Siteswap Notation V: Synchronous Patterns

As discussed in Siteswap Notation IV, at a given beat, H hands could be throwing N objects simultaneously. We said that a synchronous throw occurs if H = N > 1, and each hand is exclusively throwing one object. Among synchronous throws, the two-handed synchronous throw (H = N = 2) is of special significance: an individual juggler will usually make a synchronous throw with two hands. For the same reason, the two-handed synchronous multiplex throw (special case of N > H = 2) is also of particular interest. It is perhaps for this reason that a version of the siteswap notation evolved to specifically address these two-handed cases which we will study in this blog.

Synchronous Possibilities

I demonstrated a couple of ways of juggling the flavoured siteswap sequence “{2 1}” as a multiplex pattern (i.e., as “[2 1]”) in Siteswap Notation IV. The video below demonstrates possible ways of juggling the “{2 1}” as a synchronous pattern.

As seen in the video, there are six “basic” ways of juggling the “{2 1}” synchronously with two hands:

1. Both “2” and “1” are thrown back to the throwing hand, with the right hand throwing “2” and left hand throwing “1”.
2. Same as 1 above but with right and left hand roles reversed.
3. Both “2” and “1” cross to the other hand, with right hand throwing “2” and left hand throwing “1”.
4. Same as 3 above but with right and left hand roles reversed.
5. “2” crosses to the other hand, but “1” does not. Right and left hands take turns throwing a crossing “2” while the other hand simultaneously throws “1” to itself.
6. “1” crosses to the other hand, but “2” does not. Right and left hands take turns throwing a crossing “1” while the other hand simultaneously throws a “2” to itself.

Jack Boyce enhanced the siteswap notation to uniquely notate each of the cases listed above so that the juggler understands exactly which version is to be juggled.

Synchronous Notation

Remember that we are restricting ourselves to the case of two-handed synchronous throws for now and so by definition, N = H = 2. Therefore, the flavoured throw will have exactly two siteswap numbers within the brackets. Jack Boyce enhanced the notation thus:

1. Siteswap numbers for flavoured throws meant to be thrown synchronously are enclosed in round brackets ( ). So a {2 1} meant to be thrown synchronously will be written (2 1).
2. The first number in the brackets represents the throw to be made by the right hand and the second number represents the throw to be made by the left hand^[1]. So (2 1) means the right hand throws a “2” and the left hand throws a “1”. Unlike for multiplex throws, the order of the numbers within the round brackets matters: (2 1) and (1 2) mean different things. Often, a comma (,) is used to separate the two numbers.
3. If a throw has to be made to the other hand, this is indicated by appending an “x” to the corresponding siteswap number. For example, if we want the right hand to throw a “2” across to the left hand while the left hand throws a “1” to itself, this throw will be written as (2x 1).

With these enhancements, the 6 “basic” ways of synchronously juggling the flavoured siteswap {2 1} can now be uniquely specified as:

1. (2 1): If we “hold” the “1”, we’re effectively juggling “2-in-1-hand” in the right hand.
2. (1 2): If we “hold” the “1”, we’re effectively juggling “2-in-1-hand” in the left hand.
3. (2x 1x): The “shower” pattern, juggled anti-clockwise from the juggler’s perspective.
4. (1x 2x): The “shower” pattern, juggled clockwise from the juggler’s perspective.
5. (2x 1) (1 2x): If we “hold” the “1”, we’re effectively juggling the “3-ball cascade“!
6. (2 1x) (1x 2): The “box” pattern.

Note that in options 5 and 6 above, we had to write the sequence with period P = 2, with the second term mirroring the first term. This is because the sequence “(2x 1)” for example, cannot be sustained. This can be logically understood as follows – the sequence “(2x 1)” would mean that the right hand continuously throws crossing 2’s to the left hand while the left hand throws 1’s to itself. At this rate, all the objects will soon end up in the left hand, leaving the right hand with nothing to throw. This also means that the permutation test which said that the flavoured sequence “{2 1}” is a valid sequence, must now be modified to be able to tell that the synchronous sequence “(2x 1)” is an invalid sequence. Before we explore this modified permutation test though, let us consider one more question.

The Synchronous Hand Siteswap

What happens if not all throws of a sequence are flavoured and therefore can’t be thrown synchronously? Consider for example, the flavoured sequence “{3 2} 3 1”: The {3 2} can potentially be thrown synchronously but on the other two beats, N = 1 and a synchronous throw can’t be made.

We deal with such cases by assuming that ALL the throws are synchronous; even on beats where only one object is being thrown. In other words, synchronous siteswap notation implicitly enforces the hand siteswap sequence “(1 1)”^[2]: Round brackets are consistent with synchronous notation; the first “1” indicates that the right hand should throw again one beat later and the second “1” indicates the same for the left hand^[3].

For a synchronous throw with only one object being thrown to make sense, we assign the siteswap number “0” (empty hand) to the non-throwing hand. Let us now return to the “{3 2} 3 1” sequence with the {3 2} thrown synchronously. In Table 1, we list down the siteswap numbers from beats 0 to 9 as they occur in three different synchronous variations of this sequence (click on the option in Table 1 to see corresponding video).

Table 1: Translating synchronous throws to siteswap notation

Time (in beats)	0	1	2	3	4	5	6	7	8	9
Option 1(b)	(3x 2)	(0 3x)	(0 1x)	(2 3x)	(3x 0)	(1x 0)	(3x 2)	(0 3x)	(0 1x)	(2 3x)
Option 1(c)	(3x 2)	(0 3)	(0 1x)	(2 3x)	(0 3)	(1x 0)	(3x 2)	(0 3)	(0 1x)	(2 3x)
Option 2(b)	(3x 2)	(3 0)	(0 1x)	(3x 2)	(3 0)	(0 1x)	(3x 2)	(3 0)	(0 1x)	(3x 2)

Up to the minimal period, the siteswap sequence representing each of these patterns is given in Table 2.

Table 2: Siteswap sequences for patterns of Table 1

Juggling pattern	Siteswap Sequence	Minimal Period
Option 1(b)	(3x 2) (0 3x) (0 1x) (2 3x) (3x 0) (1x 0)	6
Option 1(c)	(3x 2) (0 3) (0 1x) (2 3x) (0 3) (1x 0)	6
Option 2(b)	(3x 2) (3 0) (0 1x)	3

Permutation Test for Synchronous Patterns

The permutation test for synchronous patterns needs to comprehend the extra information that is now encoded in the synchronous notation. Let us take the example valid sequence “(3x 2) (3 0) (0 1x)” from Table 2. Note that the 0’s in this pattern were introduced to “convert” the vanilla throws into “synchronous” throws. We add the following two rules to the permutation test for flavoured sequences before applying it to synchronous sequences:

1. No arithmetic operation is performed on the added 0’s. To avoid confusion, we write “-” instead of the “0”.
2. If an “x” is present, we perform the arithmetic operation on the number preceding the “x”. We then put back the “x” in the result, except when calculating the quotient.

With these modifications, we apply the permutation test as shown in Table 3.

Table 3: Permutation Test for synchronous sequence (3x 2) (3 0) (0 1x), Period P = 3

Given Sequence	(3x 2)	(3 -)	(- 1x)
Add	(0 0)	(1 1)	(2 2)
Result	(3x 2)	(4 -)	(- 3x)
Remainder of (Result/P)	(3x 2) mod 3 = (0x 2)	(4 -) mod 3 = (1 -)	(- 3x) mod 3 = (- 0x)
Quotient of (Result/P)	int{(3x 2)/3} = (1 0)	int{(4 -)/3} = (1 -)	int{(- 3x)/3} = (- 1)

Now it remains to interpret the remainder and quotient rows. The first term in the remainder row is (0x 2). The 0x is present in the right hand position. The “x” indicates that a siteswap number should have been present for the left hand in the original sequence at beat #0. Indeed, the 2 of the (3x 2) at beat #0 is present in this position. The “2” present in the left hand position of the term (0x 2) of the remainder means that a siteswap number should have been present in the original sequence at beat #2 for the left hand. This number is the 1x of the (- 1x) at beat #2. The second term in the remainder row is (1 -). The “1” in the right hand position indicates that a siteswap number should have been present in the original sequence at beat #1 for the right hand. This number is the 3 of the (3 -) at beat #1. The last term in the remainder row is (- 0x). The “0x” present in the left hand position means that a siteswap number should’ve been present in the original sequence at beat #0 for the right hand. This number is the 3x of the (3x 2) at beat #0. Thus, we see that each term in the remainder row is pointing to one of the non-zero siteswap numbers in the original sequence such that all non-zero siteswap numbers are accounted for. When this happens, the original siteswap sequence is said to be a valid sequence. If we hadn’t replaced the dummy 0’s with “-“, we’d have simply ended up with the appropriate beat number in the remainder row corresponding to the position of the “0”. Using “-” instead of “0” just reduces the number of calculations to be done. As usual, if we add up the numbers in the quotient row for a valid sequence, it will give us the number of objects involved in the corresponding juggling pattern, in this case: 1+0+1+1 = 3.

More examples

Let us take another couple of examples.

“(2x 1)”:

We had logically concluded earlier that this sequence of period P = 1 is invalid. Let us see if the permutation test agrees.

Table 4: Permutation Test for synchronous sequence (2x 1), Period P = 1

Given Sequence	(2x 1)
Add	(0 0)
Result	(2x 1)
Remainder of (Result/P)	(2x 1) mod 1 = (0x 0)
Quotient of (Result/P)	int{(2x 1)/1} = (2 1)

The remainder row now has 0x in the right hand position and 0 in the left hand position. This means that two terms should be present at beat #0 of the original sequence, both for the left hand. But the original sequence “(2x 1)” has one term each for the right and left hand at beat #0. Consequently, the right hand term (2x) is not accounted for by any term in the remainder row and so the original sequence is not a valid siteswap. We ignore the quotient row.

“(2x 1) (1 2x)”:

Let us see if the permutation test confirms that this variation with period P = 2 is a valid way of synchronously juggling the  flavoured siteswap {2 1}.

Table 4: Permutation Test for synchronous sequence (2x 1) (1 2x), Period P = 2

Given Sequence	(2x 1)	(1 2x)
Add	(0 0)	(1 1)
Result	(2x 1)	(2 3x)
Remainder of (Result/P)	(2x 1) mod 2 = (0x 1)	(2 3x) mod 2 = (0 1x)
Quotient of (Result/P)	int{(2x 1)/2} = (1 0)	int{(2 3x)/2} = (1 1)

We see that the remainder row has a 0 in the right hand position indicating the original siteswap sequence should have a siteswap number at beat #0 for the right hand position. This is the 2x of (2x 1). The remainder row also has a 0x in the right hand position indicating the original siteswap sequence should have a siteswap number at beat #0 for the left hand position. This is the 1 of (2x 1). Similarly, the remainder row has 1 and 1x both in the left hand position implying that at beat #1 of the original sequence, we should have a siteswap number respectively for the left and the right hands. These numbers are respectively, 1 and 2x of (1 2x). Thus, all siteswap numbers of the original sequence are accounted for by the terms in the remainder row and so, (2x 1) (1 2x) is a valid sequence. Adding up the numbers of the quotient row, we get, 1+0+1+1 = 3 objects are required to juggle the corresponding pattern.

The average theorem can be applied exactly as described in Siteswap Notation III (ignore the “x”) to get the number of objects required to juggle a valid synchronous siteswap sequence.

Synchronous Shorthand

Many common synchronous sequences have to be juggled with a sequence of throws followed by the same sequence mirrored so that all the objects don’t end up in one hand. For example, “(2x 1)” is not a valid sequence but “(2x 1) (1 2x)” is. But it is cumbersome to write out such sequences in full. Ben Beever came up with  a shorthand to make this less cumbersome. We write only one half of the sequence and then put an asterisk to indicate that the sequence is to be completed by repeating the written half but with the order of the numbers within the brackets reversed. Table 5 gives some examples of this shorthand notation.

Table 5: Shorthand for synchronous sequences

Complete sequence	Shorthand sequence
(2x 1) (1 2x)	(2x 1)*
(2 1x) (1x 2)	(2 1x)*
(3x 2) (0 3x) (0 1x) (2 3x) (3x 0) (1x 0)	(3x 2) (0 3x) (0 1x)*

Often, even the asterisk is dropped; one is just expected to “know” that only half the sequence has been written out!

Synchronous Multiplex Notation

The case of two handed synchronous multiplex throws is a simple extension combining the multiplex notation with the two handed synchronous notation. Instead of two numbers in the round brackets, there could now be two sets of numbers in the round brackets. The first set of numbers will represent the multiplex throw from the right hand while the second set will represent the same for the left hand.

The permutation test and average theorem for synchronous multiplex patterns follow along exactly the same lines as what has been discussed for synchronous patterns.

Table 6 lists some valid synchronous multiplex sequences along with the number of objects that will be needed to juggle these as calculated by the average theorem.

Table 6: Example synchronous multiplex sequences

Sequence	Period	# Objects
([2x 2x] 2x) (0 0)*	4	([2+2]+2+0+0+2+[2+2]+0+0)/4 = 12/4 = 3
([2 2] 2) (2x 0) ([2 2] 2) (0 2x)	4	([2+2]+2+2+0+[2+2]+2+0+2)/4 = 16/4 = 4
([2 2] [2 2]) (2x 0)*	4	([2+2]+[2+2]+2+0+[2+2]+[2+2]+0+2)/4 = 20/4 = 5

Note the asterisk in the first and third sequences in Table 6. This denotes that though only two beats are explicitly written out, we need two more beats to complete the sequence. Hence, the period of both these sequences is listed as 4. However, the average theorem will work out even if we add only the numbers explitly written out (yielding half the sum) and divide by half the period. This may have encouraged the practice of dropping the asterisk. However, the permutation test will in general not work out if tried on only half the sequence.

I have demonstrated the synchronous multiplex sequences of Table 6 in the video below.

Footnotes

1. The reverse convention may also be chosen, i.e., the first number corresponds to the left hand throw and the second to the right hand throw. This may even be the more intuitive choice: the first bracketed number is on the reader’s left and the second on the right. I have made the opposite choice because in my demonstration videos, my right hand will be to the viewer’s left and vice-versa.
   
2. There is a school of thought that favours the hand siteswap sequence “(2 2) (0 0)” instead. This also means that all object siteswap numbers must be even numbers. This alternative will be explored in more detail in the next part.
3. At this point, one may ask if the hand siteswap sequence can also have an “x”. This deserves a more involved answer than a footnote. For now, let’s just live with the hand siteswap sequence being “(1 1)” and interpreted as given here.