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01 Jan 2016
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# Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations

###### Accettato: 24 Jun 2022
Dettagli della rivista
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Introduction

Mathematical cognition and intelligent creation of music are regarded as one of the fundamental goals of computer music. Research in mathematical music cognition and brilliant creation is just the beginning. The traditional method of generating music by human works shows its limitations in meeting demand and broader adaptability. In the past, people kept a close eye on music generation and kept it at a distance. One of the reasons is that it is overly believed that the whole process of generating beautiful music depends on the level of human intelligence [1]. But the piece consists of the sounds of nature. The so-called “natural sounds, earth sounds, and human sounds” are all reorganized forms that people feel in their senses. It is a rhythmic reproduction associated with human emotion. In this sense, mathematical simulation of music generation should have unique characteristics. Because today's computer technology already has a relatively mature software and hardware foundation in simulating vocalization or analyzing rhythm organization. But to highly theorize the melody and rhythm of music have so far been impossible. This study treats the digitized music signal as a time series. This paper discusses the mathematical modeling, mathematical simulation generation, and related technical problems of music signal time series based on the homogeneous first-order linear partial differential.

Analysis and parameter estimation of music signal time series

We get a sequence of numbers by transforming and A / D transforming a piece of music. The series of numbers we get at this point is denoted as {xi}(i = 1, 2, ⋯⋯). All the information about the amount of music is contained in the sequence of numbers. We refer to this sequence of numbers as the music signal time series. Figure 1 shows the music signal time series of a piece of pop music I belive. This is 48,000 data records at sampling frequency 8000Hz for the first six seconds of the article [2]. Therefore, it should generally be always equal to 0. It can also be found that the curve vibrates up and down with 0 as the center. We define the following eigenvalues (Figure 1).

Autocorrelation function
$R(τ)=limN→∞1N∑i=1N(xi−x¯)(xi+τ−x¯)s2(τ=0,1,2,⋯)$ R\left( \tau \right) = {{\mathop {\lim }\limits_{N \to \infty } {1 \over N}\sum\limits_{i = 1}^N {\left( {{x_i} - \bar x} \right)\left( {{x_{i + \tau }} - \bar x} \right)} } \over {{s^2}}}\left( {\tau = 0,1,2, \cdots } \right)

The autocorrelation function gives the inter-signal correlation coefficient with a disparity τ (τ =0, 1, 2, ⋯) in the music signal time series. It is closely related to the melody of music. FIG. 2 shows the autocorrelation function curve of the music signal time series shown in FIG. 1. The big difference τ in the figure ranges from 0 to 500.

Frequency density function

The Fourier transform of the music signal time series is $F(ω)=limN→∞1N∑k=1Nexp(−2kπ ωi)xk$ F\left( \omega \right) = \mathop {\lim }\limits_{N \to \infty } {1 \over N}\sum\limits_{k = 1}^N {\exp \left( { - 2k\pi \;\omega i} \right)} {x_k}

F(ω) represents the distribution of the fluctuation component of frequency ω in the music time series.

Envelope peak and its peak period

Observing Fig. 1, it can be seen that the music signal time series has a vibration peak at every period (about 0.55 seconds) and then gradually decays [3]. Such changes are repeated. It is closely related to the weak and robust rhythm of the music. We refer to the interval between two vibration peaks as the envelope peak period [4].

Because the music signal time series can be approximately regarded as a series of symmetrical fluctuations with the origin as the center, we only need the value of the positive part of the series in the analysis. We divide the sequence into n intervals of length l. Here, the size l is set to be 2 to 3 times or more of the average period of the autocorrelation function of the original music. Find the maximum value in each interval and denote it as zk = max {x(k−1)l+1, x(k−1)l+2, ⋯, x(k−1) l + 2}. This way, we get a new sequence {zk} {k = 1, 2, ⋯, n). We can get the approximate envelope of the positive value of the music signal by connecting the points of this sequence with a straight line. We use the method described above to obtain the maximum amplitude value from {Aj} to the next vibration peak {Aj+1} from the envelope peak sequence [5]. It represents the magnitude of vibrational energy. The mountains of the envelopes vary, and {Aj} forms a fluctuating sequence. $A¯=1m∑k=1mAj SA2=1m∑k=1m(Aj−A¯)2$ \bar A = {1 \over m}\sum\limits_{k = 1}^m {{A_j}} \;\;\;S_A^2 = {1 \over m}\sum\limits_{k = 1}^m {{{\left( {{A_j} - \bar A} \right)}^2}}

We can also obtain the time interval of each adjacent peak value of the envelope: the peak cycle of the envelope and the sequence {Tj} (j = 1, 2, ⋯, m) of the peak cycle of the envelope. $T¯=1m∑k=1mTk ST2=1m∑k=1m(Tk−T¯)2$ \bar T = {1 \over m}\sum\limits_{k = 1}^m {{T_k}} \;\;\;S_T^2 = {1 \over m}\sum\limits_{k = 1}^m {{{\left( {{T_k} - \bar T} \right)}^2}}

Fluctuation characteristics in the peak period of the envelope

We stretch the fluctuation curve inside a certain envelope peak period on the time axis. At this time, it can be seen that the music signal time series presents a superposition of large low-frequency fluctuations and small high-frequency fluctuations [6]. We can filter higher frequency small fluctuations by the moving average calculation method. The moving intermediate sequence of the music signal time series{xi} is represented as follows $x¯1=1k∑i=1kx1,x¯2=1k∑i=1kxi+1,⋯⋯,x¯r=1k∑i=1kxi+r−1,⋯⋯$ {\bar x_1} = {1 \over k}\sum\limits_{i = 1}^k {{x_1},{{\bar x}_2}} = {1 \over k}\sum\limits_{i = 1}^k {{x_{i + 1}}} , \cdots \cdots ,{\bar x_r} = {1 \over k}\sum\limits_{i = 1}^k {{x_{i + r - 1}}, \cdots \cdots }

Here, the value of k is slightly smaller than that of 1/4 in the considerable fluctuation period. Relative to the moving average series, we define the small volatility as $δ=x−x¯r$ \delta = x - {\bar x_r}

The resulting sequence can be denoted δ1, δ2, δ3, ⋯ ⋯. Minor fluctuations are usually a sinusoidal function with zero mean and high frequency. We estimate the autocorrelation function of the arrangement according to the formula (1). We can use autoregressive models to make predictions on sequences.

The decay rate of large fluctuations in the peak period of the envelope

The sequence starts from the maximum amplitude, and then the wave amplitude is approximately exponentially decaying [7]. If we filter the fluctuations in the peak period of an envelope with the above method to filter out small volatility, we can see that it has relatively regular vibrations. We use the following exponential decay vibration curve model. The purpose is to approximate its continuous fluctuation characteristics. $xjk=Ajexp(−λk)cos2π ωt (k=1,2,⋯⋯)$ {x_{jk}} = {A_j}\exp \left( { - \lambda k} \right)\cos 2\pi \;\omega t\;\;\;\left( {k = 1,2, \cdots \cdots } \right)

In the formula, xjk represents the displacement of the sounding oscillator at the k time point in the j envelope cycle. λ is the decay rate of the sequence. ω is the frequency of large fluctuations during the peak period of this envelope. The series of changes in the j envelope peak period is denoted as {xij} (k = 1, 2, ⋯ ⋯). We obtain the moving average sequence ${x¯jk}(k=1,2,⋯,r)$ \left\{ {{{\bar x}_{jk}}} \right\}\left( {k = 1,2, \cdots ,r} \right) within the peak period of the j envelope according to equation (5). If the 1st to n vibration peaks are yj1, yj2, ⋯ ⋯, yjn respectively, then the decay rate λ in the peak period of the envelope can be estimated by the following formula: $λ¯j=1n−1∑ν=2n1ν−1lin(yiνyj1)$ {\bar \lambda _j} = {1 \over {n - 1}}\sum\limits_{\nu = 2}^n {{1 \over {\nu - 1}}lin\left( {{{{y_{i\nu }}} \over {{y_{j1}}}}} \right)}

Fluctuation frequency in the peak period of the envelope

ω in equation (7) is the frequency with the most considerable contribution rate in the peak period of the envelope. Frequency is what determines how high or low a scale is. The larger the ω value, the higher the pitch. The law of frequency change is one of the essential factors in music composition.

Music creation based on homogeneous first-order linear differential equations
The introduction of the differential operator method

We give the Cauchy problem with the semi-inverse operator $p−1(∂∂t+f(∂x))=e−tf(∂x)p(∂∂t)etf(∂x) (x∈Rn)$ {p^{ - 1}}\left( {{\partial \over {\partial t}} + f\left( {{\partial _x}} \right)} \right) = {e^{ - tf\left( {{\partial _x}} \right)}}p\left( {{\partial \over {\partial t}}} \right){e^{tf\left( {{\partial _x}} \right)}}\;\;\;\left( {x \in {R^n}} \right) of operator $p(∂∂t+f(∂x))$ p\left( {{\partial \over {\partial t}} + f\left( {{\partial _x}} \right)} \right) : ${∂u∂t+p(∂x)u=f(x,t)u|t=0=φ(x) (t∈R1,x∈Rn)$ \left\{ {\matrix{ {{{\partial u} \over {\partial t}} + p\left( {{\partial _x}} \right)u = f\left( {x,t} \right)} \hfill \cr {u{|_{t = 0}} = \varphi \left( x \right)\;\;\;\left( {t \in {R^1},x \in {R^n}} \right)} \hfill \cr } } \right.

Its operator solution is $u(x,t)=∫0te−(t−T)p(∂x)f(x,T)dT+e−tp(∂x)φ(x)$ u\left( {x,t} \right) = \int_0^t {{e^{ - \left( {t - T} \right)p\left( {{\partial _x}} \right)}}} f\left( {x,T} \right)dT + {e^{ - tp\left( {{\partial _x}} \right)}}\varphi \left( x \right) ${∂u∂t=a2Δnu (x∈Rn)u|t=0=φ(x)$ \left\{ {\matrix{ {{{\partial u} \over {\partial t}} = {a^2}{\Delta _n}u\;\;\;\left( {x \in {R^n}} \right)} \hfill \cr {u{|_{t = 0}} = \varphi \left( x \right)} \hfill \cr } } \right.

In the formula, $Δn=∂2∂x12+∂2∂x22+⋯+∂2∂xn2$ {\Delta _n} = {{{\partial ^2}} \over {\partial x_1^2}} + {{{\partial ^2}} \over {\partial x_2^2}} + \cdots + {{{\partial ^2}} \over {\partial x_n^2}} is the Laplace operator. If Δn is treated as a number symbol, the equation is transformed into $∂u∂=(a2Δn)∂t$ {{\partial u} \over \partial } = \left( {{a^2}{\Delta _n}} \right)\partial t . The general solution is that u(x, t) = ea2tΔn [G(x)], G(x) is an arbitrary function. And because of u(x, 0) e0 [G(x)] = φ(x), the solution of problem (1) is u(x, t) = ea2tΔn φ (x).

Definition 1: Assuming GC (I), then $emΔn[G(x)]k=0∞=mkk!Δnk[G(x)]$ {e^{m{\Delta _n}}}\left[ {G\left( x \right)} \right]_{k = 0}^\infty = {{{m^k}} \over {k!}}\Delta _n^k\left[ {G\left( x \right)} \right] exists.

Definition 2: Assuming that [G(x)] is an integrable function on Rn, then there is $H(x,m)=(4mπ)−n/2exp(−|x|4m)$ H\left( {x,m} \right) = {\left( {4m\pi } \right)^{ - n/2}}\exp \left( { - {{\left| x \right|} \over {4m}}} \right)

We write the equation of problem (11) as $(∂∂t−a2Δ)u=0$ \left( {{\partial \over {\partial t}} - {a^2}\Delta } \right)u = 0

Its eigenvalue is r = at Δ. The general solution is denoted as u(x, t) = ea2tΔ [F(x)]. ${∂2u∂2t=a2Δu (t>0,x∈Rn)u(x,0)=φ(x),∂u∂t(x,0)=ψ(x)$ \left\{ {\matrix{ {{{{\partial ^2}u} \over {{\partial ^2}t}} = {a^2}\Delta u\;\;\;\left( {t > 0,x \in {R^n}} \right)} \hfill \cr {u\left( {x,0} \right) = \varphi \left( x \right),{{\partial u} \over {\partial t}}\left( {x,0} \right) = \psi \left( x \right)} \hfill \cr } } \right.

Similar to the method of the problem (11), we obtain the eigenvalue $r1=at Δ¯$ {r_1} = at\;\bar \Delta , $r2=−at Δ¯$ {r_2} = - at\;\bar \Delta of the homogeneous equation of (13). Assuming that the solution of (13) is $u(x,t)=eat Δ¯[G1(x)]+e−at Δ¯[G2(x)]$ u\left( {x,t} \right) = {e^{at\;\bar \Delta }}\left[ {{G_1}\left( x \right)} \right] + {e^{ - at\;\bar \Delta }}\left[ {{G_2}\left( x \right)} \right] , then the initial conditions are satisfied by the following formula: ${G1(x)+G2(x)=φ(x)a Δ¯[G1(x)+G2(x)]=ψ(x)$ \left\{ {\matrix{ {{G_1}\left( x \right) + {G_2}\left( x \right) = \varphi \left( x \right)} \hfill \cr {a\;\bar \Delta \left[ {{G_1}\left( x \right) + {G_2}\left( x \right)} \right] = \psi \left( x \right)} \hfill \cr } } \right. ${G1(x)=12[φ(x)+1a Δ¯ψ(x)]G2(x)=12[φ(x)+1a Δ¯ψ(x)]$ \left\{ {\matrix{ {{G_1}\left( x \right) = {1 \over 2}\left[ {\varphi \left( x \right) + {1 \over {a\;\bar \Delta }}\psi \left( x \right)} \right]} \hfill \cr {{G_2}\left( x \right) = {1 \over 2}\left[ {\varphi \left( x \right) + {1 \over {a\;\bar \Delta }}\psi \left( x \right)} \right]} \hfill \cr } } \right. $u(x,t)=ch[at Δ¯]φ(x)+sh[at Δ¯]a Δ¯[ψ(x)]$ u\left( {x,t} \right) = ch\left[ {at\;\bar \Delta } \right]\varphi \left( x \right) + {{sh\left[ {at\;\bar \Delta } \right]} \over {a\;\bar \Delta }}\left[ {\psi \left( x \right)} \right]

In the formula $ch[at Δ¯]=12[eat Δ¯+e−at Δ¯],sh[at Δ¯]Δ¯=1a Δ¯[eat Δ¯−e−at Δ¯]$ ch\left[ {at\;\bar \Delta } \right] = {1 \over 2}\left[ {{e^{at}}\;\;\bar \Delta + {e^{ - at}}\;\bar \Delta } \right],{{sh\left[ {at\;\bar \Delta } \right]} \over {\bar \Delta }} = {1 \over {a\;\bar \Delta }}\left[ {{e^{at}}\;\bar \Delta - {e^{ - at}}\;\bar \Delta } \right] $ch[at Δ¯F(x)]k=0∞=m2k(2k)!Δk[F(x)],sh[at Δ¯]Δ¯[F(x)]k=0∞=m2k+1(2k+1)!Δk[F(x)]$ ch\left[ {at\;\bar \Delta F\left( x \right)} \right]_{k = 0}^\infty = {{{m^{2k}}} \over {\left( {2k} \right)!}}{\Delta ^k}\left[ {F\left( x \right)} \right],{{sh\left[ {at\;\bar \Delta } \right]} \over {\bar \Delta }}\left[ {F\left( x \right)} \right]_{k = 0}^\infty = {{{m^{2k + 1}}} \over {\left( {2k + 1} \right)!}}{\Delta ^k}\left[ {F\left( x \right)} \right]

Simulation generation experiment of music
Experimental method and digital transformation using MATLAB software

We used MATLAB software in the simulation generation experiments of music. This software provides a series of functions in audio processing to realize the mutual conversion of sound signal analog quantity and digital sequence [8]. MATLAB is used to recognize the analog conversion of the recording, the restoration of the digital sequence, and the playback.

Simulation generation test

After analyzing the statistical characteristics of the original music signal time series shown in Figure 1, we use the string vibration function model (9) and the autoregressive model (10) to simulate the generation of music. In the simulation generation, the parameters in the peak period of each envelope and the age of the peak period of the envelope adopt the linear autoregressive model of formula (10). The characteristic parameter autocorrelation function of the continuous change of each parameter is obtained by analyzing the original sequence.

Results
The sound effects generated by the time series of simulated music signals

FIG. 3 is an example of a music signal time series generated by simulating the aforementioned original music according to the abovementioned method. We use MATLAB's playback function Soundview to playback and feel similar to the original music [9]. We further analyze the statistical characteristics of the time series of simulation-generated music signals. At the same time, we compare it with the original music signal time series.

Autocorrelation function of analog music signal time series

The pseudo-period averages for both analog and original music are around 40. The time-series correlograms of analog music signals failed to show this feature [10] clearly. Its autocorrelation curve peaks have relatively minor differences.

Frequency-Domain Characteristics of Analog Music Signal Time Series

The frequency-domain characteristics of analog music signal time series can be understood and compared by Fourier transform of music signal time series On the whole, the Fourier transform is performed on the entire music signal time series. FIG. 4 is a spectrogram of the whole sequence of the time series of the original music signal and the time series of the analog music signal shown above. It can be seen from the figure that the overall frequency distribution range of the two pieces of music is the same, and the primary frequency is the same. At the same time, it can be seen that the frequency distribution area of the original music is relatively wide., the frequency components are richer than analog music. The spectrum distribution range of the whole sequence is another problem that needs to be explored.

(2) It is imperative to examine the contribution rate of various frequencies in the time series of music signals in different periods. We performed the Fourier transform of each vibration peak period in the music signal time series [11]. Figure 5 presents the spectrograms of the fifth vibration peak cycle in the original and analog music signal time series. It can be further seen that the frequency distribution area of the original music is still more significant than the time series of the analog music signal during the rhythm period.

Conclusion

This paper discusses various characteristic parameters of music signals based on the homogeneous first-order linear partial differential method. At the same time, we use two models to simulate and generate music signals. According to the investigation results of the simulation experiments, we find that the simulation music and the original music have some similarities.

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