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Siteswap (also called Cambridge notation in the United Kingdom) is a notation used to describe juggling patterns. It encodes the number of beats of each throw, which is related to their height, and the hand to which the throw is to be made. It is an invaluable tool in determining which combinations of throws yield valid juggling patterns for a given number of objects. It does not describe body movements such as behind-the-back and under-the-leg. The notation was developed in 1985 independently by three people: Bruce "Boppo" Tiemann at the California Institute of Technology, Paul Klimek in Santa Cruz, California, and Mike Day, Colin Wright, and Adam Chalcraft in Cambridge, England (whence the alternative name).
Vanilla siteswap
Its simplest form, sometimes called vanilla siteswap, describes only patterns whose throws alternate hands and in which one ball is thrown at a time. If we were to watch someone from above as they were juggling while walking forward, we might see something like the diagram to the right, which is sometimes called a space-time diagram.
We can describe this pattern by stating how many throws later the ball is thrown again. For instance, on the first throw in the diagram, the purple ball is thrown in the air by the right hand, next the blue ball, the green ball, the green ball again, and the blue ball again and then finally the purple ball is caught and thrown by the left hand on the fifth throw, this gives the first throw a count of 5. We end up with a sequence of numbers which denote the height of each throw to be made. Since hands alternate, odd-numbered throws send the ball to the other hand, while even-numbered throws send the ball to the same hand. A 3 represents a throw to the opposite hand at the height of the basic three-cascade; a 4 represents a throw to the same hand at the height of the four-fountain, and so on.
There are three special throws: a 0 is a pause with an empty hand, a 1 is a quick pass straight across to the other hand, and a 2 is a momentary hold of an object. Throws longer than 9 beats are given letters starting with a. The number of beats a ball is in the air usually corresponds to how high it was thrown, so many people refer to the numbers as heights, but this is not technically correct; all that matters is the number of beats in the air, not how high it is thrown. For example, bouncing a ball takes longer than a throw in the air to the same height, and so can be a higher siteswap value while being a lower throw.
Each pattern repeats after a certain number of throws, called the period of the pattern. The pattern is named after the shortest non-repeating segment of the sequence, so the pattern diagramed on the right is 53145305520 and has a period of 11. If the period is an odd number, like this one, then each time you repeat the sequence you're starting with the other hand, and the pattern is said to be symmetrical because each hand is doing the same thing (although at different times). If the period is an even number then on every repeat of the pattern, each hand does the same thing it did last time and the pattern is asymmetrical because each hand is doing something different.
The number of balls used for the pattern is the average of the digits of the name of the pattern. For example 441 is a three-object pattern because (4+4+1)/3 is 3, and 86 is a seven-object pattern. All patterns must therefore have a siteswap sequence that averages to an integer. Not all such sequences describe patterns as there is an additional constraint that no two objects may land at the same time.
It is also a convention in that a siteswap is written with its highest numbers first. For example in the siteswap 531 it is not written as 315 or 153.
State diagrams
Just after throwing a ball (or club or other juggling object), all balls are in the air and are only under the influence of gravity. Assuming you catch the balls at a consistent level, then the timing of when the balls land is already determined. We can mark each point in time when a ball is going to land with an x, and each point in time when there is not yet a ball scheduled to land with a -. This describes the state that you're in and determines what you can throw next. For instance, we can look at the state just after our first throw in the diagram, it is xx--x. We can use the state to determine what can be thrown next. First we take the x off the left hand side (that's the ball that's landing next) and shift everything else to the left filling in a - on the right. This leaves us with x--x-. Since we caught a ball (the x we removed from the left) we can't "throw" a 0 next. We also can't throw a 1 or a 4, because there are already balls scheduled to land there. So assuming that the highest we can accurately throw a ball is to a height of 5, then we can only throw a 2, 3, or a 5. In this diagram, the juggler threw a 3, so an x goes in the third spot, replacing the -, and we have x-xx- as the new state.The diagram on the left shows all possible states for someone juggling three items and a maximum height of 5. From each state you can follow the arrows and the corresponding numbers give you the siteswap. Any path which brings you back to the same state that you started with is a valid siteswap, and all siteswaps can be generated this way. The diagram becomes a lot bigger very fast when more balls or higher throws are introduced as there are more possible states and more possible throws.
Another method of representing siteswap states is having the next throw on the right and read to the left (instead of the left to right system described above), represent a ball with a 1 instead of an x, and represent a spot where there's no ball scheduled to land with a 0 instead of a -. Then the state can be represented with a binary number, such as binary 10011 for the first state in the space-time diagram above. This method makes it a lot easier to represent more objects or higher throws as it is less complicated to create.
| 7 | 11 | 13 | 14 | 19 | 21 | 22 | 25 | 26 | 28 | |
| (111) 7 | 3 | 4 | 5 | |||||||
| (1011) 11 | 2 | 4 | 5 | |||||||
| (1101) 13 | 1 | 4 | 5 | |||||||
| (1110) 14 | 0 | |||||||||
| (10011) 19 | 2 | 3 | 5 | |||||||
| (10101) 21 | 1 | 3 | 5 | |||||||
| (10110) 22 | 0 | |||||||||
| (11001) 25 | 1 | 2 | 5 | |||||||
| (11010) 26 | 0 | |||||||||
| (11100) 28 | 0 |
1-handed and 4-handed site-swap
1-handed site-swap (diabolo site-swap)
Site-swap notation in its simplest form ("Vanilla siteswap") assumes that only 1 ball is thrown at a time.
This means that these patterns are just as valid for a single hand, or for a diabolo player throwing diabolos at different heights.
4-handed site-swap
Conversely, "vanilla" site-swap patterns are also valid for a 3-handed juggler, or for 2 jugglers coordinating 4 hands, on the condition that no hands throw at the same time!
In practice, this is most easily obtained if the jugglers throw by turns.
(Left hand of juggler A, left hand of juggler B, right hand of A, right hand of B).
Synchronous siteswap
Siteswap notation can be extended to denote patterns containing synchronous throws from both hands. The numbers for the two throws are combined in parentheses and separated by a comma. Only even numbers are allowed, so throws that move to the other hand are marked by an x. Thus a synchronous three-prop shower is denoted (4x,2x), meaning one hand continually throws a low throw to the opposite hand, while the other continually throws a higher throw to the first. Sequences of bracketted pairs are written without delimitting markers.
Multiplexing
A further extension allows siteswap to notate patterns involving multiple throws from either or both hands at the same time in a multiplex pattern. The numbers for multiple throws from a single hand are written together inside square brackets. For example, *33 is a normal 3 ball cascade, with a pair of balls always thrown together.
Other extensions
Various other enhancements to the siteswap system have been proposed and enjoy some use.
- Passing:
- Synchronous siteswap repeats: synchronous patterns that repeat in mirror image can be abbreviated with a *. For example, Instead of (4,2x)(2x,4) (3 ball box pattern), you can write (4,2x)*.
There are many computer programs available which simulate siteswap patterns.
External links
"Siteswap"