Let´s start easy, funny and entertaining with vanilla siteswap, explained at the 'Nerd Nite' in Magdeburg, November 2014 by Alexander Leymann (in German, but the slides are very useful!)

If you prefer further reading, here is the link to Allen Knutsons Siteswap FAQ

If you think you got everything he said, you should be able to play with siteswap dice. The explanation as well as a download you find here. I made a set of dice for 3 ball siteswaps - if you want to play, just ask me.

Martin also made a list of all 3 objects siteswaps up to height 5 after we visited Guido de Hout´s workshop at the ejc in Toulouse. It´s here!

If you don´t want to watch the video or don´t speak german, I tried to summarize the most important points for you:

Talking juggling theory means talking 'Siteswap' (or 'Quantum Juggling' and 'Cambridge notation'). Siteswap is useful not only to describe tricks or patterns but also to discover new ones: the trick '441' was only invented after Siteswap was around. However, as Siteswap does not describe body movements such as behind the back, under the leg and crossing or uncrossing arms, there are many tricks that need additional descriptions. The simplest form, 'vanilla siteswap', describes only patterns where the hands throw alternately and only one ball is thrown at a time. Siteswap is limited to encode the number of beats of each throw. Beats are discrete amounts of time needed to throw an object. The number of beats of a throw is obviously related to the relative height, and the hand to which the throw is going. The word 'relative' comes into play as every juggler may need a different amount of time to throw objects. So, for a fast juggler, a throw with 3 beats may be shorter and therefore less high than for a slow juggler. An object thrown by any juggler with a height suitable for 3 beats will land 3 beats after being thrown, giving the juggler 2 free beats during which he may throw 2 additional objects. This is the case in 3 ball cascade (Siteswap '3'). We stop writing as soon as the sequence of beats repeats itself - the period of the pattern. If the period is an odd number, 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 possibly asymmetrical because each hand might be doing something different.

Siteswap beats are shown as relative height

0 = 'missing' [empty hand]
1 = 'zip'  very fast pass [between hands]
2 = 'hold' [object is kept in same hand]
3 = 'cascade' toss [between hands]
4 =  'fountain' toss  [into same hand]
5 = higher 'cascade' toss [between hands]

A nice thing, especially with clubs, are 'active' 2s: The 2 is just a hold of 1 object in a hand for 1 beat. Still, you can make a nice twirl or flourish with this object - just make sure it stays in the same hand (or returns to it) and it does not take longer than a beat. Still, one beat may be quite long - as the height of your other throws determine the length of the beats.

There is an easy way to estimate whether a sequence of numbers may be a valid siteswap and how many objects it will involve: The average of the sequence should be an integer, giving the number of objects. Still, this calculation does not tell you whether the sequence is a valid siteswap in this order: For example, the average of 534 is 12/3=4, so it could be a valid 4 object siteswap. To find out without juggling it, you can use a 'space-time diagram' as Alexander explains in the video, or a 'state diagram':

In a state diagram, you write your sequence for at least 3 periods and mark every beat where a ball is scheduled to land with an 'x'. There should not be more than one 'x' on the same beat as this would mean that 2 or more objects are landing in this hand at the same time (squeeze catch) but there should be an 'x' at every beat that means an object is thrown - otherwise you won´t have an object to throw. So, empty spaces are allowed only for the number of objects beats in the beginning (you start with the objects in your hands) and at a beat 0 (empty hand).

valid siteswap:   invalid sequence:
5 3 4 5 3 4 5 3 4 _ _ _ _ _     3 5 4 3 5 4 3 5 4 _ _ _ _ _  
        x x x x x x x x x             x     x     x     x    
                                           

x

x

   

x

x

     

The name 'Siteswap' comes from the 'Siteswap Operator' that allows you to change a valid siteswap sequence into another as follows: If you have a sequence ABMNYZ, you can change it to AB(N+1)(M-1)YZ. This does not change the average and it also does not change the beat each of the swapped objects is scheduled to land, so it must also be a valid siteswap sequence. If you did not get this, I can really recommend Alexander´s video, he has a nicely animated slide for it (and a more mathematical way to write).

Outside of the vanilla restriction:

Synchronous siteswaps (the hands do not throw one after each other but at the same time): These patterns are written as pairs of numbers inside parentheses and separated by a comma. Sequences of bracketed pairs are written without delimiting markers. Only even numbers are allowed, so throws that move to the other hand are marked by an x. For example, synchronous 3 ball shower is (4x,2x), and 3 ball box is (4,2x) (2x,4) or (4,2x)*.

Multiplexing (2 or more objects are collected in one hand and thrown together):  The numbers for multiple throws from a single hand are written together inside square brackets. For example, [33]33 is a normal 3-ball cascade, with a pair of balls always thrown together.