A metaphor: Consider a stack of 16 pennies. Divide these pennies into 5 stacks in such a way that the stacks are as equal as possible. For example, the following stacks
33433
each contain either 3 or 4 pennies and are as nearly equal as possible. Now imagine that each penny represents an eighth note, 16 pennies represents 2 bars of 4/4, and each stack represents a musical time interval, 3 pennies denoting a dotted quarter note and so on. In this notation, 33433 represents a familiar rhythmic pattern, bossa nova. Keep in mind that each stack represents some combination of note and rest values. The "3" in this example could be a combination of eighth note and two eighth rests, adding up to a dotted-quarter time interval. Let's generalize this. Consider a stack of n pennies to be divided into m nearly equal stacks, m < n. Using the familiar process of long division, we can write
n/m = p + r, where r = n mod m, 0 ≤ r ≤ m-1
That is, we now have m stacks of p pennies each, with r < m pennies left over. If we now distribute those r pennies among the m stacks, adding just one penny to each stack, we now have m stacks each of which contains either p pennies or p+1 pennies.
Theorem: Any stack of n pennies can be divided into m stacks of pennies, m < n, in such a way that the size of the stacks varies by no more than 1.
Different orderings of the stacks are possible, yielding different rhythmic patterns, for example:
33433
34333
43333
33334
33343
Some of these patterns may be more interesting than others. That's where the math leaves off and rhythmic feeling and art comes in. You might look not only at the slight unevenness of the stacks, but at how this unevenness is distributed.
Computer music uses the common concept of a rhythmic grid on which musical events, notes or pulses, can be placed. So instead of counting pennies, count pulses on a grid. For example, the 33433 notation from above could alternately be expressed as
100 100 1000 100 100
a sequence of on/off pulses on a grid, here, two 4/4 bars of eighth notes. This can be understood as a repeating or looping sequence,
100 100 1000 100 100 100 100 1000 100 100 ...
in which case an even better metaphor might be a roulette wheel with n slots, on which we can distribute m balls such that balls are as evenly spaced as possible.
The musical usefulness of such time divisions has been widely discussed. Just do a web search for "Euclidian rhythms". Such patterns are rhythmic approximations. For example,
332
approximately divides an 8-fold pulse into three, and approximates a triplet. Similarly, in 9/8 meter,
54
approximately divides a 9-fold pulse in two. It is important to note that these are not exact divisions but approximations of a tuplet, which yet preserve the underlying pulse. Another side effect of this slightly uneven division is a feeling of swing.
We have been looking at evening up the "stacks" or pulse groupings so that they differ by no more than 1. But we can relax the restriction such that the stacks differ by no more than 2, and still come up with some interesting rhythms. For example, the following grouping represents another familiar rhythm, the 3-2 clave used in Latin music:
33424
Take the 3-2 clave notation above, 33424, and rewrite it in the alternate grid notation,
100 100 1000 10 1000
and then rotate (or phase-shift) it by 8 grid slots so that it becomes the following, the familiar 2-3 clave used in Latin music:
00 10 1000 100 100 10
The roulette-wheel metaphor may be useful here. The pattern can start anywhere around the perimeter of the wheel, that is, be rotated against the bar lines, without changing the relationships in the pattern. The listener will hear the pattern, not the bar lines. In practice, of course, a musical composition will often nail down the "one" as a rhythmic anchor for the listener. Or a composition can be ambiguous about the "one", for example, Richard Carpenter's tune "Walkin'" as played by Miles Davis.
Ambient compositions, drones and such, can be completely ambiguous about bar lines, or even about meter. Such compositions can be pulse-based, with rhythmic groupings unfolding freely from the underlying pulse.
Euclidian rhythms represent an even distribution of events over a time interval. This could be a "frame" two bars in length, or maybe the perimeter of the metaphoric roulette wheel. In the above bossa nova pattern, 100 100 1000 100 100, five out of the 16 grid slots are "on". But this is not equivalent to a sequence where each grid slot has a 5/16 probability of being "on".
The Euclidian pattern is an even distribution of m events over n slots, that is, an even distribution within a finite frame. A Euclidian pattern generator could reset at the end of each frame and compute a new Euclidian pattern, but it will always fill m slots in the frame.
A random sequence is not based on a fixed frame, but on a probability factor, such as 5/16, for each grid slot. So any pattern at all could appear, including runs of 1s or 0s, but longer patterns would tend toward 5/16 of the slots being "on".
Instead of building a Euclidian pattern in a fixed-length frame, how about if we extended the "approximately even" distribution in a more open-ended way. For example, you could have a sequence of 2s and 3s something like
23232232323323223233323223222... or
10 100 10 100 10 10 100 10 100 10 100 100...
where each "pulse group" would be either 2 or 3 grid spaces in length. Computationally, each time a new group begins with an "on" pulse, you would flip a coin to determine whether the length of the group will be 2 or 3. Repeat each time you start a new group.
Another possibility might be to roll a die to determine the length of the next pulse group, which would now be anywhere from 1 to 6 in length.
In the tempered chromatic scale, dividing the octave into 12 equally spaced (logarithmically speaking) intervals, we can do procedures analogous to what we did with rhythms above. For example, 12 intervals can be subdivided into 5,
22323
to produce a major pentatonic scale. Divisions into 7 intervals produce the familiar major and minor scales. Division into 8 intervals produces a diminished scale:
21212121
There are at least two obvious ways to extend these procedures. The first is to look at larger spans of intervals and subdivide those. For example, take a span of 18 chromatic intervals, say from C2 to F#3, and divide it into 7, perhaps as:
22322322
So this is equivalent to the scale C-D-E-G-A-B-D-E-F#, a C major Lydian scale. I will need to explore to see if this kind of procedure comes up with something more interesting.
Another way to come up with less recognizable and familiar sounds would be to divide the octave into some other number of equal-tempered intervals, such as 18 or 19 or 24, and work with the Euclidian interval divisions of those.
There are several computer-music environments that allow you to play around with algorithmic composition, including Max from Cycling74, and Reaktor from Native Instruments. I'm presently working with Propellerhead Reason, a full-featured DAW that provides a variety of synthesizers, audio effects, MIDI effects, third-party plugins, and a virtual Control Voltage (CV) system that allows interactions among modules. I like it because I can work with lots of prebuilt components to set up, record, and edit generative music. It's maybe not as totally flexible as Max or Reaktor, but allows me to get musical results without a lot of coding. On the downside, it costs $$ to buy some of the components I want to work with, but I'm willing to pay developers to do the technical work.
Within Reason, there are a couple of modules that support either Euclidian or probabilistic generation of musical sequences. Here are a few:
Of course, you can also just manually sequence Euclidian rhythms, based on loops of different lengths.
I hope you find this writeup to be useful and are inspired to experiment with these ideas. I've been thinking about this concept for some years now. Following is an unpublished chart I made in 1987 to map out some of the possibilities:
rhythmic approximations (PDF)