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What is the mathematical relationship between the pitch of guitar, namely frequency, chord length and tension? Is there a formula?
1) No matter what kind of legal system, an octave corresponds to a relationship with twice the frequency. That is, the frequency of standard pitch a 1 is 440Hz, so the frequency of a2, which is one octave higher, must be 880Hz.

(2) We can get the theoretical pitch of each empty string sound of the guitar by calculation (the Twelve Average Laws), and then see if the pitch obtained by overtone tuning method is equal to it. For the standard pitch a 1=440.0000 Hz, the empty string sound generated by the averaging method 12 is as follows:

1 empty string, pitch e 1, frequency f = 440.0000/2 (5/12) = 329.6276 Hz.

2 empty string, pitch b, frequency f = 440.0000/2 (1012) = 246.438+07 Hz.

3-string hollow heart, pitch g, frequency f = 440.0000/2 (14/12) =195.9977 Hz.

4-string empty string, pitch d, frequency f = 440.0000/2 (19/12) =146.8324 Hz.

5-string empty string, pitch a, frequency f = 440.0000/2 (24/12) =110.0000 Hz.

6-string hollow heart, pitch e, frequency f = 440.0000/2 (29/12) = 82.4069 Hz.

As for the pure rule mentioned above, the method of generating the rule is to triple (including 1/3 frequency multiplication) and quintuple (including 1/5 frequency multiplication), and there is no other setting. The so-called triple frequency means that the vibration frequency of the string is three times the fundamental frequency. According to the formula of chord vibration frequency f = (1/2l) * (t/ρ) (1/2), it can be concluded that the chord length is inversely proportional to the vibration frequency, that is, when the chord length becomes 1/3, the frequency will be tripled, but other parameters will remain unchanged.

The position of the seventh fret on the guitar is actually 1/3 of the whole chord length, so we can calculate that the overtone frequency at the seventh fret is 3 * 329.6276 = 988.8828 Hz. The position of the fifth fret on the guitar is 1/4 full string length, where the vibration frequency of overtones should be four times that of empty strings, that is, 4 * 246.94 17 = 987.7668 Hz. If we calibrate the 1 string first, and then calibrate the 7-tone overtones and the 2-tone overtones with 1 string, the 2-tone overtones will be adjusted to 988.8828 Hz in an ideal state, which is1.160 Hz higher than the actual 987.7668 Hz. This kind of overtone tuning has the same problem when used on other strings. That is to say, due to the use of seven overtones, pure law is introduced, and 12 average law and pure law are mixed when tuning strings. Therefore, if the pitch obtained by tuning the strings in this way is "inaccurate", it should be said that it cannot be wrong.

However, we can't ignore a problem. Guitar is a musical instrument, not a physical experimental instrument. Although guitar sound contains countless physical principles, it is actually acceptable as long as the pitch deviation is within the allowable range. Moreover, the height of the pillow on the guitar will actually affect the effective vibration length of the strings, that is, it will affect the pitch. Therefore, pitch compensation still has a lot to study. According to the national standard, the pitch deviation range of ordinary guitar is 20 minutes, and the pitch deviation of advanced guitar is between+10 minutes and -5 minutes.

The so-called spectrum is to divide an octave into 1200 equal parts, and each part is called a spectrum. It is expressed by a mathematical formula: score = 1200 * log2 (f2/f 1).

Then we can calculate how many points the pitch deviation caused by overtone tuning is.

1200 * log2 (988.8828/987.7668) =1.9549 score.