Model of Markov-Based Piano Note Recognition Algorithm and Piano Teaching Model Construction

Abstract

Piano note recognition is a process that converts music audio files into digital music files automatically, which is critical for piano assistant training and automatic recording of musical pieces. The Merle spectral coefficients, for example, have been used to implement the majority of the existing examples. The piano is one of the most popular forms of student education in today’s world. Piano teachers should be aware of the implications. We can only truly adapt piano teaching to the educational purposes of higher education institutions if we implement a systematic, progressive, practical, and innovative philosophy of piano teaching. The Markov model is a statistical model that is widely used in speech signal processing. This thesis develops a set of mathematical models for piano speech recognition based on the Markov model, learns them systematically and scientifically, and achieves a better teaching effect. It is demonstrated that the Markov method detects the corresponding endpoints with an accuracy of 72.83 percent, which is 16.42 percent better than the a priori method. In terms of amplitude and phase, the Markov model shows a significant improvement. The findings of this study can be used to improve piano playing techniques taught to students in accordance with their favourite popular music, depending on the theme.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The process of converting piano audio into digital audio format by “making the computer understand the piano performance” [1] is known as piano tone recognition. The amount of multimedia information is rapidly increasing as multimedia technology and network technology continue to advance, and studying multimedia technology is becoming increasingly important to fully utilise the available multimedia resources [2]. Music has a meaning and connotation, implying the artist’s life experiences, thoughts, and emotions [3]. Music is defined as a transition between noise and pure tones of a specific frequency in acoustic analysis, allowing people to appreciate beauty and express human emotions [4]. One’s creative efficiency will be greatly improved and one’s creative inspiration will be greatly stimulated if one can use a computer to automatically recognise the music one is playing and automatically complete the creation of the score, eliminating the inconvenience of transcribing the score now [5].

In the modern era, technological advancements and the growth of culture and art have resulted in the integration of technology and culture, as well as the growth of cultural development [6]. Researchers began to use signal processing tools to analyze music scientifically and objectively and develop note extraction techniques to automatically extract music information as people’s research on music signals progressed [7]. Quality education, as the most important basic piano course for high school music majors, should be an important educational content to meet the needs of music education personnel in the new era [8]. Chinese society recognizes that piano education has not yet developed a mature development model and has been unable to realize its industrial potential because of a lack of development space [9]. As a result, in piano lessons, various piano materials can be studied and taught to play familiar popular music in tandem with learners’ preferences for popular music [10]. Alternatively, for educational purposes, deliberately introduce popular music pieces with multicultural elements and then teach the necessary piano performance techniques [11].

In the development of traditional music education, the teacher plays an important educational function as the organizer of the classroom [12]. However, the modern development of innovative pedagogy is more focused on the state of the curriculum and active participation of students [13]. Traditional pedagogy proposes a research-based approach, i.e., teacher-centered and passive-receptive teaching and learning [14]. Markov chains and model of Markovs have advantages that other methods do not have because they have a good mathematical basis [15]. Markov-based instructional models are learning activities in which the teacher initiates, facilitates, supports, and guides learners in their study of the learning object to accomplish the instructional task.

This paper has several innovative features, which are as follows:

(1) Combining Markovian models with the physical and psychological characteristics of students, it explores some feasible ways of teaching piano on the piano.

(2) On the basis of the existing research results, the model is improved, the hidden process is extended to the uppermost level of Markov chain, and a piano teaching model based on the model of Markov is established.

(3) This paper presents the idea of using the model of Markov to establish an online piano teaching environment. For a long time, piano phonetic recognition algorithm has been a democratic and free learning method.

2. Related Work

2.1. Piano Note Recognition Algorithm and Piano Teaching Mode

The overall quality of the people is improving, and so is the artistic level and artistic environment of the whole society. Based on this, music education is receiving more attention. Note recognition technology can easily realise the computer input of music scores, and it has broad application prospects in the fields of music content, computer-assisted piano education and digitization of music works. The practice of quality education is gradually changing from the level of theoretical research to the level of practical operation, and teaching methods are an intermediate stage between theory and practice. The research and discussion of teaching methods are an important part of promoting quality education.

Riska and Smirni pointed out that with the rapid development of information technology and the increasing consumer demand for piano music education, networked piano education that can provide more personalized educational programs and educational services will gradually become the development and progress of piano education [16]. Chen started to study the sensory information in music signals and built a sensory music machine similar to the human ear system. If more recognition errors are tolerated, the system increases the maximum number of notes that can be articulated simultaneously [17]. By systematically studying the whole process of piano teaching, Fang revealed “the comprehensive development of piano playing talent and the inner law of piano teachers” teaching [18]. Gong proposed the application of heuristic signal processing method and music recognition system. Based on this method, the maximum number of simultaneous organ ensemble music can be considered to be the number of seven phonetic symbols [19]. Chi documented the use of seven measures and provided a comprehensive summary of the specific implementation of the assessment of learners’ musical ability [20].

Increasingly expensive piano tutoring fees and the heavy creative workload of music creators have forced the urgent need to use computers to assist in their work, freeing people from the tedious tasks of manual training and manual sheet music recording. Thus, people can improve the teaching and creation of music. The many roles of musical notes foreshadow the important role they will play in the field of music and in the analysis of musical emotions.

2.2. Model of Markov

A large amount of multimedia data exists in many information systems, such as photos, digital videos, digital sounds and music, animations, maps, recordings, etc. These types of data are diverse and large in quantity, and the effective management and utilization of multimedia information is an important issue that needs to be addressed urgently. The model of Markov approach can incorporate individual differences into the analysis model, which is a discrete-time stochastic process with Markovian properties. The basic idea is to use only the current state information to predict the future given the current state information, and the past and future states are mutually exclusive.

Jing et al. proposed a more compact higher-order model, namely the hybrid transmission and distribution model. This model can reduce the number of parameters to be estimated in the model and improve the prediction accuracy of the model [21]. Dai divided the input notes into three parts, namely keywords, nonkeywords, and background tones, and trained a model of Markov for each keyword using an initial training algorithm [22]. Marco et al. constructed a more general MTD model that is more flexible. It can represent a wider range of relevant patterns and has good data adaptability [23]. Attal et al. performed a second-order extension and reformulation of the model of Markov and improved and modified the context-independent grammar using the Lagrange multiplier method, ultimately obtaining better results [24]. Blei and Moreno introduced the idea of multidimensional data modeling and discussed the parameter estimation of multiple higher-order model of Markov and their applications [25].

To objectively evaluate the effectiveness of the research-based pedagogy, it is necessary to exclude the influence of fundamental differences among students. Therefore, if we are to focus on extracting musical features, we need to build the model of Markov in terms of notes and display music files in terms of notes. More accurate identification of notes is an important goal of this study.

3. The Algorithm of Piano Note Recognition Based on Model of Markov and the Idea of Constructing Piano Teaching Mode

3.1. Piano Note Recognition Algorithm Based on Model of Markov

What this algorithm needs to do is to solve the problem of automatic polyphonic transcription of piano music. The Markov process is doubly probabilistic. The first probabilistic process is used to describe the statistical properties of each short-term stationary segment in a nonstationary signal [26]. To measure the quality of a cluster, the sum of squared error reference functions is defined as follows:$$\begin{array}{}\text{(1)}& E={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{x\in C_i}{x-\overline{x_i}}^2}}.\end{array}$$$\overline{x_i}$ is the average of clusters, i.e., the cluster center.

A second stochastic process is used to characterize the dynamic statistics of the transitions between the short-term stationary segments in the anomalous signal, which is implicitly included in the observation order. The final result of the processing is the feature vector of the note signal or the encoded note data [27], i.e., different types of features have different weighting coefficients, and each component within the same feature vector has a different weighting coefficient. The secondary feature extraction process is shown in Figure 1 below.

[figure(s) omitted; refer to PDF]

Firstly, the energy spectrum is estimated in the frequency domain by STFT, and then the acquired energy spectrum is smoothed and filtered to select the spectral peaks. The sampling frequency must be at least twice the original note frequency, and this threshold must be increased to account for filter performance effects. The time domain analysis of the signal usually uses a rectangular window to meet the requirements. The Hamming window function is as follows:$$\begin{array}{}\text{(2)}& \omega x=\begin{array}{c}0.54-0.46\cos \frac{2\pi n}{N-1},0\le n\le N-1,\\ 1.\end{array}\end{array}$$$\omega x$—Window function

In the time domain linear superposition model, musical audio is considered to consist of one or more individual notes connected linearly according to certain coefficient sound names. The human ear perceives audio at different frequencies differently, and the perception is linearly related to frequency as follows:$$\begin{array}{}\text{(3)}& \text{Mel}f=2595\lg 1+\frac{f}{700}.\end{array}$$$\text{Mel}$$\text{Mel}$—Mel frequency, $f$—Linear frequency

A doubly stochastic process is a Markov model. A digital IIR filter and a psychoacoustic model are used to filter and smoothen the generated spectrum. The filtered spectrum is then peak selected with a simple local maximum algorithm, ensuring that the selected peaks match the original spectrum’s strong peaks. As typical audio observation sequences are continuous and the cepstrum or eigenvectors rarely follow a normal distribution, a Markov model must be trained, as shown in Figure 2.

[figure(s) omitted; refer to PDF]

Secondly, we need to classify the spectral peak information to generate a list of possible fundamental frequencies to determine these comb models. A test input signal consists of one or more simultaneously sounding monophones that are linearly independent. If the sampling rate does not satisfy the sampling law, spectral aliasing occurs, which distorts the high-frequency portion of the signal. Therefore, if the graph model [28, 29] is limited to tree or potential function convergence, predictions can be made using Markov random fields. The goal is to find the clustering prime with the best clustering effect, even if the reference function is minimal. Therefore, the fitness function should be taken as $1/E$, i.e., the fitness function is equal to the following:$$\begin{array}{}\text{(4)}& fx=\frac{1}{E}=\frac{1}{{\displaystyle {\sum }_{i=1}^K{\displaystyle {\sum }_{x\in C_i}{x-c_i}^2}}}.\end{array}$$

However, mutually exclusive potential functions representing the differences in factors are also highly desired [30]. Thus, in signal modeling problems, linear predictive and limiting zero models solve time-invariant smooth signal or short-term smooth signal modeling problems, and the model of Markov solves time-varying smooth signal or process modeling problems.

Finally, a set of heuristics is used to select the pattern that best describes the fundamental frequency. Mono identification is translated into calculating a matrix of weighting coefficients for each note contained in the input signal so that the names of the monophonic sounds contained in the actual signal can be calculated. Despite its computational problems, MRF remains a popular, important, and intuitive model because of its potential to model simple local relationships well. The MFCC coefficients are obtained by discrete cosine transformations of the logarithmic energy of the Mel frequency domain subbands.$$\begin{array}{}\text{(5)}& c_i=\sqrt{\frac{2}{N}{\displaystyle \sum _{j=1}^N{m}_j\cos \frac{m_{ij}}{N}}}.\end{array}$$$m_j$—Band energy, $c_i$—MFCC coefficient, and $N$—The number of filters in the filter group

Given the presence of high-frequency noise, to prevent spectral aliasing caused by high-frequency noise higher than half of the sampling frequency, the note signal is usually prefiltered at sampling time to reduce high-frequency noise. One of the features of the model of Markov lies in this. In other words, it can describe both transient and dynamic characteristics. It is described as a set of probability distributions where each observation vector is represented by a different state of some probability density distribution, and each observation vector is generated by a sequence of states with a response probability density distribution.

3.2. Construction of Piano Teaching Mode Based on Model of Markov

Assessing the teaching effectiveness of the research-based teaching model solely on the basis of students’ test scores without taking into account the impact of student differences on teaching effectiveness does not adequately reflect research-based teaching’s teaching effectiveness. Educational and Mathematical Psychology is a branch of psychology that focuses on education and mathematics. The Markov model divides students’ problem-solving abilities into three states, namely analysis, solution implementation, and verification, according to educational psychology. As a result, we will use the Markov model to empirically analyze the teaching effect of research-based teaching in universities and provide useful countermeasures and suggestions for research-based teaching. The training model has detailed classroom operation procedures that correspond to various stages of cognitive development. It is shown in Figure 3 below.

[figure(s) omitted; refer to PDF]

Firstly, the probabilistic process model of Markov establishes a mathematical model to assess the effectiveness of research-based teaching methods and their implementation paths and to develop teaching and learning strategies for mastering research-based teaching methods. Creative thinking skills are mainly based on traditional methods while constantly innovating to find solutions and achieve value-added information. If the variance of the cost function is negative, the original center of mass is replaced with the noncenter of mass at the current position. Otherwise, the center remains unchanged. The fit is appropriately extended by a simulated annealing algorithm, and the fit stretching method is as follows:$$\begin{array}{}\text{(6)}& f_i=\frac{e^{f_i/T}}{{\displaystyle {\sum }_{i=1}^M{e}^{f_i/T}}}.\end{array}$$$f_i$—_{The fitness of the}$i$—_individual

Teaching strategy is a central aspect of the “project-based” piano teaching approach in practice as it is directly related to the overall course and the final outcome of the training. A model of Markov is used to describe the initial probability distribution of the system, and the transfer of states is described as a state transfer matrix. A generic stochastic process is used to represent the relationship between the hidden states and the actual sequence of observations, described in terms of the probability of the observations. Thus, the traditional first-order correlation is extended to a higher-order correlation, i.e., the observation at the current moment is only related to the previous consecutive and continuous time states and is independent of all previous states. Given that the low-dimensional eigencomponents of the eigenvectors are strongly affected by noise, the component weighting coefficients are usually rounded to a half-sine function, which is as follows:$$\begin{array}{}\text{(7)}& r_i=\frac{1}{2}+\frac{1}{2}\sin \frac{{\pi }_i}{P}.\end{array}$$$i$—Dimension of eigenvector

Secondly, in constructing the model of Markov, the application of Markov chains in studying the teaching model is considered based on two tests before and after. A transformation matrix is created by carefully analyzing the variation between the different grades of the students’ tests. The original signal is smoothed or passed through a low-pass filter to obtain the signal envelope.$$\begin{array}{}\text{(8)}& E_{0}n=\frac{1}{N}{\displaystyle \sum _{m=N/2}^{N/2-1}xn+m|\omega m}.\end{array}$$$wm$—$N$ point window

Also, reading piano music allows students to experience real emotions and more. This innovative teaching method preserves the traditional classroom approach and guides students into the essence of learning. Then, the best model of Markov for the observation sequence is determined based on the probability maximum of the model generation observation sequence, which is actually a pattern matching problem, i.e., the change in local energy determines the note endpoints, which is as follows:$$\begin{array}{}\text{(9)}& E_{0}n=\frac{1}{N}{\displaystyle \sum _{m=N/2}^{N/2-1}|{xn+m}^2\omega m}.\end{array}$$

The implicit parameters of the process are identified from a set of observable parameters, which are then used for further analysis. The need to create an environment and atmosphere in which the impact of the disciplinary techniques on musical performance can be felt in the context of listening, seeing, feeling, and touching results in creating a strong interest in learning. The high-frequency energy is weighted to improve the frequency domain analysis of the high-frequency band of the signal. The formula is as follows:$$\begin{array}{}\text{(10)}& \overline{E}n=\frac{1}{N}{\displaystyle \sum _{m=N/2}^{N/2-1}{W}_k{X_{k}n}^2}.\end{array}$$$W_k$—Frequency domain weighted window

Finally, under the assumption that teaching quality remains constant, we can represent the degree that students eventually achieve by obtaining a stable distribution of Markov models. The teacher must change the concept of teaching for the student to take the lead in teaching in the innovative pedagogy of piano teaching. The central idea of the decoding problem is that the sequence of observations determines the sequence of states that correspond to the sequence of observations. The default state sequence associated with the model has the following property: the state labels are also added (or remain the same) over time, and the state transitions from left to right. Teaching piano is an emotional experience that makes students feel happy, joyful, and blissful as they touch a piece, learn it, express it, and improve the technique.

4. Application Analysis of Model of Markov in Piano Note Recognition Algorithm and Piano Teaching Mode

4.1. Determination and Analysis of Markov State Number

In the model of Markov, a finite state space is mainly used to predict the next state using the transfer matrix after determining the initial state probability distribution. To find the probability of an observation sequence, we search for the hidden states most likely to produce an observation sequence to train a given model of Markov. For the transfer arc generation output type, each output is not associated with an arrival state but with a transition arc. Hence, the probability distribution of the output is related to the determination of the number of states. The pre-emphasis processing of the signal can be seen as a process of filtering the signal, and the Porter diagram of the pre-emphasis filter is shown in Figure 4 below when the pre-emphasis coefficients are 5, 10, and 15.

[figure(s) omitted; refer to PDF]

The structure of the model of Markov is first used, with each state of the note corresponding to one phoneme when moving from left to right without crossing the training. Since ear audio is a continuous analog signal and computers can only process digital information, the analog continuous audio signal must be discretized and sampled using the individual sampling points processed by the computer. A Markov random field can describe a distribution in which the first element is strongly negatively correlated with the other two elements, however, the second and third elements cannot be strongly negatively correlated. A DPP with correlated transferability can explain this distribution. Some piano note recognition devices work by considering the internal note output as hidden states and the sound outcome as a series of observed states, which are generated by the note process and best approximate the actual states. It is important to note that in both examples, the number of hidden states and the number of observed states may differ. By analyzing the musical and physical properties of the pitch signal, important features of the pitch are extracted and used as the theoretical basis for the algorithm. We added this quadratic function parameter optimization method to HCpoy.exe, as shown in Table 1.

Table 1

Comparison of recognition rates before and after optimization of secondary feature extraction.

Feature extraction method	Quadratic feature extraction	Adding secondary feature extraction
Multinote recognition rate	56.23%	71.98%
Recessive personality difference	0.7162	1.255

Second, based on the phoneme translation reference of the training sample notes, the total number of phonemes contained in the training sample notes corresponds to the total number of states in the model of Markov, assuming that there are no silent pauses in consecutive note segments. For an audio example, if the average energy of a particular short-time frame in that audio example is below a preset threshold, the short-time frame is determined to be silent, otherwise it is nonsilent. The calculation is small and can be implemented using a local energy endpoint localization algorithm. If the number of segments with energy values greater than the threshold exceeds 2/3 of the total number of segments, it can be considered to be the following: the current signal frame belongs to a note segment. Figure 5 shows the time domain waveform diagrams of a note frame with the number of segments as 1/3, the number of segments as 2/3, and the number of segments as 1 and its local energy distribution. Figure 6 shows the corresponding energy distribution.

[figure(s) omitted; refer to PDF]

To assume mutually exclusive relationships, Markov random fields are context-independent, and Markov random fields cannot handle this problem if the selected subset must have exact dimensionality. Music sound signals have the dual nature of sound and music signals, and their physical characteristics serve as the study’s starting point, while their musical characteristics serve as the study’s breakthrough. The edge energy of different pianos, on the other hand, varies from piano to piano and is difficult to estimate accurately. In the time domain, however, this method may be more computationally intensive.

Finally, the total number of states in the Markov model is determined by the total number of training sample notes contained in the model. Once the number of states in the Markov model has been determined, the model of Markov training and cell division can begin. The note signal is always represented by a short-duration, low-energy consonant signal, followed by a long-duration, high-energy vowel signal in the waveform. As a result, the consonant signal has a low rate of overzero, and the vowel signal has a high rate of overzero. The fundamental frequency of a single tone determines its pitch, and the size of the single tone is determined by the energy flow transmitted to the human ear by the air vibrations of that single tone. Tone formation is more complicated and is influenced by the following factor: the single tone’s spectrum. The likelihood of producing this sample is maximised by estimating model parameters. The model is most likely to encounter a set of samples while attempting to find the best parameters for a given maximum relief estimation.

4.2. Model of Markov Adaptive Analysis

The basic idea of statistical pattern classification is that the training data should be completely representative of the test data. Otherwise, recognition accuracy will be affected because of the mismatch between training and testing. In practical applications, the first-order model of Markov cannot meet the prediction needs of people. Based on the detection function obtained by various algorithms, a peak extraction algorithm is needed to finally obtain a specific endpoint location. The detection function and threshold curve are shown in Figure 7 below.

[figure(s) omitted; refer to PDF]

The constraints imposed on the left and right models have no intrinsic effect on the re-evaluation process since parameters with initial values of zero remain zero throughout the re-evaluation process. Since then, the model of Markov has been widely used for pattern recognition, lexical annotation, and word separation. The model adaptation of the model of Markov is analyzed below.

Firstly, adaptive techniques extract a small number of learners’ piano audio data and alter the acoustic models, so that each model in the model library is more appropriate for the current learner. Using the Viterbi algorithm, the optimal sequence of states is found for each observed vector sequence. The sequence of states most likely to be experienced by the system model when generating the sequence is estimated, and then the states of the model of Markov are computed corresponding to each decision phoneme. The initial values are uniformly assigned to the initial probability matrix and the transfer probability matrix, and the mean of each state and the variance of each dimension are calculated using the feature vectors in all training segments. The results of calculating the division in phonemes are shown in Figure 8 below.

[figure(s) omitted; refer to PDF]

There are differences in the effectiveness of teaching by research-based professors because of differences in students’ proficiency in subjects and other areas. Therefore, these differences need to be linked to research-based teaching processes and outcomes to successfully conduct the study to draw more accurate conclusions. The parameters of the model are not constant, however, there is some structural variation, the entire sample must be decomposed into several subsamples with different parameters, and the regression equations are subject to state changes as the data come from different production processes.

Secondly, it maximizes the possibility of adaptive data by linearly transforming the existing model set into a new adaptive model set. The a priori algorithm and the model of Markov are used to detect the corresponding endpoints, and the experimental subjects are 15 piano pieces with 6652 endpoints. The results are shown in Table 2 below.

Table 2

Comparison of accuracy of different endpoint detection algorithms.

Algorithm	High-frequency component (%)	Spectral difference (%)	Phase information (%)	Amplitude phase information (%)	Average value (%)
A priori	67.77	45.67	76.54	35.66	56.41
Amplitude phase information	87.54	56.76	81.73	65.28	72.83

As shown in Table 2, the average accuracy of the model of Markov for detecting this endpoint is 72.83%, which is 16.42% better than the A-day algorithm. Hence, the model of Markov endpoint detection algorithm combining magnitude and phase information has a high correct rate ratio.

The training Markov divides the states linearly according to the number of states in the note model, and the internal state residency time is uniform. If a student’s performance does not change in the initial stage, then the student’s performance change is stable, and if a student’s performance type improves or decreases, it is considered to be up or down. Thus, for the Markov theory-based system transfer model, the state transfer process is stochastic and certain states can be transformed into other states or into their own states, i.e., no state transitions occur.

Finally, because of differences in the training and testing environments, channel noise during the recording process, and other factors, there are some differences between the training model and the test. As a result, in practise, the introduced noise must be suppressed or removed. In all training segments, the initial values are uniformly assigned to the initial probability matrix and the transfer probability matrix, and the mean of each state and the variance of each dimension are calculated using feature vectors. For example, on the bass part of the left hand, the author uses octave superimpositions, while on the right hand, the author plays a more rhythmic and acoustic sound pattern. Alternatively, one can use vibrato with a wind instrument’s breathing effect and a rich harmonic resonance and rhythmic penetration. As a result, the error terms for various states correspond to various states and reflect various reliability criteria. For various independent variables, various sequences of dependent variables are obtained.

5. Conclusions

Note recognition is a hot topic in the field of music signal analysis and processing, with applications in computerized automatic music score recognition, music database recognition, instrument tuning, electronic music synthesis, and more. Traditional time domain and frequency domain signal processing methods, on the other hand, have flaws in detecting multiple baseband mixed music signals because of the unique nature of music signal spectrum distribution. Furthermore, the innovative piano teaching model contributes significantly to improving innovation ability, primarily through the development of students’ creative thinking skills. Since its inception, the Markov model has been widely applied in a variety of fields, yielding successful prediction results. The effectiveness of the learning training model’s application is assessed using a Markov model approach. In comparison to traditional evaluation, this one is more objective. Furthermore, because early piano learning only requires real-time robust single-tone recognition, this paper proposes to build a Markov-based piano timbre recognition algorithm and a linear model-based piano teaching model. To achieve real-time robust recognition of single piano notes, we discuss the model of Markov-based piano timbre recognition algorithm in detail in conjunction with the model of Markov. As a result, when compared to the traditional manual method, the construction of the Markov-based piano note recognition algorithm and piano teaching model, the entire algorithm is automatically executed by computer without human intervention, saving human and material resources with great development potential and good application prospects.

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