The design of an optimisation algorithm for musical performance guided by constructivist theory in musical theatre

Constructivism, which is also known as structuralism, posits that children develop their cognitive structures by gradually building knowledge about the external world while interacting with their surroundings [29-30]. Children’s interaction with the environment involves two basic processes: “assimilation” and “adaptation”. Assimilation refers to the process of absorbing relevant information from the external environment and incorporating it into the child’s existing cognitive structure (also known as “schema”), i.e., the process by which an individual integrates information provided by external stimuli into their original cognitive structure. Conformity refers to the process of restructuring and reforming children’s cognitive structure caused by changes in the external environment and the inability of the original cognitive structure to assimilate the information provided by the new environment, i.e., the process in which an individual’s cognitive structure is altered as a result of the influence of external stimuli.

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Scenario setting.The broader context in which the complete learning experience (i.e., what is learnt, how it is learnt, and how it is used) exists is the external scenario that facilitates students’ understanding and construction of meaning and promotes the connection of knowledge, skills, and experiences. Second, it is not only the physical and conceptual structure of the problem, but also the social context in which the intention of the activity and the problem are embedded. Creating a context that facilitates the construction of meaning by the performer is the most important aspect or aspect in the music performance process.

The basic idea of genetic algorithms as evolutionary algorithms is based on Darwin’s theory of evolution and Mendel’s doctrine of heredity [31-32]. From Darwin’s theory of evolution, it can be seen that all kinds of organisms in the natural world in the continuous struggle for survival, through heredity and mutation to produce new individuals, the new individuals can be retained if they are able to adapt to the environment, while those who are not adapted to be eliminated, that is, the law of “survival of the fittest”. According to Mendel’s theory of heredity, heredity exists in the form of genes in the chromosomes, which are the command codes that control individual traits. Genes have a specific location and determine a particular trait. Through crossbreeding and genetic mutation, offspring are produced that are better adapted to the environment, and through natural selection (i.e., survival of the fittest), the better-adapted individuals are retained, while the less well-adapted are eliminated.

The flowchart of the genetic algorithm is depicted in Figure 1. Given a real problem, the genetic algorithm first encodes the problem as “chromosomes”, and according to “survival of the fittest”, selects chromosomes for replication, a process known as regeneration, and then through crossover, mutation and other operations to produce new chromosomes, which have greater adaptive capacity. This process is called regeneration and is followed by crossover, mutation, and other operations to produce new chromosomes that are more adaptable. This process is repeated until the most adaptable individual emerges, i.e., the optimal solution.

Figure 1.

Flowchart of genetic algorithm

The underlying theory of genetic algorithms focuses on analysing the convergence of the problem to be solved, i.e., calculating the probability of the population converging to the global optimal solution. On the whole, it can be divided into stochastic model theory and evolutionary dynamics theory. 1)

Stochastic model theory

The theory is based on stochastic processes, and if the coding space or population is finite, a Markov chain model can be used to represent the search process. However, it should be noted that the model is discrete-time. Finally, this search process is analysed based on the already existing related stochastic process theory.

To obtain a harmonious lighting configuration scheme for a musical performance stage, it is necessary to collect color images of the lighting from different angles in the scene. Considering the usual movement paths of actors on a musical stage scene, this study tries to set up several panoramic cameras around the movement paths to capture the lighting colour images in the musical scene, and a total of 3,000 sets of stage image data are captured. The blue trajectory represents the regular movement path of an actor on a musical theatre stage, and six cameras are set up around the stage, each of which captures images from six different viewpoints: up, down, left, right, front and back, and then forms a panoramic image cube.

This study was conducted to evaluate the light colour images in the scene and to provide a harmony colour scheme for the light colours. The light colours in the scene are represented as a colour vector containing RGB colour information, each light colour vector can be represented as , and the full colour vector in the scene can be defined as V = (V₁, V₂…V_n).

In this study, a colour evaluation function is constructed with the colour vectors in the scene as the evaluation parameters and three factors as the evaluation criteria, namely: the colour harmony factor E_h, the colour emotion factor Em, and the colour contrast factor Ee. The colour evaluation function can be expressed as follows: (1)E(V)=λhEh(V)+λmEm(V)+λcEc(V)$E(V) = {\lambda _h}{E_h}(V) + {\lambda _m}{E_m}(V) + {\lambda _c}{E_c}(V)$

Here λ_h, λ_m, λ_e is the colour evaluation factor normalisation parameter, which is set to 100, 5, and 1 respectively, where the difference in λ_h compared to the other two factors is to enhance the weight of colour harmony in the overall colour scheme. The three different colour evaluation factors are defined as follows: 1)

Colour harmony factor E_h

In this study a tonal template approach is used to evaluate the colour harmony of a 3-D scene by evaluating the colour harmony of each face in each panoramic cube image and aggregating it into a colour harmony formula with weights: (2)Eh(V)=∑iλi∑j=16λjEkt(fij(V))${E_h}(V) = \sum\limits_i {{\lambda _i}} \sum\limits_{j = 1}^6 {{\lambda _j}} {E_{kt}}({f_{ij}}(V))$

λ_i in Eq. (2) represents the weight of each panoramic camera cube image and λ_j represents the weight of each face in each cube image. In this study, the weights of the four cameras are set to 0.2, and the weights of the two cameras at the corners are set to 0.1. For each cube image, the weights of the top and bottom faces are set to 0.1, and the weights of the four faces at the front, back, left, and right are set to 0.2. The colour harmony factor of each face of each cube image is defined as follows: (3)Eht(fij(V))=argminm,α∑p∥H(p)−ETm(a)(p)∥⋅S(p)180${E_{ht}}({f_{ij}}(V)) = \arg {\min _{m,\alpha }}\sum\limits_p {||H(p) - {E_{{T_m}(a)}}(p)||} \cdot \frac{{S(p)}}{{180}}$

H in Eq. (3) is the colour channel, S denotes the saturation channel, and the equation in Eq. denotes the distance on the colour swap. where T_m is the mth colour template of Matsuda. 2)

Colour emotion factor E_m

This study proposes the method of colour emotion factor, which uses PCA algorithm to extract these three components. ψ₁, ψ₂, ψ₃, ψ₄, ψ₅, ψ₆ is used in this study to represent each of these six components. Through the emotion evaluation method and combined with the music classification result labels in this study, the expression can be shown as follows: (4)Em=Emdef∑i=02Emi${E_m} = E_m^{def}\sum\limits_{i = 0}^2 {E_m^i}$

Emdϕf$E_m^{d\phi f}$ in Eq. (4) represents the user’s desired light colour emotion, which is preset in such a way that it affects the whole effect of the light colour setting. Where Emi$E_m^i$ is the emotion defined by ψ_i: (5)Emi={ (2−ψi)2/16, for active, heavy and warm (ψi+2)2/16, for passive, light and cool$E_m^i = \left\{ {\begin{array}{*{20}{l}} {{{(2 - {\psi _i})}^2}/16,}&{{\text{for active, heavy and warm}}} \\ {{{({\psi _i} + 2)}^2}/16,}&{{\text{for passive, light and cool}}} \end{array}} \right.$ 3)

Colour contrast factor E_m

In order to optimise the lighting colours in a scene, colour contrast is an important evaluation factor and many different colour contrast methods have been defined in the literature. Using global light in this study, the colour contrast factor can be defined as follows: (6)Ec=∑m,nωmn(1−(∥Lm−Ln∥/100)2)${E_c} = \sum\limits_{m,n} {{\omega _{mn}}} (1 - {(\parallel {L_m} - {L_n}\parallel /100)^2})$

Lm${\mathcal{L}_m}$ and L_n in Eq. (6) are two different light colours in the scene, and ω_nm represents the intersection area of the two light ranges.

The method used in this study is different from the traditional genetic algorithm in that the algorithm starts with the initialization of the population at the design point. The genetic algorithm calculates the fitness value of the population in the same way as the traditional method, and collects enough feature points for the optimisation model, which is thus constructed and applied to the subsequent evolutionary operations.

In this study, the genetic algorithm model is used as our optimisation model, the genetic algorithm is able to predict an unsampled point x as a global trend function f^T(x)β and Gaussian treatment G(x). Namely: (7)y(x)=fT(x)β+G(x),x∈Rm$y(x) = {f^T}(x)\beta + G(x),x \in {R^m}$

f(x) = [fo(x)…fp − 1(x)]^T ∈ R^p in Equation (7), which is defined by a series of basic regression equations. β = [β₀… β_p−1] ∈ R^p, denotes the correlation coefficient between the vectors, where T is the maximum number of iterations and m is the number of populations, which is optimized by genetic algorithms to derive the fitness value of the colour evaluation function.

In order to set keyframes during the playback of the whole music file, this study outputs 50 tags for each music clip, each tag has the corresponding output sorting, to find out the “fast” and “slow” tags, and then keyframes are set. The calculation formula can be defined as follows: (8)Tk=Tavg+ηsps−ηfpf+∑i=1k−1Ti${T_k} = {T_{avg}} + {\eta _s}{p_s} - {\eta _f}{p_f} + \sum\limits_{i = 1}^{k - 1} {{T_i}}$

T_k in Eq. (8) is the Kth keyframe, which represents the sequence of keyframes from the beginning to the end of the music, and k ∈ [1, 2…N] .Tavg is the average time interval between all the keyframes, which is calculated from the total duration of the music and the total number of music clips. P_s and P_f are the user-defined speed of the light changes, which can be manipulated to control the light rate at fast and slow tempo. n_s and n_s are the weights of the labels “slow” and “fast”, the calculation of the label weights is defined below: (9)ηj=rj/∑j=1Nj${\eta _j} = {r_j}/\sum\limits_{j = 1}^N j$

r_j in Equation (9) is the output ordering of the label among a total of all N output labels, the higher the ordering of the label, the more weight it takes. The colour of the light is obtained by interpolating between keyframes.

In order to set the light intensity for the lights of the whole scene. In this study, the output weights of “loud” and “quiet” among the 50 classification labels of the output music are used for calculation. The calculation method is as follows: (10)Ik=Idef+ηlfl−ηqfq${I_k} = {I_{{\text{def}}}} + {\eta _l}{f_l} - {\eta _q}{f_q}$

I_k in Eq. (10) represents the light intensity of the kth keyframe, l_def represents the user-defined overall colour intensity tone, η_l and η_q represent the weights of the labels “loud” and “quiet” in the whole output label sequence, and f_l and f_q are the user-defined weights of the light intensity, which can manipulate the weights of bright and grey lights. η_l and η_q are the weights of the labels “loud” and “quiet” in the whole output label sequence, and f_l and f_q are the user-defined light intensity levels, which can be manipulated to give more weight to the light in bright and grey light.

In this study, we define the music light color mixing template by the output music labels, and we divide the music labels into two basic color templates: cool templates and warm color templates. Cool templates include “classical”, “techno”, “beats”, “ambient”, and warm templates include “rock”, “electronic”, “pop”, “metal”. For each color template, different color calculations are defined:

The cold colour template includes some cold colours, such as blue, purple, etc., which are calculated as follows: (11){ C1=(λ1/255,1,0) C2=(0,1,λ2/255) C3=(0,λ3/255,1) C4=(λ4/255,0,1)$\left\{ {\begin{array}{*{20}{l}} {{C_1} = ({\lambda _1}/255,1,0)} \\ {{C_2} = (0,1,{\lambda _2}/255)} \\ {{C_3} = (0,{\lambda _3}/255,1)} \\ {{C_4} = ({\lambda _4}/255,0,1)} \end{array}} \right.$

In Equation (11), each expression in the λ₁ ∈ [0, 194], λ₂ ∈ [0, 255], λ₃ ∈ [0, 255], λ₄ ∈ [0, 102] formula represents an RGB colour component, and the cool light colour will be randomly selected from C₁, C₂, C₃, C₄. The warm colour template includes some warm colours, including red, orange, etc., which are calculated as follows: (12){ W1=(1,ε1/255,0) W2=(1,0,ε2/255)$\left\{ {\begin{array}{*{20}{l}} {{W_1} = (1,{\varepsilon _1}/255,0)} \\ {{W_2} = (1,0,{\varepsilon _2}/255)} \end{array}} \right.$

ε₁ ∈ [0, 255], ε₂ ∈ [0, 220] in equation (12), the warm light colour will change from {W₁, W₂}

In this study, each music clip is classified by musi CNN model, in which the first N classification labels of each first music clip will be output. So for the music clip K the light colour can be calculated as follows: (13)Lk=∑j=1NηjCj${L_k} = \sum\limits_{j = 1}^N {{\eta _j}} {C_j}$

η_j in Equation (13) represents the weight size of each music clip label, C_j ∈ {C_c, C_v}, whose value is defined through the jth label. With each music colour template L_k, and the harmonic light colour H_k, the mixed light colour can be defined as follows: (14)Mk=(Lk−Hk)F+Hk${M_k} = ({L_k} - {H_k})F + {H_k}$

F in equation (14) represents the mixing coefficient of the two colours, F∈[0,1]$F \in \left[ {0,1} \right]$ .

The coloured light of lamps is often divided into two categories: cold light and warm light. Cold light is mainly blue, while warm light is yellowish or reddish. Appropriate use of colored light can not only produce beautiful effects but also bring out different artistic atmospheres and ideas to be expressed in the works. Based on the colour factors mentioned in the previous section, the colour emotion factors of musical performances are statistically analysed, and the colour emotion factors of musical performances are analysed as shown in Table 1. It can be seen that among the colour emotion factors of musical performances, the red emotion factor has the highest number, with a number of 579, and the corresponding emotion types of red are passion, love, vitality, and positivity. The second color is orange, with a specific value of 483, and the corresponding emotional type is bright, spiritual, carefree, and excited. Other emotional factors related to colors are the same. Musical theatre performances are bound to have a tone, such as the use of red light to highlight the warmth of the fire and the use of blue light to set off quiet, gentle, and deep feelings. When this tone is set, the other lights are arranged for use. That is to say, the principle of each show’s hue should be a basic colour, supplemented by other colours, reflecting the sense of ups and downs, in the hue of the collocation of the hue should be just right, and the warm and cold flow of hue changes with the change in colour saturation should be appropriate. Choose the right lighting color to enhance the drama or atmosphere of the stage to achieve better stage expression.

Taking the musical “Cats” as an example, the musical “Cats” is divided into 20 keyframes, each keyframe corresponds to a “loud” and “quiet” labels, and by calculating the stage light intensity of each frame, the stage lighting effect can be intelligently adjusted. Lighting effects, η_l, η_q, f_l, f_q, l_def data are all from the source data set A. In order to explore the key frame lighting intensity of the musical, the key frame lighting intensity of the musical “Cats” is analysed based on the formula (10), and the analysis of the key frame lighting intensity is shown in Table 2.

In conclusion, the stage lighting art effect is an important component of music performance, but also a key factor that affects the effect of music performance, directly affecting the quality of music performance. Appropriate use of stage lighting can enhance the impression of music performance activities, resulting in endless reminiscence and thinking. A truly perfect musical performance not only relies on the performance of the actors but also attaches importance to the use of various choreographic techniques, of which the reasonable use of stage lighting is very important. Stage lighting should be used correctly to create the atmosphere, emotions, and service for the music performance, so that the stage art has more infectious force and attraction.

For the optimization of the color evaluation function, this paper uses a genetic algorithm to search for the optimal parameters of the color evaluation function. Genetic algorithm in the optimisation problem solving The main idea of this method is to take the probability as a guide in a given range of space, set the initial population, and then introduce the hybridisation operator, focusing on searching for the expectation of the emergence of the part of the parameter that is most in line with the fitness, which overcomes the problem of the general algorithm of searching the space is too large, and the calculation is too complex. In order to maintain the diversity of the population, then introduce the variation operator. The parameter selection for ten-fold cross-validation is depicted in Fig. 2, and the experimental analysis of the colour evaluation function optimization is carried out using the matlab libsvmmat toolbox.

Figure 2.

Ten fold cross-validation parameter selection

Step 1: Initialisation settings, set parameters f(x)$f\left( x \right)$ G(x). This parameter is from equation (7), the value range of f(x)$f\left( x \right)$ is [0,2000], the value range of G(x) is [0,2000].

Step 2: Chromosome coding design, binary coding is used.

Step 3: Calculate the fitness value in the sense of crossover validation using matlab software.

Step 4: Design the selection operator, crossover operator and variation operator.

Step 5: Abort condition setting. Output the fitness value of the colour evaluation function if the conditions are satisfied and return to step four if they are not satisfied enough.

The fitness value of the colour evaluation function is obtained by using ten-fold cross-validation analysis:

Best f=2^-9.83, Best G=2^-9.93, Fitiness=0.897

The higher the fitness value of its colour evaluation function, the higher the convergence performance of the evaluation function, and the better the effect of its stage lighting.

Based on the proposed musi CNN model in subsection 2.3.6, music labels are classified and analysed in a music dataset A, for example, the number of which is 3,000, which is divided into (2,400 sets of data) as training samples and test samples (600) according to the ratio of 8:2, and the data is normalised to the interval of [0,1] through data preprocessing, and the confusion matrix of the training set is shown in Fig. 3, and the test The training set confusion matrix is shown in Fig. 3 and the test set confusion matrix is shown in Fig. 4. From the above results, it can be seen that by musi CNN model for music label classification, the correct rate of the training set is 97.58%, and the correct rate of the test set classification is 95.50%, which greatly improves the efficiency of music label classification, relying on the musi CNN model and diversified facilities and equipment for the audience to create and perform the content of the same lighting effects, and the integration of the artistic emotion and the lighting effects into a whole, to create a unique mood, and to create a unique mood. It also integrates artistic emotions and lighting effects into a whole, creating a unique mood and incorporating more modern colors. Musical theatre stage lighting can stop the performance at a certain time, thus prompting the audience to truly feel a moment of emotional outburst on the musical theatre stage.

Figure 3.

The training set confusion matrix

Figure 4.

Test set confusion matrix