I think all jugglers will agree that the “4 2 3” is one of the simplest 3-ball juggling patterns. For many, it is the second pattern they learn, after the cascade. Or is it? I now believe that hardly anyone (including me, until recently) juggles the “4 2 3” when they say they’re juggling it. So what do they juggle?
The Background
While experimenting with some passing patterns at ChAoS, we wanted to switch from a cascade pattern into (what we all thought was) the “4 2 3” pattern. To ensure synchronization among all the jugglers, we thought of juggling to the rhythm of our background music. I had my misgivings about this idea. I felt it would confuse us more than help us. Our “music” was simply a metronome running a 4/4 rhythm with an emphasis on the first beat, so it would repeat every 4 beats. The “4 2 3” is a period 3 pattern. A 4-beat period rhythm will synchronize with a 3-beat period sequence every 12 beats, 12 being the Lowest Common Multiple of 4 and 3, i.e, LCM(4,3).
| Time (in beats) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 |
| Throw “Height” | 4 | 2 | 3 | 4 | 2 | 3 | 4 | 2 | 3 | 4 | 2 | 3 | 4 |
| Throwing Hand | R | L | R | L | R | L | R | L | R | L | R | L | R |
As seen in Table 1[1], at Time = 0 beats, we have 3 things happening: (1) the emphasized beat, (2) a throw of height “4” and (3) the throwing hand being the right hand. The earliest that this combination of events repeats is at Time = 12 beats. Let us contrast this with the 3-ball cascade.
| Time (in beats) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 |
| Throw “Height” | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| Throwing Hand | R | L | R | L | R | L | R | L | R | L | R | L | R |
In Table 2, the events at Time = 0 beats are: an emphasized beat, a throw of height “3” and the right hand throwing. This combination repeats as early as Time = 4 beats. This is expected because the cascade is a period 1 pattern and it’ll sync with the period 4 rhythm every LCM(1,4) = 4 beats[2]. Thus, with the cascade, every time the emphasized beat comes, the juggler has to take the same action. Not so for the “4 2 3”. Hence, my misgivings about trying to mix up a 4-beat rhythm with juggling the “4 2 3” .
The Mystery
I had everyone convinced with this reasoning until Felix demonstrated the following: he threw what looked like a “4” out of the right hand on the first beat. It looked like a “4” because it was getting caught again in the right hand. Then he threw what looked like a “3” out of the right hand on the second beat. This looked like a “3” because it was tossed up (unlike a “1”) and was getting caught in the left hand. He then proceeded to make the same two-throw sequence from his left hand – an apparent “4” to itself, followed by an apparent “3” to the right hand. After four beats, when the emphasized beat came again, he was back to the “4” out of the right hand (see Table 3).
| Time (in beats) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 |
| Apparent Throw “Height” | 4 | 3 | 4 | 3 | 4 | 3 | 4 | 3 | 4 | 3 | 4 | 3 | 4 |
| Throwing Hand | R | R | L | L | R | R | L | L | R | R | L | L | R |
There it was, the signature right-right-left-left throwing sequence of the 4 2 3 (the “2” being a “hold”) synchronizing with the 4/4 beat music. Every time the emphasized beat came, the same throw was being made – a right hand “4”. As easy to remember as the cascade.
But wait – the “2” throws are not showing up on Table 3 at all! And yet, Felix must be making them because 4 3 is not a valid siteswap. This is where one might say, “Well, the “2” throws are just “holds” and those are happening right there between, for example, the “4” throw at Time = 0 beats and the “3” throw at Time = 1 beat.” Let’s say this is correct and there is an additional beat between the “4” and the “3” throws where the “2” is “thrown”. But then to remain consistent, there must also be a beat between Time = 1 beat and Time = 2 beats and a throw being made on that beat. What are we throwing on that beat and with which hand? In Table 4, we represent these “hidden” beats with an “&”.
| Time (in beats) | 0 | & | 1 | & | 2 | & | 3 | & | 4 | & | 5 | & | 6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | & | 2 | & | 3 | & | 4 | & | 1 | & | 2 | & | 3 |
| Apparent Throw “Height” | 4 | 2 | 3 | ? | 4 | 2 | 3 | ? | 4 | 2 | 3 | ? | 4 |
| Throwing Hand | R | L | R | ? | R | L | R | ? | R | L | R | ? | R |
If this unexplained throw did indeed exist, then it would mean that our pattern is a period 4 pattern and not the “4 2 3”. If the unexplained throw did not exist, then how could the period 3 pattern “4 2 3” be synchronizing with a period 4 rhythm every 4 beats? Unless LCM(3,4) = 4? Or maybe even 3 = 4?
Lastly, and very importantly, the “4” throws in Felix’s demonstration seemed to be going just as high as the “3” throws and no higher. Was 3 = 4?
The Demystification
No, 3 ≠ 4. The mistake that all of us were making was to assume that the “hand siteswap” was “2”, i.e., the hands were throwing alternately, when in fact, they were not[3].
Asynchronous solution: Hand siteswap “1 3”
Let us describe in more detail the pattern that Felix was juggling in terms of what object was being thrown by which hand and which beat. We will not assume any siteswap numbers for the throws yet, but instead work out these numbers based on the evidence. “The fake 4 2 3” juggled in the below video is my rendition of what Felix had demonstrated and what we all thought was the 4 2 3.
Calling the red ball as object A, yellow as B and pink as C, we build Table 5 depicting the events for 10 beats of the pattern being juggled.
| Current Time (in beats) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 |
| Object thrown | A | B | C | B | A | B | C | B | A | B |
| Throwing Hand | R | R | L | L | R | R | L | L | R | R |
From Table 5, let’s now try to derive the siteswap sequence both for the objects as well as for the hands by subtracting the current time in beats from the time in beats when the object (or the hand) is being thrown (or is throwing) the next time. This leads us to Table 6.
| Current Time (in beats) (TC) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 |
| Object thrown | A | B | C | B | A | B | C | B | A | B |
| Time at next throw of same object (TN) | 4 | 3 | 6 | 5 | 8 | 7 | ? | 9 | ? | ? |
| Object siteswap (TN-TC) | 4-0=4 | 3-1=2 | 6-2=4 | 5-3=2 | 8-4=4 | 7-5=2 | Can’t say | 9-7=2 | Can’t say | Can’t say |
| Throwing Hand | R | R | L | L | R | R | L | L | R | R |
| Time at next throw by same hand (TH) | 1 | 4 | 3 | 6 | 5 | 8 | 7 | ? | 9 | ? |
| Hand siteswap (TH-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 |
It appears that the juggled pattern is the “4 2” using the hand siteswap “1 3”. Now the object siteswap is of period 2 and so is the hand siteswap. So it makes sense that it syncs up with the 4-beat music every LCM(2,4) = 4 beats. As we can see from Table 6, at every emphasized beat, object A (red ball) is being thrown, with the same siteswap number “4” with the same hand (right) and even the right hand siteswap number is the same (1). The only concern seems to be at Time = 8 beats where we were not able to work out the object siteswap number because we don’t have information about when object A will be thrown again. But if we assume that the sequence of events established in the first four beats of the 4/4 rhythm is repeating every four beats, then this siteswap number will also turn out to be “4”.
But that still doesn’t explain why the throws to the same hand (the “4” of the “4 2”) and the crossing throws (the “2” of the “4 2”) are of the same height. If 3 ≠ 4, then is 2 = 4? Again, no, 2 ≠ 4. The explanation for the equal heights of the 4 and the 2 throws lies in the flight and dwell times[4] of the throws. We can verify by watching the video demonstration that the flight time for “4” ≈ flight time for “2” ≈ 1 beat. The dwell time for “4” ≈ 3 beats while the dwell time for “2” ≈ 1 beat. Since the flight time for both the “4” and the “2” is the same, they also get thrown to the same height.
Synchronous Solution: Hand siteswap “(1, 1)”
Another technically correct siteswap representation of the “fake 4 2 3” can be arrived at by assuming that the two hands are throwing synchronously at every beat. In that case, the hand siteswap will be (1, 1). The round brackets indicate synchronicity. We can choose the convention that the first number in the brackets indicates the right (or left) hand and the second indicates the left (or right) hand. Now watch the below video as the demonstration moves from the “fake 4 2 3” to the 1-up 2-up pattern.
As demonstrated, the “fake 4 2 3” is a close relative of the synchronous 1-up 2-up pattern, the siteswap for which is worked out in Table 7 based on the events of the video above. We assume that the first number in the brackets corresponds to the right hand and the second number to the left hand. The “x” indicates that the throw was made to the other hand. Absence of the “x” defaults to the throw being made to the same hand.
| Current Time (in beats) (TC) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 4/4 Beat# | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 |
| Object(s) thrown | A, C | B, NA | A, C | NA, B | A, C | B, NA | A, C | NA, B | A, C | B, NA |
| Time at next throw of same object(s) (TN1, TN2) | 2, 2 | 3, NA | 4, 4 | NA, 5 | 6, 6 | 7, NA | 8, 8 | NA, 9 | ?, ? | ?, NA |
| Object siteswap (TN1-TC, TN2-TC) | (2, 2) | (2x, 0) | (2, 2) | (0, 2x) | (2, 2) | (2x, 0) | (2, 2) | (0, 2x) | (?, ?) | (?, 0) |
| Throwing Hand | (R, L) | (R, L) | (R, L) | (R, L) | (R, L) | (R, L) | (R, L) | (R L) | (R, L) | (R, L) |
| Time at next throw by same hand (THR, THL) | (1, 1) | (2, 2) | (3, 3) | (4, 4) | (5, 5) | (6, 6) | (7, 7) | (8, 8) | (9, 9) | (?, ?) |
| Hand siteswap (THR-TC, THL-TC) | (1, 1) | (1, 1) | (1, 1) | (1, 1) | (1, 1) | (1, 1) | (1, 1) | (1, 1) | (1, 1) | (?, ?) |
Now, we notice that if we consider for example Time = 0, 1 and 2 beats, then the left hand has to throw object C a “2” at Time = 0 beats, throw nothing at Time = 1 beats and then throw the same object C as a “2” again at Time = 2 beats. Object C then might as well be held in the left hand from Time = 0 beats to Time = 2 beats and an actual throw can be made only at Time = 2 beats. Note that at Time = 2 beats, the left hand must throw because at Time = 3 beats, it has to throw Object B. However, at Time = 2 beats, the right hand can now afford to just hold on to Object A instead of throwing it because the right hand doesn’t have to throw anything else at Time = beat 3 and then has to throw Object A itself again at Time = 4 beats. Another way to look at these “2” throws being intermittently treated as “holds” is to break the “2” into two “1” throws, where each “1” is a hold. The “fake 4 2 3” pattern then can be represented by the siteswap: (2 1) (2x 1) (1 2) (1 2x), where the “1” throws denote “holds”.
We further remark that in either choice of the synchronous representation of the “fake 4 2 3”, the actual throws that are happening are the “2” and “2x” throws. Now it is much more intuitive that all the objects are getting thrown to the same height!
Finally, either synchronous representation of the “fake 4 2 3” makes for a period 4 siteswap and therefore, syncs up with the 4/4 beat. The hand throwing siteswap (1, 1) is a period 1 siteswap and so does not change the LCM when taken into consideration.
Ladder Diagram Comparison
Figure 1 shows the ladder diagrams for the asynchronous “4 2 3” pattern in the centre, the synchronous (1 2) (1 2x) (2 1) (2x 1) pattern on the left and the 1-up 2-up variation (2 2) (0 2x) (2 2) (2x 0) on the right. Notice how the “4 2 3” looks just like a zoomed-in version of the others. No wonder then, that even experienced jugglers confuse them!
I now revisit a statement I made at the beginning of this blog: the second trick that novice jugglers learn is, more often than not, this variation of the 1-up 2-up and not the “4 2 3”. I also realized that a lot of tricks and tutorials that are presented as improvisations of the “2” throw of the “4 2 3”, are 1-up 2-up variations where the 2-up throw(s), one of which is “held”, are being improvised.
The Real “4 2 3”
If we continue to watch the video, we will arrive at the section where the “real 4 2 3” is demonstrated:
Now, there is a beat where neither hand throws at all – this is when the “2” is being held. We can also interpret this as the previous throw having been a “6” and the held “2” is in fact, part of the dwell time of the “6”. In that case, the object siteswap changes to “6 0 3” and the hand siteswap changes to “2 0 4” where the “0” reflects the beat where the “2” is being held which translates to “no throw is being made” in our new interpretation. This equivalence has been illustrated in the video.
The “4 2 3” is perhaps the most common trick that gets juggled inaccurately due to the juggler not keeping time correctly. But it is certainly not the only one. Another example is the siteswap “3” juggled with a hand siteswap of “1 3” which can easily be mistaken for the siteswap “5 0 4” juggled with the default hand siteswap “2” as can be seen from the below video which is associated with the second part of my series of blogs explaining the siteswap notation:
Again, the difference is subtle and becomes evident only if one starts keeping track of the beats accurately. In fact, I find it very unnatural to juggle patterns involving held “2” or empty “0” throws in strict time. The hands don’t seem to want to break the rhythm and suddenly do nothing for a beat. The result is that I end up changing the hand siteswap and technically juggling a different pattern from what I’m supposed to be juggling!
Footnotes
- R: Right hand, L: Left hand.
- Technically, we also need to take into account the period of the hand-throwing-sequence. In the scenarios occurring in this article, this period is either 2 (R L) or 4 (R R L L) or (in the sychronous case) 1. Our music is always period 4, so in either case, the music and the hand-throwing-sequence period will synchronize after LCM(4,2) = LCM(4,4) = LCM(4,1) = 4 beats. Since we’re always including the factor of 4 corresponding to the 4/4 rhythm while taking LCM with the siteswap period, we can get away with ignoring the period of the hand throwing sequence.
- See “Juggling the Hands” section of this blog.
- See “Throw Heights” section of this blog.
Someone linked your blog on FB and I replied there… copying it here:
There is not really any mystery. Siteswap just tells you that an object comes down N beats later. There is no rule saying that they have to be regularly spaced beats (we just usually prefer it that way for aesthetic reasons). The pattern can still be described as 423, but he did the (not thrown) 2 “synchronously” with the 4 (or the 3, who could tell with a hold), while having a full beat time space between 3 and 4. This gives you an unevenly spaced pattern… and no, you don’t have to insert a beat between 3 and 4 for consistency. As an aside: You can perfectly well describe any synchronous pattern with vanilla siteswap (you just set the beat length between every 2nd beat to zero). The reason this is not done is, that the same siteswap-number on the left and the right hand result in different throws, which people find confusing.
A further comment: All mystery usually evaporates like morning mist if you do the “2” as a proper throw, not as a hold. Doing it as a hold is “that lazy way that gets you into trouble” 😀
You are also adding to confusion by calling the number of beats throw “height”. Height is only very indirectly linked to siteswap in so far, as you usually require a certain hight to be able to only catch the object N beats later. But any other method of gaining that time would be fine. Also again: a certain amount of beats doesn’t directly translate into a specific time in seconds, as you can (and often do to a certain extent) space the beats unevenly in time.
Cheers,
K.
Hi Karsten, I would like to debate your point about there being no rule requiring beats to be evenly spaced. I think the siteswap notation will collapse without that “rule”. Consider the “illegal” siteswap “4 3 2”. Without regularly spaced beats, the “4 3 2” can be juggled by simply claiming, at every throw, that a beat corresponds to a different amount of time (see example at the bottom of my comment here).
My understanding is that the “rule” of evenly spaced beats is fundamental to siteswap notation and allows us to prove mathematical results like the permutation test etc. – it is not just a matter of aesthetics.
I would also be interested in your comments on the other blog referred to in this article: http://jigyasujuggler.com/2018/06/15/siteswap-notation-ii-juggling-a-vanilla-siteswap/. There, I have covered two other points that you brought up: the correlation (or not!) of the siteswap number with the throw height and the confusion of same siteswap numbers resulting in different throws.
I really appreciate your taking the time to re-post your comments here and to have this discussion!
Begin Example
How to juggle “4 3 2” if evenly spaced beats are not required:
Though not necessary, I will throw once every 0.5s. But at each throw, I will change how much time “one beat” means.
First throw @ 0 seconds: I say my beat rate is 120bpm (2 beats a second) and I throw a “4”, i.e., I will throw this object again 4 beats later. At 2 beats a second, 4 beats means 2 seconds. So I throw this object next at time 0s (current time) + 2s, i.e., at time = 2 seconds.
Second throw @ 0.5s: Beat rate is 3/2 of 120bpm i.e., 180 bpm (3 beats a second) and I throw a “3”, i.e., I will throw this object again 3 beats later. Now, 3 beats = 1 second. So this object will be thrown again at: 0.5s (current time) + 1s = 1.5s.
Third throw @ 1 second: Beat rate is 2/3 of 120bpm, i.e., 80bpm, i.e., 4/3 beats a second and I throw a “2”, i.e., I will throw this object again 2 beats later. Now 2 beats = 2/(4/3) = 1.5 seconds. So this object will be thrown again at 1s (current time) + 1.5s = 2.5s.
Thus, after I complete the first 3 throws at 0s, 0.5s and 1s, I have 3 objects again scheduled to be thrown at 1.5s, 2s and 2.5s. These throws can made similar to the first three, with the same interpretation(s) for the duration of a beat, and the pattern will continue. In fact, the events I just described is how we do the 4 2 3 (with a constant beat rate of 120bpm), but because I was allowed to change the duration of each beat, I can now call it the 4 3 2 and it is no longer an illegal pattern!
End Example.
Cheers,
Jigyasu.
Hi Jigyasu,
there is a misunderstanding here. When I say that the beat can be irregular, I meant that the time interval between beats can have a different length. So siteswap could be used (or juggled) on a “beat” like this with no conflict:
|–|—-|-|—————–|—–|———————————–|—-|——–|—| (with “-” as an arbitrary time unit).
And often people *do* juggle certain patterns in a kind of “galloping” way. The siteswap is still a correct description of that pattern (although, as with hand position in mills-mess etc., it doesn’t describe the pattern fully, as a description of the specific beat is missing).
Your example uses a “beat rate”, which again (usually) means a regularly spaced beat happening at a certain rate per time. You then apply different “beat rates” to each position of the siteswap, so that one beat overtakes another. This is not allowed. Siteswap *does* care about the order of beats (because that is what the numbers count). What it does not is care about the exact time that passes between the different beats.
Again for the 423 “mystery”: Do the “2” as a proper small throw instead of a hold. Doing a hold and skipping the throw is just a very lazy interpretation of the siteswap. That interpretation is very useful for making a pattern simpler to juggle (and thus making 552 a good exercise for 5 objects). But it is not at all good for thinking clearly about siteswaps. If you do make the throw, you will notice, that the 423 that baffled you probably has a beat either similar to this: |–|–|—–|–|–|—–|–|–|—-| , or that the 2 is thrown at the same time as the 4 or at the same time as the 3 (or that you now suddenly time that additional throw to your metronome as well and end up with |—|—|—|—|—|—|—|—|). In either of those cases, the mystery has disappeared!
PS. I had at first written something a bit different and maybe better/clearer, then closed the browser window. argh.
Hi Karsten,
I replied to your comment a few days back but was apparently missing a plug-in on my website that’d inform you that I’d replied. Hopefully, now you’ll get a notification. I’ve also edited the reply a little from what I’d originally posted:
Got your point (finally)! I agree “galloped throws” can be used to interpret the “fake 4 2 3” as a “true (but galloped) 4 2 3”. But I feel this concept (a) rather adds to the confusion as explained below and (b) is an unnecessary contrivance if the argument I outlined below is correct.
Let us look at the “real 4 2 3” (with held 2’s) as shown in this video from 2:45 to 3:05. Throw sequence is: R _ Y P _ Y R _ Y P and so on (where R = Red ball, Y = Yellow, P = Pink). Invoking galloped throws, “_“ just marks the passage of time but does not represent a beat as no throw is happening here. Eliminating these non-beats, the throw sequence is: R Y P Y, repeat. This is the siteswap 4 2 (with galloped throws). The “fake 4 2 3” as shown from 0:53 to 1:15 of the same video also has the same throw sequence: R Y P Y, repeat. This too, then, is 4 2 (but with evenly spaced throws). Is it more intuitive to apply the galloped throw concept to:
(a) the “real 4 2 3”, eliminate the held 2 and conclude that this pattern is a galloped 4 2 (the throws do look unevenly spaced after all); OR
(b) the “fake 4 2 3”, “discover” a held 2 “somewhere” and therefore, claim that the thrown 2 is in fact a 3, thus making this pattern a galloped 4 2 3 even though the throws look perfectly evenly spaced?
In this blog, the intention is not to understand how to juggle the 4 2 3 correctly, but to precisely communicate (the relative timing of) the actual pattern being juggled. In this respect, I find using the concept of galloped patterns to say that the “fake 4 2 3” is indeed the 4 2 3, is counter-intuitive and confusing as described above. Instead, saying that the “real 4 2 3” (with held “2”) is a galloped “4 2” makes more sense from my perspective. Note that in the video section for the “real 4 2 3” that I pointed to, I’m actually saying that this pattern is the “6 0 3” when the 2 is not explicitly thrown. Again, the intention is to notate what is happening so that someone else can replicate it rather than to explain how to juggle the “4 2 3” correctly.
I think galloped patterns are a contrivance because: In the paper Jugging drops and descents by Joe Buhler, David Eisenbud, Ron Graham and Colin Wright (one of the inventors of siteswap notation), they say, “… pay attention only to the times at which they (objects) are thrown, and will assume that the throw times are periodic.” They also say, “Nothing… really requires the rhythm of the juggling pattern to be constant… We retain the constant rhythm terminology in order to be consistent with jugglers’ standard model of sites swaps”, (supporting the case that constant rhythm need not be assumed). Even so, my understanding is that galloped throws were a contrivance introduced later especially to explain “irregularly thrown” patterns. Finally, I feel this is unnecessary as I briefly argued in Footnote #8 of this blog (the same one I pointed you to earlier) – in fact, I also remarked that the concept of polyrhythms is a fancy term for saying that a different hand siteswap is being used. It was with this background that I chose to look for a “better” explanation than galloped throws could provide for the scenario discussed in this blog.
best regards,
Jigyasu.
Hi,
I would just say that for a “real 423” the “2” to be thrown, because the definition says, that the object is “thrown N beats later”. A “held 2” is then a derived pattern. This makes a lot of apparent paradoxa disappear – e.g. it also answers the question what the difference between “3” and “522” is. The timing of a thow you do not make is just much more flexible than of a throw you do make, and it is also not clear “when exactly it didn’t happen”, and so very often confusion is the result.
If your goal is, to describe the timing more closely, why don’t you just use synchronous siteswap. You are already demonstrating that what people often do for 423 is similar to columns, which are a synchronous pattern. What you get using synchronous siteswap is: (4,2)(4x,2)(2,4)(2,4x) … which you can type exactly like this into siteswap simulators like http://jugglinglab.sourceforge.net/ or http://www.gunswap.co/ and they will simulate the pattern.
Cheers,
Karsten
Hi Karsten,
Again, there’re several interesting points you make:
1) Synchronous solution: Valid point. I did mention the solution “(2 1) (2x 1) (1 2) (1 2x)” in the blog. The numbers are exactly half of the numbers you wrote and though simulators don’t like it, I stand by it :-). Even the two simulators you pointed to (I’ve used JoePass! previously) insist on even numbers in synchronous patterns (two beats collapsed into one). It makes the notation compact, but causes other problems… I’m digressing – that’s for the blog I’m currently working on!
2) Why a vanilla explanation? Well, it “looks” like a vanilla pattern! The popular vanilla solution was the one you suggested: galloped patterns. I have explained why I found this approach “counter-intuitive”. My alternative – retain the regular beats “rule” but violate the “rule” that hands throw alternately. To me, this seems a more obvious description of what is happening.
3) “3” vs “5 2 2”: You are right again. Juggled separately, there’s no way to tell which is which if we “hold” the “2”. In that context, I agree that a “2” ought to be thrown. But if I’m transitioning between 3 and 5 2 2, then even with held 2’s, assuming regular beats, the notation is powerful enough to tell me that the timing has to change. If we allow galloped beats, this information gets blurred.
Given a choice, I’d prefer breaking the “rule” of “hands throw alternately” rather than the “rule” of “regularly spaced beats” – that’s basically the point of the blog :-).
Regards,
Jigyasu.
P.S.: “when exactly it didn’t happen” – very nicely put!! :-D.
Hi Karsten,
Further to the first point in my previous reply regarding the use of “collapsed beats” in simulators causing “other problems”, I have finally completed the two posts across which I’ve explained how I feel simulators are limited in what they can simulate. These posts are Siteswap Notation VI (see section on “Juggling Simulators”) and Siteswap Notation VII. In particular I have also closed the loop in the “Musings” section of the latter post with the 4 2 3 demystified post.
Regards,
Jigyasu.