Math and Music

Discuss the relationship between mathematics and music/art.In what ways do they complement each other?People more often regard mathematics as an enigma, because of the strictly ration and abstract definition that combine with numbers, formulas and calculations. On the contrary, music is presenting in the daily life simply reflects emotions and feelings that people can interact in. Mathematics is a product of highly abstraction among objective things and logical thinking while music is a spiritual and emotional aspect of the sound externalization. Both mathematic and music play a significant role in the society that cannot be ignored in a different way. However, the inside relationship between mathematic and music is more mysterious than what is commonly perceived. Numerous researches and analyses of how mathematics connected music or in what extent the music was affected by mathematics had been in progress for a long time even for several centuries. Music has some mathematical feature and characteristics in a potential way, and fortunately, the musical and aesthetic thoughts can also be found in mathematics.Mathematics and music connect with each other in many aspects, for example, mathematics bound up closely with music in tones, rhythm and beats, in melodies and harmonies, and in musical form etc. Back to the time of ancient Greeks, music was reckoned as a strictly mathematical and scientific discipline. In the research of mathematic and music, the Pythagorean ratio rules were used to explain the relationship between mathematics and tone or tuning in music (Alexander et al., 2007). Alexander and other researchers (2007, p.1521) found that:―Millennia ago Pythagoras noted a simple but remarkably powerful rule forthe aesthetics of tone combinations: pairs of tones – intervals – with simpleratios such as an octave (ratio 2 : 1) or a fifth (ratio 3 : 2) were pleasantsounding (consonant), whereas intervals with complex ratios such as themajor seventh (ratio 243 : 128) were harsh (dissonant). ‖Considering the Pythagorean ratio rules, the foundation of Western classical music, it specifically explains that musical intervals can be defined as ratios of the sounding frequencies of the two pitches. It suggests that the tone and tuning in the composition can be described as a specific and abstract concept in mathematical way. Althoughordinary people cannot notice the distinction among different ratio in the tone just by hearing music, people will feel it is unpleasant if they were given a piece of music with the complex ratios. It shows the tone and tuning in music will be affected and changed by transforming the mathematical ratios. By using the simple ratios, the tone and tuning in a composition or a piece of music will leave a pleasant and comfortable impression in audiences’ mind.With the growth and maturation of the musicality and performance, the traditional mathematical model lost its important and irreplaceable role in the composition. It seems that ―the music released itself from the mathematical domination‖ (Garland and Kahm, 1995, p.36). However, the music does not only consist of the tone, tuning, notes and harmony. More pivotal and indispensable aspects that will always tightly and deeply relate to the mathematics were rhythm and melody. In this aspect the Fibonacci number and Golden Ratio were omnipresent in majority of compositions and were widely taken into application in music. Researches on timing of compositions show that―the climax of songs is often found at roughly the phi point (61.8%) of the song‖(Olsen S, 2006). While musician creating or composing, the melody should be attractive and rhythmic that makes the audiences feel pleasant and happy. The rhythmic, well-balanced and dynamic melodies always follow the Fibonacci sequence and Golden Ration. The basis of the 3rd movement in Bela Bartok’s composition named ―Sonata for Two Pianos and Percussion” was formed in the counting of semitones by Fibonacci intervals. Another example is that in a large number of Mozart’s piano sonata, almost all the rhythm and melodies in each section were developed and divided in the rule of Golden Ratio. Not only in the classical music but also in pop and rock music that was composed in recent years the mathematical rules appeared. Such as a rock band named ―Tool‖ created a song was known as ―Lateralus‖, that was tightly related to the Fibonacci numbers. The statistics in the researches (Cheryl, 2013) show that even more than 4% of this melody entitled ―Lateralus‖ was totally relied on the Fibonacci sequence. Not just simple disciplines or regulations formed by numbers and formulas, mathematical rules and laws are tools even essential part in conforming the structure in a melody.It is obvious that mathematics has profound impact on the compositions in both superficial and fundamental ways, and it is also applied in the design of instruments. The delicate instruments was produced strictly after being calculated and designed bysorts of mathematical methods. Focus on the king of the instruments—piano. On the keyboards of the piano from a C key to the next C key is an octave, in which are thirteen keys with eight white notes and five black notes. The five black notes were divided in two groups, which one has two black keys and another has three black keys. According to the Fibonacci sequence, the number on the piano keyboard of a span which called an octave is the same with what in the beginning of this sequence. Another example for the application in instruments by mathematics is the structure of the violin. One of the most exquisite and valuable violins in the world with the delicate tonal and harmonic voice was designed by Stradivarius used the Golden Ratio. Other than piano and violin was closely linked with mathematics the design of guitar was contained the golden section as well.On one hand the music was rooted in the application of mathematics, on the other hand the music also stimulate the mathematical thinking in solving problems.The rhyme, melody and harmony in the compositions as well as the sound and voice in the music all related to the ability and function of processing information in humans’ brain. While playing a piano, the complicated and flexible skills should be combined with particular form by using one’s brain and hands. It will develop one’s spatial thinking and assist one’s inferential capability, so that the mathematical thinking will be exploited. It was demonstrated by both Henle (1996, p. 19) and Reid (1995), the music and mathematics should not be divided in two parts in education, while the musical style can stimulate the learning of mathematics. Henle found that(1996, p.28):―Students should make mathematics together (as in fact professionalmathematicians do), not alone. And finally, students should performmathematics; they should sing mathematics and dancemathematics.‖The proper method to understand and study the real means of the mathematics is regarded it ―not divine, but mortal, and not law, but taste‖(Henle, 1996, p. 28)by using the musical thinking.Considering the relationship between the mathematics and music, the way they providing assistance and complement with each other should be one of the mostsignificant parts to be analyzed. The mathematics have affected the tone or tuning in music in a long time ago by the traditional rules such as Pythagorean ratio rules. With the time passed the relationship between mathematics and music become more obvious and indivisible. Especially the application that used in both melodies and instruments that is related to Fibonacci sequence and Golden Ratio. In addition, music is an unconscious behavior with numbers, formulas and calculation in a hidden way, so that the music which in return has the feedback to the mathematics. The musical style in education will make mathematical problems to be more easily understood and solved. It is important to show the links and relationships between mathematics and music that existed in a long duration, and is also necessary to study the development and complement between each other in double sides.(1298 words)Reference:Archibald, R. C. (1924). Mathematicians and music. American Mathematical Monthly, 1-25Beer, M. (2007). Mathematics and Music: Relating Science to Arts.Cheryl, P.(2013). Mathematics and music. Journal of the Association of Teachers of Mathematics, Mathematics Teaching (233), 24-26.Foss, A. H., Altschuler, E. L., & James, K. H. (2007). Neural correlates of the Pythagorean ratio rules. Neuroreport, 18(15), 1521-1525.Henle, J. (1996) Classical mathematics. The American Mathematical Monthly 103(1): 18–29.Kramer, J. (1973). The Fibonacci series in twentieth-century music. Journal of Music Theory, 110-148.Olsen, S. (2006). The golden section: nature's greatest secret. Bloomsbury Publishing USA.Stakhov, A. (1998). The golden section and modern harmony mathematics. In Applications of Fibonacci numbers (pp. 393-399). Springer Netherlands。