This is the evaluation of the comparison of all the tricks in Daniel’s 3 ball trick list (D3TBL).

I made some observations about the 8 different systems used, I scored their support of various trick aspects, and I compared their elegance or complexity.

The systems we discuss are siteswaps, Body Trick Notation (BTN), Mills Mess State Transition Diagrams (MMSTD), Beatmap, Harmonic Throws v2 (HT) (the link goes to a description of Harmonic Throws v1, see the note about this here), Symbolics for Objects and Juggler Planes (SOJUP), Rhythmic Catches (RC) and Improved Body Trick Notation (IMBO). These systems were chosen for their abilities to support a wide range of patterns.

General observations about the systems

These are some observations I have made about the notation systems whilst going over the transcriptions of D3TBL.


Siteswaps greatest strength, simplicity, is also its greatest weakness. Over half of the tricks on the list can be written simply as 3.

Body Trick Notation

BTN can get very complex in the simplest of throws. It is quite readable when doing a throw under the legs, which is ALBOL, although it can be arbitrary if that is an under the leg throw or an under the leg catch. However, a top crossed hand to the other top crossed hand throw, which is possibly even easier to make, would probably be written as OPACACOPACAC which is not very legible.

Mills Mess State Transition Diagrams

MMSTD was doomed to fail such an extensive test, it was really only designed to deal with a small subcategory of juggling tricks. However, because it offers inside and outside throws, I am surprised how to how many tricks it was able to contribute some clarification. For example, the trick “tennis” can be described just fine, even though it does not involve any crossing hands.


Beatmap can do a lot, but it also requires a lot of space. Tables get wide very quickly, making the system sometimes hard to read or write. Also there is a lot of redundant information, on every beat there is written what both the left and the right hand do, even if only one hand is doing any action.

Harmonic Throws v2

Harmonic throws v2 (portée system) is much better and more elegant than Harmonic Throws v1 (grid system), sadly it is not yet published so you can’t yet read about it. A major difference is that in many cases the Benesh layer is no longer needed even when doing body tricks. This makes the system much more compact.

It has support for almost any feature of juggling, except for prop rotation but this is not covered in this analysis.

However, the Harmonic Throws system expects a juggler to both throw and catch on a beat, not a little before or after. This means that none of the transcriptions are actually accurate to the reference video. Ideally, I would like to have another video that performs all the tricks as how Harmonic Throws describe it, but I am not able to make this myself as this style of juggling is very difficult to execute for me.

Symbolics for Objects and Juggler Planes

This system sadly does not get to shine when transcribing D3BTL. It is designed with different orientations and rotations in mind, something that is not often relevant when juggling with balls.

With these particular tricks it was often struggling, and I was not always sure what the best way to write a trick would be. Despite its documentation being very long and detailed, it may need to be even longer and even more detailed. I was surprised to find that inside and outside are not covered. Also, despite its graphical and spatial nature, because it uses siteswap it is far from ideal to describe contact with the body in places other than the hands.

Rhythmic Catches

Rhythmic Catches was a joy to use, but I may be biased as its inventor.

When the pattern was simple, the RC was often as short and straightforward as the siteswap. However, when the pattern was more complex, the system scaled to just what I needed to make accurate descriptions.

Improved Body Trick Notation

I consider IMBO to be indeed an improvement on Body Trick Notation. Not only is it more specific than BTN, which I consider an absolute requirement to make it useful, it also is a bit more readable in my opinion.

IMBO works very similar to the TAC symbol layer in Harmonic Throws, there are only 2 tricks in D3BTL where its features are used differently. However, I noticed that sometimes the TAC was written differently by Jonathan Lardillier than how I had written the IMBO, perhaps there is a bit of ambiguity in how to read and write these patterns? For example in the Harmonic Throws factory pattern there are two body throws written, in IMBO I chose to write none.

Comparing support for various trick aspects

I have attempted to rank the quality of notation systems by scoring how well they can support certain aspects of the tricks in D3TBL. For most tricks I cover only one unique aspect, for a few I cover some more. I tried to pick aspects that are “interesting”, in the sense that not all systems would deal with them equally.

Note that this is a relatively arbitrary process, with different aspects the results would have been different, and even another person scoring the same aspects could come to different conclusions. On top of that, I am biased. I created the list of tricks, I created notation systems to solve specific problems, and I did the scoring and aspect choosing myself. A better way to do this would be to have a handful of independent experts score these systems in a similar fashion, and see where they have a consensual agreement.

However, I believe that this is the best I could do by myself, and that this evaluation is still valuable. It gives a quick overview into some of the features and failures of the systems.

Because siteswap by itself is not sufficient for most tricks, and neither are Body Trick Notation and MMSTD I have grouped these together, a pair “SS + BTN” and “SS + MMSTD”. Similarly I consider Rhythmic Catches and IMBO together to form one system that can cover both the patterns and the pathways, listed as “RC + IMBO”.

In this table below I ranked these aspects 0-3. This has the following meanings:

0 = does not support
1 = supports partially
2 = supports fully
3 = supports more specific than other systems, in a meaningful way

As much as possible I have tried to motivate my choices of 1’s and 3’s in the notes, 0’s and 2’s are often more obvious.

Link to the original spreadsheet

# Trick name Aspect SS + BTN SS + MMSTD Beat- map HT (v2) SOJUP RC + IMBO Notes
1 Cascade Can represent cascade as shown in the video 2 2 2 1 2 2 HT uses a very specific timing grid, which differs from how the cascade was juggled in the video.
2 Tennis Inside & outside 0 2 1 2 0 2 Beatmap can possibly do this but I have not found any examples of this. I consider this system not practical.
3 Mill’s mess Crossed hands 2 2 2 2 2 2 It’s worth mentioning that MMSTD, Beatmap and SOJUP do this much more elegantly than BTN, HT and IMBO
Mill’s mess Inside & outside with crossed hands 0 2 0 1 0 2 HT inside outside can be confusing when throws take place around the body center line, a crossing hand does not necessarily also cross the body center.
4 Cherry picker Claw catches & throws 0 0 1 2 0 0 Beatmap could have mentioned this, but the current transcription does not
Cherry picker Ball trajectories 1 2 3 2 3 2 BTN is very vague about the trajectory, I’m not even sure if OPACAC really is a pathway. Beatmap and SOJUP are specific about the columns, the other notations allow you to deduce them.
5 Shower Galloping, as in the video 0 0 0 0 0 2
6 Box Synchroneous throws and catches 2 2 2 2 2 2 RC is more specific about the synchronous catches than the other systems, but possibly less specific about the synchronous throws. Harmonic throws is specific about both throws and catches, but uses a very different rhythm than in the video.
7 Shuffle Slam 0 2 0 3 0 2 HT also specifies the direction of the hands, clawing. In this particular notation HT makes correct use of the outside throw notation, I am not sure why this is not also done in the transcription of the shower pattern.
8 Factory Carry path 0 0 2 3 2 0 HT allows you to draw out the path on the Benesh layer
9 Fake mess Carry path 2 1 2 2 2 3 in MMSTD I added this extra (hold) bit that’s not part of the original system. IMBO is explicit about the hold path, unlike the other systems.
10 Siteswap 441 second 4 & 1 thrown at the same time, as in video 0 0 0 0 0 2 This could arguably be written in SS, Beatmap and HT with a lot of extra effort, but the systems were not designed for this unlike RC
11 Siteswap 12345 take 4 beats for pattern, as in video 0 0 0 1 0 2 In HT this could easily be done, but the transcriber chose not to.
12 Siteswap 423 active 2’s Active 2’s 0 0 0 2 0 2
13 Three in one hand Repeat a hand, without having to write both hands 1 0 2 0 0 2 In the alternative SS this makes a lot of sense, but in vanilla the original LR notation is more common
14 Overheads Overhead position 1 0 0 2 1 2 BTN describes pathway but could also refer to blind behind the back. SOJUP is unclear about the orientation of the hands
15 Penguin catches Penguin catch 1 0 2 2 2 2 BTN describes pathway but could also refer to shoulder catch/lazy.
16 Under the leg throws Under leg throw 1 0 2 2 1 2 BTN & SOJUP describe pathway but could also refer to under leg catch
17 Behind the neck throws Behind neck throw, arms extended as in video 1 0 0 2 1 2 BTN & SOJUP describe the pathway, but could also refer to folded arm throws or catches
18 3 up Pirouette Pirouette 1 1 1 2 0 0 SS does not specify the pirouette, but using 00 for a pirouette is common practice. Beatmap signifies facing front and back, but this could also refer to two opposite direction 180s. HT seems to have a symbol for a pirouette, but until its documentation is published I can’t say for sure if it is a defined symbol or made up for this particular occasion.
19 Knee kick Knee kick 0 0 1 2 1 2 Beatmap does not specifically mention a kick, it could be seen as a very quick catch and throw. I’ve seen it used in a similar construction where a catch was made even if there was no times between beats for something to be “in the hand”. SOJUP is not very clear about how to describe this, but the ball pathway is interacting with the knee position.
20 Head roll Head roll 0 0 2 2 1 2 SOJUP is vague about how to clarify this interaction
21 Blind Behind the Back (rolling contact) Ball trajectory 2 0 0 2 2 2
Blind Behind the Back (rolling contact) Body contact 0 0 0 1 0 2 HT uses an extra written description to note the body contact
22 Monkey juggling Armpit trap 0 0 2 1 1 2 HT uses an extra diagram to explain armpit trap. SOJUP is vague about how to describe this trap.
SCORE SUM 17 16 27 41 23 45

From this table we get the following scores if we add up all the numbers, sorted from highest to lowest:

Rhythmic Catches + IMBO: 45
Harmonic Throws: 41
Beatmap: 27
Symbolics for Objects and Juggler Planes: 23
Siteswap + Body Trick Notation: 17
Siteswap + Mills Mess State Transition Diagrams: 16

Although these numbers can not represent any exact differences between systems, they clearly suggest that Harmonic Throws and Rhythmic Catches + IMBO can support much more features of juggling tricks than the other systems can.

Based on my experiences with the systems I intuitively agree with this conclusion.

Comparing elegance, complexity, simplicity

Not only are the features of each system different, also the way they express those features is very different. One of my ideals for a system is one that has very little redundant information.

This was hard to compare systematically, so instead of a scoring I will just write about my observations.

Left and right

Multiple systems, namely Beatmap, MMSTD and Harmonic Throws use the terms left and right. This has the benefit that you can specify left or right when this is important in a choreography. However, in most tricks, the mirrored version is considered equal. There is no way in these systems to share the trick without sharing information about its orientation, like whether it starts on the right or on the left.

This also has the downside that tricks that repeat on both sides become twice as long to write. A cascade consists of 1 repeating (mirrored) throw in siteswaps or Rhythmic Catches, but consists of 2 throws (a left and a right) in Harmonic Throws and MMSTD.

In Beatmap you get the best of both worlds, there is a way to write that the pattern continues mirrored using an asterisk *. However, at the recommendation of Luke Burrage, the author of beatmap, the patterns were all written out fully in both directions so they would be easier to read.

Setup information

Harmonic Throws, Beatmap and SOJUP had some setup information on the patterns, that were not relating to the throws themselves but are still required to view the patterns through the lens of the system. When you quickly want to write down a trick, this extra writing can be annoying.

Harmonic Throws has the time signature and the starting position of the balls.

SOJUP has the orientation of the juggler to the viewer, and the orientation of each diagram.

Beatmap has the list of keys, and sometimes their definitions.

A system that does not need this kind of setup may be more suitable to write down single tricks. With longer choreographies the extra effort and space for the setup becomes a smaller proportion of the whole transcription.

Default information

MMSTD and Beatmap included a lot of information that could be considered ‘default’, and could perhaps be omitted for more clarity. In MMSTD every uncrossed inside throw was specifically mentioned, whereas in the other systems this is simply assumed. In beatmap, every hand is notated on every beat, even if it is doing nothing. This added a lot of 1s to the notation. Also in beatmap a site that only performs one action in a sequence still takes up a whole column of space, as can for example be seen in the head roll.

Graphical vs textual

SOJUP and Harmonic Throws are graphical notation systems. This has the benefit of being able to compress more information in a single symbol so it could theoretically be smaller. However, as the characters are foreign or are more detailed than a text character, and because both in SOJUP and in Harmonic Throws the symbols are placed on a grid, they arguably take more space than their text based counterparts.

SOJUP is extra difficult to write as it requires the use of multiple colors, black, and grey and red.

It would require another study to find out if graphical systems can be read faster than the text based ones, but surely it would at the very least take a lot of practice to get up to speed. I find it easier to work with the text based systems.

Another benefit of text based systems is that they can be typed on a keyboard and can be sent through any messaging service. With a graphical system a pencil or graphics editor is required, making it much more of a hassle to write, edit and share notations.

Elegance/complexity/simplicity conclusion

MMSTD, Beatmap, Harmonic throws and SOJUP all have some downsides considering elegance and complexity, as mentioned above.

I consider siteswaps, BTN, Rhythmic Catches and IMBO to be better as they are no more extensive than they need to be. They have no left or right, they have no setup, they make use of default assumptions to avoid redundancy and they are textual.

To me, this elegance is a really important feature of a notation system, I will be much less likely to use one if I consider it to be a hassle to use.