Interplay between music and mathematics in the eyes of the beholder: focusing on differing types of expertise

The paper investigates long-recognized links between Western music and mathematics dating back to Pythagoras, Plato, and Aristotle. It outlines shared representational languages and symbolic notation, and shows how mathematical concepts (symmetry, patterns, ratio, division) manifest in music (intervals, rhythm, duration, tempo). Historical examples include harmonic intervals determined by small-number ratios (e.g., octave 1:2, fifth 2:3, fourth 3:4) and mathematically inspired compositional techniques (counterpoint, crab canon, palindrome, geometric symmetries). Theories of harmony, rhythm, and musical form reflect regularities akin to mathematical structures. Against this backdrop, the study aims to uncover the interconnections between music and mathematics as perceived by mathematicians, musicians, and teacher educators, emphasizing practical relevance for teacher training and instruction. The research questions ask what main categories of perceptions about the connection exist and how distinctions manifest among musicians, mathematicians, and teacher-education lecturers.

The background synthesizes research on integrating music and mathematics in teaching and learning. Studies with elementary students show positive transfer when explicit parallels are drawn via symbolic notation between musical and mathematical concepts such as patterns, symmetry, and fractions; meta-analytic evidence indicates music integration can significantly enhance mathematics outcomes, especially when music instruction is embedded in math interventions, uses calming/math-related music, and occurs at early grades. Music provides real-world context, boosting meaning, relevance, interest, and motivation in math. The review also examines professional identity of theorists versus educators and pedagogical content knowledge (PCK), highlighting how identity (subject-matter, pedagogical, didactical expertise) and affect influence teaching in both music and mathematics. It discusses emotions’ impact on learning and mathematical thinking, and contrasts theorists’ roles (integrated theoretical frameworks) with educators’ practical, context-driven choices. On integrated instruction, the review notes benefits (innovation, literacy, motivation, teamwork, creativity, problem-solving) but limited implementation due to constraints (teacher self-efficacy, knowledge, resources, training, time) and the need for deep understanding of both disciplines. Teachers often view math as a supporting component in multidisciplinary learning and cite few math areas suitable for integration. The literature supports multidisciplinary teaching yet recognizes significant challenges, especially for sustained, meaningful integration when teachers lack music-theory familiarity.

Design: Qualitative study with a 2×2 design contrasting domain (Mathematics vs Music) by professional role (Theorists vs Educators). Participants: 16 experts from Israeli universities/colleges: 4 mathematicians (theorists), 4 mathematics education lecturers, 4 musicians (theorists), 4 music education lecturers. Recruitment: Voluntary participation. Data collection: The first author conducted semi-structured interviews asking experts to articulate their views on connections between mathematics and music in general and in pedagogical application. Interviews were video- and audio-recorded and fully transcribed; interviewer maintained neutrality and probed for clarity. Data analysis: Guided by grounded theory and content analysis, researchers developed three a priori analytic lenses reflecting professional identities and literature: theory, affect, and learning opportunities. Statements were first categorized into these lenses; inductive coding then derived subcategories within each lens through iterative refinement across participants. Cross-category comparisons identified commonalities and differences. Reliability: Inter-coder validation by the first and third authors achieved approximately 90% agreement; disagreements were resolved through discussion.

- Seven main categories characterized experts’ perceptions: (1) Abstract language and structure; (2) Freedom and creative thinking; (3) Beauty and aesthetics; (4) Discovery, wonder, and emotions; (5) Integrating mathematics and music into various disciplines; (6) Music as a tool for learning mathematics; (7) Mathematics as a tool for music analysis and creation. - Theorists (both mathematicians and musicians) emphasized core structures and numerical representation, often referencing historical contexts (e.g., Pythagoras). Music theorists particularly described mathematics as instrumental for composition and analysis (e.g., fractals, counterpoint, geometric transformations). - Educators in both fields framed structure and language as means of communication and pedagogy, highlighting audiovisual connections and their use for teaching. Music educators uniquely stressed music as a tool to teach mathematics (e.g., using rhythm for fractions and temporal division). - Across all groups, participants valued structure, beauty, and order, yet also underscored freedom and creativity within structured frameworks; symmetry was associated with consonance, rule-breaking with dissonance. - Affect was central: beauty, aesthetics, discovery, wonder, and emotional/spiritual engagement were frequently reported as intrinsic to both disciplines and as drivers of learning. - Most supported multidisciplinary integration (e.g., with literature, philosophy, art, physics/acoustics), while one math theorist cautioned against direct curricular fusion. - Sample: 16 experts (4 per group); intercoder agreement ~90%.

The findings address the research questions by delineating convergences and divergences across expertise types. All experts recognize music and mathematics as abstract, structured languages linked by patterns and numeric representations. Theorists prioritize foundational structure and historical/numerical frameworks; music theorists additionally foreground composition and analytical methods grounded in mathematics, highlighting structured freedom as a creative catalyst. Educators focus on pedagogical affordances—using audiovisual and structural parallels to communicate concepts and engage learners. Beauty and aesthetics emerge from structure and from innovative departures, evoking wonder and emotional resonance that can be leveraged for learning. These components coalesce in a proposed conceptual model: abstract language/structure → beauty and aesthetics → discovery/wonder/emotions → freedom/creative thinking → learning opportunities (integration across disciplines; music-to-teach-math; math-to-analyze/create-music). The model explains differences among groups: educators emphasize pedagogical mediation and student affect; theorists stress formal systems and, for musicians, compositional technique. Resistance to integration appears limited but signals the need for careful design so disciplinary rigor is preserved.

The study characterizes expert conceptions of connections between music and mathematics across theorists and educators. While theorists emphasize structure, abstraction, and (for musicians) mathematically informed composition/analysis, educators leverage these commonalities to enhance comprehension via audiovisual and structural parallels, cultivating aesthetic appreciation, discovery, and creativity. The proposed model organizes these elements into a pathway from structure through aesthetics and affect to creativity and integrative learning opportunities. Limitations include small, context-specific sample size; future research should expand participant pools, use mixed methods to test and refine the model, and translate expert insights into integrated teaching modules for secondary and tertiary education.

- Small sample size (n=16) limits generalizability. - Participants drawn from Israeli institutions may constrain contextual transferability. - Qualitative self-reports from experts may reflect professional identity biases. - No classroom intervention or learning outcome measures; findings describe perceptions rather than causal effects. - Data not publicly available due to privacy, limiting external audit.