The siteswap is a notation of juggling which describes the rate/rhythm to launch S and thus the Trajectoire of the objects in space. It was essential like language reference to describe and communicate the sequences of interesting Jonglerie in solo and to several jugglers. The notation consists of a sequence of Chiffre S coding for each throw the number of time passing until the revival of the same object. This makes it possible to model and invent an infinity of sequences, whatever the number of objects or sites of launching (of hands for example, but not only). In spite of its mathematical roughness décriée by some Jongleur S the siteswap largely contributed to enrich the repertory by the figures of juggling.
The siteswap is particularly adapted to the techniques of to launch (balls, bludgeons and rings mainly). Nevertheless the throws can as well be interpreted like a blocking in contact on the body, a bearing on the ground, one or more rebounds on a surface, etc the siteswap can also be adapted to describe techniques of air twin wheel (siteswap with a hand) or more recently of Astrojax.
History of the notation
The simplest, often named notation vanilla siteswap in the Anglo-Saxon Countries, described the asynchronous mode of juggling with two hands. It was developed by three independent groups: Paul Klimek with Santa Cruz ( Quantic Juggling , 1983?) and Bruce Tiemann with the California Institute off Technology in California; Mike Day, Wright Hake and Adam Chalcraft with Cambridge ( notation of Cambridge 1985?).The Mailing list rec.juggling in since August 1990 activity on Usenet and the beginnings of Internet allowed the fast diffusion of these notions among the community of the scientific jugglers. rec.juggling remains the most complete information source today on the siteswap: first diagrams state-transitions, announces and often publication of the essence of mathematical work related to the notation as well as many councils for the jugglers in practice.
Quickly some conventions typographical will make it possible the notation to adapt to describe the Passing, the synchronous mode of juggling as well as the throws multiplexs.
In 1991 the notation multi-hands or mhn of ED Cartsen (multihand notation) synthesizes more the share of the known concepts. It makes it possible to describe the asynchronous one with the same notation as synchronous whatever the number of sites of launching. It will be useful for example as of 199? with the troop Gandini Juggling to carry out sequences siteswaps in Passing with an odd number of hands or towards 2002 with the diabolists whose string can be used as site to launch (mhn with only one site of launching). In 2005, the mhn is always sufficient to describe the polyrythmic mode of juggling .
Bases: the asynchronous siteswap
The most common form of the siteswap, called siteswap Vanille describes the sequences where the throws are asynchronous (alternate between the left hand and the right hand) and where only one ball is launched at the same time. If one filmed in diving a juggler successively launching each ball while alternating between the left hand and the right hand while it advances, one could see something which would resemble the diagram of right-hand side, diagram which one calls sometimes diagram of space time by analogy with his equivalent in physics.To describe the sequence, one determines for each stage how long later the object will be started again. For example, if one considers the diagram of right-hand side, to the first throw, the ball violet is launched by the right hand, then the blue ball, then the green one, again the green one, again the blue one and finally the ball violet is caught up with and started again by the left hand at the fifth time: that confers on the first throw the value of 5 . By repeating the operation for each time, one obtains a sequence of numbers which indicates for each throw the number of times passed until the revival. Since the throws alternate hand in hand, the odd numbers indicate throws from one hand to another (cross throws), while the even numbers fall down in the hand which launched them (uncrossed throws). A 3 corresponds to the number of times of a throw of cascade with three objects. A 4 corresponds to that of a fountain with 4 objects, etc
The number associated with each throw makes it possible to determine the number of times that the ball will pass in the air and thus to evaluate the relative height of the throw. This is why much the numbers like heights interpret:
-
the 3 corresponds to the height of the throws when one juggles with 3 balls (crusader)
- the 4 corresponds to the height of the throws when one juggles with 4 balls (noncross)
- the 5 corresponds to the height of the throws when one juggles with 5 balls (crusader)
and so on…
This short cut, although technically incorrect, is particularly useful in practice since it makes it possible to produce first blow the sequence without having to think of the whole of the concerned permutations. This short cut is worth only for the strictly air techniques of juggling and does not describe to in no case an absolute height (this one depend on the tempo, spacing of hands etc).
Three particular throws are noted:
-
the 0 corresponds to a time when the hand is empty and on standby
- the 1 corresponds to an object quickly last with the other hand to be started again at once (the master key being able to be done while launching the ball strictly speaking or not)
- the 2 generally corresponds to an object maintained in the hand, however it is possible to express it by a throw uncrossed
The throws higher than 9 are noted by letters, while starting with has : 8,9, HAS, B, C… (Has = 10, B = 11… as for a hexadecimal base ).
Each sequence is brought to repeat itself after a certain number of throws called period . The sequence is thus noted by the smallest representative and non-repetitive segment: the sequence of right-hand side is thus noted 53145305520 and has one period of 11. When the period is odd (as in this example), the sequence will be known as symmetrical because she will be repeated starting from the other hand. The sequences of even period are known as asymmetrical , each hand unceasingly repeating the same types of launching.
The number of objects necessary to juggle the sequence is the arithmetic Mean numbers of the notation. Thus, 441 is juggled with 3 objects because (4+4+1) /3 = 3, 86 will be juggled with 7 objects, 7531 with 4 etc Any sequence valid thus has a whole arithmetic mean, but it is only one requirement and nonsufficient.
By convention, one notes the siteswap while starting with the largest numbers. Thus the siteswaps 315 , 153 and 531 will uniformly be noted 531 .
State, transition and diagram state-transitions
A sequence siteswap determines a sequence of throws in time (diachronic point of view). But it is also possible for each time to describe the state objects in the air, i.e. to determine times to which each object will be caught up with and started again (synchronic point of view).One thus associates with each time a state by marking each point of time when the object will be caught up with by a X and each point of time when nothing is still envisaged by a - . Let us consider for example the state of the balls right after the first throw in the diagram of right-hand side. We know that a ball will land at next time, then immediately another and finally, we have just launched a ball which will land at the fifth time. State xx thus is noted--X and one reads from left to right while following the arrow of time. One can now make use of this state to determine the possibilities of the next throw. Us will start with to remove X on left (the ball which lands at next time) and to add a - on the line. X thus will be obtained--X: this corresponds at the virtual state of the balls when the juggler caught up with the ball but did not start again it yet. Thanks to this operation we can determine the possible throws exhaustively. As we have just caught up with a ball, we cannot make a 0 . We cannot make a 1 either since we have already a ball which will be caught up with for this time (the first X ). The same applies to the 4 (fourth X ). We can on the other hand carry out a 2 , a 3 or a 5 . If, as the juggler of the diagram we choose to do one 3, the following state is x-xx- (it the - in position 3 by a X is enough to replace).
In this way it is possible to make an inventory of all the possible states for a given number of objects and a maximum length of launching given and determining the possible throws starting from each state. The diagram of left shows all the possible states for 3 objects and a maximum throw of 5 . This diagram indexes all thus the possible siteswaps under these conditions. It is enough to start from a state and to follow the arrows to its own way and when one returned to the starting point one obtained a valid siteswap. All the siteswap can be indexed by this skew. However, the diagram quickly becomes extremely complex as the height of the throws increases. One then uses other types of diagram of state.
The notation of the states is in binary fact , for each time it is only two possible values: often one writes 0 and not - and 1 instead of X . Thus, the state X X - - X can be noted 11001 . One can even convert it into Decimal system and note it 25 . However, for technical reasons of lightening of calculations, when one passes to the binary system, it is of use to operate one second transformation. Since the weak bit is read on the right in the binary notation and that it is most capable to correspond to an object caught up with at next time, the direction of writing is changed. Thus, the state noted X X - - X in the preceding diagrams will be finally noted 10011 and not 11001 and will correspond to 19 and not 25 in decimal system. Thus the temporal shift operated to determine the possibilities of the next throw will correspond to a simple whole division by two (the following virtual state will be 9 or 1001 , previously noted X - - X - ). The table state-transitions uses this transformation: it contains in another form same information as the preceding diagram. The lines give all the possible throws to change state and the columns indicate the resulting state. As for the preceding diagram, when one returns to the starting state, one produced a valid siteswap.
Mathematically, one will say that what is called commonly a throw is actually a Permutation of the Ensemble which a state constitutes, whom the cardinal of this unit corresponds to the maximum length of the throws. A sequence siteswap will be thus a composition of permutations whose result is an identical permutation. One thus finds the literal direction of the word siteswap (permutation of sites).
It also should be noted that the siteswap vanilla is not limited to two hands: there is no limit with the number of sites, however the condition according to which the throws are successive makes its use difficult to describe the sequences including more than one juggler.
Siteswaps asynchronous and synchronous
The form vanilla of the notation siteswap limits to the description of asynchronous figures : the two hands launch alternatively the objects. Each hand thus launches every two times. But juggling also includes a not less vast range of synchronous figures , the throws being done simultaneously with the two hands. In this case, since for each time there are two throws, one notes the two throws between brackets and separated by a comma, for example (4,4) . For reasons of convenience and coherence, one notes the synchronous siteswap each time by doubling the length of the throw. Thus, a synchronous fountain with four objects should be theoretically written (2,2) each ball being started again every two times, but one writes (4,4) for:- To keep the correspondence with the siteswap vanilla (thus by making this correction a synchronous 4 in siteswap will be equivalent to a 4 in siteswap vanilla).
- Conserver the rule of the arithmetic mean which makes it possible to calculate the number of objects necessary to the realization of the sequence (thus for a (4,4) it will be necessary (4 + 4)/2 = 4 objects).
To determine the validity of a sequence
A sequence is valid only in the condition of being juggled with an integer of objects. But it is simply about a requirement and nonsufficient. The rule of the average will be thus a good preliminary test taking into account the simplicity of calculation, but it will be never a sufficient demonstration. For example 432 would be juggled with three balls but is not valid.
The fundamental rule requires that only one ball be able to be launched (or caught up with - two equivalent faces of the same principle) at the same time. Thus a sequence will be valid if and only if all the throws intervene in different times. Starting from this postulate, one can imagine various methods of checking.
Remark : this principle applies to the form vanilla of the siteswap and mutatis-mutandis with its synchronous form (it will be a question in this case of checking that each hand launches only one ball at the same time), but it goes without saying the rules of validation of a siteswap multiplexing will be basically different.
Rough method
The rough method will consist in making an inventory of all times of revival to make sure that to each throw will correspond a different time. Thus to check the validity of 758514 :
siteswap: 7 5 8 5 1 4 7 5 8 5 1 4 7 5 8 5 1 4 ... temps where the ball is launched: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ... temps of revival: 7 6 10 8 5 9 13 12 16 14 11 15 19 18 22 20 17 21 ... The time of the revival corresponds to the sum of the value of the throw and the time to which this throw takes place.
In this example, one notes that times all of revival are different: each ball is launched in its time and there is no collision in the hand - the siteswap 758514 is thus a valid siteswap.
For 578514, which however would be juggled with an integer of balls, one observes an inconsistency immediately:
siteswap: 5 7 8 5 1 4 5 7 8 5 1 4 ... temps where the ball is launched: 0 1 2 3 4 5 6 7 8 9 10 11 ... temps of revival: 5 8 10 8 5 9 11 14 16 14 11 15 ...
Throws 2 and 4 will be both started again at the 8th time, which contradicts a basic principle. The sequence 578514 is thus not valid.
Modulation
The traditional method will consist in avoiding making long calculations which oblige to repeat several times the sequence to make sure of the validity. One is satisfied to calculate times of revival for the p first throws (where p is the period of the sequence siteswap) that one will modulate by the period (one will keep the remainder of whole division by the period). If the unit obtained is a permutation of {0; 1; 2; …; p-1} (more simply known as: if all the numbers obtained are different), then the sequence is valid.séquence siteswap: 7 5 8 5 1 4 temps where the ball is launched: 0 1 2 3 4 5 temps of revival: 7 6 10 8 5 9 modulation by the period (here 6): 1 0 4 2 5 3
{1; 0; 4; 2; 5; 3} is a permutation of {0; 1; 2; 3; 4; 5}, the sequence siteswap 758514 is consequently valid.
séquence siteswap: 5 7 9 4 0 5 temps where the ball is launched: 0 1 2 3 4 5 temps of revival: 5 8 11 7 4 10 modulation by the period (here 6): 5 2 5 1 4 4
There is at least a repetition, consequently the sequence siteswap 579405 is not valid.
Characterization of the sequences
A sequence siteswap is known as fundamental when this sequence can be carried out directly starting from the basic state (in other words, when one can directly connect it and without transition since the basic cascade or fountain corresponding to the number of objects). 441 is thus a fundamental sequence: one can directly connect a 441 after a cascade 3 , then to return directly cascades about it 3 . Consequently, all the fundamental sequences for a number of objects given are connected between them without transition: 531 , 441 and 423 can be thus connected in any order, for example 531531441423423 .A sequence siteswap is known as excited when it is not fundamental, i.e. when it cannot be connected directly starting from the basic state and that it requires a transition to enter and leave. The shower with three objects 51 is thus an excited sequence because it is not possible to connect 3 and 51 , it is necessary to carry out a transition, for example: 3 (4) 51 (2) 3 . The transitions are given between Parenthèse S.
A sequence siteswap is known as first when it cannot be divided into several independent sequences and répétables, another manner of characterizing it is to say that it never passes twice by the same state. Thus 3 , 441 and 633 are sequences siteswap first, whereas 423 is not one since it breaks up into 42 and 3 . A sequence first can as well be excited (for example 51 or 741 ) that fundamental (for example 642 or 97531 ).
The throws multiplexing
A throw multiplexing corresponds to the simultaneous throw several objects by the same hand. The wide notation of the siteswap uses the hooks to describe the multiplexings. For example means that the hand launches two objects at the same time: one in 5 the other in 4.As for the '' siteswap '' vanilla each number is represented by a single character, the number of natures present between the hooks thus corresponds to the number of objects necessary to carry out the throw multiplexing. The jugglers use a specific nomenclature according to the number of objects implied in the multiplexing:
-
is a duplex,
- is a plywood,
- is a quadruplex system,
- is a quintuplex…
As explained the 0 previously the absence of object in the hand defines at the time of launching it. Although it is possible indefinitely to describe invisible additional balls in the hand, for example, it is of good sense to reduce these expressions to the simplest this is why 0 is not used inside the hooks, in our example is the equivalent of 4. The same good sense carries out us to exclude the use of hooks inside other hooks, for example 2] is nona direction the good writing for this multiplexing is in fact.
According to the software and the local habits the throws are sorted between the hooks by order ascending or decreasing (more running). To distinguish these throws with the oral examination and to mark the difference with the traditional throws the jugglers took the practice to name the action like a complete number. For example 23 will be marked: forty three-two times - three.
Mathematical rules
The notation multiplexing was developed to extend the siteswap vanilla but its operation can be retranscribed for the whole of the notations: synchronous notation, notation mhn. Note that in this case, the states will not be any more binary S, but written in another Numbering system (ternary, quaternary or more still).
The rules applicable to the siteswap vanillas, synchronous and mhn always applies with the multiplexings. However it is necessary to integrate the concept of launching simultaneous and avoiding the traps. Let us take the example of the rule of the average 23, I add the throws 4+3+2+3=12, I divide by the number of times 12/3=4, this sequence thus juggles itself with 4 balls. The sequence comprises 4 throws for 3 times with the difference in a sequence vanilla for which many throws and many times are equal.
The notation multiplex described in fact the superposition of two juggled layers. This observation makes it possible to work more easily on the sequences by concaténant them, for example 423 + 300 = [20] that is to say 23.
(to be finished) Checking MOD, method +p, interruption, method +n, to launch initiating
Arms crosseds
It is possible to add to this notation the placement of the arms. The principle is the following:- S for " side" represent the uncrossed arms.
- O for " over" represent the arms crosseds, the hand which launches to the top.
- U for " under" represent the arms crosseds, the hand which launches in lower part.
Examples with the sequence siteswap 3 :
- S represents simple a cascades .
- S U represents a mill by lower part .
- S U O represents a half Mills mess (see figures of air juggling): the throws are 3 but correspond alternatively to an uncrossed throw, in lower part then with the top.
- S O represents a mill over .
- S O U represents a half funky mess .
Limits of the siteswap
This notation remains a tool noting only the order of succession of the throws. Contrary to what was reproached to him, it does not limit the creativity of the jugglers, nor the artistic side of juggling. It allows contrary to the easier exchanges between the jugglers. There remain many alternatives not specified by the siteswap:
- the throws around the body (in the back, under the leg, with the top of the shoulder…) ;
- various types of launching and correction possible (in leg of cat, penguin, on the back of the hand…) ;
- gestures and acrobatics accompanying the idle periods (pirouettes, somersaults, reverences…) ;
- alternatives specific to the tackles (1 turn, 2 turns, flat, on the level parallel with body etc for the bludgeons, the throws out of crepe for the rings…)
One can see the siteswap like the Solfège of juggling. The exclusion of certain aspects and its abstraction make at the same time its force and its weakness. The siteswap does not exhaust by far all the possible figures, but it of described in advance the structure and notes with a great economy a good part of them. Inventiveness in the use of the siteswap precisely consists in developing the variations of them. In addition, various extensions were proposed to refine the descriptive character of the siteswap, as for example the siteswap generalized which adds an indefinite number of dimensions to a sequence siteswap where can consequently be noted the majority of the indications of movement, of style etc
Software of simulation
There exists a great number of Computer programs allowing to simulate sequences siteswap:- Juggling Lab animator - the program of reference, a simulator Open source writes in Java able to interpret almost all syntaxes of siteswap. It can also be included on a Web page in the form of a Applet. It manages the passing, the multiplexings, the movements of hands personalized and makes it possible to generate siteswap;
- DSSS Simulator of siteswap in flash applied to the twin wheel;
- Jaggle - Another simulator in java, in 3D, able to manage the movements of hand and to pass by again the figures with back;
- Jongl - a simulator developed for a great number of operating systems. It is particularly neat in its graphics 3D and manages the sequences of passing
- JoePass a simulator 3D for OS 9, OS X and Windows, conceived especially for the passing with a large library of sequences, the possibility of visualizing the diagrams of cause and of simulating sequences in notation multi-hand (mhn).
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