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**ICMC 94, Aarhus (Danemark) 1994**

Copyright © Ircam - Centre Georges-Pompidou 1994

Finally, our last goal is to control the ratio of non sinusoidal components (noise) induced by chaos in synthetic signals, opening a new field of fascinating research and applications. Many systems which have been used as models of physical instruments for music synthesis can produce chaotic signals in some regions of their parameter space. This is related to instability since chaos often appears when all periodic solutions become unstable. The most innovative direction is probably in the area of the control of chaotic behavior of the models and thus of chaotic sounds. In the past it may have seemed that chaotic sounds could not be of any musical interest. To our great surprise we have found the contrary: these signals exhibit very interesting properties, such as a clearly perceived pitch or an intermittent type behavior. We have also found that a signal which is mathematically chaotic can be perceived in a very different way. Actually, chaos can happen within a more or less small neighborhood of a periodic trajectory. It can cover a large space, from a sound perceived as harmonic without noise, up to an essentially noisy sound. This is a very important feature for sound synthesis since the introduction in synthesis of noise and irregularities of natural instruments has always been difficult and unsatisfactory. Our models, by themselves, effectively generate a noise component that is so important for the sound quality of musical instruments.

In more general terms it appears that one should not merely "build" models and deliver them to musicians. It is indispensable to go into the understanding of the models, to conceive abstractions of them and to propose explanations helpful to the users. In particular, this comprehension is indispensable for elaborating the control of synthesis models, which are at the same time efficient and musically pertinent.

Similarly, the linear part of Chua's circuit [Chua 90], at least for some values of the parameters, can be viewed as a band pass filter, i.e. a low-pass filter above the frequency of a pair of conjugate poles, this frequency being approximately the oscillation frequency. This suggests some relation with the dominant mode of an acoustic instrument [Rodet 93b]. This idea has also been evoked in [Wawrzynek 83] but was not fully developed. It can be shown [Rodet 93e] that there is a precise correspondence between Chua's circuit and a model of an acoustic instrument where the feedback loop is limited to the first mode(s) (the first mode could easily be replaced here by the dominant mode in the same way). Naturally, the system so obtained has a unique oscillatory solution in a certain range of parameter values. Fig. 1 shows a possible implementation and the control of each partial through a second order section.

Fig. 1. Feedback loop limited to the first mode(s).

Some properties and musical uses of the Time Delayed Chua's circuit [Sharkovsky 93] have been presented in [Rodet 92c] and [Rodet 93d]. Compared to the original Chua's circuit, the time delayed version includes a delay line which is directly related to the structure and physics of many classical musical instruments. This delay allows for much better control of the circuit in terms of musical usage. The delay can also be seen as a stabilizing feedback loop applied to the original circuit. The Time Delayed Chua's Circuit exhibits a very interesting variety of sounds which we have analysed in [Rodet 93c]. This is due to the combination of the rich dynamics of the nonlinear map together with the number of states represented by the delay line. In particular, Sharkovsky et al. [1993] have shown a remarkable period-adding property. In some regions the system has a stable limit cycle with periods 2, 3, 4, etc.... In between every two consecutive stable regions the system exhibits a chaotic behavior. We also observe the simultaneous presence of sinusoidal components and noise in the signal. By a proper affine change of variables, the invariant interval of the nonlinear function used in the circuit can be set to the interval [0,1]. For certain parameter values, the function is composed of two segments only in the invariant interval [Rodet 93a] with slopes l1 and l2. When l1 = 1.025 and l2 is varied from 1 to 2 and then to 40, we observe the progressive appearance of non sinusoidal components superimposed on the harmonic sinusoidal components (Fig. 2).

Fig. 2. Spectra showing presence of sinusoidal

components and noise, l2 = 1.7 (top) and 30 (bottom).

This property is very promising since it opens the possibility of a precise control of the proportion of noise added to the harmonic components, which is essential for a musical usage. This is the typical property that we want to extend to nonlinear functions and systems beyond the very specific two segment function of Sharkovsky 's case.

We focus here on the class of systems composed of an instantaneous nonlinearity and a linear feedback loop, which is a valid basic model of reed instruments when the mass of the reed is considered negligible. But in the case of the trumpet [Rodet 92b] for example, this assumption cannot be maintained: as we noticed in [Rodet 93c], the clarinet and trumpet would then be represented by the same model. On the contrary, in [Rodet & Steinecke 94a] we propose a minimal one mass model of the trumpet in the form of a two-loop system. This system can also exhibit chaotic behaviors.

For speech production, various models have been proposed. The most common is a two mass model of vocal folds [Flanagan 72]. Using the impedance of the opening in the glottis and the driving point impedance of the vocal tract, air flow pressure and velocity in the glottis are computed and used to drive two coupled non-linear oscillators representing a vibrating vocal fold. This model usually exhibits bifurcations and chaos often related to the desynchronisation of lower order modes, e.g. of lateral and vertical motion, or to a left-right asymmetry [Steinecke & Herzel 93], [Rodet & Steinecke 94b].

where h is the impulse response of a filter in the linear feedback loop, * is the convolution operator and g(.) is the nonlinear function representing for instance the influence of the reed in a basic clarinet model [Rodet 92a]. The system can be decomposed into an instantaneous nonlinearity g(.) and a linear element including h and a delay. This defines the class of single feedback loop systems with an instantaneous nonlinearity of which we can easily determine the stability and some oscillation properties [Rodet 93c]. Note that the only restriction on the linear element is that its impulse response be stable . In particular the continuous transfer function of the linear element does not need to be a rational function and thus can include delays. Many systems can be redesigned to fall into this class. Equation (1) can also be written:

(3)

(3) defines a map Q:Rm2+T+1 Æ Rm2+T+1. Since we are dealing with physical systems, it is natural to suppose that {hm} is causal, m1 < T and finite and m2 < . The sample n depends on the past samples n-m2-T to n+m1-T. Therefore the set of samples from n-m2 to n+T-1 depends only on the values xn-m2-T to xn+m1-1, themselves depending on the values xn-m2-T to xn-1 only which defines the map Q. Such a system may have many periodic solutions. Since they are the most important for the normal playing conditions of a musical instrument, we study here only 2-periodic solutions, i.e. fixed points of Y=Q Q with period length about 2T. But finding the 2-periodic solutions of Q and their stabilities in general is not easy. Therefore we are going to introduce another system. Let {sn} be a 2-periodic solution of (3). Assume now that {hn} is even symmetric. We have noticed in [Rodet 93c] that in this case the period 2 is of length 2T exactly, i.e. such that sn = sn+2T for all integer n. Let us take the Fourier series of one period of sn and of g(sn) and call S={Sk} and G(S)={Gk(S)} respectively their Fourier coefficients which are vectors of length 2T. Then for {sn} equation (3) can be written:

Sk = (-1)k Hk Gk(S) (4),

where the term (-1)k accounts for the time delay T in (3), and H={Hk} is the Fourier series taken after time aliasing of {hn} on an interval of length 2T and overlap-adding.

From equation (4) we define a map M:B2T Æ B2T where B C is the set of Fourier series of real periodic discrete signals of period length 2T:

Mk(S) = (-1)k Hk Gk(S), k=1,2,...2T (5)

where Gk(S) is the kth Fourier coefficient of g(s) when s is the inverse Fourier series of S. In the time domain, we call P:R2T Æ R2T the map associated with M and defined by:

P(s)=Fourier-1(M(S)) (6)

where Fourier-1 designates the inverse Fourier series, or equivalently:

P(s)=h g(s-T) (7),

where h=Fourier-1(Hk) and is the circular convolution. The reason for introducing the maps M and P is that they have the same fixed points as Y=Q Q. They are simpler than Y in the sense that they have smaller dimension and that they map exactly the signal of one time interval of length 2T into the signal of the next time interval. The dynamical system that they define is different from the original one and we call it the period-after-period (p.a.p.) system as opposed to the original one (1). Let DYx be the derivative of Y at the point x Rm2+T+1. The stability of a fixed point s of Y is given by the eigenvalues of DYs. The problem is therefore to find all the fixed points of Y or equivalently of P and M.

The portion of g which we are interested in for easy control of oscillations is that of negative slope around the origin and greater slope further from the origin [Rodet 92a]. A drawback of the choice of a polynomial g is that it is not possible to avoid the existence of other points of functioning than the origin, which is a serious inconvenience for musical usage [Rodet 93d]. We will see in section 10 that a rational function allows us to avoid that difficulty and can be studied with the same tools which we are using for the polynomials.

g(u)= -u, u R. (8)

Among the solutions of M0(s)=s-g g(s)=0, s R2T, we retain only the vectors s such that sn W for n=1,2, , 2T. The other solutions imply a value x R, x 0, such that g(x)=x which we want to avoid since it corresponds to non-zero fixed points of the original system, which give no sound. Note that for the solutions which we retain sn= - sn-T for n=T+1,T+2, 2T. When a signal sn has this odd symmetry of the period with respect to its middle, we say that the signal is odd harmonic (o.h.) because its even harmonics are null. The initial solutions for the homotopy are easy to build since they are the combinations of values from W. Another reason for this choice of M0 is as follows. The homotopy can simply be chosen as:

F(s, r) = (1-r)M0(s) + rM1(s). (9)

Therefore, when varying r from 0 to 1, we follow the evolution of the solutions according to the increase of low-pass filtering in the feedback loop, which is one of the main goals we mentioned earlier.

Starting from one of these initial solutions and from r=0, we increase r by a small Dr, estimate the new solution by the Newton method from the previous solution and iterate this procedure until r=1. Using prefixed superscripts to index successive steps of the algorithm, let 0s be the initial solution, rs the solution estimated for the value r of the homotopy variable. The solution r+Drs is obtained as follows. Let z0=rs be the initial value for the Newton method, and DF the differential of F. We iterate:

zk+1 = zk - [DF(zk, r+Dr)]-1F(zk, r+Dr), (10)

until when zk+1 - zk is sufficiently small. Then r+Drs gets the last value zk+1. It is easy to derive expressions for the coefficients of the polynomials of F and to show that these coefficients never vanish for r in [0, 1] so that no solutions escape to infinity.

Fig. 3. Equivalence classes of initial solutions

Fig. 4 shows the successive values of rs for the equivalence class C , a1 = -1.35 and r from 0 to 1. Note that 1s is a solution of the original system (1), which is otherwise not easy to find in general. However, in case of class A the numerical estimation procedure does not succeed to go further than r = 0.965 for a1=-1.1. The reason is that the arcs issued from two different initial solutions join together at that value of r and disappear for larger values as explained in section 9. But as long as we can follow a real solution s, we can find its stability by computing the eigenvalues of DYs. Since Y=Q Q and the solutions under study are o. h., it is sufficient to compute the eigenvalues of Q and to compare them to unity. Solution A for instance rapidly becomes unstable.

(11)

where Ds and Dr denote partial derivatives and l denotes arc length. If DsF(s, r) is non singular, then:

(12)

It is straightforward to verify that when s is o. h., is o. h. also. preserves this property, which implies that is o. h. as well. From (12) it is then clear that the curves A to E are composed of o. h. solutions. In consequence, any solution issued from the initial ones is o. h.. We have observed in [Rodet 93f] that a slight breaking of the odd symmetry of g can lead to the appearance of even partials, with large amplitudes if {hn} is not even symmetric.

Fig. 4. Successive values of rs for the equivalence class C of solutions, a1 = -1.35 and r from 0 to 1.

x1 = h1g(x-3) + h0g(x-2) + h-1g(x-1) (13)

x2 = h1g(x-2) + h0g(x-1) + h-1g(x0) (14)

x3 = h1g(x-1) + h0g(x0) + h-1g(x1) (15)

Using the periodicity of the solution x, the symmetry h-1=h1, and the odd symmetry of g, we can rewrite the equations in x1, x2 and x3:

x1 = -h0g(x1) - h1g(x2) + h1g(x3) (16)

x2 = -h1g(x1) - h0g(x2) - h1g(x3) (17)

x3 = h1g(x1) -h1g(x2) - h0g(x3) (18)

This can be written in matrix notation x = N g(x), (19), i.e. N-1x = g(x), (20) where:

(21)

Using g(x) = x3+a1x, equation (20) leads to

x13 - (g0-a1)x1 + g1x2 - g1x3 = 0 (22)

x23 + g1x1 - (g0-a1)x2 + g1x3 = 0 (23)

x33 - g1x1 + g1x2 - (g0-a1)x3 = 0 (24)

Let a = g0-a1-g1. Using two equations to eliminate one variable, we get:

(x1+x2)(x12+x22- x1x2-a) = 0 (25)

(x1-x3)(x12+x32- x1x3-a) = 0 (26)

The two left hand terms of (25) and (26) define a plane and a cylinder with an ellipsoid basis provided a > 0. Each ellipse can be described by a parametric equation with r = :

x1 = r cos(w + p/6), x2 = r cos(w - p/6) (27)

x1 = r cos(m + p/6), x2 = r cos(m - p/6) (28)

The solutions are found on the intersection of the planes and cylinders. The intersection of the cylinders

Fig. 5. Left hand term of (22) to (24) versus w.

is a curve obtained with m = - w or m = w - p/3. Then we can follow this curve by using the parameter w and evaluating left hand terms of (22) to (24) for zero crossings. The values of these terms versus w from 0 to p are shown on Fig. 5 for a1 = -1.1, h1 = 0.98 and h0 = 1-2h1 (two terms are identical). We observe common zero crossing for three values of w which correspond to the solutions designated previously by A , D and E . The value of the left hand term of (22) versus w is shown on Fig. 6 for h1= 1-h1/2 and h0= 1., 0.98, 0.965, and 0.955. This explains how solutions designated by D and E join together when h1 reaches 0.965 and then disappear as discussed in section 7. This occurs when a becomes negative because the ellipses collapse, leading to imaginary solutions. The other solutions are easily found in the intersections of the planes with the ellipses or between the planes. Finally we get the classes A to E previously found (Fig. 7).

Fig. 6. Left hand term of (22) h0= 1., .98, .965, .955.

Fig. 7. The five classes of solutions of the example

for h0= 1 to 0.

(29)

which in general are unstable. Furthermore, as xÆ , the slope of g Æ , which implies that the system can become unstable and escape to infinity, this is rather annoying for a musical instrument.

We correct the behavior of g by dividing the polynomial by another polynomial of degree less or equal to 3. With an even degree the odd symmetry is kept :

g(x) = (x3 + a1x)/(1 + b2x2) (30)

Then the system (1) has no spurious fixed points provided b2>1 and can be studied with the same tools as the polynomial case. This function is shown on Fig. 8 for b2 = 2 and a1=-1.7. To keep things simple, let us consider the case where there is no filtering in the Fig. 8. Rational function g for a1 = 1.8 and b2 = 2

feedback loop (h0 = 1, h1 = 0). Then the first 2-periodic solutions of (1) oscillate between points where g(x)=-x:

(31)Such a solution is stable if the derivative of (g g(x)):

(g g(x))' =g' (g(x))g' (x)=g' (-x).g' (x)=-(g'(x))2 (32)is less than 1 in modulus at these points. This happens when -4 a1 -1. Fig. 8 shows g(x), (g g(x)) and (g g(x))' for a1 =-1.7. But for approximately -6.15 a1 -4, we get two 2-periodic solutions. Finally chaotic solutions can be obtained for a1 -7.2 approximately as shown on Fig. 9.

Fig. 9. Bifurcation diagram of the rational function nonlinearity versus parameter a1.

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