You will find the wonders of mathematics on the strings. Strings of different lengths produce different wonderful sounds.

In this round of curriculum reform, "mathematics and culture" has become one of the most concerning issues for mathematics and mathematics educators. In fact, for a long time, many mathematics and mathematics education Workers are already thinking and studying this issue. In the upcoming "High School Mathematics Curriculum Standards", there is a clear requirement to integrate "mathematical culture" throughout the high school curriculum. For a series of theoretical issues involving "mathematical culture", it should be recognized It has not been discussed clearly, and there is still a lot of debate. For example, many scholars have doubts about the term "mathematical culture". We think this is normal. We recommend that the research on these issues be conducted from two aspects at the same time. On the one hand, we conduct theoretical research; on the other hand, we actively develop some examples, cases, and lessons of "mathematics and culture" to explore how to infiltrate "mathematics culture" into classroom teaching and how to make students learn from "mathematics culture" To improve mathematical literacy, on this basis, we can do some theoretical thinking, from practice to theory, and do some empirical research. The following is an example we provide - mathematics and music can also be regarded as a material, I hope Teachers working on the front line can further develop such materials and enter them into classrooms or extracurricular activities in different forms. We also hope that more people will develop such materials and hope that these materials can appear in teaching materials.

In the process of developing mathematics curriculum standards, we met some experts in the music industry. They told us a lot about the connection between music and mathematics and the application of mathematics in music. They particularly emphasized that in computers Today, with the rapid development of information technology, music and mathematics are more closely related. Mathematics is needed in music theory, music composition, music synthesis, electronic music production, etc. They also told us that in the music industry, there is some mathematical literacy Excellent musicians have made important contributions to the development of music. They and we all hope that students who are interested in music careers can learn mathematics well, because mathematics will play a very important role in future music careers.

The beautiful melody of "Butterfly Lovers", the clanking pipa sound of "Ambush from Flying Daggers", Beethoven's exciting symphony, the chirping of insects in the fields... When immersed in these wonderful music, Have you thought about their close connection with mathematics?

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, but also 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 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 pure temperament 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. , the time was described in "Guanzi. Diyuan Chapter" and "Lu Shi Chunqiu. Music Chapter" 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 book "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 have attached great importance to it.

The understanding and understanding of the relationship between them is also constantly deepening. Rational mathematics flashes everywhere in the music of feeling. The writing of music scores is inseparable from mathematics.

Look at the piano, the king of musical instruments. Keyboard, it also happens to be 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). Among them*** Includes 13 keys, 8 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, 13 They happen to be the first few numbers in the famous Fibonacci sequence.

If the appearance of Fibonacci numbers on the piano keys is a coincidence, then the appearance of the geometric sequence in music It is by no means accidental: 1, 2, 3, 4, 5, 6, 7, i and other musical scales are specified using geometric sequences. Looking at Figure 1 again, it is obvious that this octave interval is divided into 12 by black keys and white keys. A semitone, and we know that the number of vibrations (i.e. frequency) of the next C key is twice the number of vibrations of the first C key. Because it is divided by 2, this division is made according to the geometric sequence. We It is easy to 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 The height is 11062 times the pitch of that note. In fact, the same geometric sequence also exists in the guitar [3].

Mathematical transformations in music.

In mathematics There is a translation transformation. Does it also exist in music? We can find the answer through two music measures [2]. Obviously, we can translate the notes in the first section to the second section, and it will appear This is the translation in music, which is actually the repetition in music. Move the two syllables into the rectangular coordinate system, and it will appear as Figure 3. Obviously, this is exactly the translation in mathematics. We know that composers create musical works The purpose is to express one's inner emotions vividly, 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 theme of the Western music When the Saints Go Marching In[2]. Obviously, the theme of this music can be regarded as obtained through translation.

If we put an appropriate horizontal line in the staff As the time axis (horizontal axis x), the straight line perpendicular to the time axis serves 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 in Figure 4 The repetition or translation can be approximately expressed by a function [2], as shown in Figure 5, where x is time and y is pitch. Of course, we can also use functions in the plane rectangular coordinate system of time and pitch. The two syllables in Figure 2 are approximately represented.

Here we need to mention a famous mathematician in the 19th century, he is Joseph Fourier. His efforts brought people's understanding of the nature of music to its peak. He proved that all music, whether instrumental or vocal, can be expressed and described by mathematical formulas, and proved that these mathematical formulas are simple periodic sine The sum of functions [1].

Not only translation transformations appear in music, but other transformations and their combinations may appear, such as reflection transformations, etc. The two syllables in Figure 6 are reflections in music Transformation [2]. If we still think about it from a mathematical perspective and put these notes into the coordinate system, then its representation in mathematics is our common reflection transformation, as shown in Figure 7. Similarly, we can also - These two syllables are approximately represented by functions in the pitch rectangular coordinate system.

Through the above analysis, it can be seen that a piece of music may be a pair of some

The result of various mathematical transformations of basic pieces.

Mathematics in nature’s music.

The connection between nature’s music 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 linear function to express it: 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 subsequent 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. 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 that 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 understanding 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 where insects chirp. 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? Researchers report, investigate, understand, and think about the impact of such reports on students and students' reactions to such reports.

Music and mathematics have been linked for centuries. During the medieval period, arithmetic, geometry, astronomy, and music were included in the educational curriculum.

Today's new computers are extending this link.

The writing of musical scores is the first significant area where the influence of mathematics on music is demonstrated. On the music manuscript, we see tempo, beat (4/4 beat, 3/4 beat, etc.), whole notes, half notes, quarter notes, eighth notes, sixteenth notes, etc. Determining the number of partial notes in each measure when writing music score is similar to finding a common denominator - notes of different lengths must fit into the measure specified by a certain beat. The composer created music that blended beautifully and effortlessly within the tight structure of the written score. If a completed work is analyzed, it can be seen that each section uses notes of different lengths to form a prescribed number of beats.

In addition to the obvious relationship between mathematics and musical notation, music is also connected to ratios, exponential curves, periodic functions and computer science.

The Pythagoreans (585 BC to 400 BC) were the first to use ratios to connect music and mathematics. They realized that the sound produced by plucked strings was related to the length of the strings, and thus discovered the relationship between harmony and integers. They also discovered that harmonic sounds are produced by equally taut strings whose lengths are in whole-number ratios—in fact every harmonious combination of plucked strings can be expressed as an integer ratio. Increasing the length of the string by an integer ratio produces the entire scale. For example, starting from the string that produces the note C, 16/15 of the length of C gives you B, 6/5 of the length of C gives you A, 4/3 of the length of C gives you G, 3/2 of the length of C gives you F, 8/5 the length of C gives E, 16/9 the length of C gives D, and 2/1 the length of C gives low C.

Have you ever wondered why grand pianos are made the way they are? In fact the shape and structure of many musical instruments are related to various mathematical concepts. Exponential functions and exponential curves are such concepts. The exponential curve is described by an equation of the form y=kx, where k>0. An example is y=2x. Its coordinate diagram is as follows.

Whether it is a string instrument or a wind instrument produced by a column of air, their structure reflects the shape of an exponential curve.

The work of the 19th-century mathematician John Fourier culminated the study of the properties of musical sounds. He demonstrated that all musical sounds—instrumental and vocal—can be described by mathematical formulas that are sums of simple periodic sinusoidal functions. Each sound has three properties, namely pitch, volume and timbre, that distinguish it from other musical sounds.

Fourier's discovery enabled these three properties of sound to be clearly represented graphically. The pitch is related to the frequency of the curve, and the volume and sound quality are related to the amplitude and shape of the periodic function ① respectively.

Without an understanding of the mathematics of music, it is impossible to make progress in the application of computers to music creation and instrument design. Mathematical discoveries, specifically periodic functions, are essential in the modern design of musical instruments and the design of sound-controlled computers. Many instrument makers compare the periodic sound curves of their products to the ideal curves for those instruments. The fidelity of electronic music reproduction is also closely related to the periodic curve. Musicians and mathematicians will continue to play equally important roles in the production and reproduction of music.

The above figure shows the segmented vibration and overall vibration of a string. The longest vibration determines the pitch, and the smaller vibrations create overtones.

① A periodic function is a function that repeats a shape in intervals of equal length.