The piano tone signal is composed of fundamental tones and overtones. The fundamental pitch determines the pitch of a piano tone signal. Therefore, detecting the pitch period is the key to identifying piano notes [1]. The detection method of pitch period mainly includes frequency domain identification and time-domain identification. Short-time ODEs are a classic time-domain detection algorithm. It is simple to calculate and widely used, but the algorithm will suffer from fundamental-tone octave or half-frequency errors.

Since the calculation removes the part where the energy of each note is relatively concentrated in the central area, the calculation amount can be reduced, and the calculation speed can be accelerated [2]. At the same time, the algorithm avoids the above errors to a certain extent and improves the recognition rate. But this algorithm still has certain limitations. Some scholars have proposed two three-level center clipping and autocorrelation processing. However, this method increases the amount of computation and is not suitable for fast computation scenarios. In addition, when ordinary differential equations estimate the pitch period, the phenomenon of pitch period jumps between frames occurs [3]. The identification process is disturbed by half frequency points, double points, and random error points. Some scholars have proposed to apply the zero-insertion algorithm and the corresponding low-pass filter to the ordinary differential equation of three-level clipping. Some scholars have proposed to combine the three-level center clipping autocorrelation function with the cyclic mean amplitude difference function. The above algorithm can achieve a relatively satisfactory recognition rate when dealing with music with a moderate rhythm. However, the recognition rate drops rapidly when dealing with fast-paced music. This paper presents ordinary differential equations. The goal is to find ordinary differential equations on a smaller scale to accommodate fast-paced music. The algorithm avoids the missed detection, false detection, or recognition error of fast-paced music by traditional algorithms to a certain extent. This method can significantly improve recognition accuracy.

Suppose _{i}_{L}_{i}

The input and output functions of the three-level center clipping method are as follows:

That is, the output
_{i}_{L}_{i}_{L}_{i}_{L}

Because

The left and right Riemann-Liouville type fractional integral operators have semigroup properties [5]. _{1} ([_{1} ([_{1}, _{2} > 0 the following formulas are given [6].

Suppose
_{1},

Because _{1} ∈ _{1}] ×

We assume

We know from _{1} ∈ _{1}] × _{2} ∈ _{1},

Prove that

Suppose
_{1}] ∪ (_{1},

We substitute formula (10) into formula (11) to get

Assuming _{1}) =

We perform endpoint detection on an audio sequence _{i}

We set _{i}_{,max} (_{i}_{i}_{,max} where _{i}_{,max} (_{i}_{i}

In a few cases, the formant affects the signal, and there will be interference of frequency doubling waves [9]. This will result in the separation of _{i}_{,max} and _{i}_{i}_{,min} = _{i}_{,max} (_{1}. where _{1} is a constant. Record the peak sequence _{i}_{i}_{i}_{,min} and the sequence numbers _{i}_{i}_{i}_{i}_{i}

The value of _{1} needs to ensure that _{i}_{i}_{i}_{i}_{,max} and the serial number is _{i}_{,max}. Threshold 1 2 _{1} = 2 in this paper.

The correct selection of the peak point requires further threshold judgment. We take the amplitude ratio _{R}_{i}_{,max} / _{i}

After calculation, the amplitude ratios in Figs. 3(a) and 3(b) can be obtained _{R}_{,1} = 1.66, _{R}_{,2} = 1.36, respectively. It can be seen that the value of _{R}

We can find that the amplitude ratio fluctuates within a certain range after multiple translations of the data frame (Figure 4). We get the amplitude ratio sequence _{R}

Assume that the threshold _{2} is a constant. We count the number _{1} of _{R}_{2} and the number _{2} of _{R}_{2} respectively. If _{2} > _{1} then we assume _{i}_{1} > _{2} then we assume _{i}_{,max}. _{2} The value has a direct impact on the statistical results [11]. After adjustment of many songs, we take _{2} = 1.43 as the ideal result.

According to the score, the tone data files used were synthesized by the software Everyone Piano. The music files are obtained from the right-hand performance part of the piano score recorded in the stereo mix. The sound source used by the software is the mad piano. Standard frequency
_{a1} = 440 is the first international altitude. ^{1}. ^{1}. The fundamental frequency is _{s}_{s}_{k}(

Comparison of recognition results of slow music

Song (excerpt) | N | Song duration/s | Song rate v/(notes/s) | The method of this paper |
---|---|---|---|---|

little stars | 42 | 30 | 1.4 | 100 |

Castle in the Sky | 56 | 33 | 1.7 | 100 |

Faded | 51 | 26 | 1.9 | 100 |

Love Romance Maps | 46 | 21 | 2.2 | 100 |

(0.75× speed) to | 41 | 19 | 2.2 | 100 |

alice | 70 | 26 | 2.7 | 91.2 |

Canon (0.75× speed) | 74 | 27 | 2.7 | 89 |

Average value | 54.3 | 26 | 2.1 | 97.1 |

Comparison of recognition results of medium tempo music

Song (excerpt) | N | Song duration/s | Song rate v/(notes/s) | The correct rate of this method is R/% |
---|---|---|---|---|

Dream wedding | 106 | 34 | 3.1 | 100 |

Flying Bumblebee (0.25× speed) | 64 | 20 | 3.2 | 87.5 |

Faded (1.5× speed) | 51 | 17 | 3.2 | 86.3 |

To Alice (1.25× speed) | 125 | 22 | 3.2 | 80 |

Romance of Love (2× speed) | 46 | 14 | 3.3 | 95.7 |

happy farmer | 44 | 13 | 3.3 | 79.6 |

Maps | 41 | 12 | 3.4 | 97.6 |

Canon | 74 | 20 | 3.7 | 85.9 |

Little Star (2× speed) | 42 | 11 | 3.8 | 85.7 |

Average value | 65.9 | 18.1 | 3.4 | 88.7 |

Comparison of Fast Song Identification Results

Song (excerpt) | N | Song duration/s (n /s) | Song rate v | The correct rate of this method is R/% |
---|---|---|---|---|

Happy Farmer (1.25×) | 44 | 11 | 4 | 59.1 |

Dream Wedding(1.25× speed) | 106 | 25 | 4.2 | 85.8 |

Faded(2× speed)Maps | 51 | 12 | 4.3 | 78.8 |

(1.5× speed) | 41 | 9 | 4.5 | 65.6 |

Croatian Rhapsody (0.75×) | 122 | 27 | 4.5 | 59 |

Canon (1.25× speed) | 74 | 16 | 4.6 | 57.5 |

Flight of the Bumblebee | 64 | 11 | 5.9 | 51.3 |

Croatian Rhapsody | 122 | 20 | 6.1 | 47.5 |

Average value | 78 | 16.4 | 4.8 | 63.1 |

Comparison of recognition results between slow and medium and fast music

Song type | The accuracy of the three-level center clipping method is R1 | The correct rate of this method is R2 | Absolute Error Rate |R1–R2| |
---|---|---|---|

Slow | 93.3 | 97.1 | 3.8 |

Medium speed | 74.6 | 88.7 | 14.1 |

fast | 42.9 | 63.1 | 20.2 |

Average value | 70.3 | 83 | 12.7 |

The last column in Tables 1–3 is the relative error between the algorithm in this paper and the three-level clipping method. It can be seen from Table 1 that the relative error rate of the two methods is only within 5.1% when the rhythm of the music is slow [14]. This shows that the traditional three-level clipping method is close to the recognition rate of this method. But it can be seen from Table 2 that when the rhythm of the music is faster, the average relative error rate of the two methods is 20.6%. The accuracy of the improved algorithm is higher than that of the traditional algorithm. It can be seen from Table 3 that the relative error rates of the two methods are larger when the music tempo is further accelerated. Although the recognition rate of this algorithm is reduced under fast-paced conditions, the recognition rate is still significantly higher than that of traditional algorithms.

The two methods have differences in the recognition results of different songs simultaneously. As shown in Table 1, the music “To Alice” and “Canon (0.75 times speed)” with 2.7 notes per second. The recognition accuracy of the two methods is not the same. Music with a relatively fast local rhythm, “Canon (0.75 times),” has a lower recognition accuracy. Even some slow songs have worse recognition results than fast ones. As shown in Table 3, the number of notes per second of “Wedding in a Dream (1.25 times)” is 4.2. The accuracy rates of the two methods are 68.8% and 85.8%, respectively. The “Happy Farmer” in Table 2 has 3.3 notes per second. But the recognition rates of the two methods are only 53.1% and 79.6%. This may be due to the uneven rhythm of the music itself.

In this paper, an improved algorithm of pitch period extraction for ordinary differential equations is proposed. When the rhythm of the piano music is fast, the average accuracy of the algorithm in this paper is 63.1%. When the rhythm of the piano music is moderate, the average accuracy of the algorithm in this paper is 88.7%. The text algorithm has an average accuracy of 97.1% when the rhythm of the music is slow. The algorithm used in this paper has an average recognition accuracy of 83.0% for the above three groups of the slow, medium, and fast music. Therefore, the algorithm in this paper has achieved high recognition accuracy in recognizing piano sounds with different fast and slow rhythms. And the algorithm model has a significant improvement in the recognition accuracy of fast-paced piano tones.

#### Comparison of recognition results of medium tempo music

Song (excerpt) | N | Song duration/s | Song rate v/(notes/s) | The correct rate of this method is R/% |
---|---|---|---|---|

Dream wedding | 106 | 34 | 3.1 | 100 |

Flying Bumblebee (0.25× speed) | 64 | 20 | 3.2 | 87.5 |

Faded (1.5× speed) | 51 | 17 | 3.2 | 86.3 |

To Alice (1.25× speed) | 125 | 22 | 3.2 | 80 |

Romance of Love (2× speed) | 46 | 14 | 3.3 | 95.7 |

happy farmer | 44 | 13 | 3.3 | 79.6 |

Maps | 41 | 12 | 3.4 | 97.6 |

Canon | 74 | 20 | 3.7 | 85.9 |

Little Star (2× speed) | 42 | 11 | 3.8 | 85.7 |

Average value | 65.9 | 18.1 | 3.4 | 88.7 |

#### Comparison of recognition results between slow and medium and fast music

Song type | The accuracy of the three-level center clipping method is R1 | The correct rate of this method is R2 | Absolute Error Rate |R1–R2| |
---|---|---|---|

Slow | 93.3 | 97.1 | 3.8 |

Medium speed | 74.6 | 88.7 | 14.1 |

fast | 42.9 | 63.1 | 20.2 |

Average value | 70.3 | 83 | 12.7 |

#### Comparison of recognition results of slow music

Song (excerpt) | N | Song duration/s | Song rate v/(notes/s) | The method of this paper |
---|---|---|---|---|

little stars | 42 | 30 | 1.4 | 100 |

Castle in the Sky | 56 | 33 | 1.7 | 100 |

Faded | 51 | 26 | 1.9 | 100 |

Love Romance Maps | 46 | 21 | 2.2 | 100 |

(0.75× speed) to | 41 | 19 | 2.2 | 100 |

alice | 70 | 26 | 2.7 | 91.2 |

Canon (0.75× speed) | 74 | 27 | 2.7 | 89 |

Average value | 54.3 | 26 | 2.1 | 97.1 |

#### Comparison of Fast Song Identification Results

Song (excerpt) | N | Song duration/s (n /s) | Song rate v | The correct rate of this method is R/% |
---|---|---|---|---|

Happy Farmer (1.25×) | 44 | 11 | 4 | 59.1 |

Dream Wedding(1.25× speed) | 106 | 25 | 4.2 | 85.8 |

Faded(2× speed)Maps | 51 | 12 | 4.3 | 78.8 |

(1.5× speed) | 41 | 9 | 4.5 | 65.6 |

Croatian Rhapsody (0.75×) | 122 | 27 | 4.5 | 59 |

Canon (1.25× speed) | 74 | 16 | 4.6 | 57.5 |

Flight of the Bumblebee | 64 | 11 | 5.9 | 51.3 |

Croatian Rhapsody | 122 | 20 | 6.1 | 47.5 |

Average value | 78 | 16.4 | 4.8 | 63.1 |

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Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leader-follower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model