[October, 2011: Rhythmicon pages archived, no longer updated or maintained.]

The rhythmicon measure

The tempo of the fundamental is the base beat of the rhythmicon. The tempos of the harmonics are whole number multiples of the base. This can be thought of as a measure consisting of a single beat, or 1/1 time if you will, that is further subdivided into 23 distinct rhythmic patterns. A graph of a measure, with the harmonics on the vertical axis and the rhythms on the horizontal, reveals many graceful interlocking curves that describe visually the swaying or rocking pattern of sound heard when all harmonics of the rhythmicon are sounding. The accordion-like squeezing and spreading of beats are clearly evident.

Cascades

There are many possible points of entry, numerous curves and patterns that could be used for compositions based on the structure of the beats. I chose to work with the curves that begin (descending) or end (ascending) on each of the 24 beats of the highest harmonic. I call these curves "cascades" (a kind of special case arpeggio)

On the downbeat all harmonics sound simultaneously. They are vertically aligned. The first descending cascade occurs, then, on the second beat of the 24th harmonic, the second descending cascade on the third beat, etc., with the 24th descending cascade beginning on the downbeat of the next measure. Each harmonic of the cascade begins to sound along a hyperbolic curve. The curve is longer for each descending cascade, and the spread between the initial sounding of each harmonic increases from higher to lower harmonic.

In the first descending cascade, each harmonic begins within the first measure. The fundamental sounds on the downbeat of the second measure. In the second descending cascade harmonics H24 down through H3 sound in the first measure. H2 sounds on the downbeat of the second measure and H1 on the downbeat of the third measure. In the third descending cascade H3 sounds on the downbeat of the second measure, H2 at the midpoint of measure two, and H1 on the downbeat of the fourth measure. The gap between H2 and H1 spreads in succeeding cascades. This makes sense algebraically, but is nonsense to the ear. Thus I adopted the compositional (and programming) rule that in terms of the cascades H1 would never sound more than one measure by itself, thus artificially shortening the curves. Internally, however, in the longer cascades, measures may be entirely skipped over before the next harmonic begins to sound. In this scheme, then, the shortest cascade is one measure long, and the longest spreads over 13 measures.

Each beat of the 24th harmonic is also the end of an ascending cascade. The first ascending cascade begins on the downbeat of the measure and ends on the 24th beat of the 24th harmonic. The second ascending cascade ends in the second measure on the 23rd beat of the 24th harmonic, and so on in reverse of the descending cascades. The spread decreases as higher harmonics sound. Because H1 always begins on the downbeat of the first measure in ascending cascades and always begins on the downbeat of the next measure in descending cascades, descending cascades are always one measure longer than the corresponding ascending cascade.

One descending and one ascending cascade 0:30    LISTEN

The cascades represent starting and stopping points for the sounding of harmonics. A harmonic may continue to sound, sound only for a short while, or sound only once. Any subset of any cascade can start or stop at any point along any of the 48 curves. In addition, using Cowell's concept of tone clusters, any subset of harmonics can start or stop at once rather than along a cascade curve. That is, the volumes can be turned up from 0 at any time. Depending on the sound used each harmonic will be heard instantly in mid-beat at some point of its ADSR curve or as a sustained tone, or not heard until its next beat.

The rhythmicon calculator

Rhythmicon sounds as described in these pages and realized with Kyma contain scripts that determine the tempo and timing of the beats and the beat lengths. When correlating other sounds with the rhythmicon and when working with rhythmicon measures of extreme length I have often found it necessary to calculate the beat lengths and hit times manually. The Ancient Chinese Enclosing Game Compositional Matrix, an abstraction of the structual elements of the rhythmicon developed more recently, requires the same calculations. To relieve the tedium of these repetitious calculations I've written a rhythmicon calculator program that can be downloaded from this site on the Rhythmicon Calculator page.