Polyrhythmic notation

Polyrhythmic notation (also called General notation, not to be confused with Generalised siteswap) is a variant of siteswap invented by Luna Harran designed to make working with sync/async transitions, polyrhythmic juggling, and dwell time easier. 
Patterns are written as a series of pairs, as in sync siteswap, but both even and odd numbers are used, ignoring the assumed empty beat of sync siteswap, and all throws are assumed to be self throws, with an x always marking a crossing throw. Each number represents the time that a prop is "in the air" (or out of the hands) until it is next used, although the number 1 usually means held dwell time. An asterisk at the end of a pattern means to repeat the pattern with the numbers of each pair swapped.
All normal siteswap patterns have at least one equivalent polyrhythmic notation pattern, with the number used in polyrhythmic typically being one less than in siteswap, to allow the beat of dwell time. However, they can also be written without the dwell time, which means that the props spend little or no time in contact with the hands or manipulators, as is the case in bounces. This distinction is difficult to communicate using normal siteswap notation.
By comparing these no-dwell patterns to standard sync notation, the link between the two is more apparent.
=== Examples ===
Normal siteswaps in polyrhythmic notation:
Siteswap	Polyrhythmic (Standard dwell	Polyrhythmic (No dwell)
3 	(2x,1)* 	(3x,0)*
5 	(4x,1)* 	(5x,0)*
534 	(4x,1)(1,2x)(3,1)* 	(5x,0)(0,3x)(4,0)*
(6x,4) 	(5x,3)(1,1)* 	(6x,4)(0,0)*
Some examples of 3:2 polyrhythmic patterns, meaning one hand throws 3 times for every 2 throws of the other hand:
Description 	Notation
2 balls in each hand 	(3,4)(1,1)(3,1)(1,4)(3,1)(1,1)
2 balls in each hand, switching sides 	(3,4)(1,1)(3,1)(1,4x)(3x,1)(1,1)*
Repeated sync-async transition from 5 ball cascade	(4x,5x)(1,1)(5x,1)(1,4x)(5x,1)(1,1)*
Note that for side-switching patterns, the number of throws for each hand also swaps.
=== Comparison to normal siteswap and beatmap ===
One of the powerful aspects of polyrhythmic notation is how you can manipulate rhythm and dwell time. For example, the sync siteswap (2x,4x)(4x,2)(4,2x) can be written in polyrhythmic notation as (1x,3x)(1,1)(3x,1)(1,1)(3,1x)(1,1), but using polyrhythmic notation you can also find the patterns (1x,3x)(1,1)(2x,1)(1,2x)(3,1)(1,1) and (2x,3x)(1,0)(3x,1)(1,2x)(3,0)(1,1), both of which have distinct 3:2 polyrhythms of throws and catches. Neither of these patterns have a simple or convenient notation in normal siteswap, but are straightforward from the polyrhythmic notation.
Conceptually, polyrhythmic notation is similar to beatmap, but there are a few minor differences. One notable difference is the use of a 0x where one would use a 1 in normal siteswap. This is because numbers in polyrhythmic notation refer to how many beats the prop spends between being manipulated, and a quick pass spends much less than a beat in the air, to allow for the ball to be held for a dwell beat before being thrown. This is seen in the async 3 ball shower, which is given by (4x,1)(1,0x) in polyrhythmic notation, since this is the only way to have a catch and dwell beat happen asynchronously. In practise, the throw is more like a "0.2x" or something similar, but since polyrhythmic notation is meant for theoretical use, 0x is used.
=== Shorthand ===
While full polyrhythmic notation is powerful and easy to manipulate for creating patterns, the abundance of "redundant" dwell time can make reading and remembering patterns more difficult, so a shorthand can be used for communicating patterns with simple polyrhythms. Since the ratio of a polyrhythm is enough information to determine the throwing order of hands, any dwell time can be ignored, with only the "important" throws (those occuring on the "beats" of the polyrhythm) being written.
Full Polyrhythmic 	Short Polyrhythmic
(3,4)(1,1)(3,1)(1,4)(3,1)(1,1) 	3:2 (3,4)343
(5,7x)(1,1)(5x,1)(1,6)(6,1)(1,1)* 	3:2 (5,7x)5x66*
(4,5)(1,1)(1,1)(4,1)(1,5)(1,1)(4,1)(1,1)(1,5)(4,1)(1,1)(1,1)	4:3 (4,5)45454
This shorthand saves a lot of space, especially with more complex polyrhythms, and holds only the information important for performing the pattern. However, it can not be manipulated like siteswap or full polyrhythmic, as some beats are ignored.
=== Constructing/Modifying Patterns ===
Polyrhythmic notation can be maniplulated similarly to siteswap, with things like site swapping and period adding behaving as expected. However, incrementing an entire pattern by one converts dwell 1s to thrown 2s, so is not always practical. As an example, swapping the sites of the two 4xs of (4x,1)(1,4x) gives (5,1)(1,3), in the same way that swapping two 5s in siteswap 55 gives siteswap 64.
The number of objects in a polyrhythmic notation pattern is also given by summing all the numbers in the pattern and dividing by the number of beats (not the number of throws as in sync siteswap), the same as vanilla siteswap patterns.
=== Multiple hands/manipulators ===
Polyrhythmic notation can also be used to describe patterns with more manipulators, such as patterns using foot catches, or passing pattern. This is done by expanding each bracket to include more numbers, and using capital letters instead of xs to show which manipulator to cross to. These can be excluded for self throws. 
Pattern rotations can be written using a variant of asterisk notation. The hands to be swapped are written after the asterisk, separated by commas. Normal swapping for two hands would be written as *AB, meaning swap the roles of hand A and hand B
Description 	Notation
4 balls in 3 hands	(2B,1,1)*ABC
6 ball 2 count 	(2B,1,2D,1)*ABCD
6 ball 3 count 	(2B,1,2D,1)(1,2A,1,2C)(2D,1,2B,1)*AB,CD
Category:Notation