In Siteswap Notation III, I wrote about siteswap sequences where more than one object could be thrown at one beat and called them “flavoured siteswaps”. In the associated video for that post, I demonstrated how two objects could be thrown either by a single hand, or by two hands throwing one object each. In theory, at a given beat, H hands could be throwing N objects, where H, N ≥ 1.
“Interesting” Throws
Just as objects could be thrown using vanilla or flavoured siteswaps, hands could also be throwing using vanilla or flavoured hand siteswaps. The object and hand siteswaps can then be combined in several ways to make different kinds of throws. However, not all combinations are of equal interest to jugglers. For example, cases where more than one hands together throw the same (set of) object(s) are usually ignored. We might say that such cases where “hands are getting multiplexed” are “uninteresting”. The case H > N = 1 is therefore, “uninteresting”. In general, the case H > N ≥ 1 (which will necessitate multiplexed hands) is considered uninteresting. Among the remaining cases, where at a given beat, N ≥ H ≥ 1, some throw possibilities have gained more popularity than others and have been grouped into various categories as discussed below.
Vanilla Throw
This case arises when N = 1 = H. One hand throws one object at a given beat.
Multiplex Throw
This case arises when N > H = 1. One hand throws more than one object at a given beat. We will study this case in a little more detail in this blog.
Synchronous Throw
The “interesting” version of making a throw when N = H > 1 is called a synchronous throw. In this case, two or more hands each throw exactly one object at a given beat. “Uninteresting” options also exist for N = H > 1. For example, multiplexed hands throwing the same set of objects while the remaining objects are thrown as vanilla or multiplexes by the remaining hands.
Synchronous Multiplex Throw
The “interesting” version of making a throw when N > H > 1 is called a synchronous multiplex throw. Here, two or more hands each throw one or more objects at a given beat, with no two hands being involved in the throw of the same object, i.e., no hands are multiplexed. “Uninteresting” ways of making a throw with N > H > 1 again involve multiplexed hands throwing the same set of objects while the remaining objects are thrown as vanilla or multiplexes by the remaining hands.
The video above demonstrates some examples of the various types of throws that have been discussed here.
Multiplex Notation
In a vanilla hand siteswap, only one hand makes a throw at any given beat. If a flavoured object siteswap is coupled with a vanilla hand siteswap, it necessarily means that the multiple objects to be thrown simultaneously in the flavoured throw, must all be thrown by a single hand. Such a simultaneous throw of multiple objects from the same hand is called a multiplex throw.
Multiplex throws are specified by using square brackets to enclose the object siteswap numbers of the throw. For example, if we intend the flavoured siteswap sequence “{3 2} 3 1” to be juggled with multiplex throws (i.e., with a vanilla hand siteswap sequence), we will write the sequence as “[3 2] 3 1”. The flavoured “[3 2]” throw now specifies that two objects are getting thrown by the same hand at the same time such that one object is to be thrown again three beats later and the other object is to be thrown again two beats later. To which hand should these objects be thrown, depends on the specific vanilla hand siteswap being used. The video below shows the siteswap “[3 2] 3 1” being juggled with two different vanilla hand siteswap options: “3 1 2” and “2”.
Starting with a right hand throw in the “3 1 2” hand siteswap, we get the hand sequence “right, left, left” which repeats. The hand siteswap “2” on the other hand, when started with a right hand throw, results in the hand sequence “right, left” which repeats. The video goes on to demonstrate that in the [3 2] throw involving two objects, either object could be chosen to be thrown as a “3” and the other one as a “2”. In Option A, the red ball is always chosen for the 3 throw and blue for the 2. In Option B, this choice is reversed (blue always thrown as 3, red as 2). In Option C, the choice alternates between the first two options (red: 3, blue:2, then blue:3, red:2, repeat).
For multiplex throws, the arrangement of numbers within the brackets has no signifcance. For example, “[3 2] 3 1” and “[2 3] 3 1”, both represent the same pattern.
The average theorem, the permutation test and the siteswap generation algorithm apply to multiplex sequences exactly as presented in Siteswap Notation III. All that we have changed here (apart from the shape of the brackets) is that we have specified that we will use a vanilla hand siteswap, i.e., only one hand will be throwing at any given beat.
Default (Vanilla) Hand Siteswap
The default hand siteswap assumed for multiplex juggling patterns is “2”, exactly the same as for vanilla juggling patterns. This sequence involves two hands and the hands throw on alternate beats. The “rule” that an object thrown with an odd siteswap number will get thrown to the other hand and an object thrown with an even siteswap number will get thrown to the same hand again applies in this default case. If the hand siteswap is not “2”, then the hand to which an object is thrown is determined by the object siteswap number at that beat and the hand siteswap sequence being used. For example, in Table 1, we have listed the hand to which the various throws of the sequence “[3 2] 3 1” are made when juggled with the default hand siteswap “2” and the hand siteswap “3 1 2” (see video above).
| Hand Siteswap | |||
|---|---|---|---|
| Object Siteswap | “2” (Default) | “3 1 2” | |
| [3 2] | 3 (Odd) | To Other hand | To Same hand |
| 2 (Even) | To Same hand | To Other hand | |
| 3 (Odd) | To Other hand | To Same hand | |
| 1 (Odd) | To Other hand | To Other hand | |
As seen in Table 1, with the hand siteswap “2”, all the throws with odd siteswap numbers (i.e., 3 and 1) are thrown to the other hand while the throws with even siteswap number (i.e., 2) are thrown to the same hand. However, for the hand siteswap “3 1 2”, the throws with odd siteswap numbers could be thrown either to the same hand (for siteswap number 3) or to the other other hand (for siteswap number 1). The only throw with an even siteswap number (i.e., 2) happens to be thrown to the other hand in this instance, contrary to the case for the default hand siteswap.
Note that we may not generalize any of our findings regarding the hand siteswap “3 1 2” as a “rule” for that hand siteswap. For example, we can’t say that a “2” when thrown while using the hand siteswap “3 1 2” will always be a crossing throw. To see this, consider the multiplex object siteswap sequence “[2 1]” of period P = 1 juggled with hand siteswaps “2” and “3 1 2” as shown in the video below. When using the hand siteswap sequence “3 1 2”, since the object siteswap period is 1 and the hand siteswap period is 3, the overall period of the pattern is LCM(1,3) = 3.
In particular, from 1:22 onward, the video shows the hands to which the throws are made when using the hand siteswap “3 1 2”. This information is captured in Table 2 below.
| Hand Siteswap | |||
|---|---|---|---|
| Object Siteswap | “2” (Default) | “3 1 2” | |
| [2 1] | 2 (Even) | To Same hand | To Other hand |
| 1 (Odd) | To Other hand | To Other hand | |
| [2 1] | 2 (Even) | To Same hand | To Other hand |
| 1 (Odd) | To Other hand | To Same hand | |
| [2 1] | 2 (Even) | To Same hand | To Same hand |
| 1 (Odd) | To Other hand | To Other hand | |
Note that with the hand siteswap sequence “3 1 2”, both an even siteswap number (“2”) and an odd siteswap number (“1”) could get thrown either to the same hand or to the other hand. One has to be very aware of the hand siteswap being used when trying to interpret a juggling pattern in terms of siteswap notation – a crossing doesn’t always imply an odd siteswap number and a throw to the same hand doesn’t always imply an even siteswap number.
The reason I keep harping on this is that when we get to synchronous siteswap notation, one shouldn’t get perplexed on seeing that this odd/even siteswap number “rule” no longer works. It’s just that the hand siteswap sequence being used is no longer “2”.
Numbers Juggling Made Easy
A typical (and typically irritating) question asked of jugglers is, “How many objects can you juggle?” Jugglers who attempt to answer this with ever increasing numbers are said to have entered the domain of “numbers juggling”. Unfortunately for them, the going gets very hard very quickly as they have to throw the objects higher and higher with more and more accuracy (w.r.t. the angle at which they toss the objects). Multiplex juggling offers a relatively easy “cheat” to accomplish numbers juggling. Table 2 lists some “easy” multiplex siteswaps that can be used to juggle 4, 5 and 6 objects.
| Objects juggled | “Easy” Multiplex Siteswaps |
|---|---|
| 4 | [3 1] |
| [3 3] 3 3 | |
| 5 | [3 2] |
| [3 3] [3 3] 3 | |
| 6 | [3 3] |
A further “cheat” can be applied to make numbers juggling even easier! Remember that if at a particular beat, the object siteswap number matches the hand siteswap number, that throw can be made simply as a “hold”. This cheat, while using the hand siteswap sequence “2”, makes it even easier to juggle the 4-object sequence “[3 2] 3”, the 5-object sequence “[3 2]” and the 6-object sequence “[3 2] [3 2 2]”.
The reader is encouraged to apply the permutation test to all of these sequences and verify that these are indeed valid patterns involving the number of objects claimed. Or you could just watch the video below:
As seen in the video, using multiplex throws, I’m able to juggle as many as 6 balls with a difficulty level not too different from the 3-ball cascade! Unfortunately, the typical audience will also realize this and refuse to be impressed by these demonstrations. But maybe it is the sort of exasperating answer that an exasperating question like, “How many objects can you juggle?” deserves. If you can afford to exasperate your audience, that is.
Flavoured Hand Siteswaps
As mentioned in Siteswap Notation II, siteswap notation “does not comprehend hands at all”. Potentially, any given object siteswap could be juggled with a multitude of hand siteswaps. The hand siteswaps could further be themselves vanilla or flavoured siteswaps. However, a hand siteswap is rarely ever specified when communicating an object siteswap sequence. This is because a “default hand siteswap” is assumed. In particular, if an object siteswap is written as a vanilla siteswap (exactly one siteswap number per beat) or a multiplex siteswap (square brackets enclosing siteswap numbers for flavoured throws), it is assumed that the hand siteswap sequence is a vanilla sequence, i.e., H = 1 at every beat. Further, it is assumed that this vanilla hand siteswap sequence is “2”.
In the same spirit, if an object siteswap sequence is intended to be juggled with a flavoured hand siteswap (H > 1 at a beat), the indication for this is provided by further embellishments to the object siteswap and assuming a corresponding default hand siteswap. We will study these embellishments and the corresponding default hand siteswap in the context of patterns involving synchronous and synchronous multiplex throws in the next part of this series.