Introduction

The connection between Western music and mathematics has been recognized since the days of Pythagoras, Plato and Aristotle, who wrote about the overlap and parallels between the two disciplines. Both disciplines – music and mathematics – are expressed through the use of representative language and symbolic notation (Papadopoulos 2002). Mathematical concepts such as symmetry, patterns, ratio, and division are expressed in music. In music, intervals, rhythm, duration, speed, and many musical concepts are naturally represented by numbers (Bamberger and Disessa 2003). For example, the intervals between harmonic notes in music are determined by ratios of small whole numbers. When plucking two strings of the same length, the ratio between their lengths is 1:1, resulting in identical and harmonious sounds (sounds that blend well together). Moreover, differing ratios of string lengths will produce various harmonic intervals, such as the octave (string ratio of 1:2), the fifth (2:3), and the fourth (3:4).

Music was a subject of research among mathematicians such as Descartes, Kepler, and Euler, and on the other hand musicians were attracted to the possibilities inherent in the science of mathematics for analyzing works and composing (Wollenberg 2003). Compositional methods that draw inspiration from mathematical ideas included, among others, counterpoint (a second voice that appears simultaneously with the first voice in a polyphonic texture), a crab canon (the second voice is an imitation of the first voice in reverse), or a palindrome (a section that can be played from beginning to end and from end to beginning, in mathematics y = −x), and geometric designs of musical melodies (for example reflective symmetry, where a melody repeats itself in a mirrored fashion). In addition, mathematical images in Western music are expressed musically in the theory of harmony, the theory of rhythm, and the theory of forms. The theory of harmony reveals a fundamental regularity in how sounds are combined simultaneously, how they relate to one another, and how they are distributed over time. The theory of rhythm refers to how the sounds are organized in time and the continuation relationships between them. And finally, the theory of forms addresses how musical events are organized and the proportions created between musical parts (Douthett 2008; Johnson 2008; Rothstein 2006). In light thereof, the current study aims to uncover the profound interconnections between the disciplines, as perceived by mathematicians, musicians, and educators in teacher training programs. By adopting a wholistic perspective, this research seeks to highlight the practical relevance of these interdisciplinary connections both in training teachers and instructing students.

Background

The connections between mathematics and music in teaching and learning

Recent studies that cited the explicit connections between the knowledge embodied in music and mathematics indicated positive transfer effects (Akın 2023; Azaryahu et al. 2024; Azaryahu et al. 2023; Wang et al. 2024). These studies used symbolic notation to create an explicit parallel between musical concepts and mathematical concepts among elementary school students. For example, two studies conducted with 84 third graders and 86 fourth graders found that the experimental groups increased their mathematical and musical achievements over the control group following participation in a multidisciplinary program combining music and mathematics regarding patterns and symmetry (Azaryahu et al. 2023) and fractions (Azaryahu et al. 2024). The methodologies employed in these studies used symbolic notations to create an explicit parallel between temporal music concepts and numerical and spatial math concepts, such as fractions, symmetry, ratios, and patterns, demonstrating the natural alignment between these disciplines (Courey et al. 2012; Hallam and Himonides 2022; Wang et al. 2024).

A recent meta-analysis by Wang et al. (2024) supports previous research, showing a significant positive impact of integrating music into mathematics instruction. It highlights critical factors such as incorporating music instruction within math interventions, the use of calming and math-related music, and exposure at early grade levels that significantly enhance the effectiveness of music in improving mathematical learning outcomes compared to other moderating categories. Music as a pedagogical tool facilitates the connection of mathematical concepts to real-world contexts, thereby making lessons more meaningful and relevant (Erickson 2002; Nagisetty 2014). This multidisciplinary approach has the potential to enhance students’ interest and motivation in mathematics (An 2012).

In the current study, we explore the content areas that can be effectively paralleled in the classroom learning process, as perceived by musical experts and mathematicians who are actively engaged in teaching at universities and teacher training colleges.

Professional identity of theorists and educators in music and mathematics

The dynamic between theorists and educators lies in the clear distinction between the two separate realms of knowledge and practice. The academic community specializes in crafting and disseminating theoretically framed, universally applicable knowledge derived from research. In contrast, educational practitioners cultivate and enhance their knowledge while actively addressing challenges and solving problems to achieve specific objectives within a particular educational context (Bressler 1999).

Professional identity is a key underlying concept in our behavior. The development of professional identity is shaped by interactions and embedded connections with colleagues, occurring within the contexts of culture, profession, and personal experience (Hargreaves 2001). Beijaard et al. decomposed teacher identity into three categories based on a personal knowledge perspective: teacher as subject matter expert, teacher as pedagogical expert, and teacher as didactical expert (Beijaard et al. 2000). Subject matter mastery involves comprehension of the material to be taught. Didactical proficiency encompasses crafting and imparting a learning encounter, as well as evaluating the results of teaching methods. Pedagogical adeptness involves addressing students’ emotional and social needs, and customizing instruction to suit individual learning needs. Teachers are positioned within this framework according to the priority that they attribute to each form of expertise.

Pedagogical content knowledge (PCK), first introduced by Shulman in 1986, aligns closely with this framework of teacher identity. Described as the combination of teachers’ subject matter expertise with their pedagogical skills, PCK represents the integration of content knowledge and pedagogical knowledge, enabling teachers to effectively deliver subject matter tailored to the abilities and interests of learners (Shulman 1987). Some studies further define PCK as encompassing content knowledge (CK), pedagogical knowledge (PK), and knowledge of learners (Gess-Newsome et al. 2019; Liu 2013). Consequently, PCK equips teachers to address and correct students’ misconceptions about the subject matter they are learning.

According to Palmer’s (2017) perspective, adopting a comprehensive method for becoming a teacher involves intertwining the roles of teacher, student, subject, and life. He claimed that constructing a teacher identity involves recognizing one’s “inner landscapes,” which includes intellectual, emotional, and spiritual self-awareness. Therein, educators’ objectives in music/mathematics should revolve around aiding students in discovering avenues to knowing, comprehending, and enhancing their own selves.

In addition, it is widely recognized that emotions interact with and affect the learning process (Baker et al. 2010; D’Mello et al. 2014). Although it is commonly assumed that emotions play a role in musicians’ and music educators’ creative and educational endeavors (Smith 2021), the same holds true for mathematicians and mathematics educators (Bishop 2008; DeBellis & Goldin 2006; Resnick 1988; Schoenfeld 2016; Zan et al. 2006). Researchers accordingly discuss the influence of affect on “mathematical thinking” (Schoenfeld 2016), “problem solving” (Resnick 1988) and “mathematical capability” (DeBellis and Goldin 2006) among other related aspects.

Compared to educators, theorists typically present a set of propositions and theorems that are logically related to one another, providing an integrated and comprehensive way of thinking about the target phenomena (Bruscia 2005). A theory, as defined by Bruscia (2012b), is a supposition based on logically and comprehensively integrated principles derived from extensive high-quality research. Music theorists grapple with how individuals acquire the ability to derive significance from sounds. Ultimately, the responsibility falls upon the teacher to make pivotal choices regarding the optimal methods for addressing the diverse needs of students in the music classroom (Isbell 2012). In mathematics, teachers with limited mathematical knowledge might find their professional capabilities restricted, although possessing such expertise doesn’t necessarily ensure effective teaching of mathematics (Da Ponte and Chapman 2015).

In the current study, we focus on investigating the convictions held by professionals in the realms of mathematics and music who specialize in theory and education. We aim to explore their viewpoints concerning the interrelation between these two disciplines.

Integrating music and mathematics

Integrated instruction involves the presentation of varied perspectives, each with distinct aims and objectives, within a shared educational context (Park and Son 2010). The integration of various disciplines brings about numerous benefits, including fostering innovation, literacy, motivation, teamwork, creativity, and actively engaging students in the learning process while promoting essential problem-solving skills (Liao 2016; NAEA 2016; Root-Bernstein 2015). Although multidisciplinary learning is highlighted in discussions concerning 21st-century skills, its practical implementation in educational settings tends to be relatively limited as students progress through K-12 education (Boice et al. 2021).

Special attention has been paid in recent years to music’s contribution to STEM in promoting creative thinking, for example in the fields of computing (e.g., Engelman et al. 2017) and math (e.g., Azaryahu et al. 2023; Azaryahu et al. 2024). The bulk of research focusing on the music learning’s effect on mathematical achievement centers around elementary students. The primary objective of this research is to explore how music can positively influence aspects such as memory, motivation, learning, and creativity in these students (Akın 2023; Azaryahu et al. 2023; Azaryahu et al. 2020; Azaryahu and Adi‐Japha 2022; Hallam and Himonides 2022). However, few researchers have examined the effect of music on mathematical learning among high school students and at the university level (Crowther, 2012; Lesser, 2014, 2015).

This study aims to broaden the scope of integrating disciplines by focusing on middle school, high school, and music and mathematics teacher trainees at universities and colleges. Therethrough, we seek to examine the interconnections between these disciplines from the perspectives of experienced teachers and lecturers in these disciplines. The ultimate goal is to lay the foundation for developing multidisciplinary content units suitable for upper-division and university-level students.

Teachers’ perceptions of multidisciplinary education

The development of multidisciplinary curricula raises numerous questions, such as determining which content should be combined, and for which age groups or grade levels. Additionally, there is a need to explore how the integration will be applied in the classroom. Developing integrative curricula requires extensive knowledge and massive investment of both human and financial resources. As noted by Humes (2013), a teacher who undertakes the blending of two disciplines must possess a comprehensive mastery of both subjects. Moreover, blending two disciplines necessitates the respective teachers having excellent and effective didactic and pedagogic skills.

Research exploring teachers’ perceptions of multidisciplinary education is still in its early stages. However, recent studies in China, Korea, and Indonesia have shed light on the positive views of teachers regarding the future benefits to students pursuing careers that demand integrated skills (Kang 2019; Kartini and Widodo 2020; Kim et al. 2019; Rosikhoh et al. 2019). On the other hand, the obstacles to implementing such learning lie in two main areas: 1. The teachers’ sense of self-efficacy and knowledge; and 2. The teachers both being prepared for and open to the process.

Initially, it was found that teachers faced challenges in implementing multidisciplinary learning due to insufficient skills resulting from a lack of knowledge, comprehension, supportive resources, and adequate training. In many cases, teachers perceived multidisciplinary teaching merely as a series of isolated activities rather than embracing it as a wholistic approach to learning that emphasizes exploration and project-based learning (Boice et al. 2021). In the context of teaching math specifically, teachers report that there are a limited number of math areas that are appropriate for implementation in a multidisciplinary framework (Kim et al. 2019). According to their perspectives, educators believe that math is better suited as a supporting component in multidisciplinary learning rather than being the central focus. While this approach has the potential to facilitate progressive teaching and learning methods, its implementation is not fully realized to the extent it could be.

The literature presented in this section supports multidisciplinary teaching. However, it also cites significant challenges hindering its implementation, particularly concerning applications and consistent integration. These challenges stem from factors such as limited preparation time, inadequate educational resources, and the need for extensive knowledge in both music and mathematics. Moreover, it is essential to acknowledge that a deep understanding of the connections between music and mathematics often requires expertise in both disciplines. Mathematics teachers lacking familiarity and knowledge of basic music theory may encounter difficulties when attempting to establish meaningful connections between the two disciplines.

The method

The study goal and questions

The main goal of the current study is to identify the perspectives of various kinds of experts on the interplay between music and mathematics. The study’s questions are:

  1. a.

    What are the main categories of perceptions about the connection between music and mathematics?

  2. b.

    How do the distinctions between these categories manifest among musicians, mathematicians, and lecturers in teacher training programs?

The study participants

The study’s participants consisted of 16 experts (4 musicians, 4 mathematicians, 4 music education lecturers, and 4 mathematic education lecturers) who participated voluntarily in this study (see Fig. 1).

Fig. 1: Composition of Research Participants.
figure 1

Includes groups of 4 musicians, 4 mathematicians, 4 music education lecturers, and 4 mathematics education lecturers.

Full size image

Data collection

The data were collected by means of a qualitative research paradigm, through semi-structured interviews. The first author of this paper carried out all data collection. The experts were asked to think about and express their perceptions regarding the connections between music and mathematics in a general sense and from an applied pedagogical perspective during the interview. During interviews, the researcher kept a neutral attitude and did not provide any evaluative remarks. She asked the experts to clarify their ideas when they were not entirely clear and to contemplate the interconnection between their respective disciplines as it manifests in their work. Altogether 16 interviews were conducted, which were video- and audio-recorded, and all the recordings were transcribed.

Data analysis

In this study, we set out to explore the diverse perspectives of experts through three distinct lenses: theory, affect, and learning opportunities. These three distinct lenses emerged directly from the research participants – comprised of theorists and educators – and from the general literature on theory and practice (Bressler 1999). The research participants’ professional identities – whether rooted in theoretical frameworks (theory) or practical applications (learning opportunities) – can significantly shape their perspectives on the interconnection between music and mathematics (Bruscia 2005; Da Ponte and Chapman 2015; Palmer 2017). The third lens, affect, is shared by both participant types in the study – theorists and educators – given that emotions are inherent in any process of investigation, creation, or learning (Baker et al. 2010; D’Mello et al. 2014). Development of the subcategories was based on the grounded theory approach (Strauss and Corbin 1990), using content analysis of teachers’ discourse by repeated viewings of the videos of the interviews, and repeated readings of the transcripts.

First, experts’ statements about the relationship between music and mathematics were classified as fitting into one of the three categories: theory, affect, and learning opportunities. The category of theory included talk related to structures recognition, formal systems, abstract concepts and symbolic representations, and mathematical and musical creations. The category of affect included statements related to emotional engagement and enjoyment regarding aesthetic and beauty and their emotional effect on learning. The category of learning opportunities comprised statements related to educational approaches that integrate music and mathematics that enhance learning in both subjects.

Next, all of the data classified as belonging to a particular category – theory, affect, and/or learning opportunities – were categorized to create subcategories that reflect experts’ perceptions of the relationship between music and mathematics. The data analysis was inductive, first identifying the initial subcategories based on some of the data collected for a particular participant, then refining and extending the initial subcategories based on more data collected with the same participant, and finally, to refining the devised sub-categories by the analysis of data collected from all of the participating experts. Furthermore, all of the subcategories devised for the various categories of theory, affect, and learning opportunities were analyzed for commonalities and differences across the categories.

We performed inter-coder validation of the transcripts’ categorization buy the first and the third authors of this paper. We attained agreement on about 90% of utterances, while the remaining utterances were discussed by the authors and categorized based their agreement.

In this paper we present examples that illustrate experts’ perspectives on the connection between music and mathematics. The examples are borrowed mainly from the data collected from four content experts (indicated by pseudonyms) from the field of musical and mathematical theory and education, as their views were fairly common and corroborated those of many others who participated in the study: (1) a mathematics educator named Dan, (2) a music educator named Mika, (3) a mathematician named Adam and (4) a music theorist named Lisa. When any of the subcategories of theory, affect, and learning opportunities were missing in their talk, we used examples from other experts. The next section will present the outcomes derived from the insights of esteemed experts in the theoretical and educational domains of mathematics and music.

Findings

As presented in data analysis sections through directed analysis toward theory, affect, and learning opportunities, we developed various sub-categories that characterize experts’ perspectives on connections between music and mathematics. In what follows, we present excerpts from the interviews that exemplify various types of perspectives.

Theory perspectives

Abstract languages and structure

Dan and Mika are experts in their respective fields of mathematics education (Dan) and music education (Mika). When asked how they view the connecting lines between music and mathematics, they replied:

1Ma_Ed - Dan – “Both mathematics and music are semantic systems [1-1] therefore they both allow formalization [1-2]. In other words, if there is some agreement on certain rules [1-3], a musical work and a mathematical work can be combined [1-4]. Both music and mathematics in a certain sense are a means of communication between people [1-5] … It seems to me that there is some element here that is related to the transmission of messages one way or another [1-6] … In terms of the structure of these systems, they are quite similar. My point is that there is some basic structure [1-7] from which you can grow and develop [1-8]. There are patterns in both music and mathematics, and this can be represented with the help of numbers and notes” [1-9].

2Mu_Ed - Mika – “The mystical aspect in me shows that music and mathematics are two embodiments and two languages of the same principle [2-1]. The principle is the divine order in which the world was created [2-2]. It is a way of communication [2-3]. I think it is very beautiful and important to link musical theory to the structures from which it is created. Symmetry, proportions and geometry [2-4].

The excerpts contain clear indicators of Dan and Mika’s conceptions of the connection between music and mathematics as two abstract languages with common semantic systems [1-1, 2-1, later 3-1] that are expressed by musical notes and numbers [1-9]. We consider this an indicator of their views on the Theory of music and mathematics. Both disciplines allow formalization [1-2] and behave according to certain rules and order [1-3, 2-2]. Dan’s view of the connection between music and mathematics, as it relates to form and structure, including natural structures and patterns [1-4, 1-7, 2-4, later 3-3], is a clear indicator of his connecting the theory of both disciplines. In Dan’s view, the basic structures and order enables growing and developing [1-8]. It is interesting to note that the experts in the field of musical and mathematical education viewed the commonality between the two languages as a means of communication [1-5, 2-3] and transmission of messages from a pedagogical perspective [1-6].

The same question about the connection between music and mathematics was posed to the theoreticians well. Adam and Lisa, experts in mathematics and music theory respectively, replied:

3Ma_Th - Adam – “Both languages are abstract [3-1]. Numbers are connected to both of them [3-2] …When the music is good, you discover the order there. Regularity [3-3] … The first name that comes up is Pythagoras. There is the Pythagorean theorem on one side and the Pythagorean scale on the other [3-4]. The Pythagoreans viewed numbers as something musical” [3-5].

4Mu_Th – Jack – “Math is numbers and music is numbers… you could just as well replace every sound with a number and it would work just as well [4-1] … I can give a numerical representation of almost any musical element [4-2]. All music and the history of music is deeply connected and inseparable from numbers and calculations” [4-3].

While both these theoreticians view both disciplines as abstract languages, just as the educational experts do, their emphasis lies on the central foundations of these disciplines. Adam and Jack’s descriptions focus on numerical representation as a unifier between music and mathematics [3-2, 3-5, 4-1, 4-2). Also, their theoretical explanation relies on historical context [4-3] for example, the Pythagorean theorem and the Pythagorean scale [3-4].

These excerpts (1Ma_Ed Dan, 2Mu_Ed Mika, 3Ma_Th Adam and 4Mus_Th Jack) reveal that the professors’ conceptions of the connection between music and mathematics includes the fundamentals of both disciplines as abstract languages adhering to strict form and structure. We found that the direction of talk about the connection between music and math is typical for most of those who participated in this study. However, while theoreticians in both disciplines referred to the representation of music through numbers from a historical perspective and stressed the language and structure as core elements, educational experts in both fields emphasized the commonality between the languages regarding communication and the transmission of messages from a pedagogical perspective.

Freedom and creative thinking

Along with the orderly structure that characterizes the two languages, music and mathematics, the need for breaching conventions, freedom, and creativity arose in the experts’ talk. In the following excerpt, Jack described how he uses the format to compose music:

5Mu_Th – Jack – “In my composition I leave myself the freedom to do what I want [5-1]. I don’t limit myself to a certain format [5-2]. What I pour into the pattern is my business and is taken from many considerations: what instruments I write for, the tempo, the atmosphere I want to create, the harmony, the dynamic, all these elements affect what I’m going to write [5-3] … I looked, for example, at how an architect designs a house [5-4]. First of all, s/he starts drawing the outline of the house, and little by little goes into the details [5-5]. So, I create the same thing, only in music [5-6].”

Like other music theorists, Jack expands upon the structural connection between music and mathematics for the purposes of composition, and emphasizes the ability to create original music by using an ordered pattern. In the above excerpt [5Mu_Th – Jack], he describes the composition process as being analogous to an architectural work [5-4]. The structure and form, which were mentioned earlier, serve as an excellent foundation for planning his musical work [5-5, 5-6]. Creative and original content can be poured into this structure [5-3], so there is a lot of freedom within the framework [5-1, 5-2]. Lisa expands upon Jack’s line of thinking, discussing the concept of freedom in the historical context of democracy:

6Mu_Th - Lisa – “In music there’s a lot of freedom [6-1], but the Greek philosophers said that if you change things in music, then society will change too [6-2]. Music would not have developed without the development of democracy and individualism [6-3] …The existence of the individual within a society that on the one hand has very regulated structures and on the other hand allows for the individual’s personal existence [6-4]”.

In Lisa’s view, as a musical theorist, there is a lot of freedom in music, which connects to the development of democracy and individualism in ancient Greece. While in the above excerpt [6Mu_Th_Lisa], she cites a parallel line of thinking to Jack’s regarding musical freedom [6-1], her reply contains an extension connecting freedom to concepts of individualism and democracy [6-2, 6-3].

The educational experts from both disciplines also referred to the transitions between defined musical or mathematical structures and liberating departures therefrom. Emma, a mathematics education professor, and Sue, a music education professor, had this to say:

7Ma_Ed - Emma – “Both the realms of mathematics and music offer opportunities for creativity, freedom, and flexibility for our students [1] - I also find liberty of invention in mathematics [2]. [after watching the video]: Elevation! It’s charming and amazing [3] because what appears symmetrical and beautiful – an elaborate polygon blocked inside a circle – also sounds very pleasant [4]. Equal peripheral angles that rest on equal strings creating harmony [5]. As soon as you break the rules, then you have something a little more interesting; it creates dissonance [6]”.

8Mu_Ed- Sue - “In general, I see maybe something related to order at the deeper level [1]. A musical analysis that needs some sort of order in it, some kind of thinking of derivatives [2]. [after watching the video]: I saw that when there’s something symmetrical, then the sounds are also harmonious [3], and when there’s something disharmonic, then the shapes are neither completely symmetrical nor different [4]. I think it’s very beautiful and important [5]. Symmetry, proportions, and geometry are also seen as shapes, and as movement, and as color [6]. Very illustrative and helpful [7]”.

Like all of the experts who participated in the study, Emma and Sue watched a short video presenting connection between geometrical shapes and sound. After doing so, they mentioned in the above excerpts [7Ma_Ed_Emma and 8Mu_Ed_Sue] the idea that there is a visual-aural connection between the disciplines [8-6, 8-7]. Most of the participants agreed that on the one hand, symmetrical shapes were demonstrated to represent harmonized sounds [7-4, 7-5, 8-1, 8-2, 8-3], while breaking the rules of symmetry resulted in dissonant or disharmonious sounds [7-6, 8-4]. In addition, not only do both music and mathematics enable creative thinking and breaching conventions, but both disciplines offer opportunities for creativity, freedom, and flexibility [7-1], and encourage new inventions [7-2].

In addition to the theoretical aspect discussed in the above excerpts [7Ma_Ed_Emma and 8Mu_Ed_Sue], the relationship between music and mathematics and its profound impact emerged. The interplay of patterns and rules, their manifestation in geometrical forms and sounds, and the freedom that they grant for creative expression and thinking all directly contribute to the affect, which is manifested by aesthetics, beauty, and evocation of emotion [7-3, 7-5].

Affect perspectives

Beauty and aesthetics

The majority of research participants from both the theoretical and educational domains in music and mathematics cited the significance of beauty and aesthetics as a central theme:

9Ma_Ed - Dan – “Both fields are influenced by aesthetic beauty [9-1]. There’s no mathematics without aesthetic beauty, and there’s no music without aesthetic beauty [9-2]. The whole concept of aesthetics and harmony is related to both mathematics and music [9-3]. It’s very beautiful to see what you hear [9-4]. It’s visual, clear, and beautiful [9-5]. It’s interesting to explore the structure of a polygon with all the diagonals there and to hear the visual representation of all these symmetries [9-6]”.

10Mu_Ed – Sue - “Music and math are neat and beautiful! [10-1]. Patterns and connections [10-2]. If you connect a terraza and a quarta (i.e., musical intervals), you see how it also connects beautifully in geometrical space [10-3]. Mathematics is a beautiful thing [10-4]. I don’t understand it exactly [10-5] … But maybe if you find out the patterns that exists, it’s probably beautiful and they’ll understand the aesthetics better [10-6]”.

When prompted to reflect on the intersections between music and mathematics, the notion of affect arose unanimously among all the interviewees. The educational experts talked about beauty and aesthetics as tools for thinking about and understanding the disciplines. In the above excerpts [9Ma_Ed - Dan and 10Mu_Ed – Sue], the educational experts discussed the implementation of aesthetics and beauty within each discipline [9-2, 10-1, 10-4], as well as the interconnection between these disciplines [9-1, 9-3]. In addition, they cited the audio-visual connection, aforementioned in Emma’s and Lisa’s responses, which they described as giving rise to this beauty in the musical and mathematical space [9-4, 9-5, 10-3].

The theory experts approached the question of the origin of beauty and aesthetics from another aspect, citing the structural context:

11Ma_Th - Adam – “What makes beauty in music? The structure [11-1]. We cannot interpret it. You can hear a Beethoven symphony endless time. Why? Because you never get to the bottom of the guy’s mind [11-2]! That’s the complexity, that you unconsciously grasp the structure” [11-3].

12Mu_Th - Lisa – “When Beethoven writes a symphony in a way that is very regulated on the one hand and on the other hand finds his personal expression, it’s beautiful” [12-1].

Adam and Lisa argue that beauty rests on musical or mathematical form and structure [11-1, 11-3, 10-2]. In the above quotes [11Ma_Th – Adam, and 12Mu_Th – Lisa] they cited how both music and mathematics possess distinct boundaries that can be transcended or breached, and thus are difficult to understand [11-2, 11-3, 12-1, 10-5, 10-6]. In many cases, it is possible to discover the beauty and aesthetics in music and mathematics, which is accompanied by wonder and often evokes diverse emotions.

Discovery, wonder, and emotions

Beyond the sense of beauty and aesthetics associated with the musical or mathematical structure, experts from all fields linked music and mathematics to the sense of wonder when discovering a musical or mathematical phenomenon.

13Ma_Ed - Dan – “What is beauty? That’s a good question. [It’s] when we say ‘wow’ about some idea, or about something that was not clear to us and was not familiar [13-1], and suddenly it worked out so well [13-2]. I believe that both in music and in mathematics, there are such wonderful moments of discovery [13-3]”.

14Mu_Ed - Mika – “I’ve had cases where I derive aesthetic pleasure from musical structures [14-1]. Suddenly there’s a harmonious movement, and you say ‘wow, what beauty [14-2]’, and you get excited [14-3] …. I wish math teachers would try to find moments for students to experience the aesthetics of math [14-4]”.

15Ma_Ed - Emma – “The spiritual connection in my view is the main connection [15-1]. Mathematics was born, created, and developed thanks to the human spirit [15-2]. Owing to the people, the minds [15-3]. Objects with symmetrical and visually appealing qualities, such as an intricate polygon enclosed within a circle, possess an inherent pleasantness when translated into sound [15-4]”.

16Ma_Th - Adam – “A mathematical idea and a poem can affect us in the same way [16-1]. It’s a wonderful thing about music as well [16-2]. I can know that Brahms was a genius without understanding what he did [16-3]. One perceives it unconsciously as in poetry [16-4]”.

From an affect perspective, experts from all fields acknowledged the profound aesthetic connection shared by both music and mathematics. In the above excerpts [13Ma_Ed – Dan, 14Mu_Ed – Mika, 15Ma_Ed – Emma and 16Ma_Th – Adam], the participants stressed the sense of wonder [13-1,14-2, 16-2, 16-3, 16-4] and emotional experience when encountering the inherent beauty within these disciplines. Dan mentions the moment of discovery [13-2, 13-3] that evokes powerful emotions, such as pleasure or enthusiasm, and a deep appreciation for the interplay between music and mathematics [14-1, 14-3, 15-4, 16-1]. Moreover, the experts addressed the spiritual as the main connection between the disciplines [15-1]. From an affect perspective, Emma said that mathematics was born, created, and developed owing to the humans spirit and mind [15-1, 15-2, 15-3], which enable us to appreciate beauty and aesthetics, leading us to discover and be deeply moved by moments of wonder. Finally, Mika, with a music education expert’s view, mentioned her wish that mathematic instruction would enable students to learn about the aesthetics of math [14-4].

Learning opportunities

Integrating mathematics and music into various disciplines

As aforementioned, the affect perspectives inherent in music and mathematics have the power to evoke emotions, thereby facilitating deep learning experiences. Moreover, this emotional spark serves as a catalyst for meaningful multidisciplinary learning, opening up new paths for wholistic and impactful learning.

17Mu_Th – Jack - “Music doesn’t develop in a vacuum. It’s connected to nature in a wholistic way [17-1]. I would combine music and art, music and literature, and philosophy to expand thinking [17-2]. Anything that opens the mind to new things is welcome [17-3], but you have to remember that it can also scare students [17-4]”.

18Mu_Ed - Mika – “In my opinion, there’s great value in multidisciplinary learning in schools [18-1]. It can broaden students’ horizons by encouraging them to explore phenomena or objects from various perspectives and angles [18-2]. I built a lesson plan once on the subject of longing, related to infinity and the path of infinity, i.e., does a number represent what it is? And when you reach the destination? [18-3] … the entire issue of intervals and difference also relates to longing and musical, physical, and emotional distance [18-4].

19Ma_Th – Tom – “Sound is a physical phenomenon, and as we know, physics is a field where the main tool is mathematics, and there is mathematics at a very deep level that can analyze sounds [19-1], for example how musical instruments work, how waves propagate… architectural acoustics with mathematical analyses… scales, intervals, regularities [19-2]. Sound is cyclical, a wave that repeats. And it happens so fast – thousands of vibrations per second – so you don’t hear each one separately, but the pitch is the frequency [19-3]”.

From a learning opportunities perspective, most of the participants supported multidisciplinary learning in schools [17-2, 17-3, 18-1]. In the above excerpts [17Mu_Th – Jack, 18Mu_Ed – Mika and 19Ma_Th – Tom], they engaged in talk regarding the intersections of music, mathematics, and other subjects such as literature, philosophy, art, or science, and cited their potential for broadening horizons, skills, and knowledge [17-1, 18-2]. Tom expanded in his talk upon the connection between music, mathematics, and acoustics [19-1]. Beyond the physical properties of sound, he described how combining the disciplines can contribute to our knowledge of how musical instruments work, how waves propagate, etc. [19-2, 19-3]. Mika also linked a concept such as infinity to both fields, and connected it to numbers and emotions (i.e., longing) [18-3, 18-4]. This is an excellent example of the connection of music and mathematics in a theory perspective and from an affect perspective implemented in an integrated curriculum.

Only one mathematical theory expert expressed reservations about multidisciplinary learning:

20Ma_Th – Adam – I don’t think it’s necessary to combine the fields [20-1]. Mathematics is mathematics and music is music. You need to understand the Pythagorean theorem; you won’t learn something that will teach you the Pythagorean theorem [21-2]. But yes, in my opinion, music should be introduced into schools in general. It is important. It will make us better people [21-3].

From his own experience, Adam describes his resistance to the combining of music and mathematics in school [20Ma_Th – Adam]. He voiced concerns that the fusion of subjects could lead to disruption [21-2], and consequently, he believes that there is no place for such an approach [21-1]. Another professor, a musician theorist, expressed earlier that while multidisciplinary learning is welcome, it might potentially intimidate or frighten the students [17-4]. Nonetheless, Adam concluded with the notion that music is an important subject to be learned, separately, in order to “become better people” [21-3].

Music as a tool for learning mathematics

A common thread was identified only among the music education experts, who drew a connection between music and mathematics vis-a-vis teaching mathematics through the use of musical tools.

21Mu_Ed - Mika – “Only when they started working with me on rhythms, did I start to understand fractions [21-1]”.

22Mu_Ed - Sue – “Fractions and rhythms. How can one explain that twice the speed can end up reaching the same finish line as a slower speed? Division of time [22-1]. Rhythm exercises in general are a very successful way to explain topics in mathematics [22-2]. Pythagoras, the harmonic smith, divided the octave into intervals that are smaller and smaller, and the distances are mathematical. So you see the division of the musical space, which is very similar to a geometric space [22-3]”.

The music education experts recognized the potential of combining music and mathematics to enhance students’ comprehension of fractions. In the above excerpts [21Mu_Ed – Mika and 21Mu_Ed – Sue], Sue shared successful experiences in her work with children, using music as a tool to facilitate the understanding of concepts such as rhythm, division of time, and simple fractions [22-1, 22-2, 22-3]. According to Sue, there is a potential in utilizing music as a vehicle for learning fractions. In addition, Mika talked about her personal experience of understanding fractions by rhythm [21-1]. Note that except for the music education experts, none of the other participants referred to this issue.

Mathematics as a key tool for music analysis and creation

The final common thread was identified only among the music theory experts, who described mathematics as a key tool for music analysis and creation.

23Mu_Th – Jack – “I compose with a fractal method that refers to durations, proportions, and structure, and thus the micro and macro are the same as in the fractal [23-1] … I’ve been exploring this in my musical works for 20 years [23-2]. It’s a whole world, because there’s no limit to how many things you can do with it [23-3]. I wrote dozens of works using this method [23-4]”.

24Mu_Th – Zak – “Mathematics is a tool for composition [24-1]. Counterpoint, for example, is a completely mathematical matter [24-2]. Mathematical thinking is always integrated into music [24-3] … Analyzing musical works is done logically [24-4] … it reminds me that I wrote works with geometric thinking, like playing, increasing, decreasing, rotating, reflecting [24-5]”.

25Ma_Th – Adam – “Musicians think like mathematicians [25-1]. To write music or understand music, you need to understand mathematical rules and understand abstract thinking [25-2].

The theoreticians in both disciplines discussed the role of mathematical thinking in understanding and composing music. In the above excepts [23Mu_Th – Jack, 24Mu_Th – Zak and 25Ma_Th - Adam], Jack and Zak shared their own composition process resting on mathematical concepts [23-1, 24-5]. Zak cited the strong connection between composing and analyzing music using mathematical thinking and logic [24-2, 24-3, 24-4], and stated that mathematics is a tool for composing music [24-1]. Jack delved deeper into this topic, sharing his two-decade journey in composition, marked by the application of mathematical thinking [23-2, 23-3], a journey that has yielded a diverse body of musical work [23-4]. Note that only the theoreticians in both disciplines referred to this issue. In addition, Adam, a mathematical theorist who is not a composer and has no musical expertise, corroborated the others’ claims and drew a direct line between composition and mathematical thinking and understanding [25-1, 25-2].

Discussion

This paper seeks to characterize mathematical and musical experts’ perceptions of the connection between music and mathematics. Interviews with musicians, mathematicians, and educators in both disciplines were analyzed. To illustrate the main findings of this study, we chose several examples, mostly from the material generated by four particular experts (Dan, Mika, Adam, and Lisa), and added other examples to help us draw a more complete picture of experts’ perceptions of the connection between music and mathematics. The paper focuses on theory, affect, and learning opportunities as main indicators of this connection. We analyzed the experts’ discourse in light of these three categories. Based on this analysis, we devised specific characteristics of the connection between music and mathematics that reflect the experts’ perceptions. In this section, we summarize our findings, propose a model of the connection between music and mathematics, and discuss this model.

We found that experts’ perceptions of the connection between music and mathematics consist of seven main categories: (1) abstract language and structure, i.e., perceiving the core foundational structure on one hand, and their pedagogical perspective on the other, (2) freedom and creative thinking, i.e., thinking outside the box and breaching conventions, (3) beauty and aesthetics, i.e., connecting the musical and mathematical space aesthetically and structurally on one hand, and as tools for thinking and understanding on the other hand, (4) discovery, wonder, and emotions, i.e., citing the added value inherent in each discipline in expressive aspects, (5) integrating mathematics and music into various disciplines, i.e., combining the shared areas from each discipline into wholistic learning, (6) music as a tool for learning mathematics, i.e., using music to enhance mathematics, and finally (7) mathematics as a tool for music analysis and creation, i.e., using mathematics to analyze and create music. Figure 2 presents a word cloud highlighting the main words used by all participants. (Fig. 2) Fig. 3 presents a word cloud showing the main words used by different types of experts: theorists and educators from various disciplines. (Fig. 3).

Fig. 2: Word Cloud of Participants' Responses.
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The word cloud emphasizes the most frequently used words by all participants, illustrating key themes and terms.

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Fig. 3: Word Clouds by Expert Group.
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The word clouds illustrate the main words used by different types of experts: A. Mathematicians, B. Math Educators, C. Musicians, D. Music Educators, highlighting unique and common themes within each group.

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We found that musicians and mathematicians (theoreticians) recognize the connection between the two disciplines through concepts such as abstract languages and structure as core foundation. The educators in both disciplines referred to language and structure from a pedagogical perspective. The recognition of structure and language fosters an understanding of the beauty and aesthetics inherent in these domains in a structural context for the theorists, or as tools for thinking and understanding for the educators, cultivating a profound sense of exploration and wonder, while also enabling creative thinking. The conceptual combination of these four components leads to the possibility of integrating mathematics and music into various disciplines, using music as a tool for learning mathematics or using mathematics as a tool for music analysis and creation. (Fig. 4).

Fig. 4: Conceptual Model of Connections Between Mathematics and Music.
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This model depicts experts' conceptions of the interrelationships between mathematics and music, highlighting key connections and themes identified in the study.

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Two main issues that emerged unanimously among participants – both theorists and educators from the disciplines of music and mathematics through differing prisms – were abstract languages and structure and beauty and aesthetics. Similar to numerous researchers (Papadopoulos 2002; Douthett 2008; Johnson 2008), both the musicians and mathematicians discussed the historical dimension of representing music through numerical elements and expressing mathematical concepts in music. Compared to them, educational experts in both disciplines cited these languages’ shared attributes in terms of facilitating communication and conveying messages. All participants related the form and structure component in both disciplines to beauty and aesthetics. Whereas discussions among musicians and mathematicians delved into the concepts of beauty and aesthetics stemming from structured arrangements and innovative rule-breaking, educational experts from both disciplines underscored the pedagogical significance of audio-visual associations with beauty and aesthetics. They all expressed a strong urge to nurture an appreciation for these aspects in their students. These findings corroborate the research conclusions of Beijaard et al. (2000) concerning the self-identity of educators. Therein, they highlighted the shared aspects of enriching subject expertise (in terms of form and structure), devising engaging didactical approaches (including visual-auditory integration), and fostering pedagogic perspectives that facilitate the comprehension of beauty and aesthetics in both disciplines.

Two other topics that arose among the majority of participants to varying degrees were freedom and creative thinking and discovery, wonder, and emotions. In contrast to the two previous topics, the discussion thereon was not contingent upon the discipline of either music or mathematics; rather, it was shaped by theoretical or educational perspectives. Many educational experts in both disciplines cited the sense of wonder and the profound emotional journey that accompanies the experience of teaching music or mathematics, where the inherent beauty of these subjects is discovered. Liao (2016) and Root-Bernstein (2015) claimed that the integration of several disciplines brings about numerous benefits, including fostering innovation, motivation, teamwork, creativity, and actively engaging students in the learning process. From the perspective of experts in music and mathematics education, the process of learning, which encompasses understanding structural aspects on the one hand and recognizing aesthetic elements on the other, demands the cultivation of a mindset centered around exploration, discovery, and appreciation among both educators and students in these fields. Discovering the wonder inherent in both disciplines enables freedom and develops creative thinking, both significant 21st-century skills (OECD, 2019).

It is interesting that musical theorists, unlike mathematical theorists, stressed freedom and creative thinking. Most of them talked about the structural connection between music and mathematics particularly in the realm of composition, citing the ability to generate innovative musical compositions through the deliberate utilization of structured patterns. The structure and format provide a solid foundation for planning musical pieces. Therein, innovative and authentic content can be incorporated, thereby enabling substantial creative liberty. Note also that while theoretical musicians emphasized this matter, theoretical mathematicians chose not to engage in its discussion.

The examination of the four categories in this chapter (i.e., abstract languages and structure, beauty and aesthetics, discovery, wonder, and emotions, and freedom and creative thinking) paves the way to several learning opportunities. Educational experts extensively mentioned the concept of integrating mathematics and music into various disciplines, while theoreticians mentioned it to a lesser extent. Educational experts from both disciplines cited the power of wholistic pedagogy in education. They underscored this approach to education, wherein one subject can be comprehended from various perspectives. Thus, according to the experts in this study, music and mathematics can be taught along with science, literature, philosophy, or art. This kind of broad connection combines the structure of the languages, the aesthetics and beauty therein, the process of discovering the wonder of the wholistic connection, and consequently eliciting the liberty of creative thinking from each discipline.

The concept of combining music and mathematics sparked the potential for music educators to employ music as a teaching tool for mathematics, and to utilize mathematical thinking for the analysis and composition of music. Music educators acknowledged the capacity to synergize music and mathematics to enrich students’ grasp of fractions. They reported productive instances from their interactions with children when utilizing music as a tool to simplify comprehension of concepts like rhythm, temporal division, and basic fractions. While this integrated approach to learning simple fraction concepts via rhythm corroborates previous studies (Azaryahu et al. 2020; Courey et al. 2012), the educational challenge is to provide interesting and stimulating integrated lessons that could lead students to discover the wonder and beauty of these disciplines. Finally, theoretical musicians and mathematicians cited mathematics’ role in composing and analyzing music. The musicians recounted their personal experiences in employing mathematical tools and logic in the process of music composition.

Conclusions

In conclusion, when examining the intriguing relationship between music and mathematics from the perspectives of theoreticians and educators, we found that while theorists engage directly with issues related to structure, creative thinking, beauty, and the sense of wonder in discovering phenomena; educators leverage these shared aspects to enhance students’ comprehension, provide experiential learning opportunities, and deepen students’ theoretical knowledge.

The conclusions reported herein should be considered within the limitations of the study. The current study was limited by a relatively small sample size. Future research should involve a larger population of mathematicians, musicians, and education experts in both fields, and employ mixed methods to thoroughly test the model.

This study opened new doors for future research wherein utilization of experts’ insights to craft integrated study modules of music and mathematics can be explored, a pursuit that carries substantial significance. We hope to have shed light on the pivotal facets of the unique interrelation between these disciplines, a revelation poised to catalyze deep and fascinating integrated learning inspection in the times ahead.