The plan: Working with beatmap notation, investigate basic 360 pirouettes
in asynch juggling to see exactly how high you need to throw.


Objects:
balls
jugglers orrientation

Manipulators:
(Left hand, Right hand | North, South)

Rules:
180 degrees of a pirouette takes one beat to complete.
Patterns start with the juggler facing north.
A throw value must not be more than required.
A pirouette must wait until appropriate catches are made.



First experiment, find the minimal throw values needed to do pirouettes
both holding a ball and holding no balls.


1 ball, 1 hand, 1-up:

(1|1,0)+ - Holds one ball facing north
(2|1S,0)+ - Throws a ball and starts pirouette
(0|0,1N)+ - Passes half way through the pirouette
(1|1,0) - Has finished pirouetting and catches the ball1


2 ball fountain, 1 hand, 1-up:

(3|1,0)+
(1|1,0)+
(4|1,0)+
(1|1S,0)+
(1|0,1N)+
(3|1,0)+
(1|1,0)


2 ball fountain, 1 hand, 2-up:

(3|1,0)+
(1|1,0)+
(4|1,0)+
(1|1,0)+
(4|1S,0)+
(0|0,1N)+
(1|1,0)
(3|1,0)+
(1|1,0)+


Findings:
- Both 2 ball pirouettes use the same high throw value.

Questions:
- Why does a 2-up pirouette use the same throw values as a 1-up pirouette?
- How does this translate into patterns with more than one hand and more
than one ball?


Second experiment, notating 3 object, 2 hand pirouettes.


3 ball cascade, 1-up:

(2x,1|1,0)+
(1,4x|1,0)+
(1,1|1S,0)+
(1,1|0,1N)+
(2x,1|1,0)+
(1,2x|1,0)


3 ball cascade, 2-up:

(2x,1|1,0)+
(1,3x|1,0)+
(3x,1|1S,0)+
(0,1|0,1N)+
(1,2x|1,0)+
(2x,1|1,0)+
(1,2x|1,0)


3 ball cascade, 3-up:

(2x,1|1,0)+
(1,4x|1,0)+
(4x.1|1,0)+
(0,4x|1S,0)+
(0,0|0,1N)+
(1,0|1,0)+
(2x,1|1,0)+
(1,2x|1,0)



Findings:
- A 1-up pirouette uses a high throw value of 4. Once the ball is thrown,
a catch must be made in the same hand, then the pirouette, then another
throw, then the high throw can be caught.

- A 2-up pirouette uses a high throw value of 3. Two balls are thrown over
two beats and the pirouette can start immediately as the last catch is
made before the last high throw. Once the pirouette is over, no throw
needs to be made before the first high ball is caught.

- A 3-up pirouette uses a high throw value of 4. Once all three balls are
thrown high, no catches need to be made and the pirouette can start
immediately. Once the pirouette is over, no throw needs to be made before
the first high ball is caught.

- A 1-up and 3-up pirouette have the same high throw value of 4. A 2-up
pirouette has a high throw value of just 3.


New Questions:
- Does the same thing happen with 4, 5 and more object patterns?
- Is a 3 ball 2-up pirouette easier than both a 1-up and 3-up pirouette?


Third Experiment, 4 and 5 object pirouette notation.


4 ball asynch fountain, 2-up:

(3,1|1,0)+
(1,5|1,0)+
(5,1|1,0)+
(1,1|1S,0)+
(1,1|0,1N)+
(1,3|1,0)+
(3,1|1,0)+
(1,3|1,0)


4 ball asynch fountain, 3-up:

(3,1|1,0)+
(1,4|1,0)+
(4,1|1,0)+
(1,4|1S,0)+
(1,0|0,1N)+
(3,1|1,0)+
(1,3|1,0)+
(3,1|1,0)+
(1,3|1,0)


4 ball asynch fountain, 4-up:

(3,1|1,0)+
(1,5|1,0)+
(5,1|1,0)+
(1,5|1,0)+
(5,0|1S,0)+
(0,0|0,1N)+
(0,1|1,0)+
(1,3|1,0)+
(3,1|1,0)+
(1,3|1,0)


5 ball cascade, 3-up:

(4x,1|1,0)+
(1,6x|1,0)+
(6x,1|1,0)+
(1,6x|1,0)+
(1,1|1S,0)+
(1,1|0,1N)+
(4x,1|1,0)+
(1,4x|1,0)+
(4x,1|1,0)+
(1,4x|1,0)

5 ball cascade, 4-up:

(4x,1|1,0)+
(1,5x|1,0)+
(5x,1|1,0)+
(1,5x|1,0)+
(5x,1|1S,0)+
(0,1|0,1N)+
(1,4x|1,0)+
(4x,1|1,0)+
(1,4x|1,0)+
(4x,1|1,0)


5 ball cascade, 5-up:

(4x,1|1,0)+
(1,5x|1,0)+
(5x,1|1,0)+
(1,5x|1,0)+
(5x,1|1,0)+
(0,5x|1S,0)+
(0,0|0,1N)+
(1,4x|1,0)+
(4x,1|1,0)


Findings:
- with two handed juggling in a cascade or asynch fountain, when N =
number of balls, P = number of high throws, A = normal throw value and T =
high throw value -

If P=N-0 then T=A+2
If P=N-1 then T=A+1
If P=N-2 then T=A+2



New Questions:
- Using lower throw values should mean that a ball can be thrown lower and
needs less accuracy. With this in mind, is doing a 3 ball, 2-up pirouette
easier than doing either a 1-up or 3-up pirouette?


Forth Experiment, real life juggling.

I wanted to try out some pirouettes but found I was already at a level
where I could do 1-up, 2-up and 3-up pirouettes with balls with ease. I
tried with clubs, laying down an extra rule for mayself: The spin on each
club had to be A-1, a 2 was a single, a 3 a double, a 4 a triple, etc.

A 3 club 1-up pirouette was easy on triples. A 2-up pirouette was done
with doubles and didn't feel any more rushed. A 3-up with triples felt
much harder than both. Not only was I throwing higher than the 2-up, but 3
times as many high throws a the 1-up.

This told me that an (N-1)up pirouette is not half way between a (N-2)up
and an N-up pirouette in terms of difficulty. Yet after only one test, how
could I be sure? I needed another test.

I decided to make the whole thing much harder by repeating the experiment
with all the throws as backcrosses. A 1-up backcross pirouette was
obviously much more difficult than before. The high backcross on a triple
wasn't too hard but I kept trying to start the pirouette too quickly, not
leaving enough time to catch the club before spinning.

With the 2-up pirouette I didn't need to wait any time at all after the
second high backcross. The single club I needed to hold in my hand while
pirouetting had already landed so I could just spin! Also I didn't need to
make a throw after the pirouette but before I caught the first high throw.
When having to throw and catch from two different hand positions, as you
do with backcrosses, this makes a lot of difference.

I actually found the 3 club, 2-up backcross pirouette no harder than the
1-up version, and in some ways it was easier! This shouldn't have come as
a surprise as I had already worked out the numbers on paper, but there it
was.

The 3 club, 3-up backcross pirouette was too hard. I only managed it once
or twice.


Final Conclusions:

- with two handed juggling, a (N-1)up pirouette is the best value trick to
learn. It is far, far easier than a N-up and, depending on the current
skill level of the juggler, it should be no harder than the (N-2)up.

I have recently been trying 5 club pirouettes. I have had some success
with the 3-up but I have found that I spin too hastily, not waiting to
catch the last club before I spin, and I also have trouble throwing the
first club after the pirouette before catching the first high throw. From
now on I will not bother wasting time on the 5 club 3-up or the 5-up
pirouette. Instead I'm going to concentrate only on the 5 club 4-up
pirouette.

You may think that is a bit weird, but my calculations say I don't need
throw so high, or maybe to the same height but with a slower, 3-beat
pirouette, so I'm going with that option.