Report of mathematics in daily life Group 5
Report
Title: Fourier analysis in musicology Authors: Junlin Hou, Yifan Li, Huangyan Pan Abstract: The study on the relationship between mathematics and music has been an ancient but hot topic in the world. Many scientists have made researches in this realm such as Pythagoras, Kepler, Galileo, but Fourier made the biggest contribution. This report will focus on the Fourier analysis in musicology, including three parts: mathematics and music, the explanation of Fourier analysis in musicology and the process of the application.
1. Mathematics and music When it comes to mathematics, people will think that it is dull and difficult. Compared to mathematics, music seems more colorful and interesting. However, music is linked closely with mathematics. In fact, over the thousands of years, it is a hot project to study the relationship between mathematics and music in the west. Some celebrities had studied the project such as Kepler, Galileo, Fourier, Euler and so on. Leibniz said that music is the arithmetic exercise of the soul. And there are many examples to prove that music is linked closely with mathematics. A Greek composer created Algorithmic music. He took mathematical theory instead of musical thought. While he was calculating, he was also composing. And the name of his composition is similar to the mathematical formula. For example,
2.The explanation of Fourier analysis in musicology ① The function of changing sounds on the keyboard There are some buttons on the keyboard which can be used to change the sound of keyboard into other instruments such as piano, guitar and bass. One simple way to realize this function is to record all kinds of sounds of instruments and hold them in the keyboard. However, can these sounds be composed by computer? What characteristic makes the different sounds of different instruments? ② Musical quality
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In physics, all sounds come from the vibration of objects. Loudness, tone and musical quality are the three main characteristics of sound. Loudness depends on the amplitude of vibration. Tone depends on the frequency of vibration. Musical quality is the main characteristic to distinguish different sounds. As it’s known to all, a sound is a wave in essence. Thus, it can be proved in the experiment that musical quality depends on the wave form of the sound. ③ Fourier analysis The wave form of each sound can be detected by professional equipments. It can be easily observed in the image that the sound wave is a periodic function and its period is 2π. It can be expressed as a certain representation. In 1807, Fourier, a famous mathematician as well as physician, published an essay which raised a controversial idea: Every periodic function can be transformed into the superposition of sine function. Sine function is the simplest periodic function, and it comes from circle. It can be found in many realms in the world. The reason to change square waves or sawtooth waves into sine waves is that sine has an important characteristic: fidelity. A signal which was input in sinusoid will be output in sinusoid as well, and its frequency and wave form will not be changed. There is an image example:
F(t)=4sin(2πt)/π
F(t)=4sin(2πt)/π+4sin(6πt)/3π ……
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F(t)=4sin(2πt)/π+4sin(6πt)/3π+4sin(10πt)/5π+4sin(14πt)/7π+4sin(18πt )/9π+4sin(22πt) /11π+4sin(26πt)/13π And so on, the wave form of the representation: F(t)= 4sin(2πt)/π+4sin(6πt)/3π+……+4sin(2(2n-1)πt)/(2n-1)π+…… will be a square wave. If each coefficient is changed, the wave form will be changed. In the other hand, each wave form has its special representation. By professional equipments, the representation can be calculated from the wave form, and different sound can be imitated by computers.
3.The process of the application As we all know, any vector in a two-dimensional space could be expressed by the basic vectors. Such as: v= c1·v1 +c2·v2 . Any vector in a three-dimensional space could also be expressed by the basic vectors.
Such as: v=c1·v1 +c2·v2+c3·v3. In the same way, any vector in the multi- dimensional space could be expressed by its basic vectors. Such as: v=c1·v1 +c2·v2+c3·v3+…+cn·vn . Assume that v is a vector outside a space. vp is a vector inside the space which is the best approach of v . So we could get (v - vp) · vj = 0. And then we could get the following system of linear equation:
When the basic vectors are , the solution of the equation above is :
So the whole process is that we get a sound, then we get its spectrogram. We analyze the spectrogram , choose part of it and then use the result we get above. Finally, we could get music with the help of programming.
3. Reference: [1]Youhua He
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[2]Wa Bai, Fourier transform. 4. Further reading and study: Solving Applied Mathematical Problems with MATLAB. 5. Note about presentation: It would be better if we put a piece of music or video into our presentation. Materials can be more detailed and attractive. But we have a good division of labor and cooperation. In the future, we will focus more on the collection of materials and pronunciation.
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