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Optimal Research in Piano Shape Sound and Sound Propagation Model Based on Nonlinear Differential Equations

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 27 Feb 2022
Accepté: 23 Apr 2022
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Introduction

Fractional nonlinear differential equations in chain solutions such as viscoelasticity, chaos and turbulence, aerospace, polymer materials, as well as non-Newtonian mechanics and other applied science fields, has a wide range of applications. The authors using the tools of nonlinear functional analysis, in order to study the existence theory and properties of solutions of several types of fractional nonlinear differential equations (groups), Misiats O et al., strive to find the conditions that make the corresponding nonlinear equation (group) solutions exist and are easier to verify or test, and the good properties of the solutions [1]; At the same time, we strive to construct an iterative approximation sequence that converges to the solution, and give an error estimation formula for the corresponding approximate solution. Fractional calculus has almost the same development history as integer calculus, it has a history of more than 300 years since it was proposed. In the past, it has been slow to develop because it has not been promoted by the practical application background, therefore, for a long time, only the research object of curiosity-driven mathematicians, without getting enough attention, until the early 1980s, after Tan Y et al. successfully used fractional derivatives to describe damping, people gradually realized that, fractional derivative is a powerful tool to describe viscoelasticity and other materials with “memory” and “genetic” characteristics. So far [2], the research on fractional calculus and its applications has gradually attracted the attention of many physical engineering researchers. After a period of development, Kanagawa T et al. excavated its many important applications, and gradually extended to such as diffusion and transport theory, chaos and turbulence, aerospace, viscoelasticity, chain solution of polymer materials, and non-Newtonian mechanics and other application fields [3]. According to classical Newtonian mechanics, space and time are continuous everywhere, therefore, integer-order differential operators can be appropriately used to describe basic physical quantities (such as velocity, momentum, force, etc.), therefore, the development and evolution process of many phenomena in nature, it is often described by integer order differential equations, such as heat conduction equation, wave equation, Hamilton equation, etc, people have made many brilliant achievements in many fields such as electromagnetism, classical mechanics, heat transfer and so on. However, unfortunately, the integer order differential operator has no global correlation, just is only a local concept, very inappropriate to describe natural phenomena with strong “memory” and “heredity”, therefore, using the classical integer order mathematical model, in order to describe and characterize those involved in memory and inheritance of processes, when the global correlation and dependency of the path are “anomalous”, it is often very limited, at this point, the fractional differential operator provides us with a good choice, it can just make up for the limitations and deficiencies of integer order operators in describing the above phenomena. Throughout the history of sound synthesis, from the electronic analog in the 1940s and 1950s, by the 80s and early 90s FM technology and wavetable synthesis technology, then to the physical models of the mid-1990s, all reflect the increasingly mature sound synthesis technology. As far as the current research situation is concerned, involvement in the field of sound synthesis technology is relatively shallow, and mainly focus on the research of speech acoustics and hydroacoustics, among them, the research on speech synthesis is relatively mature. Because the research on sound simulation of musical instruments requires cross-disciplinary talents, there are very few researches in this area. Compared with China, the development of musical instrument sound simulation in foreign countries is earlier, and it is relatively mature, and the instrument simulation technology of physical model is the main direction of current research.

Research Methods

Since the beginning of music synthesis research in 1940, there are several simulation synthesis technologies such as Wavetable Synthesis, Spectral Synthesis, Nonlinear Synthesis and Physical Model. Wavetable synthesis, which is based on samples extracted from sounds, with little regard for their physical properties. Spectral synthesis is to generate sound according to the time-frequency characteristics of the audio itself, the synthesized parameters are derived from the description of the desired waveform. Nonlinear synthesis is the use of complex synthesis algorithms to create rich sounds [4]. In contrast to spectral synthesis, the parameters of the nonlinear model have no direct relationship to the resulting waveform. The physical model describes the sound of the object according to the vibration equation of the object. Instead of simulating waveforms, it simulates the physical behavior of strings, drums, etc. Wavetable synthesis technology (WAVE TABLE) is one of the most widely used technologies in sound synthesis technology, it is mainly composed of two parts, one is the music event database, and the other is the sound playback. The specific implementation process is to sample the sounds emitted by various real musical instruments, store it in a digital music event database. When playing back, according to the needs of the synthesized music material, send instructions to the database, so as to find out the corresponding timbre information, then it is played back after being processed by the microprocessor or CPU. The playback toolset does more than just reproduce the original material, at the same time, it also performs pitch shifting, ooping, envelope and filtering, in order to achieve the effect of rich sound. The advantages of wavetable synthesis: First, the computational overhead is small; Second, it is easy to operate. The disadvantage is that a large data storage is required, and it is difficult to present different sounds due to the influence of pre-collected data.

The upper and lower solution method combined with the monotonic iteration technique is a powerful tool for analyzing nonlinear problems, its importance and compared with other analysis tools, its uniqueness and superiority, the advantage of this method is that, not only can prove the existence of the solution of the nonlinear problem under study, and can provide a monotonic iterative sequence and error estimation current that converges consistently to the solution, this tool has also been applied to solve the existence study of solutions to fractional nonlinear differential equations, recently, by applying a top-bottom solution method combined with a monotonic iterative technique, the initial value problem of nonlinear fractional differential equations. Dαu(t)=f(t,u(t)) {D^\alpha }u\left( t \right) = f\left( {t,u\left( t \right)} \right) Here 0<a≤1 under the assumption that there is a pair of upper and lower solutions, the authors give the existence conditions for the extremum solutions of the initial value problem of nonlinear fractional differential equations. In 2013, apply the top-bottom solution method, combined with the monotonic iteration technique, a class of neutral nonlinear Riemann-Liouville fractional differential equations is studied. {Dαu(t)=f(t,Dαu(t),Dαu(θ(t)),u(t))t1au(t)|t=0=0 \left\{ {\matrix{ {{D^\alpha }u\left( t \right) = f\left( {t,\,{D^\alpha }u\left( t \right),\,{D^\alpha }u\left( {\theta \left( t \right)} \right),u\left( t \right)} \right)} \hfill \cr {{{\left. {{t^{1 - a}}u\left( t \right)} \right|}_{t = 0}} = 0} \hfill \cr } } \right. Here J = [0, T] (0 ≤ T ≤ ∞), fC(J × R3, R), θC(J, J) is the Riemann-Liouville fractional derivative and 0<a<1. Under the assumption of a pair of upper and lower solutions, the existence condition of the unique solution to this problem is obtained. Consider a class of neutral fractional differential equations whose nonlinear terms depend on low-order derivatives, namely {Dαu(t)=f(t,Dαu(t),Dαu(θ(t)),Dβu(t),Iγu(t),u(t)),t(0,T)(0<T<)t1au(t)|t=0=0 \left\{ {\matrix{ {{D^\alpha }u\left( t \right) = f\left( {t,\,{D^\alpha }u\left( t \right),\,{D^\alpha }u\left( {\theta \left( t \right)} \right),\,D\beta u\left( t \right),\,{I^\gamma }u\left( t \right),\,u\left( t \right)} \right),t \in \left( {0,\,T} \right)\left( {0 < T < \infty } \right)} \hfill \cr {{t^{1 - a}}u{{\left( t \right)}_{|t = 0}} = 0} \hfill \cr } } \right. where fC(J × R3, R), θC(J, J), θ(t) ≤ t, ∀ tJ = [0, T], Dα, Dβ are the α, β Riemann-Liouville fractional derivatives, respectively, Iγ is the γ times the Riemann-Liouville fractional integral, under the assumption that a pair of upper and lower solutions are satisfied, given the above neutral fractional nonlinear differential equation, the conditions for the existence of a unique solution, and the monotonic iterative sequence that converges consistently to a unique solution and the error estimation formula [5].

The main purpose is to study the periodic boundary value problem {x+f(t,x)=e(t),t[0,T]x(0)=x[T] \left\{ {\matrix{ {x^\prime + f\left( {t,x} \right) = e\left( t \right),t \in \left[ {0,T} \right]} \hfill \cr {x\left( 0 \right) = x\left[ T \right]} \hfill \cr } } \right. First, construct non-constant upper and lower solutions for the periodic boundary value problem, and give an estimate to the obtained upper and lower solutions, so that according to theorem A, the existence of the solution to the problem is obtained as an application, and finally the existence of the positive solution of the singular periodic boundary value problem is considered [6]. Denote C[0, T] as the space composed of continuous functions defined on [0, T], for any x ∈ C[0, T], we denote x¯=1T0Tx(t)dt \bar x = {1 \over T}\int\limits_0^T {x\left( t \right)dt} x1=0T|x(t)dt| {\left\| x \right\|_1} = \int\limits_0^T {\left| {x\left( t \right)dt} \right|} x+(t)=max{x(t),0} {x^ + }\left( t \right) = \max \left\{ {x\left( t \right),0} \right\} x(t)=max{x(t),0} {x^ - }\left( t \right) = \max \left\{ { - x\left( t \right),0} \right\} For any eC[0, T] we define E=0Te(t)dt E = \int\limits_0^T {e\left( t \right)dt} E+=0Te+(t)dt {E_ + } = \int\limits_0^T {{e^ + }\left( t \right)dt} Below we give the definition of the upper and lower solutions. function{α+f(t,a)e(t),t[0,T]α(0)α(T) {\rm{function}}\left\{ {\matrix{ {\alpha ^\prime + f\left( {t,a} \right) \ge e\left( t \right),t \in \left[ {0,\,T} \right]} \hfill \cr {\alpha \left( 0 \right) \ge \alpha \left( T \right)} \hfill \cr } } \right. Similarly, the function β ∈ C1[0, T] is the solution to the problem, which means that β satisfies {β(t)+f(t,β)e(t),t[0,T]β(0)β(T) \left\{ {\matrix{ {\beta {\left( t \right)}^\prime + f\left( {t,\beta } \right) \le e\left( t \right),t \in \left[ {0,\,T} \right]} \hfill \cr {\beta \left( 0 \right) \le \beta \left( T \right)} \hfill \cr } } \right. Consider auxiliary issues first {x+ϑ(t)=0,t[0,T]x(0)=x[T]=ξ \left\{ {\matrix{ {x^\prime + \vartheta \left( t \right) = 0,t \in \left[ {0,\,T} \right]} \hfill \cr {x\left( 0 \right) = x\left[ T \right] = \xi } \hfill \cr } } \right. Where δC[0, T] meets ξ¯=0 \bar \xi = 0 , ς ∈ R

Obviously, for any ζ ∈ R, there is a unique solution xζ: xς(t)=ς0tδ(s)ds {x_\varsigma }\left( t \right) = \varsigma - \int\limits_0^t {\delta \left( s \right)ds} Suppose there is a constant B1 ∈ R and a function C1 ∈ C[0, T] such that f(t,x)c1(t) f\left( {t,x} \right) \le c1\left( t \right) t[0,t] \forall t \in \left[ {0,t} \right] x[B1,B1+2]|η1| x \in \left[ {{B_1},\,{B_1} + 2} \right]\left| {\eta 1} \right| Where η1(t) = ϕ (t) − ϕ+ if c1¯e¯0 \overline {c1} - \bar e \le 0 There exists a solution β that satisfies B1βB1+2η11 {B_1} \le \beta \le {B_1} + 2\left\| {{\eta _1}} \right\|1

Results analysis and discussion

According to the physical sound mechanism of the piano, we can model the music simulation system, it is divided into two parts: excitation system and resonance system (as shown in Figure 1). The simulation implementation process, it is based on the superposition of a series of sine waves to simulate the vibration of strings, in order to make the sound simulation more realistic, the sound modification is carried out from two levels of time domain and frequency domain respectively. In the time domain, the envelope function obtained by fitting the time domain envelope extracted by the short-time RMS energy, in order to simulate the natural decay of the strings, so that the notes of the music can be connected harmoniously; In the frequency domain, the spectral envelope is the main manifestation of the resonant body, a filter bank for modeling the envelope of the musical audio spectrum extracted by the cepstral method, in order to achieve an effective simulation of the resonance system to synthesize musical sounds [7].

Figure 1

Simulation system model

The basic sound source of the piano is the strings, and the player excites the strings to vibrate through the hammers, resulting in sound. We know that the vibration of the strings is superimposed in a sine wave mode, which envelopes the fundamental wave and the higher harmonic components, so the author will use the additive synthesis technique to simulate the vibration of the strings [8]. Next, we need to find the fundamental and harmonic frequency components, duration and other parameters required for each tone of the song to be played. By analogy, the frequency corresponding to each note in the musical notation is calculated as shown in Table 1 below.

Frequency table of each note

Note 1 2 3 1 1 2 3
Fundamental frequency (Hz) 261.62 293.65 329.62 261.62 261.62 293.65 329.62
Note 1 3 4 5 3 4 5
Fundamental frequency (Hz) 261.62 329.62 349.22 391 329.62 349.62 392

Secondly, according to the beat identification in Table 1, 4/4 represents a quarter note as one beat, and each section has four beats. In the first and second measures, each note occupies one beat, in the third measure, the first two notes each occupy one beat, and the third note occupies two beats. Knowing the number of beats per measure, we can determine the duration of each note in the music [9]. Assuming that the time of one beat is about 0.5s, the duration of each note in the music is shown in Table 2.

The duration of each note

Note 1 2 3 1 1 2 3
Duration(s) 0.5 0.5 0.5 0.5 0.5 0.5 0.5
Note 1 3 4 5 3 4 5
Duration(s) 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Based only on the data of the fundamental frequency and duration of each note in the music obtained above, not enough to model string vibrations, because there is no overtone component added in this model, the simulated music will lack a certain sense of heaviness, it is very different from the sound of the real instrument. Therefore, to add the corresponding harmonic components to the simulated model, the harmonic frequencies of the musical sounds are almost integer multiples of the fundamental frequency, and there is a certain relationship between their amplitudes, harmonic components can be added directly using the fitted curve function. Some researchers have used the weighted Cauchy function, simulates the 5-octave resonance of a piano, the simulation of timbre was realized, and later professionals found that its overtone was insufficient, and its timbre expression was not enough [10]. Then, in the paper on the design and implementation of the piano timbre recognition and electronic synthesis system, the timbre feature matrix extraction method is used, under the condition of considering the balance of effect and efficiency, the musical simulation of the 25th octave harmonic in the timbre is realized, the sound reproduction is impeccable, making up for the lack of sound of the 5th octave of the Cauchy function. But for a piano with rich overtones, it is impossible for the number and amplitude of overtones corresponding to each tone to be the same, in Cheng Meifang's research experiment, the timbre feature matrix of a tone was extracted, tone emulation in place of the characteristics of all the tones in the song, obviously not convincing, can not show the unique spectral characteristics of each tone in the composition. Therefore, in order to grasp the spectral characteristics of each tones in the music, we collected the single-tone materials of the piano C major (do, re, mi, fa, so, la, si), fourier transform is performed, respectively, obtain the frequency and normalized amplitude of each harmonic. For the above phenomenon, we know that it is caused by the inability to smoothly transition between adjacent notes of the tones. Therefore, on the basis of the above experiments, we introduce the exponential envelope and polyline envelope functions respectively, and the function of the newly designed segmented envelope is modified in the time domain, and make a simple loop process acting on each note in the song, the length and amplitude of each envelope curve, respectively, determined by the duration and amplitude of the individual notes.

Wherein (a) is the musical tone simulated by the exponential envelope; (b) is the musical tone simulated by the polygonal envelope; (c) Tones simulated for the newly designed segmental envelope curve; (d) Simultaneously modifies the simulated tones for the segmented envelope curve and filter. The following will give a detailed analysis of each simulation effect:

It can be heard from the playing music, exponential decay technology can well eliminate the transition noise between each note, decay speed determined by playing time, makes the music sound more ups and downs, lively and powerful. But through audition judgment, it is found that there is still a sudden change in phase between adjacent notes, the resulting slight noise, the transition between the two notes is not very smooth.

The application of polyline envelope technology, it is guaranteed that when the amplitude of the vibration of the previous note is attenuated to 0, only start the vibration of the next note, which can effectively eliminate the noise between note transitions. But in the process of playing the music, careful professional musicians found that, although the polyline envelope method can make the transition between the notes of the music smooth, but the simulated music sounds more blunt, it cannot show the softness of piano music, and the amplitude change does not conform to the phenomenon of stringed instruments.

From the action diagram of the newly designed piecewise envelope curve, the envelope function curve changes softly, and the attack stage is very short in time, it can increase the vibration amplitude to the maximum, ensuring the efficient pronunciation of the piano, as time goes on, the amplitude gradually decays, and when the sound of the note ends, the amplitude decays to 0, in order to prepare for the pronunciation of the next note, it is better to avoid the noise generated by the transition between adjacent notes. Discovered by audition, the musical sound effect simulated by the piecewise envelope function, significantly better than exponential envelopes and polyline envelopes, the algorithm not only makes the transition between notes smooth, but also softens the change of each tone itself, completely eliminating the noise, at the same time, it also shows that the piano sound is pronounced from the strings being struck, the whole process of soft change to the disappearance of sound intensity.

From the newly designed piecewise envelope function, it can be seen from the effect diagram that the filter modeled by the spectral envelope is modified at the same time, this method not only makes the connection and harmony among the musical notes, but also enhances the sound intensity of some signals. The music sounds undulating and orderly, very close to a real piano.

In order to verify the effectiveness of the algorithm, this subsection further evaluates. One is to judge the readiness rate by detecting the pitch frequency of the musical tone; The second is to compare the difference between the formant frequency of the simulated music and the formant peak of the real piano music; The third is to compare the effect achieved by the analysis of the spectrogram of the simulated musical tone signal in each link.

First, the linear residual cepstrum method is used, perform fundamental frequency detection on the musical tone signal of the piano simulation, this algorithm is a research hotspot in the field of audio processing, which can well avoid the influence of channel features and noise, make the test results more prepared. The test results are listed in Table 3, and compared with the theoretical value of the pitch frequency of each note in the notation chart, get its accuracy. For simplicity, only some of the pitch frequencies of the notes are shown in the Table. Accuracy=1Detectedvaluetheoreticalvaluetheoreticalvalue×100% {\rm{Accuracy}} = 1 - {{{\rm{Detected}}\,{\rm{value}} - {\rm{theoretical}}\,{\rm{value}}} \over {{\rm{theoretical}}\,{\rm{value}}}} \times 100\%

Test results

Each note in the notation Theoretical value (HZ) Detection value (HZ) Accuracy(%)
1 216.62 262.12 99.8
2 293.65 293.01 99.7
3 329.62 330.13 99.8
1 261.62 262.12 99.8
3 329.62 331.56 99.4
4 349.22 352.7 98.9
5 390 393.74 99.5

From the data detected in the Table, it can be seen that, the musical sound signal simulated by the author's method, and the theoretical value of each note of the real piano instrument collected, very close, with an accuracy rate of more than 98%, this data proves the accuracy of the simulated piano sound.

Conclusion

Based on the analysis of the vibration and attenuation characteristics of piano strings, and on the theoretical basis of the discussion on the resonance effect of the resonance system, the model of the piano music simulation system is constructed, a class with multi-order fractional derivatives is studied, and the perturbation term of the fractional integral boundary value problem of linear impulse differential equations, and one by one for the excitation system and resonance system in the model, carry out simulation modeling and experimental verification. It is found that there is a transition noise between adjacent notes, in order to eliminate this noise, try the exponential envelope technique, the polyline envelope technique separately, improve it, then take the multi-order fractional derivative to be studied, and the solution of the nonlinear impulse differential equation with the perturbation term, transformed into the fixed point problem of the operator, then by using Schauder's fixed point theorem and Banach's compression mapping principle, given the nonlinear multifractional impulse differential equation, the conditions for the existence of at least one solution and a unique solution, although these methods can eliminate the switching noise between notes, however, this method will make the simulated music sound blunt, which is not in line with the characteristics of piano music. Therefore, according to the time-domain amplitude envelope extracted from the collected piano monophonic signal, a piecewise envelope function is obtained, this envelope was found to be the perfect interpretation of the piano's entire process from articulation to decay. Secondly, the author also combined the formant point of the spectral envelope, which reflects the important information of the piano resonance system, and the sound effect EQ to design a filter, modify the simulated musical tone signal in the frequency domain, improves the timbre of musical tones, making them sound more harmonious, close to the real instrument. Finally, from the pitch frequency detection, there are three aspects of formant frequency estimation and spectrogram analysis, the author's algorithm is subjected to subjective evaluation experiments, and the experimental results verify the effectiveness of musical sound simulation.

Figure 1

Simulation system model
Simulation system model

Frequency table of each note

Note 1 2 3 1 1 2 3
Fundamental frequency (Hz) 261.62 293.65 329.62 261.62 261.62 293.65 329.62
Note 1 3 4 5 3 4 5
Fundamental frequency (Hz) 261.62 329.62 349.22 391 329.62 349.62 392

Test results

Each note in the notation Theoretical value (HZ) Detection value (HZ) Accuracy(%)
1 216.62 262.12 99.8
2 293.65 293.01 99.7
3 329.62 330.13 99.8
1 261.62 262.12 99.8
3 329.62 331.56 99.4
4 349.22 352.7 98.9
5 390 393.74 99.5

The duration of each note

Note 1 2 3 1 1 2 3
Duration(s) 0.5 0.5 0.5 0.5 0.5 0.5 0.5
Note 1 3 4 5 3 4 5
Duration(s) 0.5 0.5 0.5 0.5 0.5 0.5 0.5

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