In my previous post, we learnt how to identify, create and manipulate valid vanilla siteswap sequences purely as mathematical constructs. In this post, we will talk about how to interpret the siteswaps so as to actually be able to juggle them. To do this, we need to determine which hand should do what on which beat.
What is a Hand?
Jugglers use “hands” to throw and catch the objects they’re juggling. That sounds like stating the obvious, but then, jugglers are often fantastically weird beings that can use virtually any part of their body to catch and throw: feet, teeth, neck, elbows, you name it. The noun “hand” is therefore used loosely here and hopefully it’ll be clear from the context when I literally do mean hand. While throwing and catching, these “hands” could be twisted around other “hands” or impossibly contorted. Siteswap notation does not comprehend such variations. Actually, siteswap notation does not comprehend “hands”[1] at all!
All that a vanilla siteswap encodes is a mapping of each beat to the object to be thrown at that beat. There is no information about the “hand” to be used for throwing. For example, in Table 1, the events for the siteswap “3” for the first 10 beats have been shown. Additionally, several options indicating which “hand” makes the throw at each beat are also shown. R and L respectively denote Right and Left. H = Hand (the literal one), F = Foot, E = Elbow, S = Shoulder.
| Current Time (in beats) (T) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Siteswap Number (S) | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| Object thrown | A | B | C | A | B | C | A | B | C | A |
| Time at next throw of object (T+S) | 0+3=3 | 1+3=4 | 2+3=5 | 3+3=6 | 4+3=7 | 5+3=8 | 6+3=9 | 7+3=10 | 8+3=11 | 9+3=12 |
| Throwing “Hand” (Option 1) | RH | LH | RH | LH | RH | LH | RH | LH | RH | LH |
| Throwing “Hand” (Option 2) | RF | LS | LE | RH | LH | RE | Neck | LF | Teeth | RS |
| Throwing “Hand” (Option 3) | RH | RH | RH | RH | RH | RH | RH | RH | RH | RH |
| Throwing “Hand” (Option 4) | RH | RH | LH | LH | RH | RH | LH | LH | RH | RH |
See the video below for a demonstration of all these options, performed to the best of my ability[2]:
In all the options demonstrated in the video, a ball is thrown at each beat and the sequence in which the balls are thrown is identical: pink, yellow, red, and repeat. If we map the pink ball to object A, yellow to B and red to C of Table 1, we see that it matches the siteswap “3”. Yet, all options look different when juggled because of the choice of “hands” used to make the throws. Let us understand this further.
Juggling the Hands
Consider Table 2, representing a juggling pattern for two objects, R and L. It maps each beat to the object thrown at that beat.
| Current Time (in beats) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Object thrown | R | L | R | L | R | L | R | L | R | L |
In Table 3, we work out the siteswap sequence represented by Table 2. Since there is no information on when objects R and L will respectively be thrown again after Time = 8 beats and Time = 9 beats, we can not write a siteswap number for the last two beats.
| Current Time (in beats) (TC) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Object thrown | R | L | R | L | R | L | R | L | R | L |
| Time at next throw of object (TN) | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ? | ? |
| Siteswap Number (TN-TC) | 2-0=2 | 3-1=2 | 4-2=2 | 5-3=2 | 6-4=2 | 7-5=2 | 8-6=2 | 9-7=2 | Can’t say | Can’t say |
If we assume that the objects R and L continue to be thrown on alternate beats even beyond Time = 9 beats, then they are being juggled corresponding to the siteswap sequence “2”. Now if object R is the right hand and object L is the left hand, and instead of “thrown” we say “triggered” to indicate that the hand is getting triggered by the brain to make a throw, then we can say that the right and left hands are being triggered according to the siteswap sequence “2”. Notice that this is exactly how the hands in option 1 of Table 1 are triggered: the right hand (mapping to object R of Table 2) is triggered at Time = 0, 2, 4, 6 and 8 beats while the left hand (mapping to object L of Table 2) is triggered at Time = 1, 3, 5, 7 and 9 beats.
Let us similarly work out the siteswap for the triggering sequence for all the “hand options” of Table 1 except hand option 2. We leave out hand option 2 because none of the hands are triggered a second time within the first 10 beats, so no siteswap number can be determined.
| Current Time (in beats) (TC) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Triggered Hand (Opt. 1) | RH | LH | RH | LH | RH | LH | RH | LH | RH | LH |
| Time at next trigger of Hand (TN) | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ? | ? |
| Hand Siteswap (TN-TC) | 2-0=2 | 3-1=2 | 4-2=2 | 5-3=2 | 6-4=2 | 7-5=2 | 8-6=2 | 9-7=2 | Can’t say | Can’t say |
| Triggered Hand (Opt. 3) | RH | RH | RH | RH | RH | RH | RH | RH | RH | RH |
| Time at next trigger of Hand (TN) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ? |
| Hand Siteswap (TN-TC) | 1-0=1 | 2-1=1 | 3-2=1 | 4-3=1 | 5-4=1 | 6-5=1 | 7-6=1 | 8-7=1 | 9-8=1 | Can’t say |
| Triggered Hand (Opt. 4) | RH | RH | LH | LH | RH | RH | LH | LH | RH | RH |
| Time at next trigger of Hand (TN) | 1 | 4 | 3 | 6 | 5 | 8 | 7 | ? | 9 | ? |
| Hand Siteswap (TN-TC) | 1-0=1 | 4-1=3 | 3-2=1 | 6-3=3 | 5-4=1 | 8-5=3 | 7-6=1 | Can’t say | 9-8=1 | Can’t say |
If we assume that the minimal period sequence established within the first 10 beats is going to repeat in the future, then we have from Table 4:
Hand option 1: Hand triggering siteswap is “2”.
Hand option 2: Indeterminate, excluded from table.
Hand option 3: Hand triggering siteswap is “1”.
Hand option 4: Hand triggering siteswap is “1 3”.
There are infinitely many such hand options possible[3]. Should, therefore, a siteswap sequence for objects to be thrown, always be accompanied by a hand triggering siteswap sequence to communicate a juggling pattern?
Default Interpretation: Hands do the “2”
Unless otherwise specified, the default hand triggering siteswap for a vanilla siteswap is assumed to be “2” (i.e., what we have been calling option 1). This implies:
- Exactly two hands are used;
- The two hands take turns to throw and so, each hand throws at alternate beats.
This default interpretation[4] leads to some interesting consequences.
The siteswap number “2”
Consider the “4 2 3” vanilla siteswap, this time with the default hand triggering sequence (“2”) also indicated alongside:
| Current Time (in beats) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Siteswap Number (for object) | 4 | 2 | 3 | 4 | 2 | 3 | 4 | 2 | 3 | 4 |
| Object thrown | A | B | C | B | A | C | A | B | C | B |
| Throwing hand (Triggering siteswap “2”) | RH | LH | RH | LH | RH | LH | RH | LH | RH | LH |
In Table 5, at Time = 1 beats, the object B is thrown with a siteswap number “2” from the left hand. This means object B should be thrown again at Time = (1+2) = 3 beats. Since the hand triggering siteswap is also “2”, the left hand will also be throwing again at Time = (1+2) = 3 beats. Between Time = 1 beats and Time = 3 beats, neither does the object B have to be thrown, nor does the left hand have to handle any other object. So why not just “hold” object B in the left hand from Time = 1 beats to Time = 3 beats? Indeed, a “2” is usually juggled as a “hold” when the hand triggering siteswap is also “2”. In general, if the object siteswap number is equal to the hand triggering siteswap number at a given beat, then that throw can be performed as a “hold”.
Odd and Even Siteswap numbers
From Table 5, we see that if the hand triggering siteswap is “2” and we start with a right hand throw at Time = 0 beats, then the right hand throws at all the even beats, i.e., at Time = 0, 2, 4, 6 and 8 beats and the left hand throws at all the odd beats, i.e., at Time = 1, 3, 5, 7 and 9 beats. Table 6 exploits this symmetry to determine which hand will make the next throw of the object being thrown at the current time, given the current time and the object siteswap number at the current time.
| Current Time (in beats) (T) | even number | odd number | even number | odd number |
|---|---|---|---|---|
| Current throw hand (HC) | Right | Left | Right | Left |
| If Object Siteswap Number (S) is | even number | even number | odd number | odd number |
| Next throw of same object is at (T+S) | even+even = even numbered beat | odd+even = odd numbered beat | even+odd = odd numbered beat | odd+odd = even numbered beat |
| Hand used at (T+S) (HN) | Right | Left | Left | Right |
| Is HN = HC? | Yes | Yes | No | No |
Thus, if the hand triggering siteswap is “2”, then, for an even siteswap number, the same object will next be thrown from the same hand. For an odd siteswap number, the same object will next be thrown from the other hand. This “rule” is often used as a quick aid in interpreting siteswaps: throw an even siteswap number to the same hand and an odd siteswap number to the other hand. Remember: this “rule” assumes that the hand triggering siteswap is “2”. Forgetting this assumption was one of the reasons for us mistaking another pattern for the “4 2 3”[5]. Indeed, the “3” juggled with hand siteswap “1 3” as demonstrated in option 4 of the video above can be similarly mistaken for the “5 0 4” juggled with hand siteswap “2” but technically, they are not the same.
Throw Height
We now know what the siteswap numbers mean and how to associate them with a hand triggering sequence to figure out at each beat, which hand should be making the throw and towards which hand the throw should go. Assuming that all throws are thrown up in the air, it now remains to be understood how high the throw should be for a given siteswap number so that the object is available in the required hand in time for its next scheduled throw.
Flight Time and Dwell Time
The time defined by a siteswap number, S, is divided into two parts:
- The flight time f is the time for which the object is in flight after the throwing hand has thrown it.
- The dwell time d is the time for which the object dwells in the catching hand before it next gets thrown.
S = f+d.
For the same siteswap number, an object may be thrown lower to get a lower flight time and higher dwell time or it may be thrown higher to get a higher flight time and lower dwell time. If the hand triggering sequence is “2”, then each hand will have 2 beats of time between its previous throw and the next throw. The object to be thrown next by a hand has to be caught somewhere within this 2-beat window. So dwell time can range between 0 beats and 2 beats if the hand triggering siteswap is “2”[6].
For the sake of simplicity and to enable us to do an indicative calculation, we assume that for the hand triggering sequence “2”, the dwell time, d, is always 1 beat. Then, f = S-d = S-1. The object will spend half of this flight time going up and half of it coming down (assuming the throwing and catching hands are at the same height). If “h” is the difference in height between the position of the hand and the top of the object’s flight, then h = (1/2)*g*t2, where g = acceleration due to gravity = 9.8m/s2 and t = time taken by the object to drop from the top of its flight to the catching hand = f/2 = (S-1)/2[7].
Let’s see how the throw heights h1 and h2 corresponding to siteswap numbers S1 and S2 will compare in this scenario.
h1 = (1/2)*g*((S1-1)/2)2.
h2 = (1/2)*g*((S2-1)/2)2.
Then,
(h2/h1) = {(S2-1)/(S1-1)}2.
For example, the ratio of heights for throws with siteswap numbers 5 and 3 would be {(5-1)/(3-1)}2 = 22 = 4. So the throw for siteswap number “5” should go four times as high as the throw for siteswap number “3”!
Remember that the calculations above have assumed:
- A hand triggering siteswap of “2” and consequently,
- A dwell time of 1 beat for every throw.
If the hand triggering siteswap is “1” for example, then this dwell time assumption will not be realistic and the throw heights for the same siteswap number will work out differently from the case where hand triggering siteswap is “2”. Similarly, if a juggler maximizes the dwell time to 2 beats, then the ratio of heights for throws corresponding to siteswap numbers 5 and 3 would be {(5-2)/(3-2)}2 = 32 = 9, i.e., a “5” will be thrown nine times as high as a “3”!
But what is the absolute height to which say a “3” or a “5” should be thrown? That depends on how long one beat lasts. There is no universal standard for this duration[8]. Usually, beginners like a slower beat and juggle with high throws thus getting higher flight and dwell times while experts can manage faster beats by making lower throws, reducing both flight and dwell times.
Musings
The hand triggering sequence “2”, indicating two hands throwing asynchronously, is, in some sense, the most efficient and therefore, natural way for humans to juggle. No matter how rapidly one hand can throw, the other hand can make a throw between every two throws of this hand. With two hands, we can also perhaps handle twice the number of objects that one hand can handle. Can’t we go faster and handle even more objects if we use even more “hands” like the feet, neck, elbows, etc.? Usually, no, because these other “hands” are significantly less dexterous than our actual hands. But why the preference to throw asynchronously with the second hand? With synchronous throws, the brain has to focus on throwing/catching two things at the same time which is harder. So the hand triggering siteswap of “2” appears to be the most natural juggling method for humans.
Footnotes
- This is debatable. Juggle wiki for example incorporates a rule, “… each of the two hands throw in turn…” in the definition of vanilla siteswap. However, the maths underlying the siteswap notation (e.g., average test, permutation test, etc.) works irrespective of how many hands there are and in what order they throw.
- The best of my ability could only make the first 6 “throws” of Option 2. But I hope it’s enough to convey the idea!
- Any siteswap sequence for objects will be compatible with any hand triggering siteswap sequence except if we get a beat where the hand siteswap number is “0” while the object siteswap number is non-zero. That would mean that an object has to be thrown but there is no hand available to throw it!
- The default interpretation is exactly what is included as part of the vanilla siteswap definition referred to in Footnote 1 above.
- See this blog on how the “4 2 3” is often unwittingly faked by even the best jugglers.
- “2” being a “hold” is equivalent to a “throw” with a flight time of 0 beats and a dwell time of 2 beats. Also, a “1” can’t be held for a dwell time of 2 beats as the “current throw” of the object is only made after the “next throw hand” for that object has already been empty for one beat.
- Using the equation of motion h = u*t + (1/2)*g*t2. “u” is the initial vertical speed of the object which is zero when the object is right at the top of its flight.
- This article speaks of “galloped patterns” and “polyrhythms” where “one beat” means different things at different points of the same pattern. I believe that for all practical purposes, these can be reduced to standard siteswap by finding a time interval small enough such that all other timings used become multiples of this “unit” time interval and/or representing the hand throwing sequence with the appropriate “hand siteswap”.