MUMT618 - Final Report Litterature Review on Guitar Body Modeling Techniques

Transcription

1 MUMT618 - Final Report Litterature Review on Guitar Body Modeling Techniques Loïc Jeanson Winter Introduction With the Karplus-Strong Algorithm, we have an efficient way to realize the synthesis The goal is to investigate as far as possible the ease and the feasibility of inputing the choices of the guitar makers in a model of the instrument. This realistic parameter coupling to guitar model would have two main interests. First, it will allow for a possible improvement of the filtering verisimilitude of the guitar body and ultimately of the verisimilitude of the timbre of the synthesized sounds. It could also help luthiers in their never ending quest of enhancing their control on the timbre of their guitars. Nowadays, only years of experience and a good sense of keeping tracks of past choices are the most common tools used during various attempts to develop the personality and balance of their various guitar models. The first step of this goal was obviously to take a closer look at the modeling techniques of the guitar body, in order to understand in which way the parameter chosen by the luthiers could be directly or indirectly implemented in the guitar models. This report reviews a series of articles related to guitar body modeling. This literature review will be organized as follow. We will go through every selected paper among the selection, they will be shortly summarized and what seemed to be the main points will be enlighten. Then a short analysis follows showing among other parameters the link with passed (and/or selected by this review) papers. The six selected papers are [A] Body Modelling Techniques for String Instruments Synthesis by Karjalainen, Smith ; 1996 [B] Plucked-string Models : From the Karplus Strong Algorithm to Digital Waveguides and Beyond by Karjalainen, Välimäki, Tolonen ; 1998 [C] New techniques to model reverberant instrument body responses by Penttinen, Karjalainen, Paatero, Järveläinen ; 2001 [D] Non Linear Guitar Body Models by Nackaers, Schiettecatte, De Moor ; 2003 [E] High-Resolution Parametric Modeling of String Instrument Sounds by Karjalainen, Paatero ; 2005 [F] String Instrument Body Modeling using FIR Filter design and AutoRegressive Parameter Estimation by Türckheim, Smit, Mores ; 2010 [G] "Numerical simulation of a guitar", by Becache, Chaigne, Derveaux, Joly ;2005 1

2 2 Litterature Review 2.1 Body Modelling Techniques for String Instruments Synthesis by Karjalainen, Smith ; 1996 Presented for ICMC 96, this article is based on work of the early and mid-90s. It explains various alternative techniques for string instruments synthesis as its title let understand. Concretely, it means the presentation of various types of digital filters. The strong contribution this article delivers is the explanation of the use of warped filters. Warped filters are the usual FIR or IIR for which the frequency scale is warped, thus lightening the computation load. The frequency warping for audio application should follow the Bark scale in order to be perceptually consistent. The warping method used in this article is the bilinear conformal warping because it is a conformal mapping that preserves order. For FIR, first, implementing the warping is not too complicated: all the unit delays are substituted by 1st order allpass sections as in (1). This is a general expression of a bilinear warping, to fit the Bark scale, we need to find appropriate values of λ. z 1 D 1 (z) = z 1 λ 1 λz 1 (1) The design of the warped FIR filters is explained in Figure 1 Figure 1: Warped FIR modeling: (a) general principle, (b) detailed filter structure for implementation For IIR the substitution is not that simple. It needs some more adjustements described by (2) 2

3 H w (D 1 (z)) = G w 1 + R i=1 α i[d 1 (z)] i (2) Implementing a filter that structure follows (2) is not possible since there is delay-free loops for λ 0. To get rid of this problem, Smith and Karjalainen propose two structures (in Figure 2). The first one uses low-pass filters instead of allpass (and was proposed by Strube in 1980). The second, more complex is a general version for a warped pole-zero structure. Figure 2: Filter structure for implementation of warped IIR filters: (a) lowpass structure that does not work with high orders, (b) modified allpass structure (warped pole-zero filter) This paper also includes the presentation of a body factoring methodology to even further reduce the computation complexity. The various modes have to be factored, splitted between the most damped and the least damped modes. The most-damped modes are then commuted and combined with the excitation in the impulse-response form. The least damped can be modeled as recursive digital filter sections. In order to make this mode factoring, various ways to estimate the mode parameters of a body are listed and shortly discussed. Another note on an easy treatment of the high frequency modes as noise is also presented. The body response in its high frequency range can be heard as a noise burst, therefore it is more convenient to model it as white noise multiplied by an amplitude envelope. The article ends in an example of the body factoring presented before. This article of the mid 90 presents way to model a guitar body, or at least to reproduce it with the help of the usuals FIR and IIR. Concerned with computational efficiency, it presents techniques allowing decreasing the computational load without altering the filter quality. To conclude on this paper, one could say that the bilinear warping reduced the order 3

4 of the design by a factor between 5 and 10, depending on desing, therefore reducing the computational complexity. The parameter λ inserted by the warping can be seen as a qualitative body size parameter that could allow the design of a familiy of instruments with an analogous geometry but varying dimensions. 2.2 Plucked-string Models: From the Karplus Strong Algorithm to Digital Waveguides and Beyond by Karjalainen, Välimäki, Tolonen ; 1998 A few years later, Karjalainen, Välimäki and Tolonen published in the Computer Music Journal a paper presenting the various evolution of the synthesis of plucked string instruments. Starting by the Karplus-Strong Algorithm,interpreted as a digitezed d Alembert s solution to the one-dimensional lossless wave equiation, the idea was to present extensions to this design. The SDL (Single Delay-Loop), in Figure 3 structure is the basic structure for the design of many musical instruments. The Karplus-Strong algorithm is a spectial caseof the SDL structure in which the excitation is given as the initial state of the delay line. Figure 3: Simplified linear string model as a single delay-loop (SDL) structure. The other, less compact or at first more explicit, way of reprensenting it, leading to equivalent results as the SDL structure has two delay lines : the bidirectional digital waveguide model is presented in Figure 4 The first extension presented in this paper is to take into account the output at the bridge. In the previous design, only the string is modeled. The input and the output are placed somewhere along its length. In reality, the string is coupled at the body at the bridge. Figure 5 shows the extende model of the bidirectional waveguide, which coupling and therefore output at the bridge. So far we have the model of the string and the one of the body. We but need a player to pluck it and to decide the timbre desired: we need a filtering due to the excitation. Our structure starts to be clearly composed of subparts. Therefore, a block building of this model was presented to explicit the various independant parts of the design and a first way, maybe not always highly efficient, to design them. After some considerations on the optimization of the design the paper presents a further extender vesion of the digital waveguide model with dedicated parts to specific functions (see Figure 6). In a similar manner, this article presents a detailled implementation of a string 4

5 Figure 4: Bidirectional digital waveguide model for a terminated string (SDL) structure. Figure 5: bridge. Dual delay-line waveguide model for a plucked string with output at the plucked with Pickup Output. This litterature review will not explain closer the differences of deisgn due to the nature of the output (physical coupling to the bridge or magnetic coupling to the pickup(s)). At the very end of the article, some other topics are discussed as the different kinds of non-linearities and some reasons for the linearities, the interaction between the strings and the body and the fact that the response of the body could be more precisely described as what this paper prensented. In order to illustrate this particuliar point, they set the previous article by Karjalainen and Smith as a reference. 2.3 New techniques to model reverberant instrument body responses by Penttinen, Karjalainen, Paatero, Järveläinen ; 2001 This next article, continues the series of Karjalainen. Only, this time, the idea is to see the body of the insturment as a reverberant structure. To describe it as reverberant 5

6 Figure 6: A schematic sound synthesis model for plucked string instruments. The delay in the string loop must be continuously variable, which can be achieved with a fractional delay filter. structure, a new type of filter is presented : Kautz filters. Implemented in the aim of body modeling they have some great properties. They reduce the computational load by reducing the order of the system and they reduce the number of parameters used in instrument body models. Kautz filter can roughly approximate the whole body response, but it lacks accuracy in the high frequency range. Because of their limitation, they can t model accurately the whole frequency range of the body. In this sense, as in the 96 paper by Smith and Karjalainen [A], the response of the guitar is split into two parts, one for the lower frequency range and, obviously, one for the higher part. Because of this need of frequency split the Kautz filter help in getting a deeper understanding of the body behaviour. They help for the identification of the limits at which the helmholtz resonator of the body and the vibrating plate see their respective range of effect. The article also shows that two Kautz filter, adjusted in different ways can model these two parts. Therefore, the adjustment of parameter of the Kautz filter to fit either the loweror higher range helps to get a better understanding of the perceptual properites of the body of the instrument. Basically a Kautz filter is a fixed poles IIR filter organized to produce orthonormal tao-output impulse responses. There is no explicit description of the nature of the Kautz filter in this paper. They reference the work of Kautz (1954) that developped the eponims filter. This being said, the methodolgy for the design of theses Kautz filters is presented. It is a two step procedure, starting with the selection of the poles. As in the paper by Smith and Karjalainen [A], it needs to start with a measured body response or a guess of the usual or average position of the poles. Then, the second step is the assignment of tap-out weights to these poles. The article then presents the methodoly used to determine the crossover frequency of the two Kautz filters used to describe the two frequency range. Based on perception tests to analyse the level of identification of a defect in the calibration of the filters and their crossover frequency. Also, since the parameter of the higher frequency Kautz filter has to be adjusted, the article alsoshortly presented the methology they used to differentiate the low frequency range calibration to the high frequency range. 6

7 2.4 Non Linear Guitar Body Models by Nackaers, Schiettecatte, De Moor ; 2003 This next paper covers a problem enlighten in the paper [B]: the non linear behaviour of the body of the instrument. In this paper it is asserted that the linear behaviour of the body of the guitar is only valid for small excitations. It is written, whitout being explicit, that there are indications that the guitar body is not entirely linear. The wooden plate tends to saturate the output due to the limited flexibility caused by the bracing at large amplitudes. Only a few papers in the past cover this issue. The present paper whishes to present a possible method to implement non-linear behavior at large amplitudes coming in the input of the body. Basically, what is presented in this paper is the modeling of a linear model combined that can be combined at some point with a static non linear model to simulate the non linearity. So it starts with the description of FIR and IIR filters models of the body, already presented in [A] and [B]. Also since [C] already presented it, the prensent paper introduced us to Kautz filter as a good way to model accurately and efficiently the body. Finally, the non-linear models are presented. In order to remain in the range of efficient models, the choice of static non-linearity has been chosen. Based on measurements made on instruments, they built Wiener-Hammerstein systems. The model using the Wiener- Hammerstein system is only used when the input is saturated. In order to estimate the linear part and the non-linear part, measurement where performed on instruments at low and high amplitude. From the low amplitude measurements, they could derive the linear behaviour that the body would have had if its response was totally linear. Then, from the high amplitude measurements, the non-linear part or the non-linear contribution can be estimated. One of the interogation that came from the reading and analysing of this article is the importance of the non-linear part. The article states that at higher amplitudes there is indeed a slight compression of the materials (in this case mainly of the wood of the top of the guitar). Also, there is no explicit definition of high amplitudes. It would be interesting to define more precisely what the limit is and how often the high amplitudes are reached in an average playing condition. Probably that some playing techniques allow for more legitimity of these considerations (manouche guitar playing or slapping on the bass guitar), but since the conceptual environment is blurry it is hard to conclude on the relevance of these considerations. 2.5 High-Resolution Parametric Modeling of String Instrument Sounds by Karjalainen, Paatero ; 2005 In this 2005 Paper, one of the other aspects of the body modelling is investigated : the modal analysis of a body response. So far, [A] presented solutions for this modal identification. Here, a new anlaysis technique is explained : FZ-ARMA. It stands for Frequency Zoom - Auto Regressive Modal Analysis. At a first sight, ARMA are in theory a bad tool to be used since the strings response to excitation is not a minimum phase signal. But ARMA techniques can be pretty efficient, although, due to iterative solution, they could lead to problems at higher order and may not converge to some stable or 7

8 useful result. In order to avoid the limitations of the ARMA method, it needed to be frequency limited, explaining the Frequency Zoom. The article explains in detail the FZ- ARMA technique, how the zooming - or the splinting in subband - is handeled so that the mapping is not missing any part of the spectrum, as well as delivering an accurate modal analysis of plucked sound recordings. T In a second time, this article presents an implementation of the data collected through the FZ-ARMA technique. in opposition to [B] where the idea was to extend the Karplus- Strong algorithm, this time, the idea is to get the whole plucked sound through one filter feeded by the parameter FZ-ARMA extracted. Even if all the modes are delivered by eth FZ-ARMA, it still is possible to separate the string- from the body modes. Ultimately, the modelling will be unified, but if one whishes to implement as in [B] a block modelling plucking variation or timbre variation by the player, each part of the modal analysis is separable. Here again, the Kautz Filters are selected because of their efficiency. The article finishes in the presentation of an example of the FZ-ARMA use from analysis to synthesis for an acoustic guitar. 2.6 String Instrument Body Modeling using FIR Filter design and AutoRegressive Parameter Estimation by Türckheim, Smit, Mores ; 2010 This before last article has a really interesting aim : allow for a specific modification of a resonnance profile, of the frequency response of a body. Before achieving this result, the article invrestigates the efficiency of FIR filters modeling and AutoRegressive (AR) Parameter estimation. As in [E], the parameter extraction from recordings and measurements are done using AR techniques. As shown in Figure 7, the present article prensents the difference between a FIR design and an AR modeling. As we can see, both orders of these designs are very high and computationaly heavy. Not designed for real time implementation, they aim, as best as possible, a fitting to a magnitude frequency response of an instrument body. Again, as [A] advocated it, in order to improve the computation load, for the AR design, a warping of the frequency scale is applied. The frequency axis warping is computed with the all pass function of equation (3). θ(ω) = arctan (1 λ2 ) sin(ω) (1 + λ) cos(ω) 2λ Explicitely, this time a value of λ is given to fit to the Bark scale : λ = 0.75 The article presents also a comparison of the efficiency of the AR design with and without warping. Figure 8showa the visual illustration of this example. For a roughly comparable accuracy we can notice the order sinificant decrease of the AR model. With this clearer view of efficiency of the AR design, the goal is achieved a desired modification of a magnitude frequency response of an instrument. Figure 9 shows the result of this modification applied to a specific frequency response and the AR filter. (3) 8

9 Figure 7: From top down, separated with 30 db offset: Original violin resonance profile, magnitude spectrum of the high-order FIR filter (order N F = 8000), and magnitude spectrum of the corresponding high-order AR model (order p = 1500) Figure 8: Top: Spectrum of an AR model (order p = 1500) without frequency warping. Bottom ( 30 db): Spectrum of an AR model which is computed with frequency warping (order p = 250) 9 Figure 9: Example of modification of the first violin corpus resonances. From top down, separated with 30 db offset: Original violin resonance profile, desired new resonance profile, spectrum of the complete AR model, and spectrum of the modified impulse response). In both cases, the AR order is p = 250.

10 2.7 Numerical simulation of a guitar by Becache, Chaigne, Derveaux, Joly ; 2005 This is probablay the most complete article of this selection.because it is really dense, it is actually pretty hard to summarize it concisely. The model presented in this paper includes : - the vertical displacement of the string - the vertical displacement of the soundboard - the acoustic pressure - the acoustic velocity field Because it mixes solid and fluid mechanics, this paper is highly vigilant about the interface between the various parts and types of calculation. Each equation in the various computational sections is bounded by either initial conditions or the other equations and equilibrium states inherent to the geometry of the guitar. 3 Conclusion From this selection, it is hard to see how close to a physical parameter driven model we are. Support for the conception of instruments bodys by maker from the numerical modeling world seem not yet complitely feasible. The option of multidimensional modeling as in [G], could be a good start in my point of view. Since it includes explicitely geometric choices as impedances along the soundboard. Also, since the acoustic pressure and velocity field is also computed, it would be possible to create perception measurement to compare with recordings. These comparisons could help improving the model and the verisimilitude of the synthesis sounds produced by the model. It was particularly rewarding, in my point of view, to see how this semester has led us to a better perception of the design of musical instruments and to an easier understanding of complex filter structures. The continuity of topics and methods developped during the 15years that covers this review was also quite rewarding: there remains a lot to be done but the way has been opened and tested in the past years. Efficient and well-spread tools have been developped and are well documented. Even if I couldn t find anything on building instrument bodies ex nihilo, the modal data evaluation and extraction as well as the ability of slightly modifying it is a known topic where great advances have been made. My investigation will be completed in the coming weeks by a closer look at what was done to evaluate the impedance/admittance functions or curves varying along the instrument. A higher priority order will be set on the soundboard, but the investigation of the other parts could have a smaller but possibly significan impact. 10