Is this piece polyrhythmic?

Anyway, what I was REALLY posting to contribute: you don't need different meters for polyrhythm, and in fact that kind isn't very prevalent nor very practical, as one conductor trying to conduct several different meters would be hard as hell, if possible at all.

Well, seeing that conducting 3 against 4 with two hands and similar things are pretty common "warm-up" exercises any conducting student goes through, it's certainly not impossible. If a percussionist or pianist can play two different meters with two hands, why shouldn't a conductor be able to?

And for the more complex cases there are always various other solutions such as click tracks, multiple conductors, playing together based on single cues, etc. It's all being done and works, even if it sometimes requires some extra effort.

I believe the phrase for when a beat is in a certain meter but feels as if it is in a different meter is hemiola, no? Do these mean the same? Hemiola and polyrhythm? Can polyrhythm also mean polymeter?

Oh, and conducting/playing 4 against 3 and vice-versa is bloody difficult. =/

Well, a hemiola is stricly speaking only a single thing: The regrouping of the rhythmic patterns in two triple time units to sound like one measure of a larger triple time (e.g. when you treat two measures of 3/4 time to sound like one measure of 3/2 time, or one 6/8 time measure to sound like a 3/4 time measure). A hemiola is therefore only a change in accentuation - it is not required that there's any other rhythm present at the same time, such as in polyrhythm.

And by all usual definitions of polyrhythm it also includes polymeter, as a more specific subform.

Personally, I think that the term "polyrhythm" only really makes sense if there's some sort of repeated patterns in the voices it consists of, OR if all information but the rhythmic one is kept quite simple and the rhythm is clearly the dominant element. In the other cases, where you have completely independant voices with non-repetitive rhythms, it will just appear more generally as polyphonic, which -always- includes somewhat independant rhythms (since it's exactly rhythmical independance that separates polyphony from homophony). To give a distinct feeling of polyrhythm, in contrast to the more general polyphony, the rhythmic aspect must therefore be somewhat emphasized.

Sure, you can of course also call other polyphonic music polyrhythmic, but it defeats the special connotations this term has if you call every Bach fugue polyrhythmic (which they -very technically- are).

So, to give a real impression of polyrhythm, you should consider the following things:

- Make the rhythmic build of the independant voices significantly more complex than their pitch content (for this it can also help to keep your melodies consciously simple).

- Emphasize the rhythms by repetition, preferably in pattern lengths that are not the same for all voices and are not integral multiples of each other. (E.g. repeat the pattern of voice one after seven 8th notes, whereas you repeat the pattern of voice two after two bars.)

- Make your rhythms distinct in either their superordinate metric groupings, or the size of their musical building blocks, respectively grids. With superordinate metric groupings I mean to emphasize accentuations that imply a certain meter which is different from the other ones (in other words: The polyrhythmic effect will seem stronger if in a section of straight 8th notes you accentuate every third one, than if you place your accentuations wildly, -even- though the latter one would be the more "complex" rhythm). With "size of musical building blocks" I mean the basic grids your rhythms are built of, the smallest unit every value is a multiple of (that may be a sixteenth, a triplet eigth, whatever). The easiest way to achieve this latter one is working with tuplets, but it's by no means the only one.

- A combination of all these elements will probably be the most effective.