Since ancient times, music and mathematics have always been linked. In the Middle Ages, educational courses included arithmetic, geometry and music. Today, with the continuous development of computer technology, this link is also expanding.
The first major influence of mathematics on music is reflected in the writing of music scores. On the music draft, we can 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 piece of music, we can see that each bar will form a specified number of beats with notes of different lengths.
Music has a close relationship not only with mathematics, but also with mathematical ratio, exponential curve and periodic function, and also with computer science.
From 585 BC to 400 BC, the Pythagorean school first connected 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 16/ 15 of C 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 2/9 of C.
Maybe many people don't know how grand piano's appearance was made. 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 the equation form of y = kx, where k > 0. For a simple example, y = 2x, and its coordinate diagram is as follows.
The shape and structure of stringed instruments and wind instruments can reflect the shape of 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. 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. This discovery of Fourier makes the three properties of sound-pitch, volume and sound quality-clearly presented on the map respectively.
If you don't know enough about mathematics in music, then computers can't make such great progress in the application of 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 play equally important roles in the production and development of music.
The figure shows the segmental vibration and the whole vibration of the string. The longest vibration determines the pitch, and the smaller vibration will produce overtones.