Marc Chemillier

Analysis and Computer Reconstruction of a Musical Fragment of György Ligeti's Melodien[*]

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Introduction

As you will see, there will be no mathematics in my presentation. Considering the title of the Symposium "Music and Mathematics", I must confess that my presentation is mostly music oriented. But I shall try to explain how it could be related to more theoretical and mathematical problems. I would like to explain too how the things I am working on could be related to the very interesting and brilliant lectures that we have already heard during the Symposium, concerning music theory. To that purpose, a classical distinction could help me clarify matters.

Xenakis defines three domains: 1) the outside time domain, 2) the time domain, 3) the inside time domain.

I think that we can relate the study of modes, diatonicism, chords structure, and so on, to the outside time domain. For this reason, the theoretical questions which have already been discussed during the Symposium are mostly concerned with the outside time domain. The time domain concerns the study of durations. My own work is related to the inside time domain. What I shall present here will then illustrate a quite different approach than the ones which have already been presented, and I hope, a complementary one.

To explain what I mean by inside time domain, let us say first that a score, for instance, belongs to the inside time domain. The most basic thing that we can say about a score, from my point of view, is that two relations can defined between events: 1) events which are simultaneous, and 2) events which are successive.

The score belongs to a level where those two relations can be clearly identified. But in the inside time domain, one can study those two relations at other levels.

By "other levels", I mean weaker representations of the musical content of a score; that is to say that you can forget some information. Let us take an example: suppose you are studying the basic rules of serialism; from this point of view, you need to know that a pitch occurs before another pitch, but you don't need to have information about durations of the pitches, nor about registers of those pitches. So you can forget durations and registers, but you need to preserve the fact that events are successive or simultaneous.

To formalize those two basic relations, I use a theoretical framework which is known in computer science as formal language theory. The corresponding algebraic structure is the free monoid, with an operation called concatenation. It is possible to extend the structure with a new operation which is called superimposition. The resulting algebraic structure is a lattice. We will not develop this point now, but we consider an example to make it more intuitive. Let us assume that two occurrences of an event a are successive; then you can distinguish these two occurrences. But if the two occurences of the same event a are simultaneous, you cannot distinguish them. For this reason, the superimposition is an idempotent operation, whereas the concatenation is not. The idempotent property is one of the basic properties of the lattice structure (for more details see [START_REF] Chemillier | Structure et méthode algébriques en informatique musicale[END_REF]).

In the traditional formal language theory, what is studied is sets of sequences which can be computed by different kinds of machines. Since the works of Turing, one has a theoretical and even mathematical classification of machines. We are interested in adapting this point of view to the study of music. That is to say, we try to considere music sequences (which belong to the inside time domain) as the result of abstract computations, and we try to study the process of such computations.

The example by Ligeti we shall study will be considered from this point of view. We shall try to discover the process which is involved in the passage we shall analyze. But there will be no theoretical considerations about this process (which is quite simple). We shall only try to represent this process by means of the programming language Lisp, which provides, as we shall see, powerfull functions which are convenient to describe the process. Gérard Assayag and I have made a computer reconstruction of the example in PatchWork which is a Lisp environment with graphical and musical interface.

General remarks about Melodien

Let us go back to the piece by Ligeti and make a few remarks about it. The diagram which is shown in Example 1 is a representation of the piece Melodien in the pitch-time domain. At the middle of the piece (measure 72), one have a C played by the full orchestra, in different registers from C1 to C8 (middle C is C4). This provides a clear cadence point, which divides the piece into two parts. These two part are quite symmetrical, because both of them ends with a beautiful solo played by two French horns coming in the foreground: the first solo takes place at the end of part one, and the second solo takes place at the end of part two. The piece begins with an introduction (from the beginning to measure 13), and ends with a coda (measure 142). A large empty space in the pitch domain can be seen in the coda, the pitches played by the orchestra being restricted to the very high and very low registers.

In the pitch-time domain represented in Example 1, one can see two special types of areas. The first one is marked by cross-hatching. This corresponds to what could be called textures. A texture is a musical surface of the pitch-time domain resulting from the superimposition of different melodic lines, in which a short motive is repeated and progressively transformed. Many examples of such textures can be found in Ligeti's compositions. The other type of areas represented in black in Example 1 corresponds to sections of the piece in which a part of the pitch space is filled chromatically. The fragment of Melodien I shall try to analyze is the texture which begins the first part of the piece, from measure 14 to measure 30.

Textures of this type have been studied in a recent paper written by Jane Piper Clendinning [START_REF] Clendinning | The Pattern-Meccanico Compositions of György Ligeti[END_REF]. This paper focuses on a few pieces composed by Ligeti in 1968-69, which are called "patternmeccanico" compositions: Continuum for harpsichord, Coulée for organ, the fifth movement of the Second String Quartet, and the eighth of the Ten Pieces for Wind Quintet. They all share a specific feature of construction, which is analyzed in detail in the paper. As we shall see, the principles involved in our analysis of the texture from measure 14 to measure 30 of Melodien are not very different from those described by Jane Piper Clendinning for the pattern-meccanico concept.

The texture from measure 14 to measure 30

Let us make a few general remarks about this texture. It results from the superimposition of four parts: namely the piccolo part, the xylophone part, the celesta part, and the violin B solo part (see the score). Along these five pages of the score, the texture is combined with a string background. Short melodic elements are progressively added to the polyphony, as we can see measure 22 in the trumpet part. Most of these melodic elements begin with a chromatic motion (for instance, the E of the trumpet measure 22 moving down to the D#). At the end of the section, the texture disappears in a diminuendo, and the musical surface changes into an intricate polyphony in which the melodic elements are combined.

From a rythmical point of view, the notes in each part of the texture have equal durations, but these durations are different from one part to the other, so that the texture consists in the superimposition of different speeds. At the end of the section, an accelerando is applied to each part of the texture.

In each part of the texture, rests are inserted to delimit motives. These motives are progressively transformed from the begining to the end of the section. At the begining, the transformations are simple: notes are progressively added to the motives; for instance, the first motive of the piccolo part is F6 A6, then it becomes Eb6 F6 A6, then Eb6 F6 F#6 A6, and so on. One can also mention that the notes of the motives are the same in the different parts of the texture: for instance in measure 17, the notes are Eb6 F6 F#6 A6 in the piccolo part, F6 A6 Eb6 Gb6 in the xylophone part, Eb6 F6 A6 F#6 in the celesta part, and A6 F6 Gb6 Eb6 in the violin B solo part. Let us consider this passage of Melodien.

The maximal number of notes in a motive is ten notes. This maximal number is reached measure 22 for the xylophone, celesta and violin B solo part. In the piccolo part, no motive has more than nine notes. After this point (measure 22), the notes of the motives in the different parts of the texture are not the same, and the evolution of the texture becomes more complicated. We notice that at the end of the process (measure 29), the motives are nearly in ascending order for the piccolo and xylophone parts, and in descending order for the celesta and violin B solo parts. We shall explain the full process in action during this section. As we shall see, this explanation relies on what we shall call virtual reference motives, that is to say motives which are not necessarily played in the different parts of the texture.

Virtual reference motives and operators

In Example 2, one can see the evolution of the xylophone part, from state 9 which occurs in measure 21 (the first state has number 0), to state 22 which is the last state of the process. The last note to be added to the motives, i.d. the tenth note, is the E5 which appears in state 10 between the C6 and the G5. For this reason, this position is marked by a nil symbol in the previous state (state 9), i.d. two empty parenthesis. Generaly speaking, we shall not consider that notes are added to the motives at the begining of the process; we prefer to say that these notes are undeleted. Actually, as we shall see, they can be deleted again during the process. When a note is deleted, it is replaced by a nil symbol at the same position. This situation introduces our first operator:

1) FIRST OPERATOR: deletes notes.

At the begining of the process, most of the notes are deleted, because the only notes which remain in the motives (see measure 16) are the following ones:

F6 A6 Eb6 ( ) ( ) ( ) ( ) ( ) ( ) ( )
and progressively notes are undeleted up to measure 22.

EXAMPLE 2. Xylophone part

When the last note E5 arrives (state 10 measure 22), a new operator is activated. As we can see in Example 2, the B5 in state 9 is decreased a half step down to the Bb5 in state 10. In the next state (state 11), one can see that some other notes are decreased: D#6 and D6 are decreased one step down to respectively C#6 and C6, and C6, E5 and Ab5 are decreased a half step down to respectively B5, Eb5 and G5. We can also notice that the delete operator is once more activated, because in state 11, the G5 has been deleted and replaced by a nil symbol. In fact, this G5 is virtualy present in the motive, because in next state (state 12), it is undeleted again, and at the same time decreased one step down to F5. This situation leads us to introduce our second operator:

2) SECOND OPERATOR: decreases notes (one step or a half step down).

As one can see in Example 2, the decrease operator always decreases notes one step or a half step down, excepted in state 12 and 14, where notes are decreased a minor third down. These are the only two exceptions to the rule in the xylophone part process.

In state 15, one can find a new transformation. In order to explain the whole process, we need to introduce a new operator. Recall that at the end of the process, the motives in the xylophone part are in ascending order. This means that the highest notes must be pushed to the right of the motives. For this reason, we have to introduce a third operator which displaces notes to the left hand side of the motives (or to the right hand side in the case of the celesta and the violin B solo parts where the final states are in descending order):

3) THIRD OPERATOR: displaces notes.

This operator is activated to produce the state 15: Bb5 is displaced so that A6 is pushed to the right. Example 2 shows the process up to state 22. The three operators are activated. For instance, the C#5 in state 13 is deleted, and appears again in state 16 as a C5, being decreased a half step down.

The four parts of the texture can be analyzed in the same manner. They all rely on a ten note motive, and these four reference motives consist of different permutations of the same ten notes. These reference motives are virtual in the sense that they do not necessarily appear in the corresponding part. For instance, as we have seen in the xylophone part, when the last note E5 arrives (state 10), the B5 is decreased a half step down to Bb5, so that the reference motive is never played in this part: EXAMPLE 3. Reference motive for the xylophone part

The same holds for the violin B solo part: when the last note of the motive arrives, other notes are decreased so that the reference motive is not played. The only part in which the reference motive is actually played is the celesta part: it appears as state 15 at the end of measure 21. In the piccolo part, something interesting should be mentionned. As we have already pointed out, the piccolo part contains no motive with more than nine notes. This means that at every stage of the process, at least one note is deleted. In fact, the reference motive for this part is represented in Example 4: EXAMPLE 4. Reference motive for the piccolo part When the E5 arrives (measure 22), the F6 is deleted (and other notes are decreased). Then the E5 is decreased down to D5 ; the F6 comes back as an Eb6, but another note is deleted. Then the E5 (which has become a D5) is one more time deleted, and will never be undeleted again until the end of the process, so that there will never be a ten note motive in this part.

Representation of the process

In order to give an appropriate representation of the process, we must point out an important distinction between the operators. The delete operator operates on the present state, whereas the decrease and displace operators operate on the previous state. For this reason, we must distinguish two levels:

1) an underground level in which all states are ten note motives. At this level, the decrease and displace operators transform the previous state to compute the current state; 2) a surface level which corresponds to what can actually be found in the score. At this level, the states of the underground level are modified by the delete operator to produce the motives played by the different instruments of the texture.

The whole process can then be represented by the following diagram (Example 5). The reference motive is denoted m0. It is transformed by the decrease and displace operators to produce states m1, m2, m3 and so on, at the underground level. The delete operator is applied to each state of the underground level to compute the sequence of states w0, w1, w2, w3 and so on, at the surface level.

EXAMPLE 5. Representation of the process with two levels

Computer reconstruction in Lisp

The simple process described in Example 5 can easily be computarized using a functionnal programming language such as Lisp. The basic elements in functionnal programming languages are functions. Our three operators can be considered as functions, taking a state of the motive as argument, and returning a new state as value. But Lisp also offers powerfull programming devices which are well adapted to our problem: second order functions, which are functions taking other functions as argument. The two basic functions of this type in Lisp are generally denoted map and reduce. For instance, the map function takes as argument a function and a list, applies the function to each element of the list, and returns as value the list of the results. By considering the process represented in Example 5, one can see that the task realized by the map function is exactly what we need to describe the action of the delete operator. In this case, the arguments of the map function are the delete operator, and the list of states at the undergroung level. The result is the list of states at the surface level, i.d. the list of motives actually found in the instrumental parts of the texture. The other classical Lisp function generally denoted reduce, corresponds to a process which is closely related to the action of the decrease and displace operators, as represented in Example 5. But in this case the reduce function must be adapted. This has been done by Gérard Assayag to provide a complete computer implementation of the process, in a quite simple manner by means of the powerfull tools specific to functionnal programming.

The Lisp environment used for the computer reconstruction of the process is PatchWork. PatchWork combines a Common Lisp interpreter, an elegant graphical interface in which functions are represented as boxes with input/output, and special boxes for music notation and Midi connexions. This system running on a Macintosh has been developped at Ircam. In Example 6 one can see the full process represented as a patch. The four virtual motives are stored in chord boxes. One can verify that the ten notes are the same in the four boxes. These boxes can be open, and an arpeggio representation of the chords would then appear, showing the different permutations of the ten notes used in each instrumental part of the texture. The process described in Example 5 is represented in this patch by the box named processus. It takes as input the virtual motive of the corresponding instrumental part, and the data for the action of the three operators, which are stored in files represented in the patch by the boxes piccolo, xylo, celesta, violon. The output value returned by the processus box is the sequence of notes played by the corresponding instrument. It is then processed by the rythme box, which computes durations for this instrumental part of the texture. The final accelerando is not computed: it is stored as data in the const boxes. The four instrumental parts of the texture are collected in the trame box. Some melodic elements have been added to the resulting polyphony: they are stored in the melodies box. Part of the string background is stored in the tenues box.

EXAMPLE 6. Computer reconstruction in PatchWork

More informations about this patch can be found in a paper written by Gérard Assayag [START_REF] Assayag | CAO : vers la partition potentielle[END_REF]. A simulation of the passage has been played on a Yamaha SY77 synthesizer connected to PatchWork running on a Macintosh.

Two remarks to conclude

The first remark concerns the melodic elements, which seem to be closely related to the texture. As we have already pointed out, the trumpet plays a short melodic element in measure 22, consisting in a chromatic motion from E5 to D#5. If we considere the two states of the xylophone part which are played at the same time, one can notice that the first one (state 10 which ends at the begining of measure 22) contains the E5, and that the second one (state 11) contains the D#5 (denoted Eb5). This means that the decrease operator has been activated (a half step down). We could then say in this case that the melodic element played by the trumpet emphasizes the decrease operator.

As a second remark, I would like to mention the fact that the process described in this presentation does not pretend to be the exact description of what Ligeti did to compose this passage. It only gives a coherent explanation of the musical content of the passage. However, Ligeti has heard the computer reconstruction of the passage (at Ircam). He said "it is correct". But he also made some comments about the way he composed Melodien. The whole pitch content of the piece was defined as an harmonic skeleton. From this harmonic skeleton, he then chose the pitches actually played by the different parts of the full orchestral score. I think it would not be too unrealistic to assume that this informal operation (to chose pitches from an harmonic skeleton) can be related to the delete operator that we have described, and to the decomposition of the process into two levels we have introduced: the underground level, and the surface level.

Addendum to the present paper

An interesting question was pointed out by Eytan Agmon during this presentation at the Symposium: is There exists a simple recursive formula which computes all the different ways a given word is obtained as a subword of another given word. The notion of subword may be adapted to formalize the relation between the motives wi occurring in the instrumental parts of the texture, and the corresponding ten note motive mi at the underground level. One can then write a program computing all the underground sequences which can be associated to the surface sequence corresponding to one part of the texture. Unfortunately, there are many thousand solutions! Our analysis of the texture should then be improved. One of the problems could be the fact that we have treated separately the four parts of the texture. A more precise analysis can be obtained by considering the surface level as the sequence of all the motives occurring in the four instrumental parts. If we compute all the underground sequences which can be associated to this new surface sequence, the program gives a surprising result: there is only one possible solution. The uniqueness of the solution means that the delete operator does not lose too much information: it is possible to restore the underground level in a unique way. This underground level produced by the computation could be considered as the "harmonic skeleton" of the passage. A full description of this analysis would exceed the limits of the present paper.

  the derivation proposed in Example 2 the only solution to explain the process? In fact, the answer to the question is no. Example 7 shows a derivation from state 15 to state 19 of the xylophone part, which is different from the one proposed in Example 2. The E5 in state 19 comes from the F5 in state 17, whereas in Example 2, it came from the E5 in state 15. The derivation in Example 7 can be substituted to the corresponding derivation in Example 2, to produce a new description of the process, different from the previous one, but correct according to the principles we have introduced. EXAMPLE 7. Another derivation (xylophone part, state 15 to state 19) It is possible to compute all the possible derivations describing the process involved in each part of the texture. This can be done by introducing the notion of subwords of a word, i.d. words obtained by deleting letters in a word. For instance, the word aba may be obtained as a subword of ababa in different ways: aba(b)(a), ab(a)(b)a, a(b)(a)ba, and (a)(b)aba.

Notes

[*] This paper is an extended version of a conference presented at the International Symposium on Music andMathematics, Bucharest, May 29-30, 1994, which was published in Muzica VI, no. 2: 34-48 (1995). Many thanks to Suzanne Winsberg who corrected my english. The computer reconstruction of Ligeti's fragment has been done in collaboration with Gérard Assayag (Music Representation Group, at Ircam).


Wind instrument provide interesting hydrodynamical systems where non-linearities are important but well localized. A simple analysis shows that these systems should undergo Feigenbaum-type route to chaos, with a cascade of period doublings. Experiments have been performed with an acoustical resonator and an "artificial" excitation (nonlinearities controlled by either analogic or digital devices); they have confirmed these predictions.

Many musical instruments are in essence nonlinear physical systems, since their role is to convert a (quasi) constant force, pressure, etc. applied by the player into an oscillatory sound wave. For a physicist, the study of wind instruments is actually nothing but a subfield of hydrodynamics, since it amounts to studying the motion of the air inside and outside the instrument; the vibrations of the walls of the acoustical cavity are known to play a negligible role in the direct radiation of sound in the air, at least for woodwind instruments [1] . Important progress has been made recently on nonlinear hydrodynamics, instabilities, turbulence, etc. (for a review, see for example ref [2] or [START_REF] Berge | L'ordre Dans le Chaos[END_REF] ) and one could wonder how these ideas could apply to the study of musical instruments. One specific property of wind instruments is that the nonlinearities are indeed important, but well localized-mostly in the excitation system [4 , 5] ; this contrasts with ordinary hydrodynamics where they are present at every point in the fluid. The aim of the present work is to study how these localized nonlinearities can drive instabilities and chaos. A related work on acoustical chaos, including sound propagation but not acoustical resonators, is described in ref. [START_REF] Yabuzaki | [END_REF] .

The physical study of musical instruments is not new and much basic work was already done in the last century [7 , 8] or 50 years ago [START_REF] Bouasse | Instrument à vent[END_REF] . More recently, a nunber of advances in our understanding of wind instruments have occurred; for general reviews, see for example ref. [START_REF] Benade | Foundamentals of Musical Acoustics[END_REF] , [START_REF] Nederveen | Acoutical Apects of Woodwind Instruments[END_REF] or [1] . Examples of important new results are the studies of the role of the cut-off frequency of hole lattices [START_REF] Benade | Foundamentals of Musical Acoustics[END_REF][START_REF] Benade | [END_REF] , and of good impedance peak co-operation [START_REF] Benade | Foundamentals of Musical Acoustics[END_REF]12] . Here, we take a model of the functioning of a woodwind instrument which does not necessarily include all these physical effects, but gives the simplest description of nonlinear generation of sound from a constant pressure. The model we use draws heavily from the work published in ref [13] and [14] (sections II-B and II-D); see in particular the short discussion of period doubling in appendix A of ref [14] . Among the instruments, we choose to concentrate on the clarinet, because it is simpler in many respects: its acoustical cavity i approximately cylindrical and the excitation mechanism uses a simple reed; double reeds for the oboe or bassoon, or flutelike excitation mechanisms are intrinsically more complicated. A clarinet reed is merely a thin, almost flat, flexible wedge of cane which moves under the effect of the pressures difference applied on both its sides (it tends to "close" or "open"); this changes the coupling between the applied pressure P 0 inside the mouth of the player and the pressure p inside the acoustical resonator. A simple way to characterize the reed is to use a nonlinear function, f, which gives the flow of air entering the clarinet as a function of the pressure difference p = P 0 -p; see fig. 1a). For a detailed discussion of the shape of the curve which gives f as a function of p, the reader is referred for instance to [START_REF] Benade | Foundamentals of Musical Acoustics[END_REF] and [START_REF] Nederveen | Acoutical Apects of Woodwind Instruments[END_REF] . We simply note here that this curve includes negative (differential) impedance parts; as is well known in electronics for example, such negative impedances are required to sustain permanent oscillations.

In the model, we include only two variables, the pressure p and the flow f, which are both functions of time t; more precisely, we define f as the flow divided by the cross-section of the cylindrical resonator (f then gives the air velocity). Since we ignore the dynamics of the reed, we simply write as a first equation 2) provide a closed system: (1) is nonlinear, but local in time; on the other hand, ( 2) is linear but includes multiple time delays, as we now discuss.

f(t) = F[p(t); P 0 ] (1)
An idealized resonator [1] can be obtained by considering a tube with constant section ending on a small acoustical impedance Z 0 , assumed to be real, positive, and frequency independent (Z 0 = c is the acoustical impedance in the open air). If any other source of dissipation (viscosity, heat conduction) is ignored, one easily obtains where is the Dirac peak function, and T= 2L/c is the time taken by a sound wave to travel twice the length L of the tube. Clearly, this form of G(t) implies multiple delays, and complicates the solution of the problem. As discussed in [13] and [14] , an elegant way to avoid this difficulty is to introduce two new variables X + and X -, the amplitudes of the outgoing and ingoing waves: Equation (2) then becomes: where R( ) is the so-called "reflexion function"; its Fourier transform R(w) is obtained from Z(w) by the homographic transformation For the idealized resonator considered above, eq. ( 3) leads to with only one time delay. The convenient variables are, therefore, X + and X -rather than p and f. Equation ( 1) can be revwritten where the graph of F is obtained from that of F (with variables p and Z 0 f) by a 45º rotation around the origin. Now, if we set and make use of ( 7), we obtain the final equation

The approximations that we have made lead to a particularly simple result: the time dependence of the variable Y = -X -(t) is given by an iteration with the F function. We shall discuss here only particular solutions (it is easy to generalize), steplike solutions which remain constant and equal to Y n in any time interval nT < t < (n + 1) T, and then jump to the next value

The corresponding geometrical construction is shown in fig. lb). When the time dependence of X -is obtained, equation ( 8) gives X + and (4) allows one to come back to p and f. The time variations obtained in this way are similar to those shown in fig. 5 of ref. [14] .

Iterations and Feigenbaum-type scenarios [15] appear often in the study of strongly dissipative systems [START_REF] Berge | L'ordre Dans le Chaos[END_REF] . This is not the case here, since is usually small (typically less than 0.1) and can even be put to zero. Another difference with the usual case is that the "control parameter" P 0 does not correspond to a "gain" on the nonlinear curve, but to a translation along a line at 45º to the axes. For small values of P 0 , the fixed point (intersection of the 45º line with the curve) i stable, since it has a slope between -1 and + 1: no oscillation occurs. When it reache the maximum of the p-f curve of fig. 1a) (that is when the differential impedance changes sign), the fixed point F become unstable: the clarinet starts to sound. The two values between which Y oscillates are obtained by intersecting the p-f curve with its symmetric with repect to the f-axes. If the curve has a steeper slope in the small pressure difference region, as in fig. 2a), small oscillations between almost equal values of Y are obtained just above threshold. On the other hand, if the opposite occurs, the bifurcation takes a different character, since the system jumps suddenly to a large level of oscillation (fig. 2b)). Fortunately, as noted by the authors of ref. [14] , the f curve of real clarinets belongs to the first class of asymrnetry [START_REF] Benade | Foundamentals of Musical Acoustics[END_REF] , which ensures the possibility for small oscillations and pianissirno playing. It is well known [15] that the stability of a 2-cycle of oscillations can conveniently be discussed in terms of the second iterate of F for which the 2-cycle corresponds to two stable fixed points Q and R shown in fig. 3. When the slope at these points exceeds 1 (in absolute value), the cycle becomes unstable and gives rise to a 4-cycle: a period doubling occurs in the system. Period doubling should, therefore, in principle be observable in clarinet-like systems. Nevertheless, in practice, with f-p characteristics similar to that of fig. 1a), our numerical calculations have shown that period doubling can only be obtained in a rather narrow range of the parameters, which might explain why the phenomenon does not seem to be known among real clarinet players.

To do an experiment, it was, therefore, natural to try to obtain more flexibility on the nonlinearity introduced by the acoustical excitation system. The principle involved is shown in fig. 4; an ordinary acoustical resonator was used, in several cases a real clarinet, but the nonlinear reed excitation was replaced by an electroacoustic equivalent: a microphone measuring the acoustical pressure p inside the tube, a nonlinear system to include controlled nonlinearities in the feedback loop, and finally a loudspeaker to create a p-dependent air velocity f. The nonlinear system was either an analog electronic device, as in ref. [START_REF] Yabuzaki | [END_REF] , or an included digital processing of the data with a computer. The latter case gives even more flexibility on the choice of the nonlinear function, but requires using analog-digital converters, which introduces time delays (of the order of 100 µs in our experiment with two converters and the 4C computer of the IRCAM). Figure 5 shows the results obtained in an experiment where the nonlinearity was obtained with an analog electronic circuit generating the function x 3 -x; the resonator was simply a plastic tube (length: 14 cm; inner diameter: 2 cm) The results shown in fig. 5 are the Fourier spectra provided by the spectrum analyser; they correspond to increasing values of the gain in the nonlinear reaction loop. A series of three successive period doublings was obtained, the last one resulting in a very low pitch sound (70 Hz), which is very unusual with such a short resonator tube. All the period doublings could be clearly heard as sounds jumping one octave below, superimposed on some random acoustical noise, especially for the third bifurcation which is rather unstable. Beyond this bifurcation, acoustical chaos could be seen on the spectrum analyser and heard in the laboratory. The number of observed period doublings might have been limited by the narrow domain of the experimental parameters in which they occur, but also by the poor efficiency of the electroacoustic components at low frequencies. The actual experiment is of course significantly different from an ideal experiment; for example phase rotations and band pass limitation are introduced by the loudspeaker. In practice, the time variations of the acoustical pressure have much smoother variations than steplike function. ALso, in the period doubling regime before the chaos limit, some random noise was already present, making the phenomenon slightly less clearly audible (and probably less interesting musically!).

Similar results were also obtained with digitally generated nonlinearities, including experiments with a real clarinet (with its mouthpiece removed). In the latter case, we never obtained more than two period doublings. This is not extremely surprising with a real clarinet, the bore is not cylindrical (later holes, bell, etc.), and the reflexion function cannot be well approximated by one delta-function. We were never able to observe any period tripling in any of the experimental conditions. A more detailed report on the experimental results obtained in various situations (e.g with different sorts of nonlinearities) will be published elsewhere. A natural prolongation of these experiments would be to use several acoustical resonators in parallel, to study for example the evolution from quasi-periodic to chaotic regimes, etc.

In conclusion, although our mathematical model and the geometrical construction used in fig. 1, 2 and 3 are limited by the various simplifications on which they rely, the experiments show that the essence of their predictions for various regims of oscillations remains valid in more realistic situations, including nonideal resonators such as a real clarinet. More generally, woodwind instruments belong to a class of nonlinear systems which exihibit interesting behaviour, especially if one allows for more flexible nonlinearities. It has been known for a long time [START_REF] Helmholz | On the Sensation of Tone[END_REF] that woodwind instruments can generate harmonic frequencies which do not correspond to any resonance of the resonator (e.g the even harmonics in a clarinet tone; the effect is simply a frequency doubling occuring in the nonlinear excitation). The effects observed here show that even the fundamental tone of the generated sound can also, fall at a frequency which is much below any resonance of the tube.

The authors are grateful to J. KERGOMARD and G. WEINREICH for many stimulating discussions and helpful advices.

Notes [1] For a real resonator, the losses are introduced by gas viscosity and conduction, and not localized at the end of the tube; can be seen as a mere (positive) convergence factor, useful to take the limit 0 + . Also, for real resonators, there are frequency-dependent end corrections, the bore diameter is not necessarily exactly constant, etc., so that the reflexion function is not a -function; nevetheless, it has a significantly shorter memory than G( ); see examples in [13] .

Fig. 1 .

 1 Fig. 1. -a) Nonlinear characteristics of the excitation system, giving the air flow f as a function of the pressure difference p = P 0 -p across the reed. When p = p c the reed closes and f vanishes. b) Geometrical construction giving the successive values of p and f, obtained by a nonlinear iteration using the function of a) after translation and a 45º rotation. O, 1, 2, 3 are successive iterates. F is an (unstable) fixed point. The case shown corresponds to no dissipation ( = o). The second equation is obtained by writing a boundary condition which expresses the fact that the acoustical wave is a purely outgoing wave from the instrument (energy flows out from the bell and the open holes). If f(cL) and r() are the Fourier transforrn off(t) and p(t), this gives

Fig. 2 .

 2 Fig. 2. -Geometrical construction of the permanent oscillation values just above threshold, obtained by intersecting the characteristic curve with its symmetric with respect to the f-axis. In fig. a), small oscillations are obtained just above threshold, but not in fig. b)

Fig. 3 .

 3 Fig. 3. -The second iterate of F;Q and R give the permanent regime of normal oscillation, until they become unstable (slope less than -l); then, period doubling occurs.

Fig. 4 .

 4 Fig. 4. -Sketch of the experimental set-up. The reaction loop includes a microphone M, a variable gain amplifier, a nonlinear system NL (either an analogic circuit or a digital device including AD converters and a computer), a power amplifier A and finally a loudspeaker LS. The signals are analysed through a spectrum analyser SA.

Fig. 5 .

 5 Fig. 5. -Results of our experiments showing the spectra obtained on the SA for increasing values of the gain in the feedback loop. a) Normal oscillation; b), c) (not shown) and d) are, respectively, the first, second and third period doublings; e) chaotic regime.


Gabriel Weinreich

Rene Caussé

Elementary Stability Considerations for Bowed-String Motion

Using the approach first pioneered by Raman, the Helmholtz motion of a bowed string is discussed as a special case of "two-velocity motions," in which a given point (at which the bow is located ) alternates, in the course of a cycle, between two constant velocities. The fact that the bow typically presents a negative resistance to the string during the "slipping" part of the cycle is adduced as a reason for the "duty cycle," that is, the fraction of the period that corresponds to slipping, to try to become as short as possible. It is shown that, for a string without dissipation or stiffness, this duty cycle can be arbitrarily low for general bow positions; data obtained with the "digital bow" illustrate this behavior. It is shown theoretically, and confirmed with computer simulations, that instabilities arising from the negative slipping resistance cannot be eliminated by assigning a finite positive value to the sticking resistance. The apparent stability of Helmholtz motion observed in real playing situations remains a puzzle.

The Ideal Model

Introduction

The "Helmholtz motion" of a string is one in which the string consists, at any instant of time, of two straight-line segments, joined by a constant slope discontinuity or "kink" that moves at the speed of transverse waves from one end of the string to the other and returns with opposite sign. At any given location, the string velocity alternates between two constant values that are opposite in sign and whose ratio is equal to the ratio of distances from the point in question to each of the string terminations. This type of motion was first observed experimentally, and discussed theoretically, by Helmholtz. [START_REF] Helmholtz | On the Sensations Of tone[END_REF] The fact that at each location on the string, including an imagined "bowing point," the Helmholtz motion corresponds to an alternation between two constant string velocities makes it easy to conceptualize it as an embodiment of the kind of"stick-slip" phenomenon that we associate with the dry friction between string and rosined bow. In particular, one usually thinks of the interval of smaller velocity (which lasts for a longer time) as "the string sticking to the bow;" the larger, oppositely directed velocity then corresponds to "the string slipping back," only to be "caught" by the bow once again.

Considerable advances have added to our understanding of bowed-string motion in recent years; [START_REF] Cremer | The Physics of the Violin[END_REF]- [START_REF] Mclntyre | On the fundamentals of bowed-string dynamics[END_REF] among them, computer simulations [START_REF] Schumacher | Self-sustained oscillations of the bowed string[END_REF]- [START_REF] Mclntyre | On the fundamentals of bowed-string dynamics[END_REF] have shown how, under various realistic conditions, Helmholtz motion, or some close approximation to it, indeed arises when a string that is originally at rest has a bow applied to it. In fact, such simulations have provided considerable fine detail on likely realistic regimes of motion. Yet, for some purposes, such approaches give too much information in that the elementry "stickslip" concept is not always easily applied to their understanding. Our purpose in this paper is to develop a somewhat different conceptual approach, which, in our opinion, can do much to clarify the discussion of why Helmholtz motion is to be expected and under what circumstances it is stable. The point of view we use has some overlap with older work of Raman, [START_REF] Raman | On the mechanical theory of the vibrations of bowed strings and of musical instruments of the violin family, with experimental verification of the results: Part I[END_REF] but is developed considerably further.

We begin by reviewing briefly the well-known simple model of the bowed string as presented, for example, by Cremer. [START_REF] Cremer | The Physics of the Violin[END_REF] For this purpose, we assume, first, that the string is linear, dispersionless, dissipationless, and terminated by absolutely rigid supports; second, that the frictional characteristic between the bow and the string can be specified as a curve that gives the frictional force as a function of relative velocity at that instant. The same curve then specifies the frictional force as a function of absolute string velocity if it is offset horizontally by an amount equal to the steady bow speed ( Fig. I ) .

Our frictional characteristic is drawn so that its "sticking" portion is not vertical but has a finite slope; that is, we do not assume that "sticking" automatically implies that the relative velocity between string and bow hair is identically zero. In fact, an infinite slope is not required for further development of the theory, but can always be inserted later as a special case. Specifically, "the digital bow" (which supplied some experimental data to be quoted below) has an adjusta-ble frictional characteristic whose slope is ordinarily not infinite.

Physically, on the other hand, the concept of"sticking" does imply that the force-velocity characteristic has an infinite slope FIG. I. Generic frictional characteristic, showing the (negative of) the force exerted by the bow on the string as a function of the string's velocity Q a a and Q g are the point corresponding lo sticking ("adhérence")and slipping ("glissage"). The line Q a Q g is horiontal and is divided by the force axis in the ratio /(1-) for Helmholtz motion or, in the case of a general twovelocity motion, in the ratio /(1-); R a and R g are the two "resistances" that is, the slope at the operating points.

Thus, for eample. it was pointed out by McIntyre [6] that one can use the bow to pull the string to one side and hold it there for a practically indefinite length of time. Yet, there are a number of factors that. in practice, may produce an "effective slope" which is not infinite. One is the torsional motion of the string; [7]-[8] another is the dynamical vibration of the bow system itself, which may make the instantaneous velocity of the bow hair deviate from the nominally constant velocity of an idealized player's wrist. In the latter connection, Cremer [8] argues that effects due to longitudinal (stretching) vibration of the bow hair themselves are probably negligible; nonetheless, other motions of the bow system, such as bending of the stick, cannot be as easily ruled out.

One should note that the assumption of a dispersionless string is equivalent to the absence of any stiffness, that is, to any contribution to the potential energy that depends on the curvature of the string. Without such an assumption, any configuration that contains a kink would imply an infinite potential energy and would thus be ruled out immediately. Under these assumptions. the Helmholtz motion (as defined in the first paragraph of this paper) is a possible solution of the free string equation, that is, a possible motion of the string without a bow [START_REF] Helmholtz | On the Sensations Of tone[END_REF]-[5]- [START_REF] Cremer | The Physics of the Violin[END_REF] . Having established this fact. we must ask whether, and how, it is also a solution of the equations that govern the system when the bow is applied.

Let L be the total length of the string, and L and (1-L) the distances from the bowing point to either string end. Since the Helmholtz motion requires the string at the bowing point to move with one constant velocity for a fraction of a period, then with a different (but again constant) velocity for the remaining 1of a period; and since we have assumed a frictional characteristic that unequivocally specifies the force as a function of the velocity; it follows that the bow will exert one constant force on the string for a fraction and another constant force for the remaining fraction 1-of the period. In fact, these two constant forces must be equal to each other; otherwise, the bow would do a net amount of work on the string in the course of a period, which is impossible if the motion is steady and there is no dissipation. On this basis, we can say that the two points marked Q a and Q g in Fig. 1, which characterize the two states normally called "sticking" and "slipping," must lie on the same horizontal line. (Since "sticking" and "slipping" have the same initial letter, we use the subscripts a and g derived from the corresponding French words "adhérence" and "glissement." ) At the same time, we know that the two velocities of the string at the bowing point must have magnitudes inversely proportional to the time spent in the respective states, so as to make the net displacement in the course of a complete period vanish. For Helmholtz motion, this means that the two velocities must be in the ratio -/(1-). Given the frictional characteristic and the speed and location of the bow, the two requirements that Q a and Q g lie on the same horizontal line, and have abscissae whose ratio is as given above, in most cases determine the location of these two points.

We ask next whether the Helmholtz motion specified by this construction is dynamically possible; for, although it is known to be a solution of the free string equation, the string is now no longer free because the bow is applying a force to it. Since, however, this force is constant in time. its effects can be taken care of by superposing upon the Helmholtz motion a stationary configuration consisting of two straight line segments with a fixed kink at the bow location, and having a slope discontinuity corresponding to the static displacement of the same string under the same constant localized transverse force. Because linearity of the string has been assumed, the superposition of this stationary kink upon the moving Helmholtz kink now constitutes a solution for a string which is free except for the application of the (time-independent ) bow force [9] .

II. Other two-velocity Motions

It should be noted that the foregoing argument, which establishes the Helmholtz motion (plus a stationary kink ) as a possible solution of the ideal string with a bow applied at location L, works equally well for any other motion of the free string which exhibits, at the bow location L, one constant velocity for some fraction y of a period and some other constant velocity for the remaining fraction I-. It is not necessary that be equal to , as is true for the Helmholtz motion itself. Of course, the construction that determined the bow speed to which such a motion corresponds would be somewhat different. requiring that the force axis divide the line segment Q a Q g (Fig. 1) in the ratio /(1-) rather than /(1-). Indeed, as was already shown by Raman, [START_REF] Raman | On the mechanical theory of the vibrations of bowed strings and of musical instruments of the violin family, with experimental verification of the results: Part I[END_REF] there do exist other possible motions of a free string for which the velocity at a given location alternates between two fixed values (although the Helmholtz motion is the only one for which such a statement can be made about every point of the string). To clarify this, we ask the following question: Given an arbitrary function of time, under what circumstances can it correspond to the velocity of a given point of the string when it is moving freely

We know, of course, that any vibration of the free string can be expanded in normal modes; and since we have assumed that there is no dispersion and that the endpoints are fixed, the period of the nth normal mode will be /n, where is the string's fundamental period. Clearly, any superposition of such modes will still have an overall periodicity , from which it follows that a function which purports to represent the time dependence of velocity of a given point must itself have that periodicity. This is an obvious necessary condition.

It this condition is satisfied, we can expand the given function as a Fourier series; we then interpret each coefficient as the strength of the corresponding normal mode at the given string position. In this way an arbitrary periodic function can be seen as all expansion in the normal modes, and hence a possible free motion of the string, except for one restriction: If the given point is an exact node of some normal mode, then the corresponding Fourier component cannot be represented in the velocity function. More specifically, if the point in question lies at a fraction m/P of the string length (where m and P are integers), the harmonics P, 2P, etc., must be absent from the velocity as observed at that point. (One may ad that under any circumstances, the "zeroth harmonie" must be absent from the velocity to prevent any secular drift of the string at least so long as the terminations are fixed.) With this understanding, we now ask what the possible free string motions are for which the velocity at location L alternates between two fixed values.

Let , as above, be the "duty cycle" of a proposed two-velocity motion, that is, the fraction of a period spend in one of the two states. [Again, to preclude a secular drift, the two-velocities must have opposite signs, and their magnitudes must be in the ratio ( I-)/ ).] We then see from the foregoing discussion that, if is not rational, any duty cycle can be constructed, but, for rational, this is not the case. We show in Appendix A that, if = m/P, where m and P are integers and the fraction is in its lowest terms, the possible values of are limited to integer multiples of 1/P. In particular, the shortest duty cycle of a free string vibration which is a two-velocity motion at = m/P is = I /P.

In physical cases, of course, the qualitative distinction between rational and irrational values of cannot be meaningful. Instead, we would expect the behavior of the string to differ between a regime of "small P" and one of "largeP," the comparison being with some "critical P" determined, perhaps, by the rounding of the Helmholtz kink due to string stiffness or bow width.

III. The "Staircase" Function

The discussion of the previous section makes it clear that the velocity function at = m/P must lack the harmonics of order P, 2P, etc. This does not in itself mean, of course, that the corresponding normal modes are necessarily absent from the string; if the dissipation is zero, as we have assumed, their amplitudes remain indeterminate. It is clear however that a bow of zero width placed at a certain location is not capable of feeding any energy into modes which have nodes at that point. Therefore, if some dissipation does exist-even if it is small one may expect those modes to decay with time, so that they would not be represented in the steady-state motion.

If harmonics of order P, 2P, etc., are removed, the shape of the force exerted by the string on the bridge ( which can be thought of as the "signal" sent by the string to the violin body) will be qualitatively modified: The displacement very near the bridge will change from a "sawtooth" to a "staircase," in which the uniform ramp of the sawtooth is replaced by P horizontal "steps." [10]-11 It is true, of course, that if P is fairly high (as it would have to be for a bow at a normal playing position rather near the bridge), the quantitative modification may not be very large. Nonetheless, there is something unphysical about a phenomenon that seems to exist when is rational but disappears abruptly when it is irrational. As we shall see the existence of other two-velocity motions will allow us to reformulate the phenomenon so as to make its dependence upon more continuous.

IV. Stability

Having shown that any two-velocity motion is a possible motion of the bowed string (provided that an appropriate stationary kink is included and that is not rational), we now imagine that such a motion has been established, and ask whether it is stable; in other words, we ask what would happen ifa small perturbation were superimposed on it. The linearity of the system other than at the bow will allow the two to coexist without interacting; at the bow, however, the perturbation will cause additional forces determined by the slopes of the frictional characteristic at the two "operating points" Q a and Q g in Fig. 1 which correspond to nominal "sticking" and "slipping." It is clear from the general nature of the characteristic that the sticking point will have a positive slope, corresponding to a "positive resistance" that drains energy from the perturbation, whereas the slipping point will have a "negative resistance" that feeds energy into it. These effects will, of course, be most pronounced when the magnitude of the resistance is comparable to the characteristic impedance of the string. In this way, the stability of any given small perturbation over the course of a complete cycle can be discussed. One immediate corollary is that, if the "sticking" slope is infinite then the Helmholtz motion Is unconditionally unstable. This fact was already obtained by Friedlander using a completely different approach [START_REF] Friedlander | On the oscillations of the bowed string[END_REF] and in-terpreted by Mclntyre, Schumacher, and Woodhouse in terms of subharmonic generation. [START_REF] Mclntyre | Aperiodicity in bowed-string motion[END_REF] With finite slopes, the question of stability requires a more detailed calculation, which does not become trivial no matter what the values of the slopes. Consider, for example, the (not particularly realistic) case in which both slopes are from which one might be led to conclude that, if the string spends a fraction of its period with a resistance R g attached, and the remaining fraction 1-with a resistance R g , the net rate of exponential decay of the amplitude of a perturbation corresponding to the nth normal mode will be In this expression, R a and R g are, respectively, the sticking slope (which is positive) and the slipping slope (which is negative); Z 0 is the characteristic impedance of the string; is the (fractional) location of the bowing point; and is the dutycycle. (For a Helmholtz motion, and are, of course, equal.) This would lead us to formulate as a stability criterion for all modes.

In fact, as we shall see in more detail in Sec. VII, this criterion is by no means sufficient. The fallacy is that it is really necessary to define the "modes during sticking" separately from the "modes during slipping" and to reexpand the perturbation first in one set then in the other, twice during each cycle. The coherent behavior of these successive reexpansions can then cause instabilities whose overall periodicity is nowhere near any of the normal modes, as in the well-established subharmonic generation discussed by Mc-lntyre el al. [START_REF] Mclntyre | Aperiodicity in bowed-string motion[END_REF] It is also possible as we shall see in Sec. VII, to construct perturbations that do have the correct fundamen-tal frequency, but that are so distributed over the string as to be essentially absent from the bow location during the sticking part of the cycle. We refer to them as "between-the-rain-drops" motions, drawing on the metaphor of a person remaining dry in the rain by "running between the raindrops." Under those circumstances, the growth due to the (nega-tive) slipping resistance is not compensated by any decay due to the (positive) sticking resistance, making the perturbation considerably more unstable than the criterion of Eq. (3) would suggest. Appendix B demonstrates how motions of this type may be constructed.

Nonetheless, the qualitative idea expressed by Eq. ( 2) that a relatively short duty cycle (that is, a short slipping time) enhances the stability of a two-velocity motion generally remains valid. Such a condition corresponds to the statement made by Cremer, 14 that "the bow tries always to maximize the sticking time," even though the explanation given by him in terms of a simple friction-driven oscillator seems to us to be insufficient.

The following specific consequence of such a condition is worth noting immediately. There is, in the common description of Helmholtz motion (or of any other two-velocity motion), an ambiguity concerning the duty cycle in that it is assumed that, of the two fractions and 1-(which, for Helmholtz motion, are the same as, and 1-), the shorter one always corresponds to slipping and the longer one to sticking. Kinematically, however, an "inverted" Helmholtz motion is equally possible. If, for example, the bow is placed at a point that lies at 10% of the string length, one can construct a Helmholtz motion corresponding to it which slips for 90% of the cycle and sticks for 10%, with the bow speed corresponding to the larger velocity. Our discussion makes it clear, however, that such an inverted motion (which, to our knowledge. has never been observed) would be very much more likely to run into stability problems.

V. Connection with Experiment

Whatever the merits of the "ideal model" that we described in Sec. I, it is surely not an exact one, so that comparison with experiment must proceed with some care. Now it is commonly stated that a bowed string in steady vibration does exhibit Helmholtz motion; nonetheless, the observed transition between sticking and slipping is not instanta-neous, giving rise to the phenomenon of "rabbit ears" in the force. Since the ideal model calls for a force that is absolutely constant, and dissipation that is absolutely zero, relatively small deviations may give rise to important modifica-tions of the theory. Specifically. the clear theoretical proof that ideal Helmholtz motion with perfect sticking is always unstable leads us to ask which specific deviation from the model produces the stability that is experimentally observed: Is it the fact that the sticking is. in effect, not perfect (due perhaps to torsion [START_REF] Mclntyre | On the oscillations of musical instruments[END_REF][8][9][10][11][START_REF] Friedlander | On the oscillations of the bowed string[END_REF][START_REF] Mclntyre | Aperiodicity in bowed-string motion[END_REF] or other effects ), or the fact that the motion deviates kinematically from Helmholtz motion? It should be added that the presence of weak dissipation (which is surely there in a real case) cannot be simply taken care of by slightly raising the point Q a relative to Q g in Fig. 1. True, that would provide a net energy influx from the bow, but the force would no longer be constant in time, so that the Helmholtz motion plus a fixed kink would no longer be a solution of the string equations.

It is interesting, in this connection, to present some relevant data from our experiments with the "digital bow." In this setup, a real string is driven, not by a physical bow, but by an electronic system that substitutes for it by electrodyna-mically exerting a force that is determined by the string's instantaneous velocity at the imagined "bowing point." Specifically, the string's velocity is measured photoelectrically and the resulting signal digitized and sent to a computer. The computer, in turn, does a table lookup according to a previously programmed "frictional characteristic," then converts the corresponding "force" to an analog current that is sent through the string. By virtue of a permanent magnet mounted at the "bowing point," this results in a corresponding physical force exerted on the string. The process is repeated at a sampling rate up to 125 kHz.I In the present experiments, the string was 1.40 m long and had a fundamental frequency near 70 Hz; the magnetic field extended over a length of approximately 2.5 cm, which is to be thought of as the "width" of the equivalent bow; and the frictional characteristic was defined by the function.

The parameters in this function, taking account of all the amplifier gains as well as the sensitivities of the converters, had the (approximate) values F 0 =0.015N, 0 =0.05m/s. (5) The string itself (made of brass) had a linear density of l.68 X l0 -3 kg/m corresponding (for this length and fundamental frequency) to a characteristic impedance of 0.329 kg/s. Figure 2 presents data obtained in such a situation. The graph shows the velocity of the "bowed" point of the string as a function of time for three placements of that point, corresponding to =0.333 (dashed), =0.250(dotted),and , = 0.291 (solid). All approximate an alternation between "sticking" and "slipping," with the "sticking" velocity the same for all three cases and equal to the programmed bow speed. But whereas the observed duty cycle corresponds roughly to in the first two cases, it is clear thatthis is not so for the third (solid) curve: Even though , is intermediate between the other two, is smaller than either one.

Figure 3(a)-(c) corresponds, respectively, to the same three cases as Fig. 2; this time, however, we show the displacement of the string as a function of time, observed at two locations: the bowing point itself (top graph), and a point very near the bridge (bottom graph) . As we indicated above, the latter can be considered proportional to the slope of the string at the fixed endpoint and thus to the lateral force exerted by the string on the bridge. In Fig. 3(a) and (b) (corresponding to = 1/3 and 1/4), we see the characteristic "staircase" function near the bridge, corresponding to the fact that certain harmonics must be missing, as discussed above. In Fig. 3(c). on the other hand, there is (when observed near the bridge) a marked enhancement of certain harmonics, the effect of which is the narrowing of the duty cycle at the bowing point, in agreement with what our considerations of stability would suggest. This motion is quite similar to the one described by Lawergren When the "bow" (that is, the magnet-photodetector assembly) is moved slowly from, say, = l/4 to a "more irrational" position like = 0.291, one observes the wave shape changing continuously from that of Fig. 3 (b) to that of 3 (c). In that way, the large quasisinusoidal admixture to the Helmholtz motion seen in the lower graph of 3 (c) joins continuously to the staircase characteristic of 3(b) [or 3(a)], making the latter intuitively more palatable.

VI. Multikink Simulations

In order to clarify the experimental results of the previous section, we attempted to simulate results such as those in Fig. 3 by constructing low-motions as superpositionsof Helmholtz motions according to the procedure of Appendix A. Our computer program accepts and as rational fractions m/P and M/P; it then superposes the appropriate number of Helmholtz kinks with the right spacing and produces the displacement as a function of time, either at the bow location , or at a point very close to the end of the string. In this way curves are produced that should simulate, respec-tively, the top and bottom graphs in Fig. 3.

Figure 4(a)-(c) shows computed results corresponding to the graphs in Fig. 3. For (a) we took m = l, P= 3, M = 1; for (b), m = l, P = 4, M = 1. For (c), was taken to be midway between the other two at 7/24, so that m = 7, P = 24; the choice M = 3 then produced the closest correspondence between experiment and simulation. In other words, Fig. 4(c) is a two-velocity motion at the bowing point whose duty cycle is somewhat less than half of its Helmholtz value, that is, 3/24 compared to 7/24. Naturally, the absolute phase in these simulations is arbitrary but was chosen to resemble as much as possible the data of Fig. 3. It should be noted that, since m = 1 for Fig. 4(a) and (b), no two-velociy motion of lower duty cycle is possible. However, in Fig. (c), is not as small as it can be. For comparison, Fig. 5(a) shows a pure Helmholtz motion for = 7/24, while 5 (b) illustrates the lowest possible duty cycle, corresponding to = 1/24 (that is, M = 1 ) for the same . Clearly, the observed motion, as simulated in Fig. 4(c), is narrower than the Helmholtz motion but not as narrow as is kinematically possible.

That there must be a dynamical lower limit onl for a given is, of course, obvious, since, as P becomes larger, the purely kinematical limit gets smaller and smaller, becoming zero for irrational values of . In attempting to explain this dynamical lower limit, three factors come to mind: the stiffness of the string. the width of the bow, and the presence of energy losses. All three constitute violations of the assumptions that we have made in defining our model.

Although we are not able, at this moment, to treat these complications in detail. We found it heuristically interesting to calculate the energies corresponding to the two-velocity motions of given , the idea being that, when dissipation is introduced. a larger stored energy may also correspond to a larger rate of energy dissipation. The larger power requirement from the bow may then make the motion more difficult to maintain.

The energy tored in a simple Helmholtz motion is easily calculated at a sum of kinetic and potential energy densities integrated over the string. If the maximum lateral displacement of the kink (which occurs when it passes the center of the string) is A 0 , the instantaneous string displacement at time t is given by where p is the linear density and T = c 2 is the static tension. When more than one kink is present, the energies cannot, of course. be simply added, since the individual kinks are not orthogonal to each other. Nonetheless, the calculation for any given superposition is straightforward, especially when done by computer.

On the other hand, the two-velocity motions for various but given , do not have the same lateral displacements; rather, they must (if the comparison is to be meaningful) be normalized to the same value of bow speed b . Since the construction that determines this depends on the frictional characteristic, as shown in Fig. 1, it cannot be given for a general case. Nonetheless, an excellent approximation is obtained by assuming, for this purpose, that the sticking slope is infinite. In that case, the bow speed is related to , the difference in velocities of the two-velocity motion, by b = (8) allowing the amplitude of the component Helmholtz motion to be determined.

Table I shows the computed results for various two-velocity motions with = 7/24 = 0.291, the case discussed above, normalized to the simple Helmholtz motion. It is seen that the energy does, indeed, rise steeply as is made smaller, making this a possible factor in putting a lower limit on duty cycles actually observed.

VII. Instability Simulations

As we mentioned in Sec. IV, the otherwise plausible idea that the (positive) sticking resistance R a , and the (negative) slipping resistance R g can somehow compensate each other in the determination of a stability criterion is not correct, since there exist possible perturbation patterns ("between-the-raindrops" motions, Appendix B) that can synchronize their appearance at the bowing point to coincide with the slipping parts of the cycle. To explore this phenomenon, we constructed a computer program to simulate perturbation of the Helmholtz motion in the linear regime, that is, when the perturbation is small compared to the Helmholtz motion itself. This is done by describing the string by an array along which the two waves-one moving to the right, the other to the left-propagate without change, except that at the point L they are scattered by a resistance R which has the value R g , for a fraction of the period and the value R a for the remaining fraction 1-. For this purpose, we use the elementry formulas that give the reflection coefficient at such a point as -R/(2+R), and the corresponding transmission coefficient a 2/(2+R), where R is, of course, measured in units of tht characteristic impedance of the string. We used a "string" of points, with = 7/23 and perfect reflection at the endoint. The initial form of the perturbation was taken, quite arbitrarily, as sinusoidal, correponding to the fundamental of a free string (that is one without a bow); other choices of initial conditions support the same conclusions.

Figure 6 shows some computed results. Each of the four graphs give the displacement of the point nearest one end of the string, plotted at the initial time and then once per period (that is, every 46 time increments) for a total time of 100 periods; they differ, however, in the values assigned to the sticking and slipping resistances. In Fig. 6(a) and (b), the slipping resistance is taken to be R g = 0, whereas the sticking resistance (expressed in units of the characteristic impedance of the string) is R a = 50 in (a) and R a = 0.5 in (b). The first shows a damped subharmonic that repeats every three periods; this is consistent with the statement made by Mclntyre et al. 14 that the possible subharmonic orders are the two integers closest to l/ , which, in our case, is 23/7 = 3.29. The damping is, presumably, due to the finiteness of R a ; we note, however, that as this perturbation dies away, the value asymptotically approached in the graph is not zero. The existence of this asymptote indicates another, undamped, perturbation which has the same values at times that are separated by one period; it takes place, in other words, at the fundamental frequency of the string. This is, of course, the type of perturbation we mentioned earlier and described as a "between-the-raindrops" motion, a conclusion that is supported by examining the shape of the perturbation at the end of the simulation computation.

In Fig. 6 (b) the slipping resistance is kept at zero but the sticking resistance is lowered to R a = 0.5. The subharmonic is now so strongly damped as to be essentially unrecognizable, yet the nonzero asymptotic value is not only still there but entirely unchanged in magnitude. as is to be epected from a perturbation that never sees the sticking part of the bow cycle at all. Finally, in Fig. 6(c) and (d), the computation is repeated after setting the slipping resistance to the mildly negative value R g = -0.05; in each figure, R a is the same as in the graph immediately to its left. It is clear that the previously constant asymptotic perturbation now becomes a growing one, since it is absent from the bow position when the bow sticks but present when it slips. The massive damping caused by R a in Fig. 6(d) has no appreciable effect on this growth rate, even though the amplification is quite small compared to this damping, and the duration of the slipping cycle is shorter.

VIII. Discussion

Effects connected with the narrowing of the duty cycle of bowed-string motion have also been observed with physical bows by Lazarus [START_REF] Lazarus | Die Behandlung der selbsterregten Kippschwingungen der gestrichenen Saite mit Hilfe der endlichen Laplacetransformation[END_REF] and Lawergren. [START_REF] Lawergren | On the motion of bowed violin strings[END_REF] The latter, in particular, interprets his observations in terms of a superposition of a pure Helmholtz motion and a single sinusoid of noninteger period which has nodes at the string endpoints and a slope discontinuity at the observation point. No theoretical reason is, however, given by Lawergren why this particular function should be preferred. In fact, Appendix A proves that, at an arbitrary location on an ideal string (one for which , is not rational), there is no limit to the narrowing of the duty cycle that is kinematically possible, and the exact nature of the observed dynamical limit is not yet understood.

It may be worth, in this connection, to mention another phenomenon observed with the digital bow by Chayé [START_REF] Chayé | Etude de l'interaction entre la corde et l'archet[END_REF] ; for the case in which is a ratio of small integers but the numerator is not unity. She found that, if = m/P, the duty cycle of the motion at the bowing point tends always to be 1/P. For example, for = 1/7, 2/7, 3/7, 4/7, 5/7, or 6/7, is always 1/7. In other words, in this case it is, indeed, the shortest that is chosen, consistent with the idea that, due to stability considerations, the motion which is preferred is the one that minimizes the time spent in the negative-resistance region of the characteristic. It is not clear to what degree this narrow-ing can be avoided on real violins by "correct" bowing; comparison with the digital bow is complicated by its relatively large width, as well as the fact that the magnitude of its force is limited by effects of string heating.

Our discussion of Sec. IV, along with the computer simulations of Sec. VII, indicates that the apparent experimental stability of Helmholtz motion remains a puzzle, in that (unlike what has been previously believed) this stability cannot be explained by changing the "effective" sticking resistance from an infinite to a finite value. As far as we can see, the truth must lie in one or more of the following three possibilities: (a) The stability comes from some other source of dissipation (such as losses at the supports); (b) the stability comes from other deviations from the "perfect" model, such as bow width. string stiffness, or perhaps bow dynamics: the latter would make it impossible to represent the frictional characteristic a an instantaneous. time-independent function: and (c) the Helmholtz motion is, in fact, not stable but its instabilities themselves stabilize at fairly small amplitudes so that casual experiments appear to suport the notion that Helmholtz motion is being executed. It should be noted in this connection that the simulations of Sec. Vll were computed in a linear regime and thus throw no light on what happens to the perturbations after they have begun to grow.

FIG. 2 .

 2 Figure3(a)-(c) corresponds, respectively, to the same three cases as Fig.2; this time, however, we show the displacement of the string as a function of time, observed at two locations: the bowing point itself (top graph), and a point very near the bridge (bottom graph) . As we indicated above, the latter can be considered proportional to the slope of the string at the fixed endpoint and thus to the lateral force exerted by the string on the bridge. In Fig.3(a) and (b) (corresponding to = 1/3 and 1/4), we see the characteristic "staircase" function near the bridge, corresponding to the fact that certain harmonics must be missing, as discussed above. In Fig.3(c). on the other hand, there is (when observed near the bridge) a marked enhancement of certain harmonics, the effect of which is the narrowing of the duty cycle at the bowing point, in agreement with what our considerations of stability would suggest. This motion is quite similar to the one described by Lawergren;'7 according to McIntyre et al.,18 it is on occasion musically useful as well.

FIG. 4 .

 4 FIG. 4. Simulted displacements at the bowing point (top)and at a point near the bridge (bottom) contructed as superpositions of Helmholtz motions, with conditions (including abslolute phase) chosen to correpond to the data of Fig. 3. (a) = =1/3; (b) = =1/4; (c) =7/24, =3/24.

FIG. 5 .

 5 FIG. 5. Simulilated displacement at the bowing point (top) and at the point near the bridge (bottom), constructed as in Fig. 4:(a) for the pure Helholtz motion at = 7/24, (b) for the shortest possible duty cycle at the same

FIG. 6

 6 FIG. 6 Perturbations computed in th linear regime for a "string" of 23 points, with = 7/23, for various value of the sticking and slipping resistances. The displacement of the point nearest one end is plotted once per period, for a total interval of 100 periods: (a) R a = 50, R g = O; (b) R a = 0.5, R g = O; (c) R a = 50, R g = -0.05; (d) R a = 0.5, R g = -0.05.

TABLE 1 .

 1 Relative energies of two-velocity motions, normalized to a fixed bow speed, for = 7/24 and varying . The energy for = is arbitrarily taken as unity. The middle column gives the number of Helmholtz kinks required to construct each motion.

	Number	Relative
	or kinks	energy

IX. Acknowledgments

We are grateful to M. Mclntyre and to J. Woodhouse for extensive discussions which not only improved this paper but also taught us a lot of good physics. We wish also to acknowledge the helpful comments of many other colleagues, among them the late Professor L. Cremer, X. Boutillon, and R. Schumacher; and to express our awestruck appreciation to Bennett Smith, who built and interfaced the digital processing equipment used for the digital bow.

Appendix A: Possible two-velocity Motions

We wish to prove the following theorem: If a free string moves in such a way that at location L, where = m/P and the fraction is in its lowest terms, the motion consists of two velocities that alternate in the course of a period, the possible values of the duty cycle are M/P, where M= 1,2,...,P-1. This result is clearly present in the work of Raman, but we find the following proof rather more accessible.

We begin by noting that = m/P is a node of all harmonics whose order is a multiple of P, so that the Fourier spectrum of the velocity observed at that point must lack those harmonics. To impose this condition, we first calculate the amplitude of the nth harmonic of a square pulse of duty cycle ; apart from an unimportant normalizing factor, it is given by

By setting A n ( ) = O, we obtain P = integer.

(10) Thus the Pth harmonic will vanish if and only if has the form M/P, where M is an integer. We note also that the vanishing of the Pth harmonic automatically guarantees the vanishing of all its multiples. as required.

If does have the above form, we know from the discussion of Sec. II that a corresponding two-velocity motion must exist. It may, nonetheless, be useful to facilitate its visualization by giving an explicit construction for it as a superposition of simple Helmholtz motions as advocated by Kubota. [START_REF] Kubota | Tables of numbers of the Helmholtzian waves on a bowed string[END_REF][START_REF] Kubota | Velocity measurement for the bowed string and plural Helmholtz waves[END_REF] The construction is as follows. Consider the multiples of m modulo P. If one such is equal to zero, it means that a certain multiple of m is equal to a certain multiple of P; and, since m and P have (by assumption) no common factors, it follows that the number by which m is multiplied to produce zero (modulo P) must itself be a multiple of P. In other words, if m is multiplied successively by 0,1,....,P-1, all of which are less than P, no two results can he equal (modulo P). On the other hand, there are exactly P numbers in the range of all integers modulo P, so that each of them must appear once and only once in the set of posible results of multiplying m by some integer and reducing the result modulo P. Let M be all integer in the set 1,2,....,P-1 . and N the integer such that Nm mod P = M. Consider now N simple Helmholtz motions, all of the same amplitude, coexisting on one string, and spaced apart by a fraction m/P of a period. In terms of a spatial distribution, it is convenient to viualize the string as an endless conveyor belt looped around a pair of pulleys located at the endpoints of the string, and moving with the speed of transverse waves on the string; half of the belt can then be thought of as carrying waves traveling to the right, the other half to the left. The N Helmholtz motions under consideration can then be represented as marks on the conveyor belt indicating the location of the kinks. Note that the total length of the belt is 2L, so that the spatial interval between successive kinks is 2(m/P)L.

Consider now the location (m/P)L along the string as measured from one end which, for convenience, we shall call the "bridge." Because the kinks are spaced apart by twice that distance, it follows that, whenever a kink passes this observation point moving toward the bridge, another will be passing it moving in the opposite direction. Since the velocity transition of a Helmholtz motion occurs whenever the kink passes the observation point, we see that in our present case each such transition will be canceled by an equal and opposite one. The two exceptions are when the first kink of the sequence passes in the direction toward the bridge, and when the last one passes in the other direction. Since there are N kinks, spaced apart by m/P of a period, the time interval between the first and the last, figured as a fraction of a period, is Nm/P. In order to reduce it to a propr fraction, we reduce the numerator modulo P; and since Nm mod P = M by construction, we see that the fraction of a period that elapes between the two transitions that do occur exactly M/P, as was required.

For the "multikink simulations" of Sec. VI, we used the above construction but modified it further by complementing each Helmholtz kink with Pnegative "subsidiary" kinks in order to produce the "staircase" behavior explained in Sec. III. This modification does not, of course, affect the motion observed at = m/P and is hence irrelevant for the theorem stated at the beginning of this Appendix. It is, however, important for describing the motion at other points and for the calculation of energies.

Appendix B: "between-the-Raindrops" Motions

The conveyor-belt picture of Appendix A is useful also for illustrating the existence of "between-theraindrops" motions, that is, possible motions of the free string for which the displacement at the bow position is identically zero except for a fraction of the cycle (which we associate with the slipping time of the bow). Now it is clear that, if = m/P, a rational number, then normal modes whose orders are multiples of P will have nodes at the bow position and thus give zero displacement at the bow at all times. We will therefore assume that is not rational, in order to avoid being confused by the existence of those modes.

Let our conveyor belt, whose total (unfolded) length is 2L, have associated with it a displacement pattern that is periodic with period 2 L; since is, by assumption, not rational, it is clear that somewhere there will be a junction point where this periodicity is broken. If we now imagine the belt mounted on the pulleys (of which one represents the "bridge") and put in motion with the correct wave velocity, the equations of the free string will be satisfied, in that the resulting displacement will contain a wave going to the right and another going to the left, each of which returns from its respective endpoint reflected with a change of sign (since the "top" of the conveyor belt going to the right turns into the "bottom" going to the left ) .

Consider now the "bow position" L. Since the two points of the belt passing it at a given moment are separated by a distance 2 L (measured around the "bridge" pulley), and since the pattern is, by construction, periodic with that quantity as period, the displacements corresponding to the left-going and right-going waves will, at each moment, be equal and opposite, resulting in a total displacement at the bow position which vanishes identically. This condition will not hold, however, when the above-mentioned junction point is passing from the "bow position" to the "bridge" and back again; and since this distance is a fraction, of the total belt length, the time for which the displacement is not zero will, be a fraction of the period. Thus the existence of between-the-raindrops" motions is proved.

It is worth noting that motions of even shorter duration at the bow position can be constructed by choosing displacement patterns whose spatial period along the belt is an integer submultiple of 2 L. In fact, it is easy to convince oneself (in analogy with the construction of short duty cycles as described in Appendix A) that, for the case when is irrational, motions exist which vanish at the bow position for all but an arbitrarily small fraction of the period.