From teaching music for a few decades I came up with some simple ways of describing it and, of course, we always had lots of examples before us in the lessons. Simply put, we have different ways of talking about time in music. There is duration, which is the length of time it takes to play a piece of music. Then there is beat, the recurring pulse that is typical of a lot of, though not all, music. Then there is rhythm, which is the pattern of long and short notes that comprise the surface of the music. Finally, there is meter, which is the way the beats are grouped. This is particularly evident in dance music where the waltz, for example, is grouped in threes while the polka is grouped in twos. The grouping is signaled by a stress on the first beat of the group, the downbeat, which is prepared with an upbeat. This obviously relates to the movement of the dancer's feet or weight. In music notation these groups are shown as measures divided off by barlines. At the beginning of the piece or movement the meter is also indicated with a time signature:
Here I have shown three different meters in succession, ending with the first again. These are all simple meters though the 5/4 one could call asymmetrical because it typically divides up into two beats plus three beats. We also find meters of seven beats, often divided 2+2+3 or 2+3+2. An example is the Precipitato from the Piano Sonata No. 7 by Prokofiev where he shows the subdivision at the beginning:
|Click to enlarge|
Polymeter comes in a couple of different forms and there are some excellent examples at the Wikipedia article. Poly- just means "many" though typically in music we would have just two different meters at once. For example, you could have three beats against four beats where the beats match up, but the downbeats would only coincide every twelve beats. Wikipedia has examples of this and other possibilities such as 5/4 and 4/4 together. Those examples are with two meters having the same note values. I have mentioned before that an interesting example of polymeter is found in Flamenco music in the Bulerías where 3/8 is superimposed on 3/4 and then 6/8. When I created the example below I couldn't figure out how to show that in the music software so I put it all in 12/8! Let me see if I can explain. The top is easy: it just consists of 3/8 all the way. But the bottom is actually a measure of 6/8, a measure of 3/4, a measure of 6/8 and another measure of 3/4. What makes it confusing is that in Flamenco music they don't start the measure with the downbeat, instead it comes at the end!
Now on to hypermeter which Wikipedia, following the theorists that discuss it, describes as the idea of beats being grouped to form measures but on a larger level. Here, measures are grouped to form hypermeasures. For example, in a quick 3/4 we might hear each measure as a single beat with four of them going together to form a hypermeasure. We run into this in Beethoven scherzos, for example. I just googled "hypermeter in Beethoven" looking for an example for you and the fourth hit that came up was a post I did a while back on the Symphony No. 9. Just go there for the example which shows a movement ending with four measures of rests to complete the hypermeter! Sorry, I didn't say where I got that example from and now I can't remember. Pretty sure it was a Beethoven movement, though.
That would be a good way to end this post: the Scherzo from the Symphony No. 9 by Beethoven. This is Daniel Barenboim conducting the West-Eastern Divan Orchestra: