Fractal Microtonal Music

Here is a 2 Megabyte mp3 file , uploaded 25 Jan 2004, with an 11 minute computer-generated tone sequence. Since it's a third version, I'll call it, "The Calliopist's Charm".

This tone sequence has two peculiarities. First, the tones are selected from an equal tempered scale that has 53 pitches per octave. Conventional music, i.e. as played by piano, uses a scale with 12 pitches per octave. So the pitches in "The Calliopist" are relatively finely spaced, or microtonal.

The other peculiarity is that the tone sequence was generated by a rather short computer program, based on a random number generator. The process generates a set of tones with an approximately fractal character, i.e. with correlations at many different time intervals.

Here is another file (5 MB) that's constructed very much like the first, except I used CSound to synthesize the waveforms. It also has a lot more variation in the duration of notes. This is smaller file (2.5 MB) where I fine tuned some of the parameters. The CSound source files for this shorter piece are here. Yet another variation is a favorite. These are from August 2005.

See also some further experiments. My most recent creations are at SoundClick.

Equal Temperament with 53 Pitches per Octave

53 might seem like an arbitrary choice for a number of pitches per octave. But one can use mathematics to narrow the choices down considerably, and it turns out that 53 is a very good number for an equal tempered scale.

An equal tempered scale uses pitches whose frequencies are proportional to 1, x, x**2, x**3, x**4, etc., where the "**" notation means exponentiation, e.g. x**2 means x*x (x multiplied by x), or x squared. Pitches whose frequencies are in a simple rational ratio sound harmonious together - this is the foundation of musical harmony, attributed to Pythagoras but probably known even before his time. The simplest non-trivial ratio is 2:1. This is the ratio between pitches an octave apart. One would generally like a scale to include an exact octave. This constrains the scale step ratio x to be 2**(1/p) for some integer p. The conventional half-step on a piano separates pitches with a frequency ratio 2**(1/12). The "Calliopist" uses micro-steps with frequency ratio 2**(1/53).

The next key ratio is 3:2, known musically as a perfect fifth. A good scale will include, for each pitch in the scale, another pitch whose frequency has a ratio very close the 3:2 with the first. Our equal tempered scale has pitches with frequencies proportional to 1, 2**(1/p), 2**(2/p), 2**(3/p), etc., i.e. 2**(q/p) for every integer q. We want there to be some value of q for which 2**(q/p) is very close to 3/2. One can use logarithms to re-express this: there should be some value of q for which q/p is very close to log(3/2)/log(2).

What makes this interesting is that q/p is a rational number, but log(3/2)/log(2) is irrational. No matter what value of p we pick, we will never find a rational number q/p that is exactly equal to the irrational number log(3/2)/log(2). On the other hand, one can work with bigger and bigger values of p to get more and more accurate approximations to the irrational target, indeed to get as accurate as we'd like. After all, for any given p, the worst error we can have is 1/(2*p).

The whole subject of approximating irrational numbers by rational ones is quite fascinating, and quite deep. The curious thing is that the error in the approximation does not steadily decline as p gets bigger. Instead, as p gets bigger, occasionally there will be a very good approximation, and then there will be a gap before some considerably bigger p finally arrives to give a better approximation.

There is a technique called "continued fractions" which can rather directly give a series of values of p which give better and better approximations q/p to an irrational number. When one applies this method to our irrational number log(3/2)/log(2), the interesting part of the sequence runs: 5, 12, 41, 53, 306. The numbers that jump out and demand recognition here are, of course, 12 and 53.

Another key musical ratio is 5:4, a major third. Again this ratio needs to be included in a good scale. Unfortunately, none of the values of p that are best for the perfect fifth are also best for the major third - the continued fraction method gives the sequence 3, 28, 59. Since the perfect fifth is more fundamental, the straightforward thing is to pick from the best p values for the perfect fifth, and from among these look for values that give reasonable approximations for the major third. The conventional piano scale does moderately well: a (just-tuned, or exact 5:4 ratio) major third is 3.86 half-steps. In the equal tempered scale with 53 pitches per octave, a just major third is 17.06 steps. With 41 pitches per octave, the just major third is 13.20 steps. The 41 pitch scale gives better perfect fifths, but worse major thirds (relative to the step size - they're still better in absolute terms, because of the finer steps). The 53 pitch scale is better both in the perfect fifth and in the major third. This makes a 53 pitch scale a particular rich scale for musical harmonies.

Just Ratios in Equal Tempered Steps
Steps in an octave
12 41 53
ratio 3/2 7.01955 23.98346 31.00301
5/4 3.863137 13.19905 17.06219

Algorithmic Sequence Generation

Here are a couple notes I posted to the MakeMicroMusic yahoo group ( first note, second note) on the algorithmic approach I used to generate the tone sequence.

Resources

Easley Blackwood, The Structure of Recognizable Diatonic Tunings (1986)

Alain Danielou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness, (Inner Traditions, 1943 & 1995)

Hermann Helmholtz, On the Sensations of Tone As a Physiological Basis For the Theory of Music, (Dover, 1885 & 1954)

Stuart Isacoff, Temperament: The Idea That Solved Music's Greatest Riddle, (Knopf, 2002)

W. A. Mathieu, Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression, (Inner Traditions, 1997)

Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rig Veda to Plato, (Shambhala, 1976 & 1978)

Harry Partch, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fullfillments, (Da Capo, 1949 & 1974)

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