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

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

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_EdMika – “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].

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