State-Siteswap-Beyond
An idea that has intrigued/haunted me for a while now is that there could be a higher 'level' of conceptual juggling, i.e. that something could exist 'above' siteswap in the same way siteswap is 'above' state. Juggling patterns are essentially collections of states transitioning between one another, therefore what might a collection of siteswaps look like? Can you express that numerically, like you can with state or siteswap?
So what I'd like to do is show the way I've found to do that. It may not be the best way, or the right way, but it works for what I want it to do. When we think about siteswap's relation to state, it helps to break it down into throws. A single throw can basically be thought of as a transformation of a state. The throw 4 on state 1236 (I'm going to be using a numerical notation for state here, by the way, it's just much denser. Each number represents the position of an object) transforms it into state 1245 (xxoxx). Therefore, what I'm looking for are numbers that can transform one siteswap into another.
Basic transformations
For this to work, I had to figure out what the most basic transformations you can apply to a juggle are. The first sacrifice to make was period-altering transforms. There's just no reliable way to change the period of a pattern. That done, what can you change about a pattern? Obviously, the values themselves could be changed, or the number of objects could be changed. These are the most basic parts of the siteswap.
To determine which transformations are 'more basic', we think about composite transformations and complete systems of transformations. Sometimes, a group of transforms can be mistaken for one. My favorite example is the "441 transform," where you take P-1 beats away from the last throw and give 1 more beat to each other throw. This works for any pattern, any period. However, you can make this transform using two others: raising each digit by one and lowering the last digit by P. Why would you want to do that? Because the two other transformations are more applicable and versatile.
An important aspect of any group of transformations, if they're to be useful for me, is that they can produce any siteswap from any other (same-period) siteswap. If you can cut out one transform out, and still reach any siteswap, you should.
Based on experimentation and intuition, the system I use is based on three transforms, which I'll use +, ^, and @ to represent.
+: raise every number by 1
^: switch the first number and any other number, then add the space in between them to the first number, and subtract it from the other. (97531 -- 67561)
@: put the first number last
The notation
Through a lot of thinking and reducing juggling to equations, and some cleverness (I think so anyway,) I eventually came to the solution that should have been obvious from the start. Each of these transforms can basically be reduced to a single value indicating how many times to do it, or in the case of ^, how far away the switched throw is from the first.
The important part about these transforms is that they only logically exist within certain boundaries:
+ is infinite, and can be any number at all
^ is less than or equal to P
@ is less than P, because at that point it becomes the same pattern
Using this, we can compress a combination of these three transforms into a single number. Here is the equation for it:
@+^P+(P-1+P2)= Transform Number
Obviously, with the inclusion of P squared, the numbers for transforms tend to be much larger than everyday siteswap values. The alphabet will be a common appearance in transform-groups, unless you want to divide the numbers another way.
Going past, going backwards, and the end.
This kind of mindset demands that we ask the question: what's beyond transformations? Shouldn't there be something that transforms a group of siteswap-transforms? And something that modifies that? in theory, yes, there should be. I've got theories about some properties that the system might have. For instance, consider the number of different numbers compressed in state, siteswap, and siteswap-transforms: State can be taken at face value, siteswap basically compresses orbits alongside a base number, and there are three transforms that we compress together. The obvious assumption I would make is that a meta-transform would have to compress four numbers, but that could just be coincidence.
Another burning question is, "Can we go the other way?" Is there something underlying state? Maybe. I'm not sure what that would be, and although I don't think it exists, I would love to be disproven.
Overall, this system of transforming siteswaps is useless for most jugglers. However, for nerds who love conceptualizing the essence of pattern structure, I think it's an absolutely fascinating way to look at the system.
If anyone wants a look at the equations or other behind the hood details, I'd be happy to show you. There's a really elegant way of looking at the three basic transforms.
RustyJuggling - - Parent #
Edit: The equation is
@ + ^P + +(P-1+P2)
Didn't think about the repercussions of labelling one of them as "+"... Whoops. I use different symbols in my notes.
RustyJuggling - - Parent #
Further corrections: I'm going to change the plus-one transform from + to ~. It's an arbitrary symbol but better than nothing. Also, the equation should actually be @ + ^P + ~(P+P2) to work properly.
Furthermore, I think there's actually a structure to how these systems tie together. Imagining a siteswap in terms of three fundamental qualities, those being order, reach, and strength, you can build a system that only effects one of these qualities at a time. Order is the position of a throw, ie when you throw it relative to the pattern. Reach is the net distance 'through the pattern' that the object goes. So the reach of the 5 in 531 is 2, where the reach of the 1 is 1. Strength is just a way of stating the number of orbits a throw makes. Going back to 531, the strength of each number would be 1, 1, and 0.
To create a more perfect system, you need to use the +period transform instead of the all+1 transform. All+1 messes with both the reach and strength of a pattern, whereas +Period only changes strength. Cycling through the pattern obviously only changes order, and swapping throws only changes reach.
With that context, we can then look at siteswap in terms of reach and strength. Siteswap can actually be reduced to just terms of reach and strength in the equation SS= R+SP, where R is reach and S is strength. Then if you contextualize state as a system of only reach, you might say that state=R.
Lining these equations up, you might notice the pattern forming. If you call reach, strength, and order qualities 1, 2, and 3 respectively, you can write them as:
J1=q1
J2=q1+q2P
J3=q1+q2P+q3(P+P2)
Where Jx is a system with x qualities.
Using this, we can guess that the system of meta-transforms above J3 would follow the structure
J4=q1+q2P+q3(P+P2)+q4(P+2P2+P3)
Because P+2P2+P3-1 is the highest value that qualities 1 through 3 could ever produce together.
Unfortunately, we don't really have any context for what the fourth quality would be, or how it would relate to J3, so it remains theoretical until we get an idea of how to transform the transforms of J3. Plus, at that point it would probably be easier to list all of the components than to compress them into a single number.
I was able to keep up with the post in the beginning, but I got lost somewhere around the middle. If this is something you just start thinking about, I am truly impressed.
kind of related to this, is there a way to express things like overthrows and underthrows reliably with siteswap?
RustyJuggling - - Parent #
Not in siteswap itself; there are ways to notate it though. In Daniel Simu's Rhythmic Catches, for example, you just use a letter i or o.
There is a lot of intermediate space between outside or inside though. At least as many spots to throw as the number of objects in a pattern, because you can theoretically use the space between any two objects, plus fully outside and an odd "fully inside" one that I don't think I've seen anyone do. I don't know of an existing way to notate all those kinds of throws, though I do have some ideas.
Would the throw that one makes under the opposite arm just before doing a chop be considered a fully inside throw? Or a back cross?
RustyJuggling - - Parent #
I think it would be, since it's thrown beyond the object that's horizontally farthest away from normal position. Although it's hard to say with a backcross just because it's so 3-dimensional.
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