Intelligent recommendation algorithm for piano tracks based on the CNN model

3.1 Design of a limiter for piano waveform characteristic factors in various complex environments

The frequency of the piano waveform eigenfactor is a very important parameter of the piano waveform eigenfactor, and the piano structure density is n ₀. We can assume that there is a small displacement Δx between the electrons and the relatively stationary ions, so there is an excess of electrons on the side where the characteristic factor of the slab piano waveform appears. However, there is a phenomenon of excess ions on the other side of the characteristic factor of the slab piano waveform. We can think of these two charge regions as thin enough surfaces with a surface charge density of n ₀ e, and the resulting electrostatic field is

where ε 0 represents the vacuum dielectric constant.

This electric field has a tendency to pull the electrons back to their original positions, which is called the restoring force. In the absence of an external magnetic field and the thermal motion of charged particles is negligible, the equation of motion for a single electron is

Among them, the electron mass is m e , and the negative sign means that the direction of electron motion is opposite to the direction of the electric field. The above equation is rewritten as the oscillatory equation:

where ω pe represents the plasma angular frequency.

Among them,

is called the electronic piano waveform characteristic factor frequency. By substituting each constant and combining formula (4) and ω pe = 2 π f pe , the following can be calculated

where f pe represents the plasma frequency.

Similarly, electrostatic oscillations of ions can be discussed. We can assume that electrons are more active than other particles, and the electrons form a uniform distribution in space during the entire oscillation period, which mainly depends on the thermal motion of the electrons themselves. The ion oscillates in the electron background, where the electron background is uniformly distributed, and the ion oscillation frequency is

Among them, m i is the ion mass. Obviously, Z i represents the charge of the ion

Due to the fact that the mass of particles is much greater than that of electrons, there exists a situation where m i >> m e , so m e can be ignored, so ω pi 2 can also be ignored.

In general, the oscillation frequency of the characteristic factor of the piano waveform can be written as the following formula, which also considers the movement of electrons and ions under the action of the electric field

3.2 Design of resonant elements

The structure of the resonance window can be considered as a metal diaphragm with a sealed solid medium. Typically, the sealing medium is low-loss glass or ceramic. Because the window is a vacuum-sealed element, the tightness of the medium and the metal should also be considered.

Figure 2 is a schematic diagram of a commonly used resonant window structure. The 10 and 3 cm bands are commonly used.

Figure 2

Schematic diagram of the structure of the resonance window.

The resonant window is a resonant element. In the waveguide piano waveform characteristic factor limiter, its equivalent circuit and frequency response curve are shown in Figure 3.

Figure 3

Response diagram of resonant tank and frequency.

The susceptance B of the equivalent circuit is

The angular frequency at B = 0 is defined as the resonant angular frequency

Because there is a relationship between the resonant frequency and the resonant angular frequency

In the formula, C is considered the equivalent capacitance of the resonant window, L is considered the equivalent inductance of the resonant window, G is considered the equivalent conductance of the resonant window, and f 0 is considered the resonant frequency of the resonant window.

The on-load quality factor of the resonant window is defined as

In the formula, ω 2 and ω 1 are the frequencies at which the susceptance B is equal to the total loop conductance ( 2 Y 0 + G ) . The susceptance at ω 2 and ω 1 are, respectively, equal to the positive and negative total conductance, and the value of Q L 2 can be obtained. When ω 2 > ω 1 , the circuit is capacitive, we obtain

When ω 2 < ω 1 , the circuit is inductive, and we obtain

By combining formulas (13) and (14), the calculation formula for Q L 2 can be obtained as follows:

The calculation of the air medium rectangular window is the basis for the design of the rounded window. The structural form of the rectangular window is shown in Figure 4.

Figure 4

Schematic diagram of the structure of the resonant window.

The rectangular window can be regarded as a small rectangular waveguide, and the characteristic impedance of the small waveguide is equal to that of the main waveguide, so the resonant wavelength of the rectangular window can be calculated

Therefore, it can be obtained

Among them, the broad side of the inner section of the main waveguide is denoted as a, the narrow side of the inner section of the waveguide is denoted as a, the broad side of the rectangular window is denoted as w, h is the narrow side of the rectangular window, and λ 0 is the resonant wavelength of the rectangular window.

However, the resonant windows used in practice are not right-angled. In order to avoid the local stress of the glass sealing, the right angle is usually changed to a small rounded corner or a semicircle. In order to calculate the rounded window, the rounded window can still be regarded as a rounded rectangular waveguide, so that the characteristic impedance of the rounded rectangular waveguide is equal to that of the main waveguide, and the resonance wavelength of the rounded window can be obtained. However, the characteristic impedance of the rounded rectangular waveguide is difficult to calculate precisely, and we have only approximate methods. The rounded rectangular waveguide can be equivalent to a right-angled rectangular waveguide under the condition that the cross-sectional area and height are equal, and the characteristics of the two are the same. According to this principle, after the rounded rectangular waveguide is equivalent to the right-angled rectangular waveguide, we can obtain the broadside dimension w e of the equivalent right-angled rectangular waveguide, as shown in Figure 5.

Figure 5

Rounded rectangular waveguide is equivalent to a right-angled rectangular waveguide.

According to the condition of equal area, it can be seen from the figure:

Therefore,

By making the characteristic impedance of the equivalent right-angle rectangular waveguide equal to the characteristic impedance of the rectangular waveguide of the main waveguide, the resonant frequency of the rounded window can be obtained

The resonant frequency of the rounded window is obtained as

When r = h / 2 , that is, when both ends of the window are semicircular, a constant value is assigned to w and r = h/2 is input into formula (19). The size w e of the broad side of the equivalent rectangular waveguide is

After sealing the resonant window with a solid dielectric, the equivalent capacitance of the window increases, and the resonant frequency decreases. At this time, the resonance wavelength formula of the rectangular resonant window becomes

In the formula, ε r is the relative permittivity of the solid medium, and we can obtain from formula (16):

Comparing the two equations, it can be seen that after the resonant window is sealed with a solid medium, it is equivalent to changing the width w of the window to ε r w , so that the resonant wavelength of the window increases and the resonant frequency decreases.

The above formula only considers the influence of the dielectric constant of the medium on the resonance wavelength and does not consider the relationship between the thickness of the medium and the resonance wavelength. In fact, the change of dielectric thickness is very sensitive to the influence of the resonance frequency. For an X-band resonant window, an increase of 0.05 mm in 7070 glass thickness reduces the resonant frequency by about 150–200 MHz. Therefore, when calculating the resonant frequency of solid dielectric windows, the effect of dielectric thickness must be considered. Therefore, it can be modified as

Therefore, it can be obtained

In this formula, γ ( ε r , t ) is the thickness factor and is related to the relative permittivity ε r of the solid medium and the thickness of the solid medium, and is the correlation function between them. The relationship curve between thickness factor γ ( ε r , t ) and thickness t needs to be obtained by testing, and this curve can be used to calculate the resonance wavelength.

According to the design requirements, the resonant gap and the resonant window should be tuned to the same frequency, but the value of the loaded quality factor Q L 2 can be the same or different. In general, the Q L 2 value of the resonant gap is always higher than the Q L 2 value of the resonant window. Usually, the Q L 2 value of the resonant gap is 46, while the Q L 2 value of the resonant window is 1–3.

Usually, 23 resonant gaps are used in the wave-guiding piano waveform characteristic factor limiter, but there are also cases where one resonant gap is used, and two resonant gaps are used in this article. The structure of the resonant gap is shown in Figure 6, in which the pheasant, the electrode, is the enemy of the concentrated electric field. Changing its size can change its equivalent capacitance C, and changing the width of the diaphragm can change its equivalent inductance L.

Figure 6

Structure of the resonance gap.

There are two parameters that characterize the resonant gap, namely the resonant frequency f 0 of the resonant gap and the on-load quality factor Q L 2 . Our design task is to determine the structural dimensions of the resonance gap according to the values of f 0 and Q L 2 . The main dimensions of the resonance gap are: the diaphragm slit width w, and the distance d between the two cones that form the discharge gap. The following will describe how to consider these dimensions. The slit width ω can be directly calculated by f 0 and Q L 2 , and the size of d is determined by the used waveband and the filling gas pressure. In general, the distance d of the discharge gap must be small, and when tuning, a small change in the distance d will cause a large detuning.

The planned inductive admittance of the symmetrical inductive diaphragm in the eigenfactor limiter of the waveguide piano waveform is approximated as

Among them, a is the width of the inner section of the waveguide, λ _go is the wavelength of the resonant waveguide, and w is the slit width of the diaphragm.

We set b L = λ go 2 a K , and in the formula, K is the susceptance parameter, substitute it into formula (27), then it can be obtained from the above formula:

The curve given in Figure 7 shows the relationship between the susceptance parameter K and the normalized slit width w/a of the resonant gap diaphragm. According to the given f ₀ (or ω ₀) and Q _L2, the susceptance parameter −K is calculated by the formula. The value of w/a can be found out using the curve in Figure 6 [21,22], so that the slit width dimension w of the resonant gap diaphragm can be calculated.

Figure 7

Relationship curve between the susceptance parameter and the normalized slit width of the resonant gap diaphragm.

In the parallel resonant circuit composed of the equivalent inductance of the metal diaphragm and the gap capacitance of the discharge electrode, the slit width w of the diaphragm determines the size of the equivalent inductance L. The inductance L increases with the increase of the slit width w. The size of the equivalent capacitance of the resonance gap is determined by the distance d between the cones. The larger the gap distance d, the larger the capacitance C. In fact, in the design process of the resonance gap, there is no need to calculate the size of d, only d is required to be less than a certain minimum value d _min. One of the empirical formulas is as follows:

In the formula, p is the pressure of the gas filled in the limiter of the waveform characteristic factor of the waveguide piano, and d min refers to the minimum gap distance of the resonance gap.

3.3 Intelligent recommendation of piano repertoire based on the CNN model

Combined with the algorithm in the second part, the intelligent recognition of piano music features is carried out. This part uses the CNN model to intelligently recommend piano pieces. The CNN model in this article is shown in Figure 8.

Figure 8

CNN model.

On the basis of statistics and analysis, this article further processes the classification results, optimizes the classification results, and obtains the classification features of piano music. The intelligent recommendation model for piano pieces based on the CNN model constructed in this article is shown in Figure 9.

Figure 9

System model.

The core of the model is based on the feature extraction ability of CNN, combined with user features and piano track features, to achieve accurate personalized recommendations. The input items include the user’s age, gender, music preferences, historical playback history, etc. These features are encoded as numerical vectors, which serve as one of the inputs to the model. The characteristics of the qin melody can be extracted from multiple dimensions, such as rhythm, melody, harmony, style, etc. These features are also encoded as numerical vectors as another input to the model. Use CNN for feature extraction of user features and piano repertoire features. CNN can automatically learn and extract advanced features useful for recommendation tasks through multi-layer convolution and pooling operations. Merge the extracted results of user features and piano track features. This can be achieved through concatenation, weighted summation, and other methods to combine the features of users and tracks, providing comprehensive information for subsequent recommendation tasks. After feature fusion, calculate the similarity between user features and each piano piece feature. This can be achieved through measurement methods such as cosine similarity and Euclidean distance. The higher the similarity, the more the track matches the user’s preferences. Sort piano pieces based on similarity calculation results and generate a recommended list. The songs in the recommended list are arranged in descending order of similarity, and users can choose based on the songs in the list.

A trained CNN can predict and classify the frequency spectrum of piano music and obtain classification feature vectors. User feature calculation is based on the relationship between piano music and classification features, combined with the relationship between piano music and users, to obtain the relationship between users and classification features, and then match piano tracks according to users’ personalized needs, and obtain highly accurate matching results, thereby providing users with more reliable recommendation results.

4.1 Test method

To verify the effectiveness of the proposed piano repertoire recommendation model (hereinafter referred to as the wave CNN model), a control experiment was designed to compare the model with existing models.

The main source of the dataset for this article is GiantMIDI Piano, MAESTRO Dataset, PianoMotion10M.

Each track contains the original audio waveform (sampling rate 44.1 kHz), manually annotated difficulty levels (1–10 levels), technical labels (arpeggios/chords/rhythms, etc.), and user interaction data.

About 1,000 users of different skill levels (beginner/intermediate/advanced) are simulated, and their historical performance data (pitch errors, rhythm stability, etc.) are recorded. The traditional CF, LSTM-based sequence recommendation model, and ResNet-18 audio classification recommendation model are selected as baseline models and compared with the wave CNN model. At the same time, accuracy, robustness, cold start performance, and computational efficiency are selected as comparison parameters.

4.2 Results

The performance comparison test results are shown in Table 1.

The robustness test is conducted on the basis of the performance test. After adding the ambient noise (snr = 5 dB), the performance decline of each model is shown in Table 2.

This article makes the following statistical analysis on the CNN classification results, as shown in Figure 10.

Figure 10

Test of recommended sample effect. The distribution of the maximum proportion of (a) the predicted classification for the test sample and (b) music test sample components.

4.3 Analysis and discussion

In Table 1, the precision of wave CNN is 0.85, recall is 0.78, which is significantly higher than other models. The precision and recall of the traditional CF model are 0.57 and 0.5, respectively, which perform the worst among these models. The precision and recall of resnet-18, which performs better in the traditional model, are 0.73 and 0.67, respectively, and have some disadvantages compared with wave CNN. The advantage of wave CNN comes from the physical modeling ability of its piano waveform feature extraction module, the cascade structure of limiter resonant elements, and accurately capturing the transient response and harmonic distribution of keystroke by simulating the acoustic characteristics of the real piano resonance plate, so as to improve the fidelity of features. The joint spectrum time domain analysis effectively integrates the harmonic energy distribution and performance dynamic characteristics of piano timbre and overcomes the problem of high-frequency overtone loss caused by traditional spectrum diagram methods (such as resnet-18).

In terms of cold start accuracy, compared with 0.22 of the CF model, Wave CNN achieved a leap in cold start performance through real-time performance feature analysis. Wave CNN uses the collected data to generate user technical portraits, which can match tracks without historical interaction records. Reasonable utilization of the matching mechanism driven by physical characteristics directly relates the user’s performance ability and tracks technical requirements through the fundamental harmonic relationship based on piano timbre in the actual recommendation.

In terms of the corresponding time, the corresponding time of Wave CNN is 123.6 ms, which is better than 97.85 ms of resnet-18 and 51.50 ms of CF, and is better than 206 ms of LSTM. In terms of the overall actual demand, the response time of these models is milliseconds, so they can meet the daily needs of users.

In Table 2, the performance degradation of wave CNN is the lowest in the noisy environment, with a decline of only 8.23%. Resnet-18 performs best in the traditional model, but the accuracy rate also decreases by 20.35%. The adaptive signal processing module of wave CNN uses a dynamic limiter to suppress the amplitude mutation caused by environmental noise and retain the core harmonic components. The frequency selectivity of the resonant element strengthens the characteristic frequency band of the piano and filters the irrelevant noise energy, so as to ensure that the performance in the noise environment is not greatly affected.

In Figure 10, 92.72% of the samples have a maximum ratio of more than 0.9, and the maximum ratio of the samples is greater than 0.5. That is to say, most of the errors still come from prediction and classification errors, and a small part of them are data errors.

Wave CNN relies on the cascade structure of limiter resonant elements to simulate the resonance characteristics of the piano, but the acoustic response of the real piano is affected by environmental temperature and humidity, nonlinear keystroke force, and other factors. The change of string tension leads to the fundamental frequency offset, and the nonlinear relationship between the actual keystroke force and timbre has not been fully modeled. In general, the error of wave CNN mainly comes from the idealized assumption of physical modeling, the instantaneous noise of user behavior, and the implicit deviation driven by data. Through dynamic parameter calibration, multi-session feature fusion, de bias loss function design, and other improvements, the robustness of the model can be significantly improved. In the future, it is necessary to further explore the cross-instrument generalization ability and real-time adaptive learning mechanism to achieve more universal intelligent music teaching support.

It can be seen from the research that the intelligent piano repertoire recommendation algorithm based on the CNN model proposed in this article has a high accuracy in piano repertoire recommendation and can effectively improve the recommendation effect of piano repertoire.