Duffing’s Equation as an Excitation Mechanism for Plucked String Instrument Models by Justo A. Gutierrez Master’s Research Project Music Engineering Technology University of Miami School of Music December 1, 1999
Purpose • The objective of this study is to provide the basis for a new excitation mechanism for plucked string instrument models which utilizes the classical nonlinear system described in Duffing’s Equation.
Advantages • Using Duffing’s Equation provides a means to use a nonlinear oscillator as an excitation • A mathematical model lends itself to user control • Removes the need for saving samples in a wavetable
Overview • Plucked String Instrument Modeling • Excitation Modeling with Duffing’s Equation • Model Performance and Analysis
Wavetable Synthesis • Method of synthesis that uses tables of waveforms that are finely sampled • Desired waveform is chosen and repeated over and over producing a purely periodic signal • Algorithm written as: Yt = Yt-p • p is periodicity parameter • frequency of the tone is fs/p
The String Model • z-L is delay line of length L • H(z) is the loop filter • F(z) is the allpass filter • x(n) and y(n) are the excitation and output signals respectively
Length of String • Effective delay length determines fundamental frequency of output signal • Delay line length (in samples) is L = fs/f0
The Comb Filter • Works by adding, at each sample time, a delayed and attenuated version of the past output
Standing Wave Analogy • Poles of the comb filter occur in the z-plane at 2np/L • This is the same as the natural resonant frequencies for a string tied at both ends • Does not sound like a vibrating string because it is a perfectly periodic waveform • Does not take into account that high frequencies decay much faster than slow ones for vibrating strings
The Loop Filter • Idea is to insert a lowpass filter into the feedback loop of the comb filter so that high-frequency components are diminished relative to low-frequency components every time the past output signal returns • Original Karplus-Strong algorithm used a two-tap averager that was simple and effective
Loop Filter (continued) • Valimaki et al proposed using an IIR lowpass filter to simulate the damping characteristics of a physical string • Loop filter coefficients can be changed as a function of string length and other parameters • H1(z) = g(1+a1)/(1+a1z-1)
The Allpass Filter • Used to fine-tune the pitch of the string model • If feedback loop were only to contain a delay line and lowpass filter, total delay would be the sum of integer delay line plus the delay of the lowpass filter • Fundamental frequency of fs/D is usually not an integer number of samples
Allpass Filter (continued) • Fundamental frequency is then given by f1 = fs/(D+d) where d is fractional delay • Allpass filters introduce delay but pass frequencies with equal weight • Transfer function is H(z) = (z-1+a)/(1+az-1) • a = (1-d)/(1+d)
Inverse Filtering • KS algorithm used a white noise burst as excitation for plucked string because it provided high-frequency content as a real pluck would provide • Valimaki et al found a pluck signal by filtering the output through the inverted transfer function of the string system
Inverse Filtering (continued) • The transfer function for the general string model can be given as S(z) = 1/[1-z-LF(z)H(z)] • The inverse filter is simply S-1(z) = 1/S(z)
Inverse Filtering Procedure • Obtain residual by inverse filtering • Truncate the first 50-100 ms of the residual • Use the truncated signal as the excitation to the string model • Run the string model
Duffing’s Equation • In 1918, Duffing introduced a nonlinear oscillator with a cubic stiffness term to describe the hardening spring effect in many mechanical problems • It is one of the most common examples in the study of nonlinear oscillations
Duffing’s Equation (continued) • The form used for this study is from Moon and Holmes, which is one in which the linear stiffness term is negative so that x” + dx - x + x3 = g cos wt. • This model was used to describe the forced oscillations of a ferromagnetic beam buckled between the nonuniform field of two permanent magnets
Modeling the Excitation • For this experiment, the coefficients in Moon and Holmes’ modification of Duffing’s Equation were adjusted to produce the desired residuals • The Runge-Kutta method was the numerical method used to calculate Duffing’s Equation
Procedure for manipulating Duffing’s Equation • Generate a waveform of desired frequency with (x, y). f10y is a good rule of thumb for starters. • Adjust the damping coefficient so that its envelope resembles the desired waveform’s • Adjust b, g, and w to shape the waveform, holding one constant to change the other • Normalize the waveform to digital maximum
Timbral Characteristics • Synthesized guitar from Duffing’s Equation very similar to that from inverse filtering • Frequency of both residuals different from pitch of synthesized stringsinharmonicity • Sonograms of both residuals also very similar
Tuning Performance (Harmony) • For individual pitches, the algorithm played fairly close to being in tune (perhaps slightly sharp). The allpass filter parameters can be adjusted to remedy this. • The C major chord played very well in tune, sounding very consonant with no apparent beats.
Tuning Performance (Range) • To test effective range of the algorithm, the lowest and highest pitches in a guitar’s range were synthesized. • Low E played in tune by itself. High E was flat. • This was more readily apparent when sounded together.
Summary of Tuning Performance • Algorithm performed as expected; it performed like Karplus-Strong; high frequencies tend to go flat, and this would have to be accounted for in the overall system.
Changing Damping Coefficient • Changing the damping coefficient can have pronounced effect on timbre of sound, specifically difference between type of pick used and type of string • The damping coefficient was adjusted to attempt to produce different sounds
Summary of Damping Coefficient Adjustments • For d = 0.2, contribution of residual made for a very hard attack, as if picked • For d = 0.5, guitar tone had much softer attack, as if finger-picked • Sonograms confirm that the latter had more high-frequency content
Production of Other Waveforms • Duffing’s Equation can be used to form a variety of waveforms • User has some control over its behavior if properties of the oscillator can be controlled to obtain the desired waveform