Polytempo refers to the simultaneous use of two or more distinct tempi in a piece of music. Polyrhythm can be viewed as a special case of polytempo; any polyrhythm can be written as a temporary tempo change.
For this post, an instance of polytempo is "trivial" if one or both of the following is true for the entirety of the polytemporal section:
- Measure onsets always coincide in all tempi (as would happen with 3/4 at 90 bpm against 2/4 at 60 bpm).
- There are only two tempi and the ratio between them (or its reciprocal) reduces to 2:1, 3:2, or 4:3.
(Note that these conditions are unrelated to meter.)
An instance of polytempo is "precise" if the relationship between the different tempi is specified exactly.
As far as I am aware, Charles Ives's Symphony no. 4 is the earliest work featuring polytempo that is both nontrivial and precise. (The polytempo in Ives's Central Park in the Dark is not precisely indicated.) Conlon Nancarrow would fully explore polytempo a few decades later. Are there any examples of explicitly-notated, precise, nontrivial polytempo before Ives? Or by composers other than Ives before Nancarrow?