Physical Modeling of Plucked String Instruments with Application to ...

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VALIMAKI ET AL. PAPERSe(n) and b(n) can be combined, as shown in Fig. 14(b). cessing point of view and as good as possible from theThe signal x(n) is then used as the input to the general- perceptual point of view. In earlier studies we have conizedKS string model. The combination of the excitation sidered three strategies [34], [37]:signal and the impulse response of the body is motivated 1) Directional filteringby the fact that then the body need not be modeled 2) Set of elementary sourcesexplicitly during real-time synthesis. The signal x(n) 3) Direction-dependent excitation.combining the excitation and the body response can be A direction-dependent digital filter may be attachedestimated by precomputing it based on some model, by to each path from the source to the listener. Moving andmeasuring it somehow, or by inverse filtering a digital rotating sources can be modeled by changing the filterrecording of an instrument, as will be discussed in Sec- parameters of the paths in a proper way (for example,tion 3.5.theLeslieeffectof a rotatingloudspeakercan be simu-We may develop the model further by explicitly mod- lated). The directional filtering method was studied foreling the excitation signal. The first step toward this the acoustic guitar. We came to the conclusion that evendirection is to extract the most prominent resonances of first- or second-order directivity filters give useful rethebody, measure their central frequencies and Q val- suits, thus leading to an efficient implementation.ues, and design a second-order all-pole filter to represent Fig. 15 depicts the modeling of direction-dependenteach resonance. These resonances must then be removed radiation of the acoustic guitar (in the horizontal plane)from the excitation signal, for example, by using nar- relative to the main-axis radiation. Shown in the figurerow-band linear-phase FIR filters,are magnitude spectra for second-order IIR filters at azi-We have successfully extracted the two lowest reso- muth angles of 90, 135, and 180°. The reference magninancesof the body of an acoustic guitar. The main ad- tude spectrum at 0 ° is assumed to be flat. The lowvantageof this approach is that the residual signal with pass characteristic is noticeably increased as the relativethe low-frequency high-Q resonances removed decays angle is greater. The measurement was carried out bymore rapidly than the original one. As a consequence, exciting the bridge of the instrument using an impulsea shorter excitation signal can be used, and the memory hammer and registering the reference response at 0° andrequirements for the model-based synthesizer are low- the related response in various directions. The measuredered. Also, the Q values and the central frequencies of reference and the directional response were fitted sepathelowest resonances are now parameterized and thus rately with first- or second-order AR models. A simpleindependently controllable,division of the models was performed to obtain thepole-zero directivity filter.2.5 Modeling Directional Properties of String Fig. 16(a) shows the transfer function measured atInstruments azimuth angles of 0 and 180 °. Fig. 16(b) gives the re-The directional characteristics of musical instrument sponse at 0° filtered with a first-order IIR directivitysound radiation are of interest in several applications filter and the actual measured response at 180 ° azimuth.using model-based sound synthesis. From the scientific Note that the spectral slopes are nearly the same, as waspoint of view the physical modeling approach serves as expected. In this example the transfer function of thea flexible tool for analysis and simulation of the directiv- directional filter isity of the instruments. Directional characteristics mustbe taken into account when model-based sound synthesismethods are used for sound generation in room acousticsR (z) - bo + biz- l1 + a]z- _ (10)simulation and other virtual reality applications [34],[36].Plucked string instruments exhibit complex sound ra- o ____
PAPERS PHYSICAL MODELING OF PLUCKED STRING INSTRUMENTSwhere the coefficient values are bo = 0.2041, bI = in parallel as there are directions to be included. This-0.1431, and a_ = -0.4389. limits the number of simulated directions, for example,When zooming to the details of the lowest resonance to the six main directions of the Cartesian coordinatemodes we notice, as described in [17], that the individual system. Even then this approach is computationallymodes behave differently. To model such details, a rela- very inefficient.tively high-order directional filter is needed. It is im- The foregoing considerations as well as our experiportantto notice that due to the critical-band frequency ments have shown that the directional filtering techniqueresolution, an auditory smoothing may be applied to the is normally the most practical one. A first- or secondresponsesbefore directional filter estimation. This helps order filter approximation is often a satisfactory solutionto reduce the order of the filter,in a real-time implementation.The use of elementary sources is based on the idea thatthe radiation pattern of an instrument is approximated by 2.6 Extensions of the String Modela small number of monopoles or dipoles. In general the The string model can be extended in several ways tomethod is computationally expensive if a large number more carefully simulate a vibrating physical string. Inof paths to the receiver is needed since each new elemen- the following we present three examples. For a moretary source adds a new set of path filters, detailed discussion, see [23].In the case of commutative excitation [such as theplucked string instrument model shown in Fig. 14(b)] 2.6.1 Sympathetic Vibrationsthe directivity filtering may be included in the composite Sympathetic vibration is a phenomenon where someexcitation in a way similar to the inclusion of the (early) modes of a string are excited by the vibration of otherroom response [7]. The problem with this method is that strings primarily due to couplings through the bridge.the same number of string model instances are to be run It can be simulated by feeding a small fraction of theoutput of each string model to the other strings [4], [11].This simplified technique does not take into account thefrequency dependence of sympathetic couplings, but itmostill addsrealismto the synthetictones.Sincethereis90 feedbackviaallstrings,thevaluesofthecouplingcoefficientsmust be carefully adjusted so that they are smalls0enoughnot to makethe systemunstable.Anotherand70 theoreticallymorecorrectwaytoincorporatethesympatheticvibrations is to represent the bridge by a separate60 filterthatis commonto allstringsin a waveguidemodelof an instrument [7]. This approach adds, however, c0m-50 putationalcomplexityto the overallmodel.40 2.6.2 Dual-Polarization String Model30 Physicalstringsvibratein boththe verticaland the0 I 2 3 4 5 6 7 8 9 10 horizontal directions. If the effective length of the stringFreque,cy(mz) is not the same in the two polarization planes, the mixing(a)of the two subsignals of slightly different frequenciesil°Ilcreatesbeatsin thesound.Thishasbeenfoundto bean]00H[]_[_ll.I _'importantfeature in the sound qualityof the kantele[20], wheretheexplanationforthis effectis thepeculiar9°_'_[[1__ _a A_, Fi't_redr_fer_nce. 180° way in which the strings have been terminated. In otherstring instruments, however, the effective length in the80 two directions can differ due to changes in the driving-7o point impedance of the bridge. This can be incorporatedin the model by using two basic string models with60 different L for each string [11]. Fig. 17 illustrates acomplete model, including dual-polarization string,5owavetableexcitation,andsympatheticcouplings.40 In thecaseof themandolin,beatingis causedbythefact that there are four pairs of equally tuned strings.30 This can be modeled using two separate string models0 _ 2 3 4 5 6 7 s 9 10 for each pair.Frequency (Hz)(b) 2.6.3 NonlinearitiesFig. 16. Modeling of guitar sound radiation with first-order In strings the polarization of vibration may changeIIR filters. (a) Transfer functions measured at azimuth angles0 and 180°. (b) Response at 0° filtered with directivity filter continuously in a complex manner since there are weakand measuredtransfer function at 180°. nonlinearities transferring energy between modes andd. Audio Eng. Soc., Vol. 44, No. 5, 1996 May 339m · · ....
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