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 •

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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

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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

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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)

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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.

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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

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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

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−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

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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). ib µ f =µ(v -vb) fr= (vr-vb) l l f =2Z(v -v ) {fr=2Z(vr-vhr) { l l hl

A two point bowed string model

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0.2 Calculation of the playability 0

Amplitude −0.2

4000 4500 5000 5500 6000 6500 7000 7500 8000

0.2

0

−0.2 What does playability mean?

4000 4500 5000 5500 6000 6500 7000 7500 8000

0.2

0 −0.2 “Playability” can be defined as the region of a 4000 4500 5000 5500 6000 6500 7000 7500 8000 0.2 multidimensional space given by the parameters of the 0 −0.2 model where a good tone is obtained, where by good −0.4 4000 4500 5000 5500 6000 6500 7000 7500 8000 Time (samples) tone, we usually mean the Helmholtz motion. Simulation results from top to bottom: basic model, basic model with torsional waves, basic model with torsional waves and string stiffness, model with torsional waves, string stiffness and bow width. 1 5

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The string sticks to the bow for most of the time, • slipping backwards just once per vibration period.

23 24 String motion: Other kinds of motions •

0

String velocity (pictures from the computer simulated model)

Time Multiple slips • The bow force is not high enough to allow the bow to

Bridge force stick throughout the nominal sticking period of the

Time Helmholtz motion. Motion of a bowed string during an ideal Helmholtz 0.25 0.2 motion. String velocity at the bowed point and 0.15 0.1 trasverse force exerted by the string on the bridge. 0.05 0

-0.05

-0.1

-0.15

-0.2

-0.25 0 500 1000 1500 2000 2500 3000 3500

Anomalous low frequency • A periodic note with a lower pitch can be produced with a control of the bow force that makes the transversal waves to miss a slip opportunity few times in the stick-slip motion.

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3

2

1

0

-1

-2

-3

-4

-5 0 500 1000 1500 2000 2500 3000 3500

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Non-periodic motions Toward a Measure of Playability • Irregular motion obtained when the bow force is too high.

1.5 The Schelleng diagram (1973)

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0.5

0 1 Maximum bow force -0.5

-1 RAUCOUS Brilliant -1.5 0.1 0 500 1000 1500 2000 2500 3000 3500 Minimum bow force Sul ponticello NORMAL Relative force 0.01 Sul tasto 0.001 HIGHER MODES

0.01 0.02 0.04 0.06 0.1 0.2 Relative position of bow, ß

Inside the two straight line is the region where the Helmholtz motion is obtained.

27 28 Formulas for the Schelleng diagram Simulation Parameters

2Zvb In the first simulations we examine a cello D string, f = • 2 max (µ µ )β with bending stiffness B =0.0004 N m . s − d The string, starting from rest, is excited by a constant 2 Z vb • bow velocity vb =0.05m/s. fmin = 2 2r1 · (µs µd)β − The Schelleng diagram is computed by varying the where µ =coefficient of static friction s • bow force f from 0.005 to 5N, and the normalized µ =coefficient of dynamic friction b d distance β of the bow from the bridge is varied v =bow velocity b between 0.02 and 0.4 (where 0.5 would be the middle β=bow position of the bow). Z=characteristic impedance of the string, Z = √T ρ A classifier routine examines the shape of the • r1=term that represents losses of the string established waveforms.

In the second simulations we fix the bow position to 0.08, where 0 represents the bridge while 1 represents the nut, and we vary bow velocity and bow force.

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Playability space of the basic model Playability waveforms of the basic model

0.5

0

-0.5

-1

-1.5

-2

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x axis=bow position, y-axis=bow force x axis=bow position, y-axis=bow force

31 32 Velocity versus force playability space of the 3D playability space of the basic model basic model

0.5 1

0.8

0 0.6

0.4 -0.5 (velocity)

0.2

-1 0 1 0.5 0 0 −0.5 -1.5 −0.5 −1 −1 −1.5 −1.5 −2 log10(force) −2 -2 log10(beta)

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x-axis=bow velocity, y-axis=bow force

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Velocity versus force playability space of the

0.5 damped model

0

0.5 -0.5

-1 0

-0.5 -1.5

-1 -2

-1.5 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x-axis: bow position, y-axis:bow force -2

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x-axis= bow velocity, y-axis= bow force

35 36 Playability space of the model with torsional Waveforms playability space of the model with waves torsional waves

0.5

0

-0.5

-1

-1.5

-2

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x axis=bow position, y-axis=bow force x-axis=bow position, y-axis bow force

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Velocity versus force playability space of the 3D playability of the model with torsional waves model with torsional waves

0.5 1

0.8

0 0.6

0.4 -0.5 log10(velocity)

0.2

-1 0 1

0.5 0 −0.5 -1.5 0 −1 −0.5 −1.5 −1 log10(force) −2 -2 log10(beta)

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x-axis=bow velocity, y-axis=bow force

39 40 Velocity versus force playability of the model Playability of the model with stiffness with stiffness

0.5

0

-0.5

-1

-1.5

-2

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 x-axis= bow position, y-axis= bow force Playability of the model with stiffness, x-axis=bow velocity, y-axis=bow force

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3D playability of the model with stiffness Conclusions

Refined models of a bowed string are nowadays • possible. 0.9 0.8 In this document we did not present a model for the 0.7 • body of the instrument. 0.6 0.5 The bow hair compliance can be further improved by 0.4 • log10(velocity) 0.3 creating a finite-difference bow-width model. 0.2

0.1

0 1 0.5 −0.6 0 −0.8 −0.5 −1 −1 −1.2 −1.4 −1.5 −1.6 −2 log10(force) −1.8 log10(beta)

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