Overview Impact of String Stiffness on Virtual Bowed Strings Acoustics of bowed string instruments. Stefania Serafin, Julius O. Smith III • CCRMA (Music 420), Modeling a violin string May, 2002 • Modeling the role of the bow Center for Computer Research in Music and Acoustics (CCRMA) • Department of Music, Stanford University Coupling the bow and the string Stanford, California 94305 • Analysis of the playability of the model February 5, 2019 • 1 2 Acoustics of bowed string instruments Elements of a Basic Bowed String Model The sound produced in a bowed string instrument is Bow-string interaction • • obtained by drawing a bow across one of the four Transverse waves on the string streched strings. • Losses at the bridge, finger, and along the string To produce sound, energy from the vibrating string is • • transferred to the body of the instrument. vh=vin+vib f= µ(v-vb) f= 2Z(v-v ) BRIDGE { h NUT v on vob v µ in (.) vib von=vib+f/(2Z) vob=vin+f/(2Z) Bow− Bow to string bridge Bow to nut delay interaction delay String String −1 BOW Bridge Body Input parameters (nut side) (bridge side) vb pb vb Control parameters Structure of a basic model of a bowed string. A bowed string instrument and the corresponding simplified block diagram of its model. This model supposes that the bow is applied to a single point pb in the string. When vb = v, bow and string stick together, otherwise they are sliding. 3 4 Bow-String Interaction v = von + vin = v + v ob ib f Slipping backward The contribution of the reflected waves vin and vib are summed at the contact point: f=2Z(v-vh) vh = vi + vi n b v-vb Bow string interaction is represented by the following Slipping forward relations: f = 2 Z (v v ) − h f = γ (v vb) f − Once this coupling has been solved, the new outgoing waves von and vob are calculated by the following equations: v-vb f=Z(v-vh) f von = vib + 2Z f vob = vin + 2Z 5 6 If three intersections occur, the result is determined by Modeling String Losses the following hysteresis rule: The system follows its current state (stick or slip) as In an anachoic room, we have recorded strings plucked in • long as possible. five different positions, as shown in the figure below: The system will never fall in the middle of the three NUT BRIDGE • intersections. 1 2 34 5 FINGERBOARD Five different positions in which the string has been plucked. Plucks have been made using a hard-plastic pick, in order to excite also high frequencies. 7 8 Analysis by Energy Decay Relief (EDR) la1.wav − Energy Decay Relief 40 20 0 Violin G string −20 −40 Magnitude (dB) −60 −80 10 −100 0 8 5 6 SOL1.wav − Energy Decay Relief 10 4 15 2 20 0 25 Time (frames) Frequency (kHz) 40 20 Plucked violin A string. 0 −20 −40 la1wf.wav − Energy Decay Relief −60 Magnitude (dB) −80 −100 10 −120 40 0 8 5 6 20 10 4 15 0 2 20 0 −20 25 Time (frames) Frequency (kHz) −40 −60 Magnitude (dB) Plucked violin G string. −80 −100 10 −120 0 8 5 6 sol1wf.wav − Energy Decay Relief 10 4 15 2 20 0 25 Time (frames) Frequency (kHz) 40 20 Plucked violin A string, with the nut side damped. 0 −20 −40 Magnitude (dB) −60 −80 10 −100 0 8 5 6 10 4 15 2 20 0 25 Time (frames) Frequency (kHz) Plucked violin G string, with the nut side damped. 9 10 Calculation of the decay rate of the Accounting for Torsional Waves partials Torsional waves can be modeled as an additional couple of waveguides whose speed is about 5.2 times the We calculate the slope of every partial in a db scale, transverse wave speed. • to get the decay rate. The decay time is used to obtain the low-pass filters v = v + v + v + v • that estimate losses. h in ib int ibt f v = v + on ib 2Z f v = v + ob in 2Z f vont = vibt + 2Zt f vobt = vint + 2Zt String String Bridge −1 (nut side, (bridge side, trasversal waves) torsional waves) BOW String String −1 Bridge (nut side, (bridge side, torsional waves) trasversal waves) vbpbvb Control parameters Structure of the basic model with filter in the nut side and torsional waves. 11 12 where f = applied force Modeling the stiffness of the string Z = string transverse wave impedance The stretching of the frequencies of the partials can be Zt = string torsional wave impedance, calculated as: Torsional waves facilitate the establishment of Helmholtz 2 fn = nf0√1+ Bn motion because they are more damped than the 3 4 where B = π Ed =inharmonicity factor, transversal waves. 64l2T Their contribution at the bow point can be modeled in f0 =fundamental frequency of the string. two ways: E=Young modulus of elasticity, d=diameter. 1. Changing the slope of the straight line: 10000 9000 1 8000 Z = 7000 s 1 + 1 2Z 2Zt 6000 5000 2. Changing the inclination of the friction curve 4000 3000 f 2000 1000 0 0 5 10 15 20 25 30 35 40 45 50 Shift of partials for a cello D string, f0=147 Hz,B=3e-4. x-axis: partial number, y-axis: frequency (Hz) v-vb 13 14 Physical constants of a Dominant violin string, from Designing allpass filters for dispersion Pickering simulation String E(N/m) d(m) l(m) T B E 15.7 0.000307 0.565 72.56 5.1627e-15 We choose a numerical filter made of a delay line delay A 7.470 0.000676 0.59 56.25 6.5418e-14 τ τ0 D 6.4 0.000795 0.567 43.72 1.5707e-12 line q − , and a n-order stable all-pass filter: G 6.035 0.000803 0.567 44.57 1.4963e-12 n 1 H(q)= q P (q− )/P (q) Physical constants of a gut violin string, from Schelleng. where String B n 1 n P (q)= p0 + ... + pn 1q − + q E 1.5598e-05 − A 4.8527e-05 D 2.4841e-04 and τ and n are appropiately chosen. G 1.3e-3 We minimize the infinity-norm of a particular frequency weighting of the error between the internal loop phase and its approximation by the filter cascade: δD = min WD(Ω)[ϕd(Ω) (ϕD(Ω) + τΩ)] p1,...,pm k − k∞ jΩ where ϕD(Ω) =phase of H(e ), WD(Ω) = frequency weigthing (WD(Ω) is zero outside the frequency range), i.e. [Ωc, ΩN ]. 15 16 Erreur frequentielle finale sur les partiels Results 10 8 The figure below shows the results given by the 6 algorithm, for a cello string (147 Hz) with B =3e 4. 4 − 2 Erreur frequentielle finale sur les partiels 0 10 −2 8 −4 6 −6 4 −8 2 −10 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 −2 Frequency error (in cent) after the allpass filter approximation, for a violin A string, with −4 B = 4.8527e 5. − −6 −8 String String Bridge −1 Dispersion (nut side, (bridge side, trasversal waves) torsional waves) −10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 BOW String String Frequency error (in cent) after the allpass filter approximation, for a cello D string.. −1 Bridge (nut side, (bridge side, torsional waves) trasversal waves) vbpbvb Control parameters 17 18 The friction models used in these simulations are: a plastic model recently proposed by Jim • Woodhouse, who discovered that friction does not an hyperbola • depend only on the relative velocity between the bow (µs µd)v0 and the string. µ = µd + − v0 + v vb − 0.1 4 µ =0.3 , µ =0.8. 3.8 d s 0.05 3.6 0 a sum of decaying exponentials: 3.4 -0.05 3.2 v vb v vb -0.1 3 0−.01 0−.1 µ =0.4 e +0.45 e +0.35 -0.15 2.8 2.6 -0.2 2.4 -0.25 2.2 -0.3 2 0.17 0.172 0.174 0.176 0.178 0.12 0.13 0.14 0.15 0.16 0.17 Time Time 0.75 1.16 0.7 1.14 1.12 0.65 1.1 0.6 1.08 0.55 1.06 0.5 1.04 0.45 1.02 0.4 1 0.12 0.13 0.14 0.15 0.16 0.17 -0.4 -0.3 -0.2 -0.1 0 0.1 Time Velocity 19 20 Ak (T ) Accounting for the width of the bow µ = y sgn(v) N where von=vib+f/(2Z) vob=vin+f/(2Z) A = contact area between the bow and the string v on vob N = normal load v v =v +v in h in ib vib µ ky(T ) = shearyieldstressasafunctionofthebow-stringcontact f= (v-vb) {f= 2Z(v-vh) In the plastic model the key state variable governing Model with a single bow hair • the friction force is the temperature of the contact region. To model two bow hairs, it is necessary to double the The notion of coefficient of friction is kept, but bow-string interaction and the incoming wave velocities. • instead of depending on sliding speed it depends on contact temperature. von=vib+fr/(2Z) vob=vin+fl/(2Z) v v The contact temperature, in turn, depends on the on ob v =v +v *h • vin hr in ib vhl=vin*h+vib v sliding velocity (recent history).