One day 500 years ago, the ancient Greek philosopher Pythagoras went out for a walk and passed a blacksmith's shop. He found that the sound of striking the iron was more harmonious and pleasant than other blacksmiths. He walked into the workshop, measured the size of the hammer and anvil, and found a rule that the harmony of sound was related to the volume of the sound generator in a certain proportion. Later, he did experiments on strings, and further found that a vibrating string can produce pleasant intervals as long as it is divided in proportion: for example, 1:2 produces octaves, 2:3 produces fifth degrees, 3:4 produces fourth degrees and so on. In this way, Pythagoras discovered the connection between music and mathematics for the first time in the world. He then discovered the qualitative differences of sound (such as length, height, severity, etc.). ) is determined by the difference in the number of pronunciation styles. For thousands of years, studying the relationship between music and mathematics has been a hot topic in the west. From the Pythagorean school in ancient Greece to modern cosmologists and computer scientists, they are all more or less influenced by the concept that the whole universe is harmony and numbers. Kepler, Galileo, Euler, Fourier, Hardy and others all devoted themselves to studying the relationship between music and mathematics. Mathematical geometry and philosophy go hand in hand, permeating all spiritual life and all artistic fields of westerners, which has become a major feature of western art. St. Augustine even left a famous saying that "numbers can also turn the world into music that is connected with our hearts." Modern composers Bartok, Schoenberg, Cage and others all made bold experiments on the combination of music and mathematics. The Greek composer Kaisenakis (1933~) founded "algorithmic music", which uses mathematical methods to replace musical thinking, and the creative process is also a calculus process. The name of the work is similar to a mathematical formula, such as "S+/10-1.080262" for10 musical instrument. Makah developed Stockhausen's idea of "chart music" (music for reading and watching), and made "geometric music" by rotating geometric figures.
Mathematics is a science that studies the quantitative relationship of spatial forms in the real world. It has changed from the science of counting to the science of formal symbol system. The use of symbols makes mathematics highly abstract. On the other hand, music is the art of studying the sound forms in the real world and controlling them. It also uses the symbol system, which is the most abstract art of all arts. Mathematics gives people the feeling of being monotonous, boring and indifferent, while music is rich and interesting, full of feelings and fantasies. On the surface, music and mathematics are "insulated" and irrelevant, but they are not. Leibniz, a famous German philosopher and mathematician, once said: "Music, on its basis, is mathematical; In terms of its appearance, it is intuitive. " Einstein said humorously, "Our world can be composed of notes or mathematical formulas." Mathematics is a permutation and combination with numbers as the basic symbols, which is a quantitative abstraction of things and reveals the internal laws of the objective world through various formulas. Music is an arrangement and combination with notes as the basic symbols, which is an abstraction of natural sounds, and is organized through the grammar associated with these symbols to summarize various activities in our subjective world. It is in this abstraction that music and mathematics are linked, and they both reflect and grasp infinity through finiteness.
Mathematics and music are located at the two extremes of human spirit, and all creative spiritual activities of human beings are carried out in these two opposing ranges, and everything created by human beings in the fields of science and art is distributed between them. Music and mathematics are wonderful flowers in the abstract kingdom. With these two flowers, we can grasp the spiritual wealth created by human civilization. Pythagoras, a Greek philosopher and mathematician known as the father of number theory, said: "Music is sacred and sublime because it reflects the relationship between numbers as the essence of the universe." Where there are numbers in the world, there is beauty. Mathematics, like music and other arts, can arouse people's aesthetic feeling and taste. In the creative activities of mathematicians, there are also aesthetic factors such as emotion, will, belief and hope. Concepts, axioms, theorems, formulas and laws created by mathematicians, like poetry, music, painting, sculpture, drama and movies, can make people intoxicated and enjoy beauty from them.