-A brief analysis of the physical principles of violin sounding and tuning.

First, the significance of the topic

According to the textual research of Mr. Wu Nanxun, the earliest physical historian in China, the first artificial physical instrument in the world is a flute that digs holes in animal bones or bamboo pipes and can blow out sounds. This is both a musical instrument and an acoustic instrument. In ancient China, the study of * * * tone, string vibration and pipe timbre was carried out through musical instruments; Pythagoras, a Greek philosopher, found that the length of strings has a certain relationship with pitch; From the development of modern physics, acoustics still occupies a very important part and is closely related to our lives. ……

Many students will learn some musical instruments, but they can't debug stringed instruments. Based on the vibration knowledge we have learned, this paper analyzes the sounding principle of violin, a western stringed instrument, and provides theoretical basis and experimental results for the players to check and debug.

2. Related physical knowledge

The actual musical sound consists of three parts: fundamental frequency, harmonic (overtone) and partiality. Every musical tone, that is, periodic vibration, can be decomposed into the superposition of many simple harmonic vibrations with different frequencies, phases and amplitudes. Simple harmonic vibration, that is, the sound wave generated by the propagation of sinusoidal vibration or cosine vibration, is called pure tone. The actual musical sounds such as singing and instrumental music are not simple pure tones, but the superposition of many pure tones. Among these simple harmonic vibrations, the lowest frequency is called the fundamental frequency, and the energy of the fundamental frequency is often the largest. Vibrations whose frequency is an integer multiple of the fundamental frequency are called harmonics, and other high-frequency vibrations are called partials. Modern analysis shows that there is infrasound below the fundamental frequency. Therefore, physically speaking, musical sound should be composed of three parts: musical sound, noise used in music (such as percussion instruments with no fixed tone, such as gongs, drums, sand hammers and bangzi, and effect sounds such as waves, running water and wind) and some ultrasonic waves that have an impact on timbre.

Generally speaking, the higher the frequency of body vibration, the higher the tone people make; The lower the vibration frequency of the generator, the lower the sound made by people. But there is no strict proportional correspondence between tone and frequency. It is generally believed that every time the frequency is doubled, the tone sounds an octave higher, which is limited to the middle frequency band. The high-pitched part has low hearing, that is, the frequency is doubled, which sounds not eight degrees higher, but low, so the frequency should be increased to adapt to people's hearing. The bass part is on the high side, so the frequency needs to be lowered.

Music sounds strong or weak, that is, the loudness of sound, which is the second subjective quantity of music. The greater the energy and intensity of sound, the louder the sound. However, the two are not directly proportional.

As for timbre, it's a subjective feeling. Traditionally, the main factor that determines timbre is frequency spectrum, so we often imitate various timbres according to frequency spectrum. However, according to the data, practice shows that the instantaneous state of the beginning and end of sound, that is, "beginning" and "end", is also closely related to timbre. Timbre is not only related to the composition of the spectrum (that is, the number, length, relative intensity, disharmony and transient state of partials), but also related to the position of the fundamental frequency and harmonics in the listening area, because the loudness response of human ears to various frequencies is different. Timbre is also related to the distance between the listener and the sound source, because the attenuation of various components in a sound is different.

3. Related music knowledge

An interval is the distance between two notes. Musically, intervals are expressed in degrees. Several degrees are exponential initials, and there are several names along the scale. There is a semitone difference between two adjacent keys (including black keys) on the piano, and two semitones are equal to a whole tone. This is also a way to express intervals. Interval and frequency are basically one-to-one correspondence.

Arrange the sounds between two octaves in order into a scale. The mathematical method of specifying the source of each sound in a scale and its precise pitch is called a music system.

The three most commonly used legal systems are the law of twelve averages, the law of five degrees and the law of purity. Each note in the scale has a name. Because of the different methods of producing melody, the frequency of homophones (such as all) produced by different music systems is also different.

The law of twelve averages was first invented by Zhu Zaiyu, a scientist in Ming Dynasty in China, which was decades earlier than that in Western Europe. He divided an octave (frequency ratio 2) into twelve parts according to geometric series and got twelve laws. Today's pianos, all keyboard instruments and stringed instruments with "goods" are made by this legal system.

Mathematical expression: the frequency ratio between two adjacent sounds is: starting from any sound, multiply the frequency of one sound in the upper half of the pitch by the frequency of that sound; A sound that is half a tone lower than a sound, whose frequency is the frequency of the sound multiplied by it; By analogy, the frequencies of all sounds can be obtained.

The law of twelve averages has many advantages, for example, it is easy to transfer tones and simplifies the relationship between rising and falling semitones in different tones.

In violin, if the chord length of sound is taken as the benchmark, then a set of fixed ratios can be determined according to the frequency relationship between the chord lengths corresponding to a group of small characters (in which the ratio is two octaves higher) according to the law of twelve averages.

Four. Research and experiment

The string of a violin is a thin steel wire fixed at both ends. The wave train generated during plucking and wiping strings is reflected and superimposed by two fixed ends to form a standing wave, but it contains waves of many frequencies. Here, only the fundamental frequency vibration that determines the pitch level is analyzed and studied.

The fundamental frequency vibration of the standing wave corresponds to the vibration with the longest wavelength, that is, the chord length. The distance between the lifting string and the fingerboard is very small, and the deformation of the strings caused by fingers pressing different positions on the fingerboard is very small and can be ignored. It can be considered that the mass linear density of the tension chord on the string is constant, and the wave velocity in the string is approximately constant. Therefore, it can be considered that the following proportional relationship holds:

Experimental process: a violin, after being tuned by a professional musician, is played by a student with many years of playing experience. Many students with keen sense of music and professional training listen together and proofread each pitch with other musical instruments. The recorded and calculated data are shown in the following table. The values of k in the table are defined as follows:

Two pitches that differ by a semitone correspond.

Two pitches that differ by a whole tone correspond.

Serial number n pitch-to-sound ratio

Interval difference chord length /mm total length: k value above 320.0 mm.

Theoretical error rate of the first, second and third average calculation values

1

The whole tone is 243.0 243.8 243.7 243.51.1.1.39%.

2

Full tone 220.0 220.9 219.2 220.01.131.120.25%

three

Semitone195.5196.195.0195.51.071.061.

four

Whole tone182.5181.9183.182.51.12/kloc-0.

five

Whole tone162.5162.0162.31.131.120.48%

six

Full tone143.8143.8144.2143.91.1.021.00%.

seven

Semitone130.0129.8128.7129.51.051.06 0.79%

eight

124.0 122.4 123.2 123.2

Among them, the column of string length is the distance between the high-voltage finger corresponding to each sound on the violin string and the two fixed points of the piano code (the four strings are called,, and in turn from coarse to fine, which refers to the empty string sound of the string), that is, the part of string length that participates in vibration.

As shown in the above data, the average error rate is 0.74%, which is basically in line with the previous theoretical analysis.

Verb (abbreviation of verb) conclusion

We summarized the self-tuning method of a violin (adjacent strings are separated by five degrees):

Based on the empty chord sounds of chords, such as chords, the length of the corresponding chord on the same chord is calculated according to the pitch relationship. The pitch is the same as the empty string sound (this is the requirement of violin making). After adjusting the tightness and length of the string in turn, the pitch on the string is calculated as the empty string sound of the string. ..... in the same way.

This method is suitable for all kinds of violins and guitars, but it should be noted that:

(1) For sounds much higher than empty string sounds, the calculation method has a large error. In the experiment, multiple sets of data are measured on one string just for the convenience of calculation, comparison and conclusion. In practice, adjacent strings should be proofread in turn.

② The adjacent chords of cello and guitar differ by four degrees, and the data should be paid attention to when calculating.

I hope our research can help music lovers who play stringed instruments.