Siteswap notation reduces a juggling pattern to a short sequence of numbers. For the basic notation, we assume that the two hands throw alternately at equally spaced intervals or beats. No notice is taken of where the hands are throwing from (behind the back, under the leg etc), and multiplex throws (more than one ball at a time) cannot be described. Despite these restrictions, the notation is useful and it has the advantage of being very concise. Other juggling notations include ladder diagrams and causal diagrams.
Each throw is then given a number, which is equal to the number of beats before it is to be thrown again. Since the hands throw alternately, even numbered throws are caught in the same hand, and odd numbered throws are caught in the opposite hand. A normal throw from a three-ball cascade is denoted ‘3’. A ‘2’ is a ball that will be thrown again from the same hand in two beats time. Since this time is very short, and that hand doesn’t need to do anything else in between, ‘2’ signifies holding on to a ball when one would normally be thrown. A ‘1’ must be throw on the very next beat by the other hand. To accomplish this, a ‘1’ represents a quick pass across from one hand to another. Finally ‘0’ is used to denote an empty hand which hasn’t got a ball to throw. This is not so obvious, but (like 0!=1) is consistent with everything else.
A juggling pattern is then described by writing down the numbers that correspond to each throw in the order in which they are made. Once the numbers start repeating themselves you stop. To juggle a siteswap pattern you simply pick up enough balls and follow the numbers, making the right throw for each one, and remembering to alternate your hands. If you get to the end of the numbers before you drop, you go back to first number and carry on from there.
The observant may have noticed that not every sequence of numbers represents a possible juggling pattern. For examples, a sequence containing -5-4- cannot work since in four beats time two balls need to be thrown at once. For sequences which do lead to valid juggling patterns, it can be shown that the number of balls required is precisely the average (arithmetic mean) of the siteswap numbers in the sequence.
A multiplex is when multiple balls a thrown from the same hand at the same time. To show this in siteswap, the throws that are at the sime time are enclosed in square brackets.
To notate patterns where throws from both hands happen simultaneously, we put each pair of throws in parentheses, and separated by commas. Conventionally, the first throw listed in each pair is from the left hand. Now, each "beat" consists of two throws, so we only use even numbers to represent throw heights. An 'x' after a throw is used to indicate that it switches hand, otherwise it returns to the same hand. A * at the end of a siteswap means to repeat the whole thing but with the hands swapped.
Juggling software is not only useful to discover and learn new patterns, it can also be fun to play around with. Here are some good juggling animators:
Juggling lab has an animator which will juggle siteswaps. There are cool options like the ability to choose the hand movement of the animator. It also has a pattern generator, among other things.
Joepass! is an animation software for juggling patterns. You can input your patterns using siteswaps, causal diagrams, and multiple hand notation. It can animate solo and passing patterns. It also has several other features - try it out!