#	A	B	G	D#
C	D	A	A#	F#	G#	E	F	C#	D#	B	G
B	C#	G#	A	F	G	D#	E	C	D	A#	F#
D#	F	C	C#	A	B	G	G#	E	F#	D	A#
C#	D#	A#	B	G	A	F	F#	D	E	C	G#
F	G	D	D#	B	C#	A	A#	F#	G#	E	C
A	B	F#	G	D#	F	C#	D	A#	C	G#	E

Here is a midi statement of the original row, the retrograde, and the inverse. In this recording you’ll hear that the pitch durations are also determined based on the digits of e (a pitch arrived at by going up 2 half steps is twice as long as one arrived at by going up 1 half step).

I’m currently working on two things in connection with this: an actual composition based on the tone sequence, likely a two-part invention or fugue (which I hope to generate completely out of the number itself), and some software to search through the digits of pi, e, and other transcendental numbers in search of digit sequences that will produce tone rows, and therefore matrices, and (hopefully, eventually) inventions, fugues, and other types of music.