SC(F, L, N)

I just danced cha-cha for the first time, and I must say I approve. Unfortunately, I missed the class tonight, which would have been my first. Fortunately, I know a girl who leads very well, so I was able to dance it just fine -- as a follower, that is. Not only do good (well, decent in my case) following skills let you dance moves and patterns you've never seen before, but they also apparently help with dances you've never danced before. Rock on! Sadly, the minute she tried to lead me into any syncopated patterns I faltered, but that's something I'm still working on even in my salsa/rueda following.

Now, what in the heck does the post title mean? I had a stunning revelation on the drive home. It struck me that marching in place, the salsa basic and the cha-cha basic are all successive members of the same self-similar (fractalomgwtf!) recursive sequence.

Wait, what?

This is awesome, so let me break it down. I'm a little lost as to what notation to use to make this clear, so feel free to comment if something doesn't make sense. If I could diagram it out it'd probably flow quite a bit more easily.

I'm going to define a function SC(F,L,N) (SC for salsa/cha-cha) that describes all of the above basic steps, as well as an infinite number of increasing ridiculous "alternative dance styles". The function takes an interval (of length L) and divides it into one or more steps (precisely, 2*N+1), starting with a given foot (F=0 for the left, F=1 for the right). That probably doesn't make much sense yet.

Here's the base case -- SC(F,L,0). Simply take "L" beats to step with the foot "F". Marching in place, once step per beat, would just be the sequence [ SC(0,1,0) SC(1,1,0) ] repeated over and over.

The salsa basic is usually stated as "quick-quick-slow". That's a step that takes one beat, followed by a step that takes one beat, followed by a step that takes two beats. This happens to be SC(F, 4, 1), but I haven't explained why yet, have I?

The function SC(F, L, N), for N >= 1, is defined as follows:
SC(F, L, N) := [ SC(F, L/4, 0) SC(!F, L/4, 0) SC(F, L/4, N-1)]
with, as already stated...
SC(F, L, 0) := [ F (L beats) ]

So, the actual salsa basic is SC(F, 4, 1). Expand that out and you get [F(1 beat) !F(1 beat) F(2 beats)], the same "quick-quick-slow" described above. The alternating sequence you actually dance, which is actually two basics, can be written as [SC(0, 4, 1) SC(1, 4, 1)], the same "quick-quick-slow" action done starting on the left, and then starting on the right.

Now, where does cha-cha come in? Would you be surprised if I claimed it was the alternating sequence [SC(0, 8, 2) SC(1, 8, 2)]? Would I have asked that question if it wasn't? You could say it as "slow-slow-quick-quick-slow", with that last "quick-quick-slow" being the trademark cha-cha-cha action, as well as an internal instance of the salsa basic!

SC(F, 8, 2) =
[F(2 beats) !F(2 beats) SC(F,4,1)] =
[F(2 beats) !F(2 beats) F(1 beat) !F(1 beat) F(2 beats)] =
"slow-slow-quick-quick-slow"

There's a bit of a style issue to bring up here, to be fair. Some people will dance salsa as more of a "quick-quick-quick-pause", some as a "quick-quick-quick-tap" and some as a "quick-quick-slow". The timing is really the same, but it can differ in how to punctuate that last step -- a sudden stop, or a gradual roll-through. While I see all of them used in salsa, I'm not so sure about cha-cha. As stated, my experience with it at this point is pretty minimal. Regarding the recursive expansion, this has to do with the interpretation of the very last step -- should it strike early and fast, followed by a wait, or should it be spread out over the entire remaining interval?

Anyway, here's a cheer for SC(F, L, N), allowing us geeks to find all new pleasures in dancing!