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.

We get a sequence of numbers by transforming and _{i}

The autocorrelation function gives the inter-signal correlation coefficient with a disparity

The Fourier transform of the music signal time series is

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 _{k}_{(k−1)l+1}, _{(k−1)l+2}, ⋯, _{(k−1)} _{k}_{j}_{j}_{+1}} from the envelope peak sequence [5]. It represents the magnitude of vibrational energy. The mountains of the envelopes vary, and {_{j}

We can also obtain the time interval of each adjacent peak value of the envelope: the peak cycle of the envelope and the sequence {_{j}

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{_{i}

Here, the value of

The resulting sequence can be denoted _{1}, _{2}, _{3}, ⋯ ⋯. 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 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.

In the formula, _{jk}_{ij}_{j}_{1}, _{j}_{2}, ⋯ ⋯, _{jn}

We give the Cauchy problem with the semi-inverse operator

Its operator solution is

In the formula,
_{n} is treated as a number symbol, the equation is transformed into
^{a2tΔn} [^{0} [^{a2tΔn}

Definition 1: Assuming ^{∞} (

Definition 2: Assuming that [^{n}

We write the equation of problem (11) as

Its eigenvalue is ^{a2tΔ} [

Similar to the method of the problem (11), we obtain the eigenvalue

In the formula

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.

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.

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.

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.

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.

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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