In the process of formulating the mathematics curriculum standards, we met some experts in the music industry. They told us a lot about the relationship between music and mathematics and the application of mathematics in music. They particularly emphasized that with the rapid development of computer and information technology, music and mathematics are more closely linked, and mathematics is needed for music theory, composition, music synthesis and electronic music production. They also told us that in the music industry, some musicians with good mathematical literacy have made important contributions to the development of music. They and we all hope that students who are interested in music can learn math well, because math will play a very important role in the future music career.
The beautiful melody of "butterfly lovers", the clank of pipa, Beethoven's exciting symphony, the chirping of insects in the field ... When immersed in these wonderful music, do you think they are closely related to mathematics?
In fact, people's research and understanding of the relationship between mathematics and music can be said to have a long history. This can be traced back to the 6th century BC, and the Pythagorean school used the ratio [1] to connect mathematics with music. They not only realized that the sound produced by plucking a string is closely related to the length of the string, but also discovered the relationship between harmony and integers. In addition, it is found that the harmonics are emitted by the same taut string with an integer ratio. As a result, Pythagorean scale and modulation theory were born and occupied a dominant position in western music circles. Although C. Ptolemy (about 100- 165) reformed the shortcomings of Pythagoras scale and got the ideal pure scale and the corresponding modulation theory, it was not until the appearance of tempered scale and the corresponding modulation theory that the dominant position of Pythagoras scale and modulation theory was completely shaken. In China, the first complete legal theory is the three-point profit and loss method, which is about the melody of Guan Yuan's essays and Lu Chunqiu's essays in the middle of the Spring and Autumn Period. In the Ming Dynasty, Zhu Zai (1536- 16 10) outlined the calculation method of the twelve-average law in his music work "New Theory of Law". This is the first time in the world to discuss the theory of the law of twelve averages, and accurately calculate the law of twelve averages, which is exactly the same as today's law of twelve averages. Therefore, in ancient times, the development of music was closely related to mathematics. Since then, with the continuous development of mathematics and music, people have deepened their knowledge and understanding of the relationship between them. Rational mathematics flashes everywhere in the music of feeling. The writing of the score is far from perfect.
Look at the piano keyboard, the king of musical instruments, which also happens to be related to Fibonacci sequence. As we know, on the piano keyboard, from one key C to the next key C is an octave in music (as shown in figure 1). Among them, * * * contains 13 bond, 8 white bonds, 5 black keys and 5 black keys, which are divided into 2 groups.
If it is a coincidence that Fibonacci numbers appear on piano keys, it is by no means a coincidence that geometric series appear in music: 1, 2, 3, 4, 5, 6, 7, I and other scales are all defined by geometric series. Looking at the picture 1, it is obvious that this octave is divided into 12 semitones by black keys and white keys. Moreover, we know that the vibration frequency of the next C key is twice that of the first C key, because it is divided by 2, so this division is done according to geometric series. We can easily find the frequency division ratio x, which obviously satisfies x 12= 2. By solving this equation, we can get that X is an irrational number. About 1 106. So we say that the pitch of a semitone is 1 106 times that of that sound, and the pitch of the whole sound is 1 1062 times that of that sound. In fact, there is the same geometric series in the guitar [3].
Mathematical transformation in music.
Does mathematics have translation transformation, and does music have translation transformation? We can find the answer through two music sections [2]. Obviously, the notes in the first bar can be translated into the second bar, and there will be translation in music, which is actually repetition in music. Move the two syllables into a rectangular coordinate system, as shown in Figure 3. Obviously, this is a translation in mathematics. We know that the composer's purpose in creating music works is to express his inner feelings to the fullest. The expression of inner feelings is expressed through the whole music and sublimated at the theme, and the theme of music sometimes appears repeatedly in some form. For example, Figure 4 is the theme of western music, when saints enter [2]. Obviously, the theme of this music can be regarded as translation.
If we take a suitable horizontal line as the time axis (horizontal axis X) and a straight line perpendicular to the time axis as the pitch axis (vertical axis Y) on the staff, then we have established a time-pitch plane rectangular coordinate system on the staff. Therefore, a series of repetitions or translations in Figure 4 can be approximated by a function, as shown in Figure 5, where x is time and y is pitch. Of course, we can also throw the ball in time.
Joseph Fourier, a famous mathematician in the19th century, needs to be mentioned here. It was his efforts that made people's understanding of the essence of music reach its peak. He proved that all music, whether instrumental or vocal, can be expressed and described by mathematical formulas, and these mathematical formulas are the sum of simple periodic sine functions [1].
There are not only translation transformations in music, but also other transformations and their combinations, such as reflection transformations. The two syllables in Figure 6 are the reflection transformation in music [2]. If we still consider these notes in the coordinate system from a mathematical point of view, then their mathematical expression is our common reflection transformation, as shown in Figure 7. Similarly, we can approximate these two syllables with functions in the time-distance rectangular coordinate system.
From the above analysis, it can be seen that a piece of music may be the result of various mathematical transformations on some basic pieces.
Mathematics in natural music.
The connection between music and mathematics in nature is even more magical, which is usually unknown to everyone. For example, the chirping of crickets can be said to be the music of nature, but I don't know that the frequency of crickets' chirping has a great relationship with temperature. We can express it by a linear function: c = 4t–160. Where c stands for the number of crickets chirping per minute and t stands for temperature. According to this formula, as long as you know the number of crickets chirping every minute, you can know the temperature of the weather without a thermometer!
There is also perceptual music in rational mathematics.
Starting from a trigonometric function image, we only need to segment it appropriately, form appropriate segments, and select appropriate points on the curve as the positions of notes, and then we can make music piece by piece. In this way, we can not only compose music with golden section like Hungarian composer Bela Bartok, but also with pure functional images. This is the successor work of mathematician Joseph Fourier. It is also the reverse process of his work. One of the most typical representatives is JosephSchillinger, a professor of mathematics and music at Columbia University in the 1920s. He once drew a ups and downs commercial curve in The New York Times on a coordinate paper, then transformed each basic segment of the curve into music according to proper and harmonious proportion and interval, and finally played it on an instrument. The result turned out to be a beautiful piece of music, which is very similar to Bach's music works. His student george gershwin even innovated and created a system of composing music with mathematics. It is said that he used this system to create the famous opera "Bogey and Beth".
So we say that the appearance of mathematics in music and the existence of music in mathematics are not accidental, but a reflection of the integration of mathematics and music. We know that music expresses people's feelings or attitudes towards nature and life by playing a series of notes, that is, music expresses people's feelings, reflects people's own inner world and feelings about the objective world, so music is used to describe the objective world. Only in an emotional or more personal way. Mathematics describes the world in a rational and abstract way, which enables human beings to have an objective and scientific understanding of the world and express nature through some simple, beautiful and harmonious formulas. So it can be said that both mathematics and music are used to describe the world, but the ultimate goal is to better serve the survival and development of mankind, so they are both used to describe the world.
Since there is such a wonderful connection between mathematics and music, why not immerse yourself in the beautiful melody of Liang Zhu, or settle in the field where insects chirp and think about the internal connection between mathematics and music? Why don't we continue to explore their internal relations with confidence in the clank of pipa or exciting symphony?
Above, we have provided some materials related to mathematics and music. How to "process" these materials into the content of "mathematics education"? We put forward several questions for textbook writers and teachers working in the front line to think about.
1) How can such materials be processed and infiltrated into mathematics teaching and teaching materials?
2) Whether these materials can be compiled into a "popular science report", and in extracurricular activities, report, investigate, understand and think about the influence of such a report on students and the students' reaction to such a report.
Music and mathematics have been linked for centuries. In the Middle Ages, the educational curriculum included arithmetic, geometry, astronomy and music. Today's new computers keep this connection going.
Music score writing is the first important field to show the influence of mathematics on music. In the music draft, we see the speed and beat (4/4 beat, 3/4 beat, etc. ), whole notes, half notes, quarter notes, eighth notes, sixteenth notes, etc. Determining the number of some notes in each bar when writing a score is similar to the process of finding the common denominator-notes of different lengths must fit into the specified bar in a beat. The music created by the composer is very beautiful and effortlessly integrated into the tight structure of written music scores. If we analyze a finished work, we can see that each bar uses notes of different lengths to form the designated beat per minute.
Music is not only related to mathematics, but also to proportion, exponential curve, periodic function and computer science.
The Pythagorean School (585-400 BC) was the first to connect music with mathematics through proportion. They realized that the sound produced by plucking the strings was related to the length of the strings, thus discovering the relationship between harmony and integers. They also found that harmony is produced by the same taut string with an integer ratio-in fact, every harmonious combination of plucked strings can be expressed by an integer ratio. Increasing the length of strings by an integer ratio can produce the whole scale. For example, starting from the string that produces the note C, the length of C 16/ 15 gives B, 6/5 of C gives A, 4/3 of C gives G, 3/2 of C gives F, 8/5 of C gives E, and 16/9 of C gives D and 2 of C.
Have you ever wondered why grand piano was shaped like that? In fact, the shape and structure of many musical instruments are related to various mathematical concepts. Exponential function and exponential curve are such concepts. The exponential curve is described by an equation in the form of y = kx, where k > 0. An example is y = 2x. Its coordinate diagram is as follows.
Whether it is a stringed instrument or a wind instrument with an air column, their structures all reflect the shape of an exponential curve.
/kloc-the work of John Fourier, a mathematician in the 0 th and 9 th centuries, pushed the study of musical characteristics to a climax. He proved that all musical sounds-instrumental music and vocal music-can be described by mathematical formulas, which are the sum of simple periodic sine functions. Every sound has three attributes, namely pitch, volume and sound quality, which are the characteristics that distinguish it from other music.
The discovery of Fourier makes these three properties of sound can be clearly expressed by graphics. Pitch is related to the frequency of the curve, and volume and sound quality are related to the amplitude and shape of the periodic function ① respectively.
If you don't understand the mathematics of music, you can't make progress in the application of computers in music creation and musical instrument design. Mathematical discovery, especially periodic function, is very important in the design of modern musical instruments and voice-activated computers. Many musical instrument manufacturers compare the periodic sound curves of their products with the ideal curves of these 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 segmental vibration and the whole vibration of a string. The longest vibration determines the pitch, and the smaller vibration produces overtones.
① A periodic function is a function that repeats shapes at equal intervals.