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Page 00000082 Generating pitches in transients by a percussive excitation Patrick Fourcade, Sylvain Mangiarotti and Claude Cadoz E-mail: Patrick.Fourcade@imag.fr A.C.R.O.E. I.N.P.G. - Institut IMAG 46, Av. Felix Viallet, 38031 Grenoble - France http:\\www-acroe.imag.fr Abstract: Studying the excitation by percussion in the musical instruments universe, we make known a musical application of the physical modelling for sound synthesis. We have managed to generate pitches in the sound percussion transients. The percussion model employs a time rheologic representation postulating the Newtonian mechanics. We show, manipulating parameters in a modal analysis, how to get a pitch transient and how to tune the relative interval between the transient pitch and the pitch of stroked sound structure. We give a numerical application for a quinte transient. Some melodies sentences can be produced by adding successive strikers on the sound structure. 1. Introduction Today, the computer offers to musicians a large catalogue of sound synthesis tools. When composers have to use a new synthesis tool, they often say: "can we control the hearing pitch with great accuracy?" The attack transient of the real percussion sounds do not contain any hearing pitch. On the one hand, the duration of the transient is too short, on the other hand the transient spectra is near from noise spectra. Using the CORDIS-ANIMA formalism for physical model simulation of vibrating systems, in the framework of a general sound synthesis tool, we study a striker model allowing a pitch perception of the attack transient. Applying to a vibrant object a succession of percussive excitator elements, it is possible to synthesize transients of percussive sounds containing for example an arpeggio or a pentatonic scale. 2. Model of Percussion The percussive object, in this physical model, is a punctual mass m2 (Cf. figure 1). The vibrating structure is a first-order oscillator 01 with an inertia ml, a stiffness K1 and a viscosity Z1. All movements are on a one dimensional space. An interaction between m2 and 01 is introduced as a repulsive force produced by a viscoelastic element of stiffness K2 and viscosity Z2. The interaction is such that F= K2(Y Yo)+2(- Yo) if Y, < Yo, F= 0otherwise (eq. 1). Yp is the altitude of the mass m2; YO is the altitude of the vibrating structure impact mass. The striker mass is shot with an initial velocity and 01 is initially at rest. The percussion model has been studied exhaustively in [Fourcade and Cadoz 1996]. Yo Yp 01 K1 m2 Figure 1 - Model of percussion. The mass m2 strikes the mass ml of the 01 oscillator, with the interaction law: F= K2(Y - YO)+ Z2 - o) if Y, < Yo, F= Ootherwise. 3. Pitches and Modal Analysis When the striker mass altitude, Yp, is smaller than the oscillator mass altitude, YO, the system is on a transient phase. We studied the frequency modes of the system during its transient phase (linked system). The viscosity.Z and Z2 are neglected. We note -82 - ICMC Proceedings 1999
Page 00000083 fr r'- and f.i ~1- If we note 2trljm1 2t inm2 F+and F_ the owners frequencies of the linked system, introducing for convenience the variable 7, y=, itcomes: F2 ~A+f2~ (eq. 2) a =(~2 + 62)2 4fi~f2 Because of the non-linearity condition, F, mode will not be perceived as a pitch. Except indeed in. a specific case (high rate m2/ml and strong stiffness K2), the interaction has got only one contact and the striker inns away. So, the linked phase duration is quite F./2, an oscillation half period. This mode acts upon the rise-time (time duration between the mmn and the max amplitude reached in the transient signal). Parameters min, f1, EF.and FL define the model on the frequency space. We determine others system parameters from in, f1, F4and F Observing from (eq. 2) that ftf2 = FY+, and that f12 +,2 = p2? + F,2, we have: f2' f12 f14 F;' F, F'F,2 f2F, (eqi. 3) in2 = m1(y-1) F> c o11---< b FLf F4 b and c are constant, function of 7: b = d2y-1-2 ~2..yb and c 62y-1+ 21 T'-7 'trise.4ime Figure 2 - Example of a "percussive" attack: grain on an harmonic oscillator. Frequencies result from an analytic modal analysis 4. The CORDIS-ANIMA formalism The CC)RDIS-ANIMA system developed at the ACROE fCadoz, ct al. 19931 allows the computerbased modelling and simulation of physical objects that can be seen, heard and handled (with forcefeedback gestural control device). An object is a modular assembly of elementary mechanical components picked up amongst a limited number of types with very simple associated elementary simulation algorithms. Tfhe description is based on the fundamental laws of Newtonian Mechanics. The percussion model has been realised thanks to CORDIS-ANIMA for computer simulations and synthesized sounds. I[CMC Proceedings 1999-83 - 83 -
Page 00000084 5. Numerical Application: a quinte transient It is possible to create an attack which generates the pitch intervals perception from: - the oscillator's inertia mi, - the resonance frequency fi, - the frequency rate between F+ and F (interval), - the number N of F+ mode periods during the transient phase. In the case of a quinte percussion E5-A4, we give mi = 1Kg, fl = 440Hz, N high enough to perceive a pitch, N = 10 and F+/fl = 1.5 (quinte of the Pythagorean scale). We want F_ - F+/2N. So, F+= 660Hz and F. = 33Hz. From (eq. 3), we obtain y = 20.1975, m2 = 19.1975Kg, f2 = 49.5Hz. Explaining with stiffness, we have kg = 4r2f2n2m2; we obtain k2 = 1.8573E6Nm"1; same here, k1 = 7.6412E6Nm-1. By applying the formula for simulation algorithm parameters (algo suffix), that take into account the time quantification and the causal algorithms necessity, neglecting the oscillator viscosity effect, we have: Fe K alg f cos-1 1--ag,m = Malgo with 2;r 2 Malgo Fe = 44100Hz (cf. [Incerti 1996]). Thus, finally we have Klalgo = 3.929E-3 and K2algo = 9.55E-4. The resulting sound has the following reference [Sound example A]. 6. Play with Pitch transients We made a serie of elementary percussions. Adding to the three parameters (mi, ki, vi) describing one striker i with inertia mi, stiffness ki, and velocity vi, we get Ti, the time when striker i reaches the 0 altitude without the oscillator presence. We note N the total number of strikers. From the equations system showed in (eq. 3), it is possible, neglecting the viscosity influence, to build attacks that induce pitch variations. In the sound example [Sound example B], a major arpeggio is perceived; we can listen successively B3, G#3, and E3. In the same manner, in the example [Sound example C], we perceived a melody in arch at the attack; it listens successively at three octave: G#, A, A#, B, A#, A, G# and E. 7. Singular Sounds It is difficult to make expectations on the oscillator behaviour after a percussion. One oscillator can be stop moving with the last percussion in a series (cf. [Sound example D]). Viscosity of the striker can operates in a particular case modifying the transient pitch (cf. [Sound example E]). This phenomenon appears for a light striker's inertia (ml/10) and a weak stiffness link (kl/100). The perceived attack pitch is lightly under the oscillator pitch (70 cents). The striker mass stays a long time (50 ms) under the oscillator mass and vibrates in cooperation with it. Timbre evokes drop water. References [Fourcade and Cadoz 1996] P. Fourcade and C. Cadoz, "Sound Synthesis by Physical Modelling: an Elementary Striker," Forum Acusticum, Antwergen, 1996. [Cadoz, et al. 1993] C. Cadoz, A. Luciani and J.-L. Florens, "CORDIS-ANIMA: a Modeling and Simulation System for Sound and Image Synthesis - The General Formalism," CMJ, vol. 17, 1993. [Incerti 1996] E. Incerti, "Synthese de sons par modglisation physique de structures vibrantes: application pour la creation musicale par ordinateur," These de docteur ingenieur sp6cialite informatique, Grenoble, 1996. -84 - ICMC Proceedings 1999