Polyrhythmic Siteswaps - A need to extend the siteswap notation system?

(by @don_kuehleon)

1) Intro

When I first started looking into polyrhythmic siteswaps, my goal was to develop ways to find patterns besides the base pattern. When I figured out a way it quickly changed to finding a way to notate these patterns in a way it’s useful if you actually try to juggle them. This document focuses on the notation part.

With the 4-ball 3:2 fountain as an example, the current notation systems are either through:

• sync-notation - (8,c)(0,0)(8,0)(0,c)(8,0)(0,0)
• sync-async-transition-notation - (6,4)4640
• short-sync-notation - (4,6)(4,0)!(0,6)!(4,0)
• or even alternative notation systems like Polyrhythmic_notation

For visualisation all siteswaps are linked to JugglingLab.

All these notations are ok for 3:2, but quickly become incomprehensible for other polyrhythms, especially for practical use (like for example, what the hell is (8x,cx)(0,0)(0,0)(ax,0)(0,ax)(0,0)(cx,0)(0,0)(0,8x)(cx,0)(0,0)(0,0)*??). Perhaps polyrhythmic siteswaps in real life are so rare because their current notation is so hard to read^[a]. This could change with a better alternative.

The goals of this new system:  

• numbers should be a proxy for the throw heights^[b][c]
• numbers should correspond to the heights of sync and async notation
• more readable than current notations
• built upon the standard siteswaps notation-system
• easily convertible from and into the currently most used sync notation
• no empty "phantom" beats

I couldn't figure out anything until I accepted that one hand needs to be able to throw off-beat. This is the new and unusual part. The hand throwing on-beat needs to be the faster one (otherwise it gets weird) and will function as the beat-reference-system. The off-beat hand is the slower one.

For sync or async, only focusing on one hand shows the identical picture: “Throwing” the ball to the nearest beat (of the same hand) is a 2 and looks like a hold. Throwing it to the second nearest beat is a 4, and so on. This will also be true for our faster hand in a polyrhythmic juggling pattern. For the slower hand, this will not be the case and the values of a hold or a “4” will be dependent on the given polyrhythm.

2) First Example

Let’s look at an example and try to transform the sync-notation to a new notation fulfilling all the goals mentioned above.

Example 1: 4 ball 3:2 fountain, notated as (8,c)(0,0)(8,0)(0,c)(8,0)(0,0). Check the table below for better comprehension. The faster hand throws 8’s (and 0’s). The 8 appears because there is one empty “phantom” beat after each throw of this hand. If you juggle this pattern it feels like throwing 4’s. If we set the beat reference system to this hand and get rid of the phantom beats the 8 will transform to a 4. As a general rule this transformation works if we transform every number in the sync-notation, by dividing it by the lower number of the given polyrhythm. This also applies for the beat values. For 3:2, this means dividing everything by 2. This automatically gets rid of all the phantom beats of the faster hand. As a consequence, the throws of the slower hand transform from c to 6. This is actually really close to the height-feeling of the throw (with a small variance due to dwell time, more on that later). If you try to juggle this pattern all you need to know is:

• faster hand throws 4’s
• slower hand throws 6’s
• the specific rhythm - each cycle includes 3 fast hand throws and 2 slow hand throws spaced out evenly in each hand. Circle starts with sync-throw. Here polyrhythm-websites will help to get used to the rhythm (e.g. https://4four.io/ploop)

The most confusing part here is that there are only two balls in the slower hand, not three, even when you repeatedly throw 6’s. But that’s because the slower hand only throws every 3rd beat.

Table 1: Transformation of beat and siteswap numbers for a 3:2 polyrhythm (FH = Fast Hand, SH = Slow Hand). The transformation is done by dividing the sync-values with the lower number of the polyrhythm (2) to obtain the new values.

3) New Notation

Now, what is the best way to notate this? There are so many different ways to do that. As a start, I worked out a version for me. This version can then be examined and revised by other jugglers.

If you look again at the table above, the new values are three 4’s as faster-hand-throws and two 6’s as slower-hand-throws for each cycle.

These are the numbers now used for the new notation. As a base I will use the sync notation (x,x) to emphasize that the left and right hand throws need to happen at the same time. The new thing will be the bracket symbol “{}”, now representing polyrhythms. If sync and multiplex deserve their own bracket, I think polyrhythms deserve one too :)

The new notation for the 3:2 4 ball fountain is:

({4,4,4},{6,6})

It indicates that the left hand throws three 4’s in the same time the right hand throws two 6’s. (By using advanced siteswap notation this could be even further simplified to ({4^3},{6^2}). But for the next examples I’ll use the non-simplified versions)

This is the best I could come up with. It may be a bit confusing at first, especially since the order of the numbers are a bit complicated to read at first. But it works really well once you come adjusted to it. Here is an overview:

• The {}-bracket gives you the information that it's a polyrhythm
• The count of numbers inside the two {}-bracket gives you the information about the specific polyrhythm (here 3:2)
• It assumes that you already know the correct rhythm of a given polyrhythm
• The ()-brackets illustrates that the throws of the left and right hand side have to be thrown in the same time frame. But not all throws simultaneously like in a multiplex, which would be notated with a []-bracket
• It highlights one cycle of the polyrhythm, now embedded in one sync bracket (x,x). If a polyrhythmic pattern runs for more than one cycle, you also have to write more of these brackets. For example, a pattern that repeats after two cycles would be notated as: ({x,x,x},{x,x})({x,x,x},{x,x})^[d][e]
• Numbers are now a proxy for the throw heights. For slight variation due to different dwell times of the faster and slower hand see chapter 6
• Numbers now correspond to the heights of sync and async
• More readable than alternatives, once you understand the notation
• No empty phantom beats  
• Built upon the standard siteswaps notation-system and easy convertible from and into current sync notation (by dividing the sync-notation by the lower number of the polyrhythm)

4) More Examples

Let’s go through some more examples.

Example 2: 97531 in 3:2 - notated in sync as (ix,kx)(0,0)(c,0)(0,6x)(4x,0)(0,0). (Shout out to @jamesmcozens for this pattern.) Transforming everything by dividing by 2 and putting it in the new notation system results in:

({9x,6,2x},{ax,3x})

or

({9x,6,2x},{10x,3x})

This is way more readable and shows it’s doable for experienced jugglers. The left hand throws a crossing 9, a non-crossing 6 and a crossing 2, while the right hand throws a crossing 10 (or a) and a crossing 3.

Here I decided to use x to notate crossing throws for even and uneven numbers. That also means that an uneven number without an x does not cross. That’s confusing at first, but not less confusing as the alternative decision of using x for uneven numbers to mean non-crossing. This would especially get messy when a fraction starts to pop up like in the next example. Also the switch to the alphabet for numbers > 9 will get weird for fractions and is not really necessary anymore since all numbers are separated by a comma.

Example 3: 4 ball 4:3 fountain - (c,g)(0,0)(0,0)(c,0)(0,g)(0,0)(c,0)(0,0)(0,g)(c,0)(0,0)(0,0). Transforming everything by dividing by 3 and putting it in the new notation system results in:

({4,4,4,4},{5.3,5.3,5.3})

or

({4^4},{5.3^3})

This means the left hand throws four 4’s while the right hand throws three non-crossing 5.3’s (rounded from 5 ⅓). Check out the table below for better comprehension.

Table 2: Transformation of beat and siteswap numbers for a 4:3 polyrhythm (FH = Fast Hand, SH = Slow Hand). The transformation is done by dividing the sync-values with the lower number of the polyrhythm (3) to obtain the new values.

Example 4: 4:3 pattern - (gx,6x)(0,0)(0,0)(c,0)(0,gx)(0,0)(cx,0)(0,0)(0,g)(i,0)(0,0)(0,0). Transforming everything by dividing by 3 and putting it in the new notation system results in:

({5.3x,4,4x,6},{2x,5.3x,5.3})

The sync-notation is basically unreadable. The new notation is complex, but doable. While the left hand throws a crossing 5.3, a non-crossing 4, a crossing 4 and a non-crossing 6, the right hand throws a crossing 2, a crossing 5.3 and a non-crossing 5.3.

To summarize: For me the central parts are the values and the count of the numbers. Meaning a 3:2 4 ball fountain should be notated with 5 different numbers (for one cycle) with the values being 4’s and 6’s. How to exactly create a notation out of that is up for debate. My system is just a suggestion.  

Here are some questions that could be up for debate in this new notation system:^[f]

• x as a symbol for crossing throws for even and uneven numbers? Benefit: This keeps x’s from sync-notation in the same spots
• How to round fractions?
• How to notate fractions for throws higher than 9? ‘a.3’ as a 10.^[g]3 looks a bit awkward. Since every number is separated by a comma, the need to switch to the alphabet for numbers higher than 9 would disappear.^[h][i]
• Is separating numbers inside the {}-bracket with ‘,’ the best option?^[j]
• Is the general form of ({L1,L2,L3},{R1,R2}) ^[k][l][m][n]the best option?

5) Validity

We are used to the sum of all numbers, divided by the number of beats/throws being equal to the objects juggled. Is this true here? We have to be careful with this new notation system. The faster hand is set as the beat-reference-system. It does not matter how many throws the slower hand does. As an example: The 3:2 4 ball fountain has only 5 throws per cycle, but as the faster hand as the beat-reference-system there are 6 beats per cycle. The sum of all values of ({4,4,4},{6,6}) is 24. 24 Divided by 6 is 4. This checks out.

6) Dwell times - And the need to adjust the relative throw heights

This chapter is extremely useful if the goal is to juggle the patterns in real life. Also it is important for improving animations.

The numbers of a siteswap do not describe the heights, but the beats until the ball is thrown again. Yet, they are a really good proxy for it and jugglers got used to the meaning of the numbers as heights. For sync and async patterns you can relate the different heights without any problems to the numbers, since both hands have the same dwell times. For polyrhythms it's a bit less accurate. Since you have two different bps for each hand you can get different dwell times. The slower hand can throw a 6 or 5.3 a bit lower than expected relative to a 4 of the faster hand, because it can hold onto the ball a bit longer. And this is usually what your body automatically does. So as a general rule: Throws that land on the slower hand can be thrown a bit lower. How much depends on how your body adjusts. For perfect equal dwell times you can throw the exact numbers of the new notation system as heights. On the other hand, if you keep the time equal your hands are empty, you have to adjust the heights of the throws landing into the slower hand. Check out the graphic below for visualisation:

Graphic 1: Difference between non-adjusted and adjusted versions. Green brackets represent the same dwell times on the left and same empty hand times on the right. Red lines represent the sound of the sync catches, yellow lines the sound of the async catches. Also included is the typical phrase “cold cup of tea” commonly used as a memory help for a 3:2 polyrhythm. Check out the corresponding animation via JugglingLab.

The right version is how my body automatically adjusts trying to juggle the pattern. The longer empty-hand-times on the left feel awkward to me. When focusing on the sound, which is very important trying to juggle these patterns, the two versions are different. The sound happens mostly on the catch of the balls. Doing the adjusted version, the sync catch sound, which emphasizes a new cycle of the polyrhythm, happens right after the sync throws. This works great as an auditory orientation. On the non-adjusted version the sync catch sound happens after the last throw in the cycle and before the sync throws. Since regular dwell-times are around 1.3, the catches are closer to the throws before, then the throws after, and therefore feel more connected to the former. This auditory difference, plus the awkward empty-hand-times of the non-adjusted version, makes the adjusted version more useful for me. But for someone else it could be different.

For an A:B polyrhythm and in the sync-notation, the adjustment can be done via subtracting the throws landing in the slower hand by:

2*(A-B) 

For the new notation via subtracting by:

2*(A-B)/B

This sounds complicated but hopefully becomes more clear with the examples below.

The effect of the adjustment is very strong if the fraction of A and B is high, like 5:2, and weak for fractions values close to 1, like 6:5. For me, the felt heights are almost identical to the adjusted values. But since siteswap is a beat-based system, with focus on the beat-length, not the throw-heights, there is no elegant way I can think of to get rid of this problem, without changing something fundamental or over-complicating the notation. My suggestion is that the new adjusted notation, even if it breaks the beat-based rule, can work as a great addition next to the “official” non-adjusted version.

A good example to visualize how the adjustment works is by focusing on a ‘hold’ of the slower hand in a 3:2 pattern. The slower hand throws every 3rd beat. This means a hold would be notated as a non-crossing 3. But it will look like a normal hold, which is normally notated as a 2. Using the formula above, the 3 will change into a 2, exactly how the ‘throw’ would feel like. To benefit the juggler, an adjusted notation besides the other notation would help a lot to be able to try juggling the pattern. For me, realizing this height adjustment was a game changer.

As another example, let's take the relatively easy to juggle 3:2 3-ball pattern:

(8,4x)(0,0)(8x,0)(0,6x)(ax,0)(0,0)

The new notation would be:

({4,4x,5x},{2x,3x})

Using 2*(A-B)/B on the throws landing in the slower hand, the adjusted notation would be:

({4,3x,4x},{2x,3x})°

The subtraction-values also represent the value of the holds of the slower hand. In this example, and for 3:2 polyrhythms in general, the slower hand has short 1-beat holds before each throw.

For being able to distinguish between the adjusted and non-adjusted notations, the adjusted notation should have a symbol at the end. As a placeholder I choose ‘°’. When trying to juggle this pattern, people will probably be able to learn it quicker with the adjusted notation.

Using 2*(A-B), that adjusted version in sync looks like:

(8,4x)(0,0)(6x,2)(0,6x)(8x,0)(0,2)

Notice how the throws landing on the right hand land there one sync beat earlier (compared to the first sync notation) and the balls then get hold for one sync beat. That's what subtracting with 2*(A-B) in the sync notation means. You can subtract the heights landing in the slower hand with 2, because you add a hold before. The (A-B) part makes sure you can apply this pattern to all polyrhythms. For 5:3 you can add two holds in the slower hand and subtract the relevant throws by 2*(5-3) = 4. This adjustment ensures that both hands have the same amount of empty beats (0’s) after each throw and therefore the same empty-hand-times. For the new notation a division by B gets added to the formula 2*(A-B) like for everything else in the transformation:

2*(A-B)/B.

With that, you can also calculate the dwell-time of the slower hand. A regular dwell-time is 1.3 beats. This would be true for the faster hand. For the dwell time of the slower hand you need to add the value derived by the formula above::

1.3 + 2*(A-B)/B

As an example this means the dwell-time of the slower hand in a 3:2 polyrhythm is 2.3.

Validity:

The adjusted notation does not have the validity mentioned in chapter 5 anymore, because the values of the short holds disappear from the notation. It really should just be used as an additional information for the juggler. To be able to check the validity of the adjusted version you have to re-add the subtracted values. As an example:

({4,3x,4x},{2x,3x})°

In the new beat-reference system a 3:2 cycle is 6 beats long. The sum of all values in the pattern above is 16. Since there are 2 slow hand throws, the 2 subtracted values (2*(A-B)/B * 2 = 2) have to be re-added (16 + 2). 18 divided by the number of beats (18 / 6) equals 3. This checks out.

Exception:

One exception to the adjustment rule is when a siteswap-number would be transformed to a 0, or below 0. Using the new notation, for 3:2 there can be a 1x landing into the slower hand. Using the adjustment formula would transform this into a 0. This is clearly wrong. So the minimum value numbers can be transformed into has to be above 0. If this happens, and a number can’t be transformed like the formula would suggest, the original number needs to stay the same, or be transformed into a number above 0 (not sure what’s the best option right now). In these patterns the characteristic rhythm of the catches will be off. That’s not a unique problem. Throwing a 1 in a regular async pattern will also break an even async rhythm, assuming common dwell-times. Here people subconsciously will break their even throw rhythm to achieve an even catch rhythm (most of the time).

Conclusion:

For practical use, focusing on the number as throw heights, the adjusted and the non-adjusted notations are important, since both versions are perfectly valid ways to juggle polyrhythms. The catch sounds of both will produce the characteristic rhythm. It’s up to the juggler if equal dwell-times or equal empty-hand-times are preferred. The rhythm will be off, if none of the two are the same. But my guess is that most jugglers will prefer the adjusted version.

7) List of patterns

I started to gather all the interesting patterns I came across in the list below. This includes some symmetrical ones, where the faster and slower hand switches after each cycle. Because of this switch you have to be very careful on which throws to adjust.

Polyrhythmic Siteswap List

8) TODO: Generate polyrhythmic siteswaps

What’s missing in this document is how to derive polyrhythmic patterns^[o]. There are plenty of possibilities to do that. As a short overview:

• You can do it by hand: Drawing a diagram, connecting the throws and using the beat template (like in the tables above) to calculate the values (shout out to @gelebki who built an app for himself to be able to do this automatically, like a few days after I showed him this method at the EJC). A diagram based approach is also used at this site: https://www.jacos.nl/jongleren/polyrhythm/
• You could wait until @jamesmcozens releases his tool which includes transforming async siteswaps to its polyrhythmic twin (if I understood correctly). Update: Early version is out: https://siteswapvisualiser.jamescozens.co.uk/?cv=1
• I derived a lot of my patterns by selecting sync siteswaps with one (or more) empty beat(s) and transformed it to its polyrhythmic twin. For example: (6x,2x)(4,6x)(4x,0)(6,4) will be transformed to ({5.3,4,4x,6},{2x,5.3x,5.3}). For each polyrhythm I first had to create a transformation-template. This was a bit tedious. But once the template was done it worked great. I could go more into detail on how this works, but I’m pretty sure this method will be obsolete if better tools are available, so I didn’t include it here. But if anyone is interested just send me a message :)