Introduction

This is part 1 of “Siteswap and rhythm”, some solutions to these problems described in this post can be found in part 2: “Siteswap and rhythm: Solutions”.

For many years, I had assumed that siteswaps patterns were typically thrown in an even left-right rhythm. I practiced juggling to get to what I thought was metronomic precision, without ever fully considering what that meant. I have now come to conclude that many of my previous understandings of siteswaps and rhythm were wrong, and I feel very disillusioned.

This article will go over how siteswaps and rhythms can be perceived, and how this aligns with optimal practice. It will also go into some other challenges that you face with siteswaps and rhythm, such as polyrhythms and varying tempos. I hope that reading this can save you the pain I felt when I realized I juggled 441 different from how I thought I did for so many years.

We will use 441 as an example, but the lessons we learn will apply to any pattern with 1s (also known as zips) in it. Also, there are similar pattern distortions with 0s and 2s which are not discussed here, and even with patterns without any of these siteswap throws there is a lot of variation in their optimal rhythms.

Throws, catches, and zips

What does it mean when we think of siteswaps as an even rhythm? Does that mean that the throws are equally spaced in time from each other, or the catches, or both? We will have a look at all three of these options.

I will be simulating the juggling in Juggling Lab which also generates a ladder diagram, and I manually add an extra timeline for the throwing (T, green) and catching (C, blue). In the ladder diagram we see two repetitions, the pattern starts one time on the right side and one time on the left side. We consider the pattern to have 3 evenly spaced beats for one repetition.

Without any pattern modification, this is how Juggling Lab normally juggles 441:

441
Unmodified 441, only throws spaced evenly

You can see in the green column that all the throws are spaced evenly in time, and in the blue column it is shown that the catches are not spaced evenly.

Both spaced evenly

We can try to modify the pattern so that both throws and catches are spaced evenly. In the pattern above there are no catches made between the second 4 and the 1, and there are 2 catches made between the 1 and the first 4. We can not move the first of those two catches up, as it is thrown on the 1 and would then have to be caught before it was thrown, which is impossible. We can not move the second of those two catches up, as there already is a ball in the same hand at that time and holding multiple balls in one hand would change the pattern. So instead we have to move the catch between the two 4s down, and then also move the second catch between the 1 and 4 down to replace the one we just moved. The result is this:

both equal
Both throws and catches spaced evenly

With the throws and catches spaced evenly the dwell time is always the same between every catch and throw. However, in this pattern the dwell time is very short. It has been observed that the ideal dwell time is about 1.33 beats in normal cascade and fountain patterns with any number of balls1. In this particular pattern the dwell time is less than 1 beat, leaving the hands empty for more than half of the time. This makes it very unnatural and difficult to juggle, and also forces the balls to be thrown higher if the same tempo is kept.

To me, this rules out using both evenly spaced throws and evenly spaced catches as a good strategy for interpreting a siteswap pattern.

Throws spaced evenly

Let us return to the original simulation with the throws evenly spaced and see if this is a good option.

441
Unmodified 441, only throws spaced evenly

This is how I (wrongly) assumed I juggled 441 myself. But it turns out, also this is a difficult and unnatural way to execute the pattern. As a result of the even spacing a hand is empty after the second 4 for more than a whole beat. Compensating for this lost time is the short dwell time before the first 4 which is shorter than a whole beat. As mentioned before the ideal dwell time is about 1.33 beats, which makes the empty hand time 0.67 beats.

We will improve upon this in the following variations. If we don’t have both evenly spaced catches and throws the dwell can never be the same between all the catches and throws. But we will be able to get rid of the unnecessary empty hand time. And that is good, the longer the hand is empty, the higher the balls need to be thrown, making the juggling more difficult.

Catches spaced evenly

This version is how I typically juggle 441 if I try to stay in an even rhythm. The catches make noise, and the extra audible feedback makes it much easier to notice what is going on and what the tempo is of the pattern.

Compared to the “throws spaced evenly”, this one only has the 1 moved upwards, it is being thrown earlier.

caught equal
Only catches spaced evenly

There is still a hand empty for longer than a whole beat, which is now after the 1. Also there is a short dwell time before the 1. However, this is a lot better than the “throws spaced evenly” for two other reasons:

The short dwell before the 1 isn’t bad at all, it does not make the juggling more difficult. A 1 throw takes very little energy, in fact it can almost be done as a bouncing action from the ‘throwing’ hand into the catching hand.

The long empty hand after the 1 isn’t bad at all either, as the 4 that is being thrown after the empty space still gets a normal amount of dwell time. Unlike in the pattern above, where the long rest was followed with a hurried throw.

I think this is the optimal way of juggling 441 rhythmically, and it is how I believe most jugglers perform 441 when asked to perform it with an even rhythm.

It turns out that as a rule, the 1 aka the “zip” always happens much earlier than how I imagined it originally. It feels as if the zip lands on beat 2, when I imagined it should only be thrown at beat 3.

Where does this myth of the evenly spaced throws even come from?

Optimal 441

Before we get into the myth, let’s try and make the ‘optimal’ 441 pattern.

The empty hand time can be filled if we catch the first 4 earlier. This means we also need to throw it less high, making the pattern easier. I think this is close to the most optimal juggling pattern, with optimal meaning that it is easiest to juggle.

optimal
Optimal 441 juggling, not to a rhythm

You may notice now that if you try to count 3 beats for this pattern it looks like the first beat has 1 throw, the second has two throws at roughly the same time and the third beat has no throws. Or similarly you could also see one catch, two catches and no catches.

This leads me to wonder, if this is the optimal 441, and we tend to think of siteswaps as even rhythm throws, should we even notate this trick as 441 or since the hands operate nearly synchronously should it perhaps be some synch siteswap? This will be discussed in the section on 0x in part 2. But first let’s finally get to the sources of the even rhythm myth.

Sources of the even rhythm myth

I’m certainly not alone in my previous and misleading assumption that siteswaps should be thrown evenly spaced. It is how I have been explained juggling by various teachers in professional circus schools, it is how all siteswap simulators that I know of simulate juggling, and it is how many sources explain siteswaps.

In the book The Mathematics of Juggling by Burkard Polster, page 8, the very first rule of how juggling will be simplified for mathematical observation is:

“The balls are juggled to a constant beat; that is, the throws occur at discrete equally spaced moments in time." 2

In the Gandini Siteswaps DVD, Sean Gandini opens with:

“There are a couple of assumptions we need to make for the siteswap system to work. The first assumption is that the hands are always alternating, and they are throwing at an even tempo at equally spaced intervals." 3

And in the Numberphile siteswap video with Colin Wright, he says, referring to the throws:

”…and I’ve got this rhythm going about the whole thing, this constant metronomic beat going on here.". This video at 2:00.

There are many many more sources that use similar explanations, but you get the point.

I consider these three above to be very important sources, yet they all present a misleading simplification of siteswaps as one of its core rules. Misleading as it does not represent how people juggle at all.

Other rhythm issues

Other than the problems above regarding the 441, there are also some better known issues regarding how siteswap can’t describe rhythm.

Different patterns, different speed

Because the pattern 441 is not very collision prone, and because 1s (zips) can be thrown very shortly after a catch, 441 can perhaps be juggled even faster than a 3 ball cascade. It is not uncommon to juggle them both at a different tempo. These kind of tempo changes can not be expressed in siteswaps. This is not a big issue when describing a single trick, but it would be nice to have such tempo writing tools if we want to write juggling choreographies.

Similar to this is the speed of a base pattern, typically 7 clubs are juggled faster than 3 clubs, and 7 balls faster than 7 clubs.

Because these differences can be subtle, it is difficult to put various juggling pattern tempos on a single grid, no matter how far we subdivide it.

High throws, low throws, and effort.

Even when not dealing with the weird siteswap throws such as 1s, but also 2s and 0s it can still be less than optimal to throw at an even rhythm.

A higher throw requires more power or effort, and could thus use more time to be thrown well. Coming out of a stable 5 ball cascade, I believe most jugglers start their 744 at quite an even rhythm. However, those who can juggle it for multiple runs in a row must have noticed that it is more comfortable to shift the rhythm a bit.

On the left you see 744 with evenly spaced throws and catches, as is the default in Juggling Lab. On the right we see it modified so that the 7 spends more time in the hand.

744
Even spaced throws & catches
744
Uneven rhythm

I think if we become more aware of these uneven optimal rhythms we can become better jugglers. For example one could already do some setup throws in a 5 ball cascade in preparation for the 744 rhythm. But we don’t have a notation system to describe these rhythms, we don’t even have the language or a good mental model required to point out the subtle differences between the two patterns above.

Polyrhythms

It is not impossible to describe polyrhythms in siteswaps, but it is far from ideal. The polyrhythm pattern below is 2 over 3, the right hand juggles throws 3 times while the left throws 2, in the same amount of time.

This image was generated with Polyrhitmic Fountain.

2 over 3
2 over 3 polyrhythm

This could be notated with siteswap if we space our events a bit more, like (8,c)(0,0)(8,0)(0,c)(8,0)(0,0) or in this case even as (4,6)(4,0)!(0,6)!(4,0) with short synch notation (!), but it does tend to become impractical as the numbers do often not relate to the throw heights we’re used to, and you need to write a lot of empty beats. For example with 4 over 5 you’d start with throwing (8,a)(0,0)(8,0)!(0,a)!... (this is only a quarter of the whole pattern) even with the obscure short synch notation.

Odd timings

Sometimes we don’t continuously throw. We might skip a beat, often described with a 2 or a 0. Jugglers have found ways to describe different rhythm variations with siteswaps, for example 522 is a slower variation of 3. However, they can both be considered equal. If siteswap did not have any rhythmic information, there would be no observable difference between a performance of 522 or 3 (assuming the 2s are held).

The fact that we do use 522 or similar tricks suggests there is a desire to describe rhythms and tempos with siteswap. Similarly, there is a desire to describe heights. Although technically a 5 doesn’t give any information about the height of a throw, we assume that 522 is a “high cascade”.

It’s not a bad thing that siteswap can be fit to the purpose of defining heights or rhythms to some degree, but once one tries to play with these one quickly notices some severe limitations. For example it can be easy to make a crossing throw that takes 3 beats (522) instead of 1 (3), but a 2 beats high cascade becomes more complex to transition into as you suddenly need to switch to synch notation ((4x,2)*). Working out the transition into and out of that is quite complex.

It gets even worse when you want to combine 2 different heights such as a 1 beat throw and a 2 beat throw. This “galloping cascade” could be written as (1x,3)!4x1x, which seems like way too many symbols for such a basic variation on the cascade.

Conclusion

I believe that it is widely known that siteswap is not ideal for polyrhythm and that it can not describe a varying tempo in a choreography. However I believe jugglers are much less aware that an even rhythm is not ideal either. This even rhythm myth is proponed by many important learning resources and has trickled into our mental model of juggling, but we might not even be aware of what it means. This discrepancy becomes obvious when looking at simulators, they all put throws at an even rhythm when I believe that jugglers tend to put catches at an even rhythm. And neither might be optimal when one aims to make the juggling as easy as possible.

In my experience, juggling to music or rhythms leads to confusion and discussions about the meaning and timing of siteswaps. By understanding what actually happens within juggling I hope our mental models can be improved so that we can communicate with each other about what we mean, and juggle together at an agreed upon rhythm.

All the things I wrote in Throws, catches and zips sound to me now like they should be obvious, yet they were not, at least not to me. My mental model of how juggling worked needed to be reconfigured in order to accept that I don’t throw the balls at an evenly spaced rhythm when performing a 441 pattern

In Siteswap and Rhythm: Solutions (part 2) we’ll look at some solutions to the problems at hand: New and existing notation systems that attempt to solve the issues that are being described here. Will they solve the mysteries of juggling rhythms? Read on to find out!

References


  1. Kalvan, J. (1996). Optimal Juggling, The Analysis and Over-analysis of Juggling Patterns. Juggling Information Service. http://www.juggling.org/papers/OJ/ ↩︎

  2. Polster, B. (2003). The Mathematics of Juggling. Springer Publishing. ↩︎

  3. Media Circus (2006). Siteswaps. ↩︎