1 Introduction

The most typical style is one in which dance moves change based on the music. Various dance styles convey different aspects of the music concept. Adapting dance works to the type of music is known as dancing activity syncing technology to the music composition [1]. Retrieving dance videos in an efficient manner can help dance instructors plan and support their dance education initiatives. Musical evaluation and dancers' feelings are closely related [2]. It is also challenging to discern the varied or dispersed organization and development of the group of dancers from the execution and layout of the music [3]. One of the most important types of sensory information found in multi-modal is music [4]. Computers have examined music in great detail as a powerful analytical tool. In the current digital music era, when music is consumed digitally, computers have studied music in great detail as a powerful analytical tool [5]. Performance is the standout aspect of the music performance major. The main objective is for students to become stage performers, regardless of whether they are studying dance, vocal music, or different instrumental genres [6]. Music and dance are frequently regarded as separate art genres. However, their relationship is as complex as it is close. The two are inseparable in many cultures due to their intense interdependence [7].

A kind of artistic expression that is stronger than words but not as strong as images is music. The music emotion model can be extended to other pattern recognition domains as well as machine learning to detect emotions [8]. Intelligent human–computer interface and domestication technologies have developed quickly. Using combining optimization a simulation, these dance and music videos are analyzed, and naturally associated movement steps sections are produced [9]. This technique not only lessens the workload for choreography experts but also makes it easier to retrieve music and dance video data and automate dance arrangements [10]. An optimization method used to match the complicated motions of dancing and music, based on evolutionary concepts [11]. It initially ascertains the genre of the music, extracts the fundamental style elements of both dance and music, and examines the interactions among these elements to eliminate characteristic pairings that are repeated [12].

Dance art embodies the creative desire for dance beauty, which aggressively and dynamically searches for goodness, truth, and beauty in human social life while also disseminating these attributes [13]. Additionally, it meets their aesthetic needs by evoking various dance images, establishing a connection with the audience, and expressing aesthetic consciousness and dedication to movement through a wide range of motion forms [14]. The efficacy of the fuzzy neural network model in approximating functions has led numerous scholars to use the BF and fuzzy approximation model for engineering design [15]. The ability to implement both linear and nonlinear mappings for the invisible layers and the outcome of the network layer is the action function [16]. When the source of the input signal from the network's model arrives near the center of the network model's central assortment, the buried layer node gives a significant and noteworthy output [17]. The concealed layer node generates a local reaction to the input signal. It is essential to note that the fuzzy neural network framework is taken into account when assessing the training process [18]. The key contributions are summarized as follows:

This study investigates the potential of neural networks and fuzzy logic for interactive verification in the context of dance and music instruction. The method's efficacy in identifying interaction rates and offering recommendations for curriculum design and planning is evaluated by analyzing datasets relevant to collaborative course development and efficient music syllabus construction. The study classifies student interactions as obligatory or trivial by applying neural networks for additional validation and refinement. Many parameters are used to formulate equations that calculate interaction rates and evaluate performance. The goal is to create an all-encompassing system for assessing and bettering IL experiences.

Course development and student–teacher interaction are essential to any music or dance educator's work. Teachers and students alike must undergo this process if they want to enhance their abilities and results. Researchers have studied the use of technology and new approaches to help improve teaching methods. The development of courses and their impact on students can be better understood by comparing and contrasting data from various studies. Datasets that assess the reliability and excellence of classes have been studied by researchers, with a focus on dance and music instruction. Course design and its ability to accommodate students with varying degrees of expertise (from complete novices to seasoned pros) are under the microscope in these studies. This study recommends changes to the course outline, length, content, and students' general happiness based on the data they collect.

This article presents a briefing on interactivity validation for music and dance teaching for course development. Therefore, this method harmonizes fuzzy and neural networks for interactivity classification and saturation identification, respectively. Post the briefing, the performance of this method is discussed using suitable datasets from renowned publications with a comparative study.

2 Related Works

Dou et al. [19] developed instructional techniques to improve remote music classroom students' focus in rural schools. This method provides a creative and innovative viewpoint for its investigation. Through empirical analysis, it establishes two types of improvisation dance groups, limited and unlimited. The purpose of these groups is to investigate different improvisation forms. The developed method strengthened the capacity to connect creative and inventive thinking. The study’s limitation lies in its focus on rural schools, which may limit the generalizability of findings to other educational settings.

Hu et al. [20] proposed a dance for learners with exceptional educational challenges using machine learning. The goal of the research is to examine the degree of psychomotor skills and physical development of students. A Bayesian network model (BNM) of a teacher has been created to evaluate the degree of physical fitness during various training phases. The proposed model enhances general physical health and should be thoroughly investigated in future studies. The study’s reliance on a Bayesian network model may overlook individual variations in student learning needs and preferences.

Catalano et al. [21] suggested an analogous assessment of population-based education and performing arts foster multiculturalism in future educators and the regional refugee communities. The application of arts and community-based (ACB) techniques to intercultural teacher education is examined in this method. The authors investigate how project participants discuss their experiences in the workshop by employing metaphor analysis. The approach promotes multiculturalism in teacher preparation. The study’s focus on teacher education programs may overlook broader societal factors influencing intercultural understanding.

Mangiacotti et al. [22] analyzed a continuing education program for music therapists who work in nursing homes on cognitive processes. The goal of this research was to examine the advantages of a continuing education program based on cognitive neuropsychology. A mixed-methods approach comprising a quantitative questionnaire and a semi-structured interview was used to evaluate this course. The method increases awareness of music cognition in general. The study’s reliance on self-reported data and a small sample size may limit the generalizability of findings.

2.1 Innovative Approaches in Music and Dance Education

Du et al. [23] developed speech recognition and the use of multiple difference feature networks in dance instruction. A model based on the dual evaluation of gender and perception is created using the MIML algorithm and vector machine. The method breaks down the entire system into three layers, the use function, the logic function, and the data function layer. The method displays the loss value change as well as the intermediate effects recorded in the model training. However, the study’s emphasis on technical aspects may overlook the pedagogical implications of integrating technology into performing arts education.

Shu et al. [24] proposed a 5G portable sensor technology used for speech detection and multimedia voice music instruction. Remote network control is the primary method used by the majority of low-power network technologies in-home wireless sensor network management systems. Technical techniques like sleep time can significantly lower the traditional network system’s running power consumption. The method enhances the interaction experience significantly. The focus on technical aspects may impact privacy concerns and ethical challenges associated with data collection in educational settings. Wilson et al. [25] suggested connections of uncertainty and spontaneity in the teaching of music. The activity-centered analysis and design (ACAD) framework describes the dimensions of complex learning environments. The adapted conceptual model incorporates these dimensions and brings the temporal aspects of the ACAD framework to light for further investigation. The suggested approach enhances the field's understanding of the relationship between sound and learning. The study’s dependency on ACAD setting affects the cultural and contextual factors and influences learning experiences. Jiang et al. [26] introduced instructional techniques to improve remote music classroom students' focus in rural schools. Qualitative analysis and categorization were applied to text data obtained from observations and interviews. The method additionally provides the assigning tasks in the classroom that are clear and interesting. The method significantly enhances students' focus when taking distance music classes. The study may lack of quantitative measures of learning outcomes.

Iqbal et al. [27] developed an acceptability of the virtual reality and technology acceptance model (TAM)-based dance education system. Quantitative analysis was used to determine the key variables influencing user acceptance. Validation of the developed and installed AR-based dance training system was achieved through evaluation and testing. The study’s reliance on quantitative analysis may overlook qualitative aspects of user experience and engagement.

Wu et al. [28] proposed an asymmetric development algorithm-based app user interface design for a pop music singing education system. The technical side employs the cognitive wireless sensor network principle, while the product design incorporates the interactive design philosophy. The online popular music singing teaching platform is considered to have the best user experience when the user enters a state of flow. The proposed approach improves the platform's interaction design. The technical aspect may overlook usability challenges and preferences along with their motivations.

2.2 Online Learning and Creative Thinking

Li et al. [29] suggested a personalized teaching strategy utilizing massive open online courses (MOOC). The components influencing MOOC learning adaptation face-to-face one-on-one were the basis for the surveys that the researchers gathered. SPSS was utilized to analyze the gathered questionnaire data for this investigation. The approach offers better guidelines for MOOC teachers. The survey data overlook factors that influence learner preferences and motivations. Wen et al. [30] introduced the effect of Internet technology on student's capacity for original thought. The purpose of this method is to evaluate how online technologies affect student’s creative thinking and to create an interactive online music education course. A questionnaire survey was employed as a data-gathering method for the experimental investigation. The method enhances the student’s capacity for innovative thought. Questionnaire survey impacts behavioral aspects of technology use and its impact on learning outcomes.

Ma et al. [31] proposed an application of adaptive picture information processing technology in dance classrooms. These methods can raise students' dancing proficiency and self-confidence while also assisting them in better understanding dance conventions and techniques. Dynamic picture data processing technology can be applied to a wide range of unique application scenarios and modes. The proposed method enhances student’s learning experiences by evaluating students' dancing gestures. The identified research gap may overlook the role of teacher training and pedagogical practices in effectively integrating technology into dance education.

By including insight discussions, the revised literature review provides a more critical analysis of each study’s contribution and research limitations, offering valuable guidance for future research directions and education practice in dance and music teaching,

Research on dance and music education that takes a fresh approach is covered in the literature review. Methods such as arts-based intercultural strategies, cognitive processes in music therapist training, improvised dance groups, and machine learning for exceptionally challenged dancers are highlighted. Some of the other subjects covered are speech recognition for dance classes, mobile sensor technologies for music classes, and massive open online courses (MOOCs) as tools for individualized education. Focus, intercultural understanding, and cognitive processes in music therapy are the targets of these research efforts. Music and dance education rely on various methods, yet a unified framework is lacking to incorporate these many perspectives. Addressing these limitations, the proposed study offers a novel and comprehensive approach to improve teaching effectiveness and student learning results by integrating fuzzy logic and neural networks to validate interaction in music and dance instruction.

By referencing these studies, the proposed methodology aligns with the existing research and offers innovative solutions to improve teaching effectiveness in music and dance education. The FNN Model-based IVD in the fields of dance and music teaching that has been proposed has a basis that is provided by the relevant literature analyzed from [19,20,21,22,23,24,25,26,27,28,29,30,31] in this research. The reviewed information that has been inserted provides insights on a variety of facets of music and dance education, including the incorporation of technology, teaching methods, augmented and virtual reality usage, and user experience design. The findings of these research contribute to a better knowledge of the techniques used to improve learning outcomes, enhance engagement among students, and optimize the development of courses in the subject of teaching in the field of music and dance.

The proposal for the IVD-FNN intends to harness sophisticated technology and novel methods of teaching to create a learning environment that is more interactive and engaging for students. This will be accomplished by incorporating the findings from these research investigations. The core idea of research helps to provide evidence in favor of the significance of decision-making based on data, adapted learning experiences, and interactive learning environments in educational institutions. The highlights of this research proposal are due to its potential for increasing the performance of students, create dynamic curriculum platforms, and enable practical suggestions for teachers that would boost the standard of education including achievement of students in music and dance teaching courses.

To improve engagement among learners, innovative thinking, and learning results in the field of music and dance education, this in-depth examination demonstrates the necessity of embracing innovation in teaching methods, the incorporation of technology, and interdisciplinary approaches.

3 Methodology

In music and dance education, fuzzy logic categorizes student interactions as non-trivial or trivial according to specific criteria. This approach permits degrees of truth rather than rigid true or false values. It aids in detecting trends and patterns in student behavior that could otherwise go unnoticed by more conventional binary classification approaches. The patterns found by fuzzy logic categorization are analyzed using neural networks designed to mimic the structure and function of the human brain. Students' interactions and performances in dance and music classes can be better understood using neural networks and fuzzy logic. While neural networks build upon fuzzy logic's original categorization framework, the former uses found patterns to modify and improve it. This powerful methodology combines the best features of the two methodologies to provide a more thorough understanding of how students behave and how well they succeed.

The study suggests fuzzy logic and neural networks as tools for interactive verification in dance and music education. Neural networks can discover intricate patterns from data, whereas fuzzy logic concerns human interaction's inherent ambiguity and imprecision. A systematic way to evaluate student involvement, performance, and the efficacy of a course is provided by this multidisciplinary approach that merges cognitive research, educational theory, and computational intelligence. Fuzzy logic and neural networks have recently found a new home in the field of performing arts education, specifically in the realms of music and dance. Incorporating real-time feedback and iterative course improvement procedures based on interaction data analysis, the suggested technique integrates a more nuanced understanding of student engagement and learning outcomes, building upon previous methods.

In a case study of Improving a Music Appreciation Class, a strategy to increase participation in a music appreciation class is being implemented at a university. Using a neural network and fuzzy logic model, this approach examines the relationship between student engagement and achievement. These are evaluated through the use of online quizzes, group projects, and presentations in class. The system uses predetermined criteria to classify student interactions as significant or inconsequential. Course materials and pedagogical approaches can be fine-tuned in real time depending on input instructors receive on student involvement and learning preferences. As a result of more opportunities for interaction and more tailored learning experiences, students are happier and do better in school.

3.1 Interactivity Validation Method (IVM) Using Combined Fuzzy Neural Network (FNN)

A combination of fuzzy neural networks and interactive learning methods is introduced for both online and offline music and dance teaching. In this paper, interaction among the student and the teacher is defined to improve the presentation. The performance is improved in this case, by deploying a significant interaction rate. The interaction rate is used to determine the performance of the different students. An illustrative representation of IVM-FNN is given below.

Interactivity Validation Method using a Combined Fuzzy Neural Network is a way to evaluate interaction in dance and music instruction that uses a combination of fuzzy logic and neural networks. Using interaction rates to measure student performance, the approach examines student–teacher interactions to improve performance. Using neural networks allows for more precise classification and better predictions, while fuzzy logic sorts interactions as either major or minor. Student performance and engagement trends can also be uncovered using this strategy. Course recommendations and design, false data detection and correction, and interaction rate analysis are all areas that benefit from using neural networks. By analyzing student interactions and performance levels, the methodology uses modern computational tools to improve music and dance teaching. Insights and suggestions for bettering pedagogical methods in these areas are the intended outcomes of the research. Insightful suggestions for enhancing pedagogical practices in these domains are designed to be produced by the technique.

3.2 Input Interactions

The education course is given as input for analysis based on their interaction rates. This interaction rate varies with the course levels and their consistencies. As the interaction is high, fuzzy classification is introduced; the saturation for low/high interaction is identified. Therefore, the consistency metric (from the data) is incorporated for saturation detection. This detection is used to provide suggestions for different course levels (Fig. 1). Here, the proposed method includes course design for the development of music and dance. A neural network is used to improve the false data in the training layer and provide better validation. The validation is carried out to distinguish the interaction rate for the different students. Fuzzy is used to classify and identify the mandatory and trivial sets. The preliminary step is used to find the interaction rate for both the music and dance online and offline. Equation (1a) is derived for is interaction rate

Fig. 1
figure 1

Illustration of IVM-FNN

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$${\text{H}}=\frac{1}{\sum_{{{\text{t}}}_{0}}\left({{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}\right)}*\left[\left({{\text{t}}}_{0}+{\text{P}}\right)*{{\text{e}}}_{{\text{a}}}-{({\text{v}}}^{\mathrm{^{\prime}}}+{\text{L}})\right]+{\text{D}}\left({\text{L}}\right)*{{\text{t}}}_{{\text{n}}}.$$
(1a)

The interaction is found in the above equation that deploys the music and dance for the student. The interaction rate is a way to quantify the amount or strength of interaction. It is determined by adding two factors, \({c}_{m}\) and \({a}_{c}\), over a given time frame. Constraints are represented by variables \(v^{\prime}\) and \(L\), and the outcome is efficiency and effectiveness. Another factor that affects the rate of interaction is the long-term impact of learning or growth. An aggregate metric of interaction can be obtained from this equation. The computation, process here is used to differentiate the performance and improve the interactivity. In this paper, both the online and offline classes are monitored, and observe the activity and performances and it is derived as \({v}{^\prime} \,{\text{and}} P\). Here,\({\text{H}}\) denotes the interaction finding, for the number of students \(\left\{{t}_{0},...{t}_{n}\right\}\). In this equation, music and dance are represented as \({{\text{c}}}_{{\text{m}}}\mathrm{and }{{\text{a}}}_{{\text{c}}}\), whereas, \({e}_{a}\) is described as Artistic education. The activity of the student is observed regularly and it is represented as \({{\text{v}}}^{\mathrm{^{\prime}}}\), the learning is described as \({\text{L}}\). The student-level understanding is represented as \({\text{D}}\). Thus, the interaction is found by equating the above equation, and from this observation step, identification of mandatory and trivial is carried out. Based on this process, the interaction is calculated, and furthermore, the fuzzy plays a vital rate in the classification of mandatory and trivial.

3.3 Fuzzy Logic Classification

The classification model for fuzzy is established to differentiate mandatory and trivial for the different students. If it is mandatory, then the interaction rate is measured accordingly, and in another case, it is trivial then the significance is not involved in this method. Based on this computation step, fuzzy is used to define whether it is mandatory or not this includes trivial accounts. Equation (1b) is used to identify the mandatory and trivial in this process

$${\text{S}}=\left.\begin{array}{c}{\prod }_{{{\text{t}}}_{0}}^{{{\text{t}}}_{{\text{n}}}}\left(\left({{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}\right)*\mathrm{\varnothing }\right)-({\text{D}}+{\text{P}})+\frac{\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{L}}\right)}{{\text{H}}},\forall \,Mandatory\\ \frac{\left(\mathrm{\varnothing }+{\text{P}}\right)}{\sum_{{\text{P}}}\left({\text{H}}*{{\text{t}}}_{0}\right)}*L(D+{{\text{t}}}_{0})-\varphi ,\forall\, Trivial\end{array}\right\}.$$
(1b)

In the above given equation, fuzzy-based classification is performed for the mandatory and trivial. Here, several students attending the classes online or offline have the quality of performances. The performance degradation means that the interaction rate decreases in this case mandatory is necessary. It includes learning from the interaction and it is represented in this equation \(\frac{\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{L}}\right)}{{\text{H}}}\). Thus, the classification is defined the mandatory and trivial where it is established in two cases; the coordination is represented as \(\mathrm{\varphi }\). The equation consists of components \(S, \sum ,\, and\, \varnothing\), representing classification outcomes, interaction quality, and learning factors. The ratio of these factors to \(\varnothing\), where \(v\) and \(L\) represent constraints, and \(\varnothing\) represents interaction rate, is crucial for determining mandatory aspects. The equation also includes \(\forall\), applicable to mandatory categories, and \(\varnothing\), calculating their significance. The fuzzy classification and its corresponding analysis are presented in Fig. 2.

Fig. 2
figure 2

Fuzzy classification and its corresponding analysis

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The course period observed \(V{^\prime}\) for \(H\) in designed course recommendations for improving \(P\). In the classification process, \(P\) is identified as high or low based on \(D{^\prime}s\) impact. If \(D\) is less, then \(P\) is not considered; the classification thus is performed for \(P={\text{low}}\) or \(P={\text{High}}\). Therefore, the \(D\)(based on star ratings) for each grade is independently assessed. In this case, the circled regions for \(H\), D, and \(P\) are induced for classification due to low performance and low \(D\) (Refer to Fig. 2). The first case indicates a mandatory process that improves the interaction, whereas the second case states the trivial which defines no significance in the computation. The trivial process states the performances for an interaction and it is represented as \(\mathrm{\varnothing }\). Here, trivial is neglected in this classification and it is described as \({\text{S}}\). From this fuzzy classification, the next step is analyzed, to design the course which is built for artistic education that incorporates multi-point interactions and monotonous presentation. Building upon the principles of adaptive instruction discussed earlier in the context of course design, by introducing a below mathematical model aimed at optimizing the learning experience for students with diverse needs. This is derived in Eq. (2)

$$C = \underbrace {{\left[ {\left( {D + \frac{{\left( {t_{0} + v^{\prime}} \right)}}{{t^{\prime}}}} \right)*P\left( {e_{a} } \right) + \left( {D*\varphi } \right)} \right]\underbrace {{\left( {L + S} \right)*\frac{{\mathop \sum \nolimits_{\varphi } \left( {L + H} \right)}}{{t_{0} }}}}_{{{\text{Monotonous}}}}}}_{{{\text{Interaction}}}} + (v^{\prime} *D) - t^{\prime}.$$
(2)

The course design is determined for the different students and deploys a better understanding in both online and offline classes. Here, the course design is built for a better understanding and deploys the activity and coordination. The verification is done to deploy the multi-point detection for the mandatory. The coordination is examined for the students based on the fuzzy classification. Here, \({{\text{t}}}^{\mathrm{^{\prime}}}\) describes the mandatory from the fuzzy classification. The fuzzy classification here states: the mandatory and trivial and improves the performances and it is derived as \(\left({\text{D}}+\frac{\left({{\text{t}}}_{0}+{{\text{v}}}^{\mathrm{^{\prime}}}\right)}{{{\text{t}}}^{\mathrm{^{\prime}}}}\right)\). The understandability is done for the students which estimates the classification and learning, and the course design here is done for better student performance that in reverse improves the interactivity.

3.4 Interactivity Validation Method (IVM)

If the interactivity increases, and the performances also increase, the course design is built for artistic education which incorporates interaction and monotonous presentation. In the above equation,\({\text{C}}\) represents the course design, Post to this method, validation is done for the interactivity which enhances the efficiency. Equation (3) is used to equate the validation process for the interactivity

$${\text{V}}=\left\{\begin{array}{c}\left({\text{C}}*\frac{\left({{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}+{\text{L}}\right)}{\left(\mathrm{\varphi }(\mathrm{\varnothing })\right)}\right)+\left[\left({\text{H}}*{\text{D}}({{\text{t}}}_{0}\right)\right]\\ =(C+I)*\sum_{{{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}}\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{P}}\right){{\text{t}}}^{\mathrm{^{\prime}}}*S\end{array}.\right.$$
(3)

The validation is performed in the above equation and it is represented as \({\text{V}}\), and in this first derivation, music and dance are observed in their coordination. From this, it derives the activity from the students and improves the performances. In this fuzzy classification is done and finds the interaction and performance. The main purpose of this validation is to analyze the course design which is built in the previous equation computes the learning methodology and incorporates the interaction and it is equated as \(\left({\text{C}}*\frac{\left({{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}+{\text{L}}\right)}{\left(\mathrm{\varphi }(\mathrm{\varnothing })\right)}\right)\). Thus, the music and dance performance is analyzed based on the activity and classification in fuzzy. To enhance the efficiency the interaction is monitored periodically in both the online and offline and the identification is represented as \({\text{I}}\). Thus, both the efficiency and interaction are validated and checked it is mandatory or not. If it is mandatory, then the interactivity is improved in this validation method. From this validation rate, the trivial is observed whether it is significant or not in Eq. (4)

$${\text{V}}({{\text{a}}}^{\mathrm{^{\prime}}})={\text{H}}+\frac{\sum_{{{\text{t}}}^{\mathrm{^{\prime}}}}\left({\text{C}}+\mathrm{\varphi }\right)}{{{\text{t}}}_{0}}*{\sum }_{{\text{P}}}^{{\text{V}}}\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{D}}\right)*{\text{M}}-{\text{I}}+\left(\frac{\mathrm{\varphi }}{{{\text{t}}}^{\mathrm{^{\prime}}}}\right).$$
(4)

In Eq. (4), validation is performed for the interactivity of both the online and offline streams. Here, the interaction is found in Eq. (1a), which deploys the mandatory and trivial and it is described as \({{\text{a}}}^{\mathrm{^{\prime}}}\). The understandability is defined to distinguish the interaction and monotonous and it is represented as \({\text{M}}\). In this approach, the coordination is monitored for the trivial and finds whether it is significant or not. The interactivity rate classification analysis is provided in Fig. 4.

The \(H\) analyses for 12 months (course period) for \(P={\text{low}}/{\text{high}}\) and \(D={\text{low}}/{\text{high}}\) is presented in Fig. 3. The above classification is based on \({\partial }^{\prime}\) and \(\psi\) for distinguishable \({t}_{n}\). In the fuzzy process, the available classification combinations are identified for triviality extraction. In the above process, the saturation detection is expected to be identified under less period for reducing further triviality. However, the fuzzy chances are ¼ or 2/4 for distinguishable \(P\) outcomes. Therefore, the consistency check for maximum interaction is required for this classification. Thus the fuzzy classification relies on a neural network for saturation detection by reducing 1 or 2 failing combinations. The fuzzy classification is established for the validation of trivial and analysis of its significance. If it is significant only the validation is further forwarded to the next computation level it is neglected from the fuzzy classification. Thus, it is computed in the above equation, next comes the neural network where the false data are collected and trained for better performances. This training is done on the hidden layer; the proposed work focuses on one hidden layer where the saturation factor output is examined. It might be high or low if it is high then, the interaction level is increased whereas, if is low then, the interaction and performance rate are identified and determine the computation in a neural network.

Fig. 3
figure 3figure 3

Interactivity rate classification analysis

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Fuzzy logic uses two parameters, student activity (SA) and learning effect (LE), to gauge the degree to which students are involved in their learning. SA refers to how actively students are engaged in learning, whereas LE measures how much progress students have made in content comprehension and application. According to SA and LE, interactions are either essential or unimportant. If the two parameters exceed the thresholds set, the contact is necessary, and the students were actively engaged and learned a lot. Minimal influence is indicated by SA or LE falling below certain levels, indicating a minor interaction. Empirical methods can establish SA and LE thresholds, such as reviewing relevant literature, analyzing educational standards, or consulting with subject-matter experts. Additional linguistic factors allow the system to account for human behavior and learning outcome uncertainty and variability.

3.5 Neural Network Implication

The neural network is defined as the interconnection of n-number neurons, in which the input is fetched and processed and provides the output. This processing step defines whether there are false information/data if it is identified then, it is fed to the training neuron and provides better results. Thus, the false information is trained to provide better results in the neural network. Equation (5a) computes the hidden layer

$$\left.\begin{array}{c}{t}_{0}({c}_{m}+{a}_{c})={\sum }_{V}^{C}(s+{t}^{\prime})*P(1)+\varphi \\ {t}_{1}({c}_{m}+{a}_{c})={\sum }_{V}^{C}(s+{t}^{\prime})*P(2)+\varphi \\ \vdots \\ {t}_{n-1}({c}_{m}+{a}_{c})={\sum }_{V}^{C}(s+{t}^{\prime})*P(n-1)+\varphi \end{array}\right\}.$$
(5a)

In Eq. (5a), the hidden layer is designed for the interaction and improves the performance. Here, the input is fetched from the interaction rate from which the fuzzy classification is computed. In this process, the hidden layer acquires the student activity that includes both dance and music. The process is forwarded to the n-number of layers in the neural network, which includes the course design and validation method. The n-number of the process includes the performances regarding the number of students in the first cycle. If it is trivial, then the output is forwarded to the second layer as the training data to improve the computation step. Thus, the hidden layers work for the number of students based on their performances. The following equation computes the classification from Eq. (1b) and hidden layer output is derived for artistic course design. This equation is used to enhance artistic education in the hidden layer and provides reliable output from the failing factors

$$S\left( \varphi \right) = P\left( {n - 1} \right)*D + \left[ {\frac{{{\raise0.7ex\hbox{${\emptyset + e_{a} }$} \!\mathord{\left/ {\vphantom {{\emptyset + e_{a} } {V*t_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${V*t_{0} }$}}}}{{\left( {t^{\prime} + C} \right)}}} \right] - V.$$
(5b)

The classification for coordination is established in Eq. (5b); here, the performance is measured for the n-number of layers in a neural network. The computation step describes the understandability for the students and evaluates the validation for this process. This equation is equated to the course design and improves the failing factors and it is represented as \(\left[ {\frac{{{\raise0.7ex\hbox{${\emptyset + e_{a} }$} \!\mathord{\left/ {\vphantom {{\emptyset + e_{a} } {V*t_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${V*t_{0} }$}}}}{{\left( {t^{\prime} + C} \right)}}} \right]\). From this evaluation, saturation factor is distinguished in Eq. (6)

$${\text{X}}=({\text{P}}*{{\text{e}}}_{{\text{a}}})+\sum_{\mathrm{\varphi }}{\text{V}}*{{\text{t}}}_{0}+{\text{C}}\left({\text{D}}\right)-{\text{S}}.$$
(6)

The examination is carried out for reliable understanding and deploys better computation based on the performances. The interaction states the efficiency factor and defines the saturation factor for the different students. The saturation factor here deploys high and low which deploys the verification phase. In this saturation factor, the neural network involves a higher or lower interaction rate. The saturation factor estimation using neural learning is illustrated with analysis in Fig. 4.

Fig. 4
figure 4

Neural learning process for saturation factor with analysis

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In Fig. 4, the correlation between the analysis and the combination of reducing layers is presented. The initial combination is performed for \(\left({t}_{n-1}, ac\right)\) and \(\left({t}_{n-1}, {C}_{m}\right)\) possibilities. The term \({C}_{m}\) and \(ac\) is used to define the coefficients of neural learning process that can influence the neural network’s behavior or performance of the model.If the combinations are reduced, then \(\psi\) validations (i.e.)\(\left({t}_{n-1}+{V}^{\prime}\right)\) and \(\left(V*D\right)\) are used for monotonous differentiation. In this process, \(\psi =0\) (due to \({t}_{n-1}\) achieving \(-V^{\prime}\)) and \(\psi \ne 0\) (due to \(V^{\prime}\) or \(D\ne 0\)) are identified for low and high combination outputs. Therefore, the change from \(\psi =0\) to \(\psi =1\) and vice versa is used for identifying the saturation point. If the \(\psi =0\) (or) \(\psi =1\) persists for a longer period of \({t}_{n-1}\), then saturation is identified. The \(\psi =0\) is referred to as low saturation for which development suggestion is provided. Based on this process, performance is improved for the better understandability. Thus, the examination is described as \({\text{X}}\) and indicates the saturation factor for the student performances. Post to this examination process, the differentiation of high and low is equated in Eq. (7) as follows:

$${\text{T}}=\left\{\begin{array}{c}{({\text{c}}}_{{\text{m}}}+{{\text{t}}}^{\mathrm{^{\prime}}})*\sum_{{\text{P}}}({\text{V}}*{{\text{t}}}_{0})-\mathrm{\varphi }=0\\ \frac{{\text{D}}}{{{\text{t}}}_{0}+{\text{P}}}*I(\varphi )+C-{{\text{e}}}_{{\text{a}}}\ne 0\end{array}.\right.$$
(7)

The differentiation is provided for the high and low interaction rates; the first case indicates that there is a high saturation factor. Therefore, it is equal to zero, whereas it has a low saturation rate and that indicates not equal to zero.

3.6 Training and Validation

Thus, the differentiation is done for the different students to provide validation and enhance cooperation. The differentiation is described as \({\text{T}}\); in this equation, the saturation factor is defined and provides the interaction and monotonous presentation. From this differentiation, high and low are estimated, post to this verification is done by fine-tuning the saturation factor. It is equated in Eq. (8a) as follows which is used to verify whether its saturation factor is up or low for the student performances:

$${\text{Y}}=\frac{\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{P}}\right)}{\mathrm{\varphi }}*\sum_{{\text{T}}}\left({\text{C}}*{\text{L}}\right)+\left[\left({{\text{g}}}_{{\text{n}}}+{{\text{w}}}_{{\text{e}}}\right)\right]*{{\text{e}}}_{{\text{a}}}-{\text{V}}.$$
(8a)

The verification is performed in the above equation that includes the course design and learning method of the students. The possible way to quantify learning \(L\) by seeing how much learner’s performance on a particular task or skill improves over the course of time. One way to accomplish this would be to compare the initial and final performance measures, or to monitor the changes in performance that occur over the various stages of learning. Behavioral patterns, attitudes, and beliefs can all be altered as a result of learning for certain people. The practice of watching and measuring developments in behavioral patterns, ways of making decisions, or other observable behaviors could be considered a method for quantifying learners in this research.

Here, the saturation factor is used to determine the high and low levels in neural networks. In this case,\({{\text{g}}}_{{\text{n}}}\mathrm{and }{{\text{w}}}_{{\text{e}}}\) are represented as high and low.

3.7 Modification and Course Development Suggestions

The suggestion ratio for development under low and high differentiations is presented in Fig. 5.

Fig. 5
figure 5

Suggestion ratio under low and high differentiations

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The suggestions presented above are derived from the provided datasets for classifying distinguishable saturation points. Based on the classification rate, the suggestions are classified based on \(T=0\) or \(T\ne 0\). In this case, the \(Y\) estimation performs two differentiations for \({W}_{e}\) and \({g}_{n}\). Therefore, as the classification increases saturation increases regardless of \({g}_{n}\). In this process, \(X\) for all distinguishable \(\left({t}^{\prime}+G\right)\) is validated for multiple suggestions. Thus, the proposed method is optimal for maximizing (retaining)\(X\) (refer to Fig. 5). Here, performance is derived for artistic education and deploys the mandatory process. The mandatory process is derived for the better saturation level in a neural network. The input is acquired from the neural network and forwarded to the saturation factor. In this saturation factor, the identification is done for the high and low of interaction. The scope of this work is to improve the interaction and performance; in this case, if the interaction increases, then the high saturation factor is estimated. If is low, then the training set is used to address the fault and provide the desired result. In this equation, verification is performed for the students and it includes the course design that derives the better classification method. Equation (8b) is used to compute the interaction rate and performance of the student

$${\text{U}}=(\mathrm{\varnothing }+{\text{P}})*\sum \left({\text{M}}+{{\text{t}}}_{0}\right)*\left({\text{L}}({{\text{t}}}^{\mathrm{^{\prime}}}\right)+{{\text{e}}}_{{\text{a}}}-{\text{I}}.$$
(8b)

From Eq. (8a), the saturation factor is estimated; in this block, both the high and low are identified. Based on this identification, the course design is evaluated and forms a better understandability for the students. Here, the computation is defined for the validation that deploys artistic education. The mandatory process is carried out for the students and estimates the better resultant as interaction. The learning defines the mandatory method for artistic education that is built from the course design. Here, computation is carried out for the monotonous validation which includes fuzzy classification and combines neural networks. This equation,\({\text{U}}\) describes computation which is used to find the interaction rate and performances using the saturation factor. Equation (9) is used to modify the interaction which is a failure in the neural network using current course development

$${\text{O}}={\text{U}}*\frac{\left({\text{P}}+{\text{V}}\right)}{\mathrm{\varnothing }}+\sum_{{\text{X}}}\left({{\text{c}}}_{{\text{m}}}+{\text{L}}\right)*{\text{S}}\left({{\text{t}}}_{{\text{n}}}\right)-\mathrm{\varphi }+{\text{C}}.$$
(9)

The modification is done for the course development which derives the fuzzy classifications to improve interaction and performance. The evaluation is carried out for the number of students. The neural network is used to train the failure and provide the coordination and performance and it is equated as \(\sum_{{\text{X}}}({{\text{c}}}_{{\text{m}}}+{\text{L}})\). The coordination and interaction increase to improve the course development. This paper fuzzy is related to the classification of mandatory and trivial, and from this equation, modification is done if there is any failure occurs during the current course development suggestion. In this equation,\({\text{O}}\) represents the modification; from this, the upcoming process is a course development suggestion and it is equated as follows:

$$N = \frac{{\left( {\omega *C} \right)}}{{P/t_{n} }} + \left( {e_{a} + t^{\prime}} \right)*U - \varphi .$$
(10)

The course development is derived for a better fuzzy classification and provides high interaction and performance. The modification in Eq. (10) states whether there is a failure factor is detected in the neural network. From this saturation factor is built for the student performance and improves the efficiency by deriving the course development suggestion. The course development suggestion is carried out from the high interaction and shows better performances. The proposed work defines the interaction that eliminates the significant process, the suggestion is represented as \({\text{N}}\). The seven suggestions presented in Fig. 6 are analyzed for their recommendation ratio in the below analysis.

Fig. 6
figure 6

Recommendation ratio of the given suggestions

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3.8 Recommendations and Analysis

In Fig. 6 analysis, the 7 suggestions (S1–S7) presented in Fig. 5 are validated for their ratio over \(U\) and \(P\). In both cases, the “low” classification requires high suggestions (including replacement), whereas the “high” classification demands varying ratios. The neural network saturation estimation conceals the stability of the outputs, such that classification is revisited. In this case, the \(H\) improvements are mandatory. The combination of fuzzy classification and neural networks is used to enhance the interaction and efficiency of both streams. By computing this artistic course development suggestion is done for the students. The suggestion is used to enhance a high rate of interaction.

Music and dance educators can benefit from data-informed decision-making, improved curriculum design, individualized learning experiences, and ongoing improvement by incorporating fuzzy logic and neural networks into their practices. Educators can use this approach to build adaptive curriculum frameworks that cater to student's unique learning styles and preferences by tracking and analyzing student involvement and performance. It also allows teachers to keep tabs on their students' progress and interactions in real time, so they may tweak their lessons accordingly. The methodology equips teachers with practical insights that can be used to make strategic decisions that will improve the quality of education and the success of students.

4 Results and Discussion

4.1 Dataset Description

The pre-discussion introduces the datasets and their connectivity for assessing the proposed method. The data from [32] provide collaborative course development validations using consistency and attributes. In the second data from [33], an efficient course construction for music syllabi is introduced. This is performed under four different categories and the linkage between the two data is provided in Fig. 7.

Fig. 7
figure 7

Data linkage representation

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In the above representation, categories and course levels are utilized from [33] that are related to music teaching. The interaction and consistency rates are obtained from [32] for classification and saturation detection. Based on these links, the final suggestions are validated across multiple courses pursued for different levels. The course levels are obtained for beginner, intermediate, standard, and professional training through teaching. This study draws on data sets from two articles that discuss effective methods of creating music education syllabi and collaborative course building. The dataset on collaborative course development sheds light on how educational courses, especially those about dance and music, are developed through teamwork. Course characteristics, consistency criteria, and educator collaboration patterns are all part of it. The efficient syllabi construction dataset for music education covers many features of syllabus production. These features include course levels, categories, and duration. It provides helpful data for gauging student happiness, curriculum development, and course preparation. The project aims to improve music and dance education through more interaction and performance evaluation, and these datasets align with that approach. The quality of educational programs is influenced directly by the dynamics of collaborative efforts in course planning and development, which may be better understood using the collaborative course development dataset. The datasets add to the body of knowledge by supplying facts and practical understandings of the intricate world of dance and music education. The recommendations are provided for curriculum, course planning, duration (period), and category satisfaction (Fig. 1).

4.2 Comparative Study

The comparative study is performed using interactivity, classification rate, development-based recommendation, evaluation ratio, and evaluation time for different course levels (I, II, III, IV). In this study, the above factor's impact for 12 months (course period) and classification rate (1) are studied.

The project aimed to create a model that could use the music and dance datasets to predict different types of dance moves based on musical characteristics. About three-quarters of the datasets were designated for training, while the remaining quarter was split evenly between validations and testing. This study tried various feature representations for music and ran tests with and without dancing data to see how the model performed. K-fold cross-validation was used for robustness in addition to classification accuracy, precision, recall, F1-score, confusion matrix analysis, and other validation methods. Data preparation, feature extraction, modality-specific processing, model training, evaluation, and analysis of results were all steps in the experimental workflow.

4.2.1 Study 1: Interactivity

In Fig. 8, the interactivity increases by deploying course duration and classification. The fuzzy classification is used to improve the interaction, which is further divided into mandatory and trivial. Equation (1b) is used to evaluate better analysis for improving music and dance performance. Here, the processing deploys the \(\frac{1}{ \sum_{{{\text{t}}}_{0}}\left({{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}\right)}\) To ensure reliable computation, students are involved in this interaction. Interaction in this evaluation step results in improved fuzzy classification that determines the significant output. The high and low interaction rates among different students are determined, which provides the course design. The course design is utilized to enhance the interaction among students, which impacts the validation process. The validation method is used for the significant identification that includes mandatory and trivial and it is equated as \((D+P)+\frac{\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{L}}\right)}{{\text{H}}}\). Thus, the computation declares the saturation factor and approximates the classification process. To determine the verification time, a fuzzy-based model is employed, regardless of whether it has a high or low saturation factor.

Fig. 8
figure 8

Interactivity study

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4.2.2 Study 2: Classification Rate

The classification rate for the proposed work increases with varying course periods and recommendations. In Fig. 9, the classification exhibits a high value, which estimates the neural network for enhanced interaction between dance and music. The identification process is responsible for artistic development, which includes the course design, which is determined by Eq. (3). A neural network is employed in this computation phase to enhance the training set for students and demonstrate superior performance. The saturation factor is defined by the performances here, which indicate both high and low values. The course design is represented as \({\text{P}}({{\text{e}}}_{{\text{a}}})+\left({\text{D}}*\mathrm{\varphi }\right)\) in this coordination. The interaction rate for the proposed work deploys the activity and performances and it is equated as \(\sum_{{{\text{c}}}_{{\text{m}}}+{{\text{a}}}_{{\text{c}}}}\left({{\text{v}}}^{\mathrm{^{\prime}}}+{\text{P}}\right){{\text{t}}}^{\mathrm{^{\prime}}}\). The efficiency of computation among students is determined by their comprehension. Fuzzy classification is used to classify monotonous validation and determine the interaction between the course period and recommendations. Estimating the saturation level is the purpose of the recommendation.

Fig. 9
figure 9

Classification rate study

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4.2.3 Study 3: Development-Based Recommendation

The proposed work's classification is improved by the development-based recommendation. The activity is examined and a course design is created for artistic development in this instance. The interaction is done for the monotonous and examines the mandatory and trivial for the students. The dance and music are used to determine the better course design and it is represented as \({\text{M}}-{\text{I}}+\left(\frac{\mathrm{\varphi }}{{{\text{t}}}^{\mathrm{^{\prime}}}}\right)\). The activity is described as monotonous and validating, with the performances being related to artistic development. The identification is employed to connect with the recommendation process. The recommendation is determined by the classification and saturation factors related to efficiency. The deployment of activities and performances that determine artistic education leads to efficiency. The representation here concerns the difference between high and low values, which coordinates, and it is compared to Eq. (5a). Music and dance determine the performance, which shows better interaction between the number of students (Fig. 10).

Fig. 10
figure 10

Development-based recommendation study

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4.2.4 Study 4: Evaluation Ratio

In Fig. 11, the evaluation ratio can be improved through changes in course length and classification rate. Here, the performance is used to determine artistic education and explore validation. The validation is done during learning and increases monotony. To estimate the mandatory and trivial, the performance is used here. The mandatory and trivial is used to define \({\text{M}}-{\text{I}}+\left(\frac{\mathrm{\varphi }}{{{\text{t}}}^{\mathrm{^{\prime}}}}\right)\). The suggestion is to classify mandatory and trivial tasks, which use identification. The identification is done for coordination and estimates the artistic education. The learning rate is defined for monotonous learners and provides suggestions and recommendations. The recommendation is given to the students, and the saturation factor is estimated. The saturation factor is used to determine the course design for performance evaluations, which determine whether they are mandatory or trivial. The computation process is used to examine the coordination for the varying classification and course periods. Thus, the performance has improved with this development of recommendations.

Fig. 11
figure 11

Evaluation ratio study

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4.2.5 Study 5: Evaluation Time

The evaluation time for the proposed work decreases which is shown in Fig. 12, for the varying course period and classification rate. Here, the evaluation process is deployed for the mandatory and trivial and estimates the artistic education and it is represented as \(\left[\left({{\text{g}}}_{{\text{n}}}+{{\text{w}}}_{{\text{e}}}\right)\right]*{{\text{e}}}_{{\text{a}}}\). The activity and performance are used to state the efficiency of the proposed work. The examination is done for the high and low based on the saturation rate. The monotonous validation is done by deploying the activity and providing the course design, and it is computed as \(\left({\text{M}}+{{\text{t}}}_{0}\right)*\left({\text{L}}({{\text{t}}}^{\mathrm{^{\prime}}}\right)\). Thus, the recommendation is performed for the varying course periods and shows the better classification ratio for the interaction among the students. The computation step is used to define the suggestion based on the saturation factor. The modification is performed for the different students in which the efficiency rate is increased. Artistic education is used to design the course and find the period of processing. The classification is done for the mandatory and trivial for the analysis of performances.

Fig. 12
figure 12

Evaluation time study

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The comparative studies show that the suggested techniques improve efficiency, classification, recommendation, interaction, and evaluation in dance and music education. Course duration and categorization were found to increase interaction in the study significantly. The fuzzy classification method, in particular, improved interaction by differentiating between important and unimportant components. Computation based on neural networks improved performance on the training set, and the classification rate rose with changing course durations and suggestions. The development-based suggestion method's better classification achieved efficiency in artistic education, which focused on creative development and identified mandatory and trivial features. The evaluation ratio study improved artistic education and offered useful advice by showing that modifications to class size and duration had a favorable effect on the evaluation ratio. The approach increased productivity through activity and performance measurements, as the assessment time research demonstrated. The scalability and long-term effects of these strategies in other types of schools could be the subject of future studies.

The proposed model IVM-FNN can be used for performance comparison in contrast with the conventional works such as BNM [20], ACB [21], and TAM [27] using metric as accuracy and interactivity level for performance prediction of the implemented model.

As a metric for evaluation, accuracy shown in Fig. 13 assesses the average accuracy of the proposed IVM-FNN methodology's recommendations by computing the ratio of instances calculated for 12 months course that were successfully predicted to the entire number of examples that were reviewed in dance and music education in comparison to existing three models. It provides a general evaluation of the performance of the technique in terms of correctly recognizing or recommending strategies for the development of courses.

Fig. 13
figure 13

Accuracy analysis

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The duration of the course is taken for analysis up to 12 month period shown in Fig. 14 analyzed for interactivity level of the proposed IVM-FNN is comparatively high in comparison to the existing benchmark model, and during that time, learners will engage in an all-encompassing learning experience that will cover a variety of fields related to the education of music and dance. Students delve into subjects ranging from fundamental concepts to advanced techniques over the course of this length, each of which is facilitated by an organized curriculum that is geared to match the varied requirements and ability levels of students. Students are able to get significant insights including real-world expertise in the areas of theory of music, dancing instructions, performing abilities, and instructional methods through an assortment of theoretical classes, hands-on training, and interactive sessions. The duration of 6 months makes it possible to have a well-rounded and engaging educational adventure, which encourages participants to gradually enhance their skills, engage in self-reflection, and further their development as they advance through the course.

Fig. 14
figure 14

Interactivity level analysis

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5 Conclusion

The design of music and dance teaching courses requires better interactivity between students and teachers to improve the latter’s performance. For such feat, this paper proposed and discussed the interactivity validation method backboned by fuzzy combined neural network. The interactivity rate is first identified as trivial or non-trivial using fuzzy classifications depending on the actual performance exhibited by the students. These classifications are used by the neural network for identifying saturation as either low or high. The need for saturation identification is the recommendation for providing suggestions for course improvements. In the neural network process, the impact of various teaching grades, performance evaluation, and interaction rate variation are considered. Based on the learning output, the low saturation classifications are induced for new recommendations/ suggestions for course development by rectifying the previous flaws. Such suggestions are decisive in incorporating a new curriculum in music and dance teaching.