Polyrhythms are two or more independent rhythms played over top of one another. More specifically, they refer to the same amount of time divided into different equally spaced divisions played over top one another. Polyrhythms like harmonic ratios, are notated in Just Intonation. In fact, Harmonic ratios such as 5/4 are actually rhythms themselves played so fast that we hear it as a pitch rather than rhythm. This hints at the fact that we can indeed produce scales, chords, and perhaps even something akin to melody with rhythm alone. This has been a part of african music for quite a long time and it serves as a good example of such musical possibilities.
Like harmonic ratios, polyrhythms are notated as ratios. Do not be confused though, they are NOT time signatures!! Polyrhythms are merely the same amount of time divided differently and played over top one another. With a polyrhythm of 3/2 we have some abstract amount of time divided into 3 equal parts and the same time divided into two. the beats are then played over top one another. This is a very common rhythm in western popular music as well as many cultures around the globe. Unfortunately, because much of the west has been so restrictive to strict use of time keeping, it may initially be difficult for you to hear and feel the different polyrhythms, especially in a scale or piece of music. But as you expose your brain more and more to these rhythms, you will begin to see their potential.
Rhythmic Limit[]
Rhythmic Limit is how complex a rhythm or rhythms are in a piece or a set. Typically in much the west, we are used to 3-limit rhythmic ideas which involve both rhythms of 2 and 3. 3/2 occurs quite a bit in popular music as well as 2/1. 4/3 also occurs but is not as common as 3/2 possibly because it is utonal. Generally 5-limit and 7-limit rhythms or anything higher in complexity than 3-limit feels awkward and is hard to follow for a large portion of western musicians and is even generally rare around the world.
This makes it worthwhile even to expand horizons to 5-limit rhythms which are more texturally complex and exotic. 7-limit rhythms sound ridiculously complex and even alien to many people and therefore may be harder to work with than 5 limit rhythms. Still, even 11 limit rhythms deserve exploration, but they're harder to do anything with.
Levels of Complexity[]
The limit of a polyrhythm determines it's complexity. The complexity of a rhythm is always 1 less than the highest number in the ratio. For example, a rhythm of 3/2 has a complexity of 2 meaning there are two possible note durations in the polyrhythm.