As the drawing shows, the length of a guitar string is 0.628 m. The frets are numbered for convenience. A performer can play a musical scale on a single string, because the spacing between the frets is designed according to the following rule: When the string is pushed against any fret j, the fundamental frequency of the shortened string is larger by a factor of the twelfth root of two (122) than it is when the string is pushed against the fret j - 1. Assuming that the tension in the string is the same for any note, find the spacing between fret 6 and fret 5.
Our professor discussed the velocity of guitar strings, but nothing about spacing. Is the space between the frets a wavelength? Any ideas on what equation to use?
No. The spacing between a fret and the end of string is a half wavelength.
To find the spacing between fret 6 and fret 5, we need to consider the relationship between the fundamental frequency and the length of the string.
When a string is pushed against a fret, it effectively shortens the vibrating length of the string. The fundamental frequency of a vibrating string is inversely proportional to its length. In other words, as the length of the string decreases, the frequency increases.
According to the information provided, each fret increases the fundamental frequency by a factor of the twelfth root of two (1/12 of an octave). This means that moving from one fret to the next doubles the frequency.
Let's start by determining the fundamental frequency for the open string (no fret pressed):
Length of open string = 0.628 m
Now, we need to find the length of the string when fret 6 is pressed. We can use the fact that moving from fret j-1 to fret j doubles the frequency to calculate the length:
Length at fret 6 = (1/2)^6 * Length of open string
= (1/2)^6 * 0.628 m
= 0.019625 m
Similarly, we can find the length of the string when fret 5 is pressed:
Length at fret 5 = (1/2)^5 * Length of open string
= (1/2)^5 * 0.628 m
= 0.03925 m
The spacing between fret 6 and fret 5 is the difference between these two lengths:
Spacing = Length at fret 6 - Length at fret 5
= 0.019625 m - 0.03925 m
= -0.019625 m
Note that the negative sign arises because fret 6 is closer to the guitar's nut compared to fret 5. Therefore, the spacing is 0.019625 meters (19.625 mm) in this case.
It's important to mention that the spacing between frets is not directly related to wavelength. Wavelength depends on the frequency of the vibrating string. In guitar playing, the frets are spaced based on the mathematics of music theory to produce harmonic intervals.