#### Rhythm Space

under construction!!!

Another side to this argument concerns the notion of a "rhythm space" -- a metric space for rhythms. One would imagine the construction of this space to be facilitated by the inclusion of quantized tatum-grids in the representation. Specifically, in that case the space would be essentially discrete, aside from "feel factors" which would provide small continuous variations of a fixed underlying rhythmic pattern. But this would not be general enough. Consider the possible placements of five clave hits over the span of 2 seconds -- a subspace of rhythm space. The range of all such rhythms is continuous, clearly, and the representation should reflect that. Thus the relationships among, say, various Afro-Cuban claves and various West African bell patterns are not lost because of quantization issues.

So imagine a rhythm space, for a monophonic voice, as follows: A particular point in the space (i.e., a particular rhythm) is characterized by

• duration `t`
• number of notes `N`
• a set of note-on times:
`t_on[i], i = 1 to N, t_on[i] < t_on[i+1] < t ` for all `i`
• a set of note-off times:
`t_off[i], i = 1 to N, t_on[i] < t_off[i] < t_on[i+1] < t` for all `i`
• an underlying pulse or beat-scheme, when applicable
Among the relevant conceptual distances between two rhythms would be:
• difference in durations
• differences in number of notes
• some weighted average of all the differences between note-on times of all possible ordered pairs of notes (This would have to try all relative displacements of the two rhythms)
• some other term involving note durations
• similar measurements after scaling one rhythm to have the same duration as the other
• differing underlying beat-schemes may also be considered for tempo and relative placement
Thus, for example, identical rhythms at different tempi would be close to each other, but so might two different-length rhythms at identical tempi, one of which "contains" the other. Also, notewise-identical rhythms with different relationships to a main pulse would be close but not identical in this space.

We could imagine a psychoacoustics experiment similar to the timbre-space multidimensional scaling, that would help us find other relevant dimensions.