This is the first page of a document that is shaping up to be quite large.
The 3-4-5 describes the three groupings used in this family. This can be organised as:
4's (semi-quavers) grouped in 3 and 5 simultaneously,
5's (quintuplets) grouped in 3 and 4 simultaneously, and
3's (triplets) grouped in 4 and 5 simultaneously,
So far in my practice I've been playing them over common forms: rhythm changes, blues etc, as well as through various transpositions of some of Carter's 12-note, all-interval chords (see the Elliott Carter harmony book of John Link's dissertation on harmony in "Night Fantasies").
I have been playing these rhythms as they are presented here, as well as swapping the groupings between hands after a full cycle (when the groupings end up on the down-beat again). Once I'm feeling comfortable I've been swapping the parts between hands when they hit together part-way through a beat. These points are shown with accents in this picture. The second line of semi-qauvers shows this, with the 3's and 5's swapping between hands at each accent. I haven't written this out for all of them, as it's reasonably easy to figure out how that will work.
If I can do this last step and keep a form, I know I have the rhythm under my belt.
Here's an audio snippet from about a year ago. It's me with the Anton Delecca quartet playing a tune in 15/8, obviously this is too-good an opportunity to miss to play the 4's and 5's grouping.
You may also notice these are called '3-part polyrhythms', well, that's because we count the underlying pulse as a rhythm. This is probably near impossible to do on any other instruments expect the piano and drums, but by playing the pulse and these two groupings you get three speeds happening simultaneously. These is, then, 12 different ways to play these grouping between two hands on the piano.
where x=3, y=4 and z=5
x/y in RH and z in LH, and vice-versa
y/x in RH and z in LH, and vice-versa
y/z in RH and x in LH, and vice-versa
z/y in RH and x in LH, and vice-versa
x/z in RH and y in LH, and vice-versa
z/x in RH and y in LH, and vice-versa
I will expound on the 'easy long-range poly-rhythms' part of this post later, as with every new investigation I do, I'm finding more a more interesting properties.
However what I will say is this: the methodology of writing out 11/13 or 19/23 is the same as these ones above: writing out groupings using a common division. The difference between them and these however is that they are much harder to grasp aurally. I am currently working on breaking their resultant rhythms (more later) down into a small formula that will be easy (easier) to follow mentally, making large poly-rhythms more useful for improvisers.
