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Model System Study of Accordion Score Based on Fractional Differential Equations

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 26 Feb 2022
Accepté: 30 Apr 2022
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Magazine
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2444-8656
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01 Jan 2016
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2 fois par an
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Anglais
Introduction

In modern society, people's demands for spiritual and cultural life are increasing. Computer automatic composition has also become an increasingly important topic. The research on automatic composition has important implications for automation technology, computer application, artificial intelligence, multimedia technology, music therapy, music technology, logic, psychology, and other theories. Development plays an essential role in promoting. It has important practical significance in music creation's efficiency, adaptability, and diversity [1]. Automatic composition is an attempt to use an inevitable formal process. This allows for minimal intervention in the use of computers to create music. At present, the recognized influential technologies in automatic composition mainly include Markov chains, stochastic processes, rule-based knowledge base systems, music grammars, artificial neural networks, and genetic algorithms. Some scholars have conducted a detailed analysis of these techniques that appeared in the literature before 2005. This document illustrates the features and shortcomings of these techniques [2]. At the same time, it summarizes the main problems existing in the research of automatic composition. The content includes knowledge expression issues in music, creativity, human-computer interaction issues, music creation style issues, and quality assessment issues for system-generated works. This article is goal-oriented towards gentleness. The report takes the pitch melody element as the basic operation unit. Mainly research the fractional differentiation algorithm for the automatic composition of an accordion.

Research on Mixed Noise Denoising Based on Fractional Differential Equation

In recent years, the models of denoising algorithms at home and abroad are assumed to be the following denoising models: u0(x,y)=u(x,y)+n(x,y) {u_0}\left({x,y} \right) = u\left({x,y} \right) + n\left({x,y} \right) u0 is the observed noisy image. u is an excellent idea that needs to be restored. n is random noise with zero mean and δ2 variance. (x, y) defines the space for the photo and (x, y) ∈ Ω, Ω represents the image area.

The global variational model of the bounded variation space is defined as TV(u)Ω|u(x,y)|dΩ TV\left(u \right)\int\limits_\Omega {\left| {\nabla u\left({x,y} \right)} \right|d\Omega}

It satisfies the constraints of Ωu(x,y)dxdy=Ωu0(x,y)dxdyΩ12|u(x,y)u0(x,y)|2dxdy=σ2 \int\limits_\Omega {u\left({x,y} \right)dxdy =} \int\limits_\Omega {{u_0}\left({x,y} \right)dxdy\,\int\limits_\Omega {{1 \over 2}{{\left| {u\left({x,y} \right) - {u_0}\left({x,y} \right)} \right|}^2}\,dxdy = {\sigma ^2}}}

Using the Lagrange multiplier method, the image noise removal problem can be defined as the depreciation of the energy functional as: u^=argminu{E(u)}=λ2Ω(uu0)2dxdy+Ωux2+uy2dxdy \hat u = \arg \mathop {\min}\limits_u \left\{{E\left(u \right)} \right\} = {\lambda \over 2}\int\limits_\Omega {{{\left({u - {u_0}} \right)}^2}dxdy +} \int\limits_\Omega {\sqrt {u_x^2 + u_y^2} \,dxdy}

The first item is the data item. The second term is the regularization term. Parameter λ is the regularization parameter. Its value depends on the noise level of the image. It plays an essential role in the image's balanced denoising and smoothing process [3]. When λ =0, the edges of the image are blurred due to excessive smoothing of the picture.

Conversely, the restored noisy image at λ = ∞ will satisfy uu0. At this moment, the image oscillates violently. Therefore, the regularization parameter E value cannot be too small or too large.

The Euler-Lagrange equation corresponding to equation (4) can be deduced by the variational method as follows: λ(uu0)x(uxux2+uy2)y(uyux2+uy2)=0 \lambda \left({u - {u_0}} \right) - {\partial \over {\partial x}}\left({{{{u_x}} \over {\sqrt {u_x^2 + u_y^2}}}} \right) - {\partial \over {\partial y}}\left({{{{u_y}} \over {\sqrt {u_x^2 + u_y^2}}}} \right) = 0

The gradient descent flow equation corresponding to the variational problem equation (5) is: ut=x(uxux2+uy2)+y(uyux2+uy2)λ(uu0)=div[u|u|]λ(uu0) {{\partial u} \over {\partial t}} = {\partial \over {\partial x}}\left({{{{u_x}} \over {\sqrt {u_x^2 + u_y^2}}}} \right) + {\partial \over {\partial y}}\left({{{{u_y}} \over {\sqrt {u_x^2 + u_y^2}}}} \right) - \lambda \left({u - {u_0}} \right) = div\left[{{{\nabla u} \over {\left| {\nabla u} \right|}}} \right] - \lambda \left({u - {u_0}} \right)

div is the divergence operator. 1/|∇u| is the TV stream. TV flow is stable and has a globally optimal solution, a diffusion function between forwarding and reverse diffusion [4]. We can get a better diffusion process and image edge information preservation effect. Since the ROF model has good properties of the TV flow, the feature information of the image can be better preserved in image denoising. Therefore, we can obtain a better denoising effect. This paper proposes a weighted mixed noise model: u0(x,y)=u(x,y)+μ[u(x,y)][n(x,y)]+(1μ)n(x,y) {u_0}\left({x,y} \right) = u\left({x,y} \right) + \mu \left[{u\left({x,y} \right)} \right]\left[{n\left({x,y} \right)} \right] + \left({1 - \mu} \right)n\left({x,y} \right)

u0(x, y) is the observed noisy image. n(x, y) is random noise with mean zero and variance δ2. u(x, y) is the original noise-free image that needs to be restored. μ is the weighting factor [5]. The noise removal of this mixed noise model can be reduced to the minimization problem of the following energy functional: E(u)=λ2Ω(u0uμuμ+1)2dxdy+Ωux2+uy2dxdy E\left(u \right) = {\lambda \over 2}\int\limits_\Omega {{{\left({{{{u_0} - u} \over {\mu u - \mu + 1}}} \right)}^2}dxdy +} \int\limits_\Omega {\sqrt {u_x^2 + u_y^2} dxdy}

This paper uses the gradient descent method to solve this differential equation. We regularize |∇u| to |u|τ=|u|2+τ2 {\left| {\nabla u} \right|_\tau} = \sqrt {{{\left| {\nabla u} \right|}^2} + {\tau ^2}} . The gradient descent flow equation corresponding to minimizing the energy functional is ut=div[u|u|τ]λ(u0u)(μu0μ+1)(μuμ+1)3 {{\partial u} \over {\partial t}} = div\left[{{{\nabla u} \over {{{\left| {\nabla u} \right|}_\tau}}}} \right] - \lambda {{\left({{u_0} - u} \right)\left({\mu {u_0} - \mu + 1} \right)} \over {{{\left({\mu u - \mu + 1} \right)}^3}}}

Commonly used numerical schemes for finite difference methods include explicit, implicit, and semi-implicit difference schemes [6]. The precise difference method is chosen for numerical implementation among these typical schemes. We employ a detailed differencing system. Equation (9) can be expressed as un+1=un+Δt2λdiv(un|un|τ)Δt(u0un)(μu0μ+1)(μunμ+1)3 {u^{n + 1}} = {u^n} + {{\Delta t} \over {2\lambda}}div\left({{{\nabla {u^n}} \over {{{\left| {\nabla {u^n}} \right|}_\tau}}}} \right) - \Delta t{{\left({{u_0} - {u^n}} \right)\left({\mu {u_0} - \mu + 1} \right)} \over {{{\left({\mu {u^n} - \mu + 1} \right)}^3}}}

We replace the partial derivative with the difference quotient to get: Δ±xui,j=±(ui±1,jui,j),Δ±yui,j=±(ui,j±1ui,j)Δ0xui,j=(ui±1,jui1,j)/2,Δ0yui,j=(ui,j±1ui,j1)/2Δt2λ[ΔxΔ+xui,jn(Δ+xui,jn)2+(Δ0yui,jn)2+τ2+ΔyΔ+yui,jn(Δ0xui,jn)2+(Δ+yui,jn)2+τ2]Δt(u0un)(μu0μ+1)(μunμ+1)3 \matrix{{\Delta _ \pm ^x{u_{i,j}} = \pm \left({{u_i}_{\pm 1,j} - {u_{i,j}}} \right),\Delta _ \pm ^y{u_{i,j}} = \pm \left({{u_{i,j \pm 1}} - {u_{i,j}}} \right)} \hfill \cr {\Delta _0^x{u_{i,j}} = \left({{u_i}_{\pm 1,j} - {u_{i - 1,j}}} \right)/2,\,\Delta _0^y{u_{i,j}} = \left({{u_{i,j \pm 1}} - {u_{i,j - 1}}} \right)/2} \hfill \cr {{{\Delta t} \over {2\lambda}}\left[{\Delta _ - ^x{{\Delta _ + ^xu_{i,j}^n} \over {\sqrt {{{\left({\Delta _ + ^xu_{i,j}^n} \right)}^2} + {{\left({\Delta _0^yu_{i,j}^n} \right)}^2} + {\tau ^2}}}} + \Delta _ - ^y{{\Delta _ + ^yu_{i,j}^n} \over {\sqrt {{{\left({\Delta _0^xu_{i,j}^n} \right)}^2} + {{\left({\Delta _ + ^yu_{i,j}^n} \right)}^2} + {\tau ^2}}}}} \right] -} \hfill \cr {\Delta t{{\left({{u_0} - {u^n}} \right)\left({\mu {u_0} - \mu + 1} \right)} \over {{{\left({\mu {u^n} - \mu + 1} \right)}^3}}}} \hfill \cr}

un is the image after the n diffusion. Δt is the time interval or time step. Generally, take a positive number less than 0.2. The regularization parameter λ should take 1σ2 {1 \over {{\sigma ^2}}} at the beginning of the operation [7]. Then the regularization parameter value λ needs to be updated every N step. The value of the time interval Δt will directly affect the convergence of the numerical calculation process. The time interval required for this iteration. We can calculate by the result of the previous iteration, namely Δt((p,q)N(i,j)Δp,qn+Δi,jn2h2+λ)1 \Delta t{\left({\sum\limits_{\left({p,q} \right) \in N\left({i,j} \right)} {{{\Delta _{p,q}^n + \Delta _{i,j}^n} \over {2{h^2}}} + \lambda}} \right)^{- 1}}

h is the pixel spacing and Δi,jn \Delta _{i,j}^n is the diffusivity.

Experiment Design and Results Analysis
Fractional differential equation model construction

After experimental verification, it is found that the Markov chain is more random when generating new notes. So we designed and implemented the fractional differential equation model on this basis (Fig. 1).

Figure 1

Fractional differential equation music recognition model

Let's take “Ode to Joy” as an example. We observed the melody of this piece and found that only the last two notes are different in measures 1–4 and 5–8. All other sounds are the same. Therefore, their melody fluctuations are the same. However, compared with bars 13–16, there is only one tone difference in bars 6 and 14. Their melody is the same. We select subsections 1 to 4 and 9 to 16 to conduct two sets of comparative experiments [8]. The MIDI code sequence corresponding to the note sequence in bars 1 to 4 of “Ode to Joy” is “646465676765646260606264646262”. Use Kslider for sound source input in the fractional differential equation model program. After the model training is completed, it is shown in Figure 2.

Figure 2

Audio input in the differential equation model

We generate a new series of pitch values with a Markov chain model. We choose a pitch consistent with the beginning of the original pitch value sequence as the starting point. We use the MIDI code as the “64” note starting point [9]. At this point, we can record the three new pitch value sequences generated separately (Table 1). The length of each group's new pitch value sequence is consistent with the size of the original pitch value sequence.

Three sets of new pitch value sequences generated by the fractional differential equation model

Group No The resulting sequence of new pitch values
1 646465676765646260606465676765
2 646567676564626060626464656767
3 646567676564626060626464626060

The MIDI code sequence corresponding to the note sequence in bars 9 to 16 of “Ode to Joy” is “62626460626465646062646564626062556464646567676564656260606264626060”. The fractional differential equation model training results are shown in Figure 3.

Figure 3

Results after entering bars 9 to 16 in the fractional differential equation model

The fractional differential equation model generates results as “2, 4, 5, 4, 0, 2, −5, 4, 4, 5, 4, 5, 4, 2, 0, 0, 2, −5, 4, 4, 4, 5, 7, 7, 5, 4, 2, 0, 0, 2, 4, 5, 4, 0”. We used the most basic drawing functions in the R language to make line graphs from the data generated by the two groups of experiments (Fig. 4, Fig. 5). Figure 4 shows the results of the first group of experiments.

Figure 4

Line chart of fractional differential equation generation results (sections 1–4)

Figure 5

Line chart of fractional differential equation generation results (sections 9–16)

Although the fractional differential equation model can better imitate the original music's melody trend when generating a new melody, it still has certain randomness [10]. There are also a few outliers in the developed theme that affect the coherence of the composition. If the ISM algorithm is combined with the fractional differential equation model to create the pitch value sequence, the melody line of the generated music can be improved. This makes it more coherent and closer to the original piece's melodic character and musical style. Therefore, we combined the ISM algorithm with the fractional differential equation model to create the melody based on the above-mentioned second group of experiments. This will verify the conjecture. This experiment uses the 9th to 16th bars of “Ode to Joy” as the training data. First, take this set of data “62626460626465646062646564626062556464646567676564656260606264626060” as the input to the ISM algorithm.

The optimal solution of the interval subsequence with melodic features found by the ISM algorithm is “002002”. It means that a pitch subsequence consisting of three tones starts from the first pitch and goes up two semitones and then goes up two semitones. From the input sequence of pitch values, we can easily observe that this sequence is actually “606264”. The sequence does appear three times. After mining this interval subsequence, we can consciously observe and control the generation of the sequence in real-time in generating the pitch sequence by the fractional differential equation. The composer can change the state of the Markov chain when appropriate [11]. The composer can actively control the “ C ” sound when the “C” sound appears. We generate the subsequent “DE” tone and then continue to generate the pitch sequence by the fractional differential equation randomly. This forms the process of human-computer interaction. In this experiment, we use this pitch sequence excavated to experiment again and properly control generating a new pitch sequence with the fractional differential equation model of the second group of experiments. The obtained experimental results are “2, 4, 5, 4, 0, 2, 4, 5, 4, 4, 5, 4, 2, 0, 0, 2, −5, 4, 4, 4, 5, 7, 7, 5, 4, 2, 0, 0, 2, 4, 5, 4, 0, 2”. FIG. 6 is a line graph drawn from the experimental results.

Figure 6

Experimental results of ISM algorithm combined with fractional differential equations

It can be seen from Figure 6 that the new melody line generated by the combination of the ISM algorithm and the fractional differential equation model is more consistent with the original melody. And the composer's proper control of the Markov stochastic process in the process of human-computer interaction [12]. This keeps the development of the theme under control. At this time, it is more efficient and convenient to generate new works that are more consistent with the style of the original musical works.

Exploratory Experiment - Generating New Music Pieces

We verified the effectiveness of the ISM algorithm combined with the fractional differential equation model for algorithm composition and then conducted an exploratory experiment. We used this method to create a more complex piece to examine its utility. We choose the melody part of Bach's No. 2 in C natural major as experimental material. Usually, in piano music, the high-pitched part is the melody. This makes the piece have a clear and beautiful line. The bass part is the accompaniment. It is mainly to make the music full of rhythmic changes. The major scale is simply referred to as the major scale. It consists of seven tones. It includes natural, harmonic, and melodic primary three. The natural major scale is the most common of the major scales. The key of C natural major is composed of seven tones “CDEFGAB.” The ending note of a piece is usually “C.” The musical score (18 bars in total) of the musical composition “Bach's Creative Variations” generated by this experiment is shown in Figure 7.

Figure 7

The score created by the ISM algorithm combined with fractional differential equations

Conclusion

The interval subsequences mined in experiments for fractional differential equations are often shorter. In particular, when the input pitch value sequence is long, the excavated melody line is still sharp. It is not easy to grasp the overall melody style of the music.

Figure 1

Fractional differential equation music recognition model
Fractional differential equation music recognition model

Figure 2

Audio input in the differential equation model
Audio input in the differential equation model

Figure 3

Results after entering bars 9 to 16 in the fractional differential equation model
Results after entering bars 9 to 16 in the fractional differential equation model

Figure 4

Line chart of fractional differential equation generation results (sections 1–4)
Line chart of fractional differential equation generation results (sections 1–4)

Figure 5

Line chart of fractional differential equation generation results (sections 9–16)
Line chart of fractional differential equation generation results (sections 9–16)

Figure 6

Experimental results of ISM algorithm combined with fractional differential equations
Experimental results of ISM algorithm combined with fractional differential equations

Figure 7

The score created by the ISM algorithm combined with fractional differential equations
The score created by the ISM algorithm combined with fractional differential equations

Three sets of new pitch value sequences generated by the fractional differential equation model

Group No The resulting sequence of new pitch values
1 646465676765646260606465676765
2 646567676564626060626464656767
3 646567676564626060626464626060

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