Siteswap Notation
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Siteswap Notation is a way to describe aspects of a juggling pattern by using numbers. In this case, "site" means a hand and "swap" refers to how the props and hands relate. It is a useful tool for communicating between jugglers, discovering new patterns, and finding different transitions between patterns. Many Juggling Animators use siteswap notation as their pattern input. The JugglingLab animator associated with this website is a siteswap illustrator.
[edit] Basic notation
A siteswap pattern or "a siteswap" is a sequence of numbers. Each number represents a type of throw to be done (explained below) at that time or "beat" in the sequence. The code is hexadecimal, so that 10 is 'a' and 11 is 'b' and so on.
It is assumed that the sequence shown as defining a pattern repeats forever, so that 5 Ball Cascade, which is tossed as '55555', is notated as only '5'. By this convention too, the code 12345 would be "...1234512345..." allowing a variety of starting points for the notation. For this reason, '12345' is the same pattern as '23451' or '34512' and so on. If no number is shifted in the sequence, and the whole sequence is shown, the pattern shown is the same.
Basic (sometimes called "vanilla") siteswap includes these assumptions:
- All tosses are made to a strict beat so that every throw is on a beat.
- Each hand throws in turn, right-left-right-left (called "asynchronous").
- Each toss and catch is of only one prop.
(but see also: Multiplex Siteswap Notation and Synchronous Siteswap Notation)
Because time and height are related by gravity, the notion that the numbers show relative heights, with '5' being higher than '3', is natural. For the theory, however, a number represents the beats before the ball is tossed again where the beats are not specified as to actual duration.
For '0' there is no toss, because there was no prop in that hand at that beat.
When a hand is empty at toss, it was also empty at catch, because no toss was down yet.
'0's may be used in a pattern for reasons of 'averaging' as explained below,
either to make some toss sequence valid such as 780
or to change the number of balls, such as 4-ball 534 changing to 3-ball 5340. (see the 2-ball section)
The pause in tossing may be called "a gap" and is not preferred for being a break in rhythm.
Starting then with '1' which means for its definition a '1' toss continuing as '1111...':
That will be possible if the left hand gets the ball in time to make the toss on the next beat.
For that to be possible, the '1' toss is directly to the other hand, and continuing on.
Adding a '1' will change the code average as does a '0' while maintaining a tossing rhythm.
It too alters the symmetry of the pattern, e.g. one-sided as 4-ball 53 or changing sides as 3-ball 531.
For the toss to not be repeated for '2' beats, it would land in the same hand:
For landing so soon in the same hand, the '2' is not considered as tossed, it is a "hold" only.
Its duration is still two beats, during which it is often moved in various ways in various patterns.
For the classic creative use of a '2' see The Machine with its high carry.
Because a '3' will be tossed three beats later, there is time for two other tosses, here the same:
This is the classic and common beginner pattern called 3-ball Cascade.
No matter what else happens in the pattern, a '3' toss will be tossed again after three beats.
Here is 423 showing that:
Note that with the '2' for one hand, the other hand can make two successive tosses.
Basic Siteswap, as treated here, does not state within its code anything about a catch, except other or same hand, as odd/even.
So where the catch is made, is up to the juggler and the pattern style.
Here is the same '3' for 3-ball Cascade shown above, but with toss and catch positions reversed:
For one hand catching on the other side but still being a '3' as the other hand, see 3-ball Reach Under.
For a further stretch of the same plain old '3' for siteswap, see 3 Ball Back Toss.
The '4' shown above continues the logic of '2' for landing in the same hand
and of '3' by providing time for three other tosses of some nature, here all the same:
The practical notion that higher numbers mean higher throws but is "really" a matter of beats
can be seen in 3-ball 531 with all crossing tosses but each to a different height:
In the same manner as tossing four props determines what a '4' is, other numbers are so defined,
with odds tossed like '3' and evens like '4', always assuming right-left-right-left for beats and tosses.
Although each next number of balls does require more height, so that 6 Ball Fountain is a lot higher than 5 Ball Cascade,
the whole pattern rather than the number of balls will actually set height.
3 Ball Shower is an example of how that works:
and the other hand has to have a ball in it to do that, which takes time too.
So this pattern, even though three balls like Cascade, is made with '5's as '51', not '3's,
no matter how short your speed might be able to make it.
That's what "beats" are about.
[edit] Valid Siteswaps and Averaging
What also follows from beat restrictions is that not just any strings of numbers are juggleable. We saw above the pattern '531'. We saw that it is the same as '315' and '153' because the tosses are in the same sequence within '531531...'. Changing the sequence order, however, to '513' makes it impossible to juggle. You could toss a '5' then a '1' (as "Shower" shown above). The code says then toss a '3' which you could do with the ball that was just caught from the '1'. But then what? You've made only two tosses after the '5', meaning only two beats. So the '5' won't be back down for another two beats, when the '3' would also arrive in the same hand -- uh-oh, not allowed in this form of siteswap. The JugglingLab animator will report "Pattern is not valid" (but will not suggest a remedy).
There have been algorithms for decades for calculating all the possible "valid" siteswaps, even to searching all the known digits for pi for the longest valid string. See Matt Hall's Monster List. One critical aspect is not included in those calculations: how to have all the balls in the air without collisions. That is what jugglers are still struggling with (and what this website might serve to help with), especially as more people are tossing more props, and tossing higher because of the growth of siteswap.
People do enjoy trying to devise numerical patterns on their own, and there is a simple calculation that you can do before submitting a string to an animator or flailing away with some number of balls. That is, finding the number of props a code string would require. The siteswap '5', as we said, takes five balls to complete as '55555'. It would take only four balls, however, to do 55550 and three to do 55500. Not because there are only that number of '5's shown, but because there are five digits in the three valid codes.
That is the process of averaging, and it works this way. You add the values of the code string, and divide by the number of digits. So "5 / 1 = 5 balls" for the first case (or "5+5+5+5+5 / 5 = 5"); then "5+5+5+5+0 / 5 = 4 balls" for the second case; then "5+5+5+0+0 / 5 = 3 balls" in the third case. This 'average' of the digits must be a whole number (tossing the partial remains of an apple is not within this form of siteswap); but the string of numbers may still not be juggleable, as above.
In the earlier days of siteswap, people used a Ladder Diagram, aka space-time diagram etc., to work out possibilities. With animators such as JugglingLab already supplied with all theoretically possible strings to compare to, the siteswap problem remaining for basic solo toss juggling is devising a siteswap code to match what you think you are doing or might want to do. For that, people are using videos that can be stopped and restarted easily. And siteswap notation itself is always having to play catch-up with what someone has managed to do for juggling, including bounce and passing.
Siteswap is still more for jugglers than for audiences, though patterns devised by siteswap such as 441 and 97531 are increasingly becoming part of standard performances.
See Siteswap Category with many other pages on the general subject
also:
- Aidan Burns' Siteswap Made Simple
- Allen Knutson's Siteswap FAQ
- Matt Hall's Monster List
- WikiPedia page on Siteswap
- and everything else about siteswap you would ever want to know
Download the JugglingLab animator (1+ Mb three formats).
