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In fact, people’s research and understanding of the connection between mathematics and music can be said to have a long history. This can be traced back to the sixth century BC, when Pida The Gorathians used ratios to connect mathematics and music [1]. They not only realized that the sound produced by the plucked strings was closely related to the length of the strings, thus discovering the relationship between harmony and integers, and It was also discovered that harmonics are produced by equally taut strings whose lengths are in integer ratios. Thus, the Pythagorean Scale and tuning theory were born, and they occupied a dominant position in the Western music world. Although Toller C. Ptolemy (approximately 100-165 years) reformed the shortcomings of the Pythagorean scale and came up with a more ideal just scale (the Just Scale) and the corresponding tuning theory, but Pythagoras The dominance of Lars scale and tuning theory was not completely shaken until the emergence of the tempered Scale and the corresponding tuning theory. In our country, the earliest complete temperament theory produced was the law of thirds. , which was described in "Guanzi. Diyuan Pian" and "Lu Shi Chunqiu. Music Pian" respectively in the middle of the Spring and Autumn Period; Zhu Zai (1536 - 1610) of the Ming Dynasty wrote about the Twelve Equal Temperament in his music work "New Theory of Rhythm" The calculation method of is summarized, and the theory of twelve equal laws is discussed in "Lulu Jingyi? Nei Chapter", and the calculation of twelve equal laws is very accurate, which is exactly the same as the current twelve equal laws. This is in This is the first time in the world. It can be seen that in ancient times, the development of music was closely linked to mathematics. From then to now, with the continuous development of mathematics and music, people's understanding of the relationship between them and The understanding is constantly deepening. Rational mathematics flashes everywhere in the music of feeling. The writing of music scores is inseparable from mathematics.

Look at the keyboard of the piano, the king of musical instruments. It also happens to be It is related to the Fibonacci sequence. We know that on the piano keyboard, from one C key to the next C key is an octave in music (Figure 1). There are 13 keys, 8 There are 3 white keys and 5 black keys, and the 5 black keys are divided into 2 groups, one group has 2 black keys, and one group has 3 black keys. 2, 3, 5, 8, and 13 happen to be the famous Fibonacci The first few numbers in the geometric sequence.

If the appearance of Fibonacci numbers on piano keys is a coincidence, then the appearance of geometric sequence in music is by no means accidental: 1 , 2, 3, 4, 5, 6, 7, i and other musical scales are specified by the geometric sequence. Looking at Figure 1 again, it is obvious that this octave interval is divided into 12 semitones by the black keys and the white keys, and we know the following The number of vibrations (i.e. frequency) of a C key that makes a musical sound is twice the number of vibrations of the first C key. Because it is divided by 2, this division is made according to a geometric sequence. We can easily find the division ratio x, Obviously x satisfies x12= 2. Solving this equation shows that x is an irrational number, about 1106. So we say that the pitch of a certain semitone is 1106 times the pitch of that note, and the pitch of the whole tone is the pitch of that note 11062 times. In fact, the same geometric sequence also exists in the guitar[3].

Mathematical transformation in music.

There is translation transformation in mathematics, and in music Is there also a translation transformation? We can find the answer through two music measures [2]. Obviously, we can translate the notes in the first measure to the second measure, and translation in music appears. This In fact, it is repetition in music. If the two syllables are moved to the rectangular coordinate system, it will appear as Figure 3. Obviously, this is exactly the translation in mathematics. We know that the purpose of composers to create musical works is to express themselves vividly One's own inner emotions, but the expression of inner emotions is expressed through the entire music, and is sublimated at the theme, and the theme of music sometimes appears repeatedly in some form. For example, Figure 4 is the Western music When the Saints The theme of GoMarching In[2]. Obviously, the theme of this piece of music can be

It can be seen as obtained by translation.

If we take an appropriate horizontal line in the staff as the time axis (horizontal axis x), and the straight line perpendicular to the time axis as the pitch axis (vertical axis y ), then we have established a time-pitch plane rectangular coordinate system in the staff. Therefore, a series of repetitions or translations in Figure 4 can be approximately represented by functions [2], as shown in Figure 5. where x is time and y is pitch. Of course, we can also use functions to approximately represent the two syllables in Figure 2 in the plane rectangular coordinate system of time and pitch.

Here we need Mentioning a famous mathematician in the 19th century, he was Joseph Fourier. It was his efforts that brought people's understanding of the properties of musical sounds to the pinnacle. He proved that all musical sounds, Whether it is instrumental music or vocal music, it can be expressed and described by mathematical formulas, and it is proved that these mathematical formulas are the sum of simple periodic sine functions [1].

Translation transformations are not the only ones that appear in music. There may be other transformations and their combinations, such as reflection transformation, etc. The two syllables in Figure 6 are reflection transformations in music [2]. If we still think about it from a mathematical perspective, put these notes into the coordinate system , then its performance in mathematics is our common reflection transformation, as shown in Figure 7. Similarly, we can also approximately represent these two syllables as functions in the time-pitch rectangular coordinate system.

Through the above analysis, it can be seen that a piece of music may be the result of various mathematical transformations on some basic pieces of music.

Mathematics in nature music.

大 The connection between music in nature and mathematics is even more magical and is usually not known to everyone. For example [2], cricket chirping can be said to be the music of nature. However, the frequency of cricket chirping has a great relationship with the temperature. We can use a Expressed as a linear function: C = 4 t – 160.

Among them, C represents the number of times the cricket chirps per minute, and t represents the temperature. According to this formula, as long as we know the number of times the cricket chirps per minute, we can know the temperature of the weather without a thermometer!

In rational mathematics There is also perceptual music.

Starting from a trigonometric function image, we only need to segment it appropriately to form appropriate sections, and select appropriate points on the curve as the location of the notes. Then we can compose a piece of music. It can be seen that we can not only use the golden section to compose music like the Hungarian composer Bela Bartók, but we can also compose music based on pure function images. This is mathematics The follow-up work of Joseph Fourier is also the reverse process of his work. The most typical representative is Joseph Schillinger, a professor of mathematics and music at Columbia University in the 1920s, who once wrote the New York Times An undulating business curve was described on graph paper, and then each basic segment of the curve was transformed into a piece of music according to appropriate and harmonious proportions and intervals. Finally, it was played on an instrument, and it turned out that it turned out to be a beautiful tune. , pieces of music that are very similar to Bach’s musical works [2]! The professor even believed that according to a set of criteria, all musical masterpieces can be transformed into mathematical formulas. His student George Gershwin even more He innovated and created a system for composing music using mathematics. It is said that he used such a system to create the famous opera "Porgy and Bess".

So we say, in music The emergence of mathematics and the existence of music in mathematics is not an accident, but a manifestation of the integration of mathematics and music. We know that music plays a string of notes to express people's joys, sorrows and joys or to express their feelings about nature and life. That is, music expresses people's emotions and is a reflection of people's own inner world and feelings about the objective world. Therefore, it is used to describe the objective world, but in a perceptual or more perceptual way. It is carried out in a personal and subjective way. Mathematics describes the world in a rational and abstract way, allowing human beings to have an objective and scientific understanding and knowledge of the world, and through some concise, beautiful and harmonious formulas To express nature. Therefore, it can be said that mathematics and music are both used to describe the world, but the description methods are different, but the ultimate goal is to serve human beings for better survival and development, so there is an inherent connection between them It should be a natural thing.

Since mathematics and music have such a wonderful connection, why not let us immerse ourselves in the beautiful melody of "Butterfly Lovers" or place ourselves in the fields of chirping insects. Why don’t we think about the inner connection between mathematics and music? Why not let us continue to explore their inner connection with confidence in the sound of the pipa or the exciting symphony?

Above, we We have provided some materials on the connection between mathematics and music. How to "process" these materials into the content of "mathematics education"? We raise a few questions for the consideration of textbook writers and teachers working on the front line.

1) How to process and infiltrate such materials into mathematics teaching and mathematics textbooks?

2) Can these materials be compiled into "popular science reports" and be used in extracurricular activities to promote music and mathematics hobbies? Report, investigate, understand, and think about the impact of such reports on students and students’ reactions to such reports.

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