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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?

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    6/8 and 2/4 both have two beats per measure, so if 6/8 and 2/4 have different b.p.m. then the measures have different durations. Perhaps 3/4 and 2/4 would be a better example.
    – phoog
    Nov 16, 2021 at 11:11
  • @phoog You're right. Thanks for pointing that out.
    – texdr.aft
    Nov 16, 2021 at 11:21
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    Donald Byrd's page on the extremes of conventional music notation simply cites Gardner Read's Modern Rhythmic Notation (1978) when it discusses polymeter. From the description, Read's book does appear to contain examples, so it might be worth perusing it if you have access to it; but there may not be much specifically talking about polytempo. Nov 16, 2021 at 12:26
  • @MichaelSeifert I have the book. It discusses polytempo only in passing; there's no dedicated section for the topic. But it's a great resource for polymeter.
    – texdr.aft
    Nov 18, 2021 at 0:37

2 Answers 2

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The first piece that comes to mind is Johannes Ockeghem's missa prolationum. This piece is entirely written on mensural canons: different parts sing the same notes at different speeds. As the example in the Wikipedia article shows, the first Kyrie is a double mensural canon. There are two written parts, each of which is sung by two voices, the ratio of their tempi being 2:3. The slower voice of one part is effectively as fast as the faster voice of the other part, more or less as though one is in 3/2 and the other in 6/4 with the measures having the same duration, so you could say that the ratios are 4:6:6:9.

Another piece that comes to mind, but which may not satisfy the requirements of the question, is the Act 1 finale of Mozart's Don Giovanni, which begins as a minuet in 3/4. While that continues, one group of instruments begins playing another dance in 2/4, and then a third group switches to a fast 3/8. The quarter note of the 2/4 and the dotted quarter of the 3/8 are equal to the quarter note of the 3/4, so the measures have different lengths. I almost did not include this example because it's three different meters at one tempo, so arguably not polytempo, but I decided to mention it because you say that polyrhythm is a subset of polytempo, and this polymeter does result in polyrhythm.

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  • The Missa meets the criteria. I'm more interested in examples after modern metric notation came into being, but it seems there aren't any.
    – texdr.aft
    Nov 18, 2021 at 0:27
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Ives is the first obvious one - although relatively unknown at the time, so others came up with the same idea independently, rather than being influenced by him.

However - you could include just about all African drumming - abeit not notated.

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  • I downvoted because this doesn't provide any new information.
    – texdr.aft
    Nov 16, 2021 at 10:44
  • It talks of travelers here, could that include notated music as well britannica.com/art/African-music ?
    – Emil
    Nov 16, 2021 at 11:32

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