The writing of music scores is the most obvious place where mathematics shows its influence on music. In music notation, we can find the time signature (4:4, 3:4 or 1:4, etc.), the beat of each measure, whole notes, half notes, quarter notes, eighth notes, etc. Composing music so that it fits the number of beats per syllable is similar to the process of finding a common denominator - in a fixed time, notes of different lengths must make it fit into a specific beat. However, when the composer created the music, he was able to integrate it with the strict structure of the score beautifully and effortlessly. Analyzing a complete work, we will see that each syllable has a prescribed number of beats, and that various notes of appropriate lengths are used.

In addition to the obvious connections between mathematics and musical notation mentioned above, music is also associated with proportions, exponential curves, periodic functions, and computer science. The followers of Pythagoras (585-400 BC) were the first to use proportions to combine music and mathematics. They discovered that there was a close relationship between the coordination of musical sounds and the integers they recognized. The sound produced by plucking a string depended on the length of the string. They also discovered that consonances are given by taut strings whose lengths are in whole-number ratios to the length of the original string. Virtually every harmonious combination of plucked strings can be expressed as an integer ratio. The entire musical scale can be produced by increasing the length of the string to an integer ratio. For example, start with a string that produces the note C, then 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, and 3/2 of the length of C gives you F. , 8/5 of C gives E, 16/9 of C gives D, and 1/2 of C gives the bass C.

You may be surprised, why does the grand piano have its unique shape? In fact, the shapes and structures of many musical instruments are related to different mathematical concepts. Exponential functions are one of them. For example, y=2x. Instruments, whether string or wind, reflect the shape of an exponential curve in their structure.

The study of the nature of musical sound reached its peak in the works of the French mathematician Fourier in the 19th century. He proved that all sounds, whether instrumental or vocal, can be described by mathematical expressions, which are sums of simple sinusoidal periodic functions. Every sound has three qualities: pitch, volume and timbre, which distinguish it from other musical sounds.

Fourier's discovery allowed people to describe and distinguish the three qualities of sound through diagrams. The pitch is related to the frequency of the curve, the volume is related to the amplitude of the curve, and the timbre is related to the shape of the periodic function.

Few people are proficient in both mathematics and music, which makes it difficult to successfully use computers to synthesize music and design musical instruments. The discovery of mathematics: periodic functions are the essence of modern musical instrument design and computer sound design. Many musical instruments are built by comparing the image of the sound they produce with the image of the ideal sound for those instruments and then improving upon it. The faithful reproduction of electronic music is also closely linked to periodic images. Musicians and mathematicians will continue to play equally important roles in the production and reproduction of music.