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Assumptions

Assume, for now, that a rhythm is specified as a set of pulses within an interval of time. Assume that a pulse is an instant of attack. Call the total interval of time a period. 'Cyclyclical rhythms' as defined here are distinguished by the exact spacing of pulses within a period; the tempo or tempo change of the period is not (yet) relevant to the cyclyclical rhythm. In our examples, the magnitude of the duration of the period will remain fixed.

The durations in cyclyclical rhythms are specified not in absolute terms of time interval (minutes, seconds, beats of a metronome), but relative to the period, and thus expressed as a (unitless) proportion. For example, '1/2' (or '0.5') will represent a duration (interval of time) of exactly half the duration of the period.

We are concerned with durations that are shorter than the duration of the period; i.e., greater than or equal to zero (no interval) and less than one (period). We can easily convert numbers outside that range by adding or subtracting 1 until they are in the range. This is tantamount to using a Modular arithmetic with a modulus of 1. (Clocks and twelve-tone theory use a modulus of 12.)

We can use the metaphor of a timeline, assuming that a line segment (representing a period) can be broken up into smaller line segments (durations or intervals) as delineated by the placement of points (pulses):

mr_line.png
Furthermore, we can emphasize the cyclical nature of our arithmetic if we bend the line segment into a circle (drawing a point at the top for 0/1):
mr_cycle.png
When we want to refer to an interval from zero, which also specifies a single pulse within a period, we will use unadorned expressions (e.g. a and 1-a). When we want to talk about an interval from anywhere, emphasizing only the magnitude of it, we will enclose it within vertical slashes | |

Generators

Cyclyclical rhythms are calculated by taking multiples of a single interval, called the generating interval or generator. When one interval is called a generator interval relative to a period, a family of cyclyclical rhythms is specified. When how many multiples and which multiples are specified, a single cyclyclical rhythm is specified.