Is it possible to devise a single juggling notation system that can describe any juggling pattern with any number of objects, any number of hands or other manipulators, any number of timing variations, any multiplex throws and any squeeze catches?

Answer yes or no and then show your workings to prove your method.

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Yes.

To do this we need to base the notation system, not primarily on a mathematical process, but on juggling in the real world. One way to do this is to notate, at every beat in time, what each hand is doing.

A Beat Based Juggling Notation System

The basic rules:

- A beat describes a regularly spaced point in time, every time an object leaves a hand or appears in a hand.
- Every hand is noted on every beat.
- A hand must do exactly as noted, no more or less.
- A number indicates how many beats until a ball is next noted.
- A letter following a number directs the ball away from that hand into a different hand.
- Reduced repeating information to a single repeating phrase.

Syntax:

- All the information for each beat is contained in brackets.
- All the different hands within the brackets are separated by a comma.
- Each hand is allocated a unique letter. This is used to direct a ball to it from a different hand.

One handed patterns:

(1) - There is one hand that contains the same ball on every repeated beat.

(2)(0) – There is one hand that contains a ball for one beat and no ball the next beat.

(1)(1)(1)(2)(0) – One hand contains a ball for four beats, then is empty for one beat.

(2) – One hand is holding two balls in turn, one per beat.

(11) – One hand is holding two balls on every beat.

(21)(1) – One hand is holding one ball continually, at the same time another ball is being held only every other beat.

(32)(0)(1) - One hand. Two balls leave the hand on the first beat, the hand is empty for one beat, holds one ball in the third beat.

(31)(2)(0)(11) – One hand. One ball leaves the hand, then the second ball leaves the hand, then the hand is empty, then two balls appear in the hand again.

As you see from these examples, holding two balls is just as possible as holding one ball. Also catching two balls in the same hand between on one beat is also just as possible as catching one ball in one hand.

A hand can’t do any more or less than instructed and a ball can’t be held again until it is next noted. This means that:

0 – is an empty hand 1 – hold a ball in the hand until the next beat. If you were to throw a 1 it would introduce a new interval that would have to be notated across all hands. A 1 must spend no time out of the hand in a one handed pattern. 2 – a ball must leave all the hands for at least ONE beat. A throw is the traditional way to do this, for it to spend time in the air instead of a hand. 3 – must be in the air for at least two beats. 4 – must be in the air for at least three beats.

Keeping in mind all the above rules and look at these two-handed patterns:

(1,1) – There are two hands that hold the same two balls on every beat.

(2,2)(0,0) – Two hands throw one ball each on one beat, the next beat both hands are empty.

(2,2) – Two hands throw one ball each every beat, resulting in a quick four ball pattern.

(3,3)(0,0)(0,0)(1,1) – Two hands throw a ball each that lands three beats later, which they hold for one beat then throw again.

(3,3)(1,1) – A four ball pattern, ball spends longer in the air each throw, and more time being held each catch than in the (2,2) pattern.

(3,1)(1,3) – A four ball pattern where each hand takes turns to throw a ball while the other hand holds a ball.

(4,0)(0,4) – A four ball pattern, each hand throws while the other hand is empty. The balls are not held for a whole beat before being thrown again, spending very little time in the hand.

Modifying throws so they land in a different hand is simple with two balls. The rules say that a number followed by a letter must not return to the same hand. As there is only one other hand to direct a ball towards when using two hands, a single letter is all that is required. This letter can be x, meaning “don’t land here”.

(1x,0)(0,1x) – One ball being passed between two hands. Remember a 1 must have no notable airtime.

(2x,0)(0,0)(0,2x)(0,0) – Like the above pattern, but with the ball having some notable airtime.

(2x,1x) – A very fast paced three ball shower.

(2x,1)(1,2x) – A traditional three ball cascade.

(3x,0)(0,3x) – A three ball cascade with very little dwell time for each ball, a high energy pattern.

(4x,1)(1,4x) - A five ball cascade.

(22x,1)(1,22x) – Five ball splits, a multiplex pattern.

(23x,11)(1,11)(11,23x)(11,1) – A variation on five ball splits where the crossing ball is thrown higher than the non-crossing ball.

Juggling patterns with three or more hands:

(1,1,1) – Three hands, each containing the same ball on every beat.

(2,1,2,1,2,1) – Six hands, three holding a ball every beat, the other three juggling two balls.

(3,1,3,1)(1,3,1,3) – Four hands each juggling two balls.

Using more than two hands means that a single letter is not enough to direct a ball away from a hand. You are saying “Don’t land here” but there is now more than one option.

Each hand is assigned a capital letter. This can be any letter as long as it stays the same throughout the pattern. A simple way is to use something like:

(A,B,C,D,E,F) – 6 hands, starting from one juggler’s right hand and working around the hands in a clockwise direction.

To direct a ball from any hand, simply follow a number with that letter.

(2B,1,2D,1)(1,2A,1,2C) – two people juggling a three ball cascade each.

(2D,1,2B,1)(1,2A,1,2C) – four hand 2 count.

Not just hands can be notated.

(3,1,2)(1,3,0) - A four ball fountain while bouncing a ball on the head. Notice the balls in the hands have dwell time, held for a beat in the hands after each catch. The bouncing ball has no dwell time.

If you wanted to make the balls move between different hands and your head, you could label them L, R and H for left, right and head.

(2H,1,0)(1,1,0)(1,1,2R)(1,2H,0)(1,1,0)(1,1,2L) - A slow three-ball cascade with each ball bounced off the head. Again notice the dwell time of the balls in the hands compared to the head bounce.

With all the above we can notate any number of objects across any number of hands. For easy of use, throw durations can now be between “2” beats, through “9” and “a” beats up to “z” beats. Also the individual hands can be labeled from “A” to “Z” and the almost any other character you need from then on.

When dealing with lots of jugglers we can make the notation easier to read. We know that when dealing with a single pair of hands we can use an “x” to say “throw to the other hand”. If we pair hands in a multi-hand pattern together, an “x” will mean “throw to the other hand in this pair”.

We separate pairs of hands by using “|” instead of a comma.

(A,B|C,D|E,F|G,H) - four pairs of two hands.

(2x,1|2x,1|2x,1|2x,1)(1,2x|1,2x|1,2x|1,2x) – four pairs of hands juggling 3 balls each.

(2D,1|2A,1|2x1)(1,2x|1,2x|1,2x)(2F,1|2x,1|2A,1)(1,2x|1,2x|1,2x) – 3 juggler, 2 count, 9 club feed.

Many club passers do a lot of two person patterns. Here are some extra, optional notations that can be used ONLY for two jugglers facing in opposite directions (face to face or back to back):

# – to the same hand #x – to your other hand #p - a straight pass #px - a crossing pass

8 club 2 count all on doubles: (3p,3x|3p,3x)(1,1|1,1)

6 club pass-pass-self: (2p,1|2p,1)(1,2p|1,2p)(2x,1|2x,1)(1,2p|1,2p)(2p,1|2p,1)(1,2x|1,2x)

For most club passing patterns, and solo club juggling, a good guide to use is that a 2 is a single, a 3 is a double, 4 is a triple, etc. Each juggler has their own taste but it seems to work for many patterns.

It is possible to describe patterns where one hand is throwing at one rate and the other hand is throwing at a faster or slower pace.

Imagine one hand is juggling three balls to the same height and within the same length of time as the other hand is juggling just two balls. First we must work out how many beats are in the pattern before it repeats. There may be only five throws in total but some of the gaps between throws are more than one beat long, and a in that time a catch is made.

We multiply the beats in one hand with the beats in the other. 3 balls in the left and 2 balls in the right leaves us with 6 beats in total. The balls are being thrown to the same height we would normally juggle a six ball fountain:

(5,1)(1,5) – a six ball fountain

The new pattern would look like this:

(5,5)(1,0)(5,1)(1,5)(5,0)(1,1)

As you see there are 5 throws over the 6 beats and two beats have no throws at all.

Sometimes it is possible that a pattern is so long and complicated that it doesn’t fit all on one line. If a notation ends and there is more to come, “+” is added to the end.

This is a 5 juggler, 10 hand, 15 club, 2 count, star shaped, all-feed-all passing pattern:

(2D,1|2H,1|2B,2|2F,1|2x,1)(1,2x|1,2x|1,2x|1,2x|1,2x)+ (2F,1|2J,1|2D,1|2x,1|2B,1)(1,2x|1,2x|1,2x|1,2x|1,2x)+ (2H,1|2B,1|2x,1|2J,1|2D,1)(1,2x|1,2x|1,2x|1,2x|1,2x)+ (2J,1|2x,1|2H,1|2B,1|2F,1)(1,2x|1,2x|1,2x|1,2x|1,2x)+ (2x,1|2F,1|2J,1|2D,1|2H,1)(1,2x|1,2x|1,2x|1,2x|1,2x)

Using a “more to come” character is also useful for showing how a pattern may start without notating enough for it to be able to be repeated. If you see a “+” you can look on the next line for more notation or, if juggling the pattern, simply stop and catch all the balls.

Another character that can be added to an incomplete pattern notation is “*”. Adding “*” to the end is a “Repeat but with the information for the pair of hands swapped over” command.

This means that patterns that repeat in the right then left hand can be notated much more quickly. At a glance you can see that once you have worked out so much, you just need to mirror it in the other hand for the next repetition.

The three ball cascade can be either:

(2x,1)(1,2x) or just (2x,1)*

The four ball asynchronous fountain can now be:

(3,1)(1,3) or just (3,1)*

One final piece of optional notation that can be added to a pair of hands is a “cross hands” character. If a hand has “~” added, for that beat the arms must be crossed with that hand above the other.

A cross armed cascade may now be:

(~2x,1)(~1,2x)

Mills mess may now be notated as:

(~2x,1)(1,2x)(2x,~1)(1,~2x)(2x,1)(~1,2x)

Or if you want, by using a “*”, not only are the numbers swapped to the other hand but the “~” is too:

(~2x,1)(1,2x)(2x,~1)*

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For this answer you score 24/25 points. Well done.