Siteswap synonyms

Every juggling pattern can be written in siteswap notation an infinite number of different ways.
==3 ball slow cascades==
There are twelve different ways to notate the 3 ball cascade (defined as a 3 ball pattern where the two hands alternately make crossing throws at a fixed height on an even rhythm) in siteswap with a maximum throw value of 9:
*3
*(4x,2)*
*522
*(6x,2)(2,2)*
*(6x,0)*
*72222
*720
*(8x,2)(2,2)(2,2)*
*(8x,2)(0,2)*
*9222222
*92022
*900
All of these siteswaps represent essentially the same pattern, but the different ways of writing it imply different heights and dwell times. If higher throws, uneven rhythms, or varied throw heights are allowed, then there are even more ways to write the pattern, such as b22222222, 5520, and 722520.
: Video: 65 variations of (4x,2)* with active 2s
==Synonyms for other basic patterns==
The 4 ball asynch fountain can be written with a maximum throw value of 8 as:
*4
*(6,2)*
*822
*(8,0)*
The 4 ball synch fountain can be written with a maximum throw value of 8 as:
*(4,4)
*(6,6)(2,2)
*(8,8)(2,2)(2,2)
*(8,8)(0,0)
The 5 ball cascade can only be written two ways without exceeding a throw value of 9: 5 and (8x,2)*. Other ways to write this pattern using higher throws include:
*(ax,0)*
*b22
*d20
*(ex,2)(2,2)*
*f00
*(gx,2)(0,2)*
*h2222
*(kx,0)(0,0)*
==Repetitions, rotations, and reflections==
A siteswap can also be written in different ways by repeating the sequence of numbers: 744, 744744, 744744744, etc. all represent the same pattern, but a siteswap pattern is normally expressed in the shortest possible form (no repetition).
Patterns containing more than one throw height, such as "...123451234512345...", can be written several different ways without repetition, since a variety of starting points are possible for the notation. 12345, 23451, 34512, 45123, and 51234 all represent the same pattern, since they all look the same when written as an indefinitely repeating sequence.
Switching the numbers in each pair of throws in a synchronous pattern, for example changing (6x,4)(2,4x) to (4,6x)(4x,2), results in another way of writing the same pattern. If you consider the number to the left of a comma to always be a left hand throw (or if you consider the number to the left of a comma to always be a right hand throw), then the new notation represents the mirror-image version of the original pattern.
==Synch, asynch, and one-handed patterns==
A shower pattern can be written either in asynchronous notation or in synchronous notation. There is not much difference between the asynch patterns 51, 71, 91, and the synch patterns (4x,2x), (6x,2x), (8x,2x), and in practice the shower is commonly done with a rhythm in between synch and asynch. (Jugglers usually use a high dwell time, which makes it impossible to do 1s with a perfect asynch rhythm.) Sprung patterns, which are normally written in synchronous notation, can also be written as asynch patterns ( e.g. 612 instead of (4,2x)* and 912 instead of (6x,2x)* ).
One-handed patterns are exactly the same whether they're written in asynch notation or in synch notation since one hand isn't doing anything, so there are no throws to be synchronized. If the 0 in the siteswap for a one-handed pattern is replaced with a 2, there is still no difference in the pattern except that the hand that isn't juggling is now holding an object. So 2 in 1 hand can be written as 40, (4,0), 42, or (4,2), 3 in 1 hand can be written as 60, (6,0), 62, or (6,2), and 4 in 1 hand can be written as 80, (8,0), 82, or (8,2).
==Synch conversions==
Asynchronous siteswaps can be converted into synchronous ones, and vice versa.
===Standard Method===
The generally used way to convert siteswaps between asynch and synch is detailed in Ben Beever's book on siteswap. The method is as follows:
# Ensure that the siteswap has an even number of digits. If it doesn't, you can fix it by adding a basic throw on the end (unless it's an excited state siteswap).
# Group the numbers into pairs using parentheses, putting a comma in the middle of each pair.
# Choose one side of the pattern to slide back. On this side (either the right or left side of each pair), increase odd throws by 1 and add an x (to indicate that the throw still crosses). On the other side, decrease odd throws by 1 and add an x. Even throws remain the same.
For example, 7445 turns into (6x,4)(4,6x), which can be shortened to (6x,4)*. 7562 turns into (6x,6x)(6,2).
===Isolation Method===
One way to convert odd numbers is to alternate between dragging either side of the pattern backwards. This also creates one throw in the pattern that doesn't move at all, which ends up 'isolated'. This is the method:
# Ensure that the siteswap has an odd number of digits.
# Group the pattern in pairs using parentheses, with a comma in the middle of each pair.
# Determine how many pairs complete their throws in the time it takes a throw to complete its arc, including the isolated throw as a pair. For example, in 534 isolating the 5, three pairs make their throws in the duration of the 4 (the 5, the 34, and the 5 again). Double these numbers and replace the original values, adding an x to any throw that was originally crossing.
# Add a 0 in the isolated number's pair (ensuring the hand order is preserved), and add an asterisk.
For example, 534 isolating the 5 turns into (0,6x)(4x,6)*.
===Cut method===
Another way to convert odd numbers is to remove a 2 from the pattern. The method is as follows:
# Ensure the pattern has an odd number of digits and includes a 2.
# Group the pattern into pairs, except the cut 2.
# Count how many pairs complete their throws during the duration of each throw, not counting the cut 2 as a pair. For example in 612, two pairs complete both throws (the 61, twice). Double these numbers and replace the original throw values, adding an x to throws that were originally crossing.
# Remove the cut 2 and add an asterisk.
For example, 612 converts to (4,2x)*.
thumb|left|480 px|3 ball 42 by Robert Wood
Category:Patterns
Category:Siteswaps
Category:Theory