As you can see from the title, we are going to talk about some new concepts and add new vocabulary into our musical terminology usage.
Polyrhythm, is the juxtaposition of two different beat divisions happening at the same time. This will create an interesting composite rhythm of the combining beat divisions. Let’s take a look:

Above is a piece of music with a lot of beat divisions working together to create one polyrhythm. We can see that the C4 pitch is played on every down beat of the 3/4 measures, but the arpeggiation (breaking apart of a chord) of the C major triad below is in a division pattern similar to a 6/8 time signature. Play it out and you will hear it.
In the melody line, we see four notes in equal length of a dotted eighth-note. Within the 3/4 time signature, we have four notes of even value going against the quarter-note pulses and 6/8 division grouping arpeggiation. So that too is another addition to the overall polyrhythm.
A hemiola, is similar to a polyrhythm in that it goes against the conventional beat division – but it is more defined of when normal groupings of three become groupings of two:

Here is another exaggerated version of a hemiola, going from odd to even groupings:

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