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Inspired Flight's "An Outlet" is an example of what I mean.

For the first few measures, a melody of 16 eighth notes is established in middle register (if I have it right: A E A E; A F G C; A E E- E; A F G C). Then, starting at 0:13, a repeated melody of 9 eighth notes is introduced in a higher register (F C A; F C A; D C A). Since 9 and 16 have no common factors, there are a lot of dynamics that arise during the theoretical 144 eighth notes that it takes for the two to reach a common repetition.

Although I didn't realize this kind of quasi-periodicity was happening until I attempted to transcribe it, it seems like a neat technique for creating texture. Does it have a name?

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The technique is called "phasing", and is often attributed to Steve Reich, who wrote a series of phasing pieces.

Two examples of Reich's phasing pieces can be found in this answer: What's “species counterpoint”? Are there any other types of counterpoint?

A similar-sounding technique, but not "phasing", because it's not intended to be periodic is Terry Riley's "In C", which is considered to be the first Minimalist composition.

The broadest relevant terms are "polymeter" and "polyrhythm", which are discussed in Polymeter vs Polyrhythm

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