When a player's finger presses a guitar string down onto a fret, the length of the vibrating portion of the string is shortened, thereby increasing the string's fundamental frequency. The string's tension and mass per unit length remain unchanged.

If the unfingered length of the string is l= 65 cm, determine the positions x of the first six frets, if each fret raises the pitch of the fundamental by one musical note in comparison to the neighboring fret. On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 2^[1/12]

2^1/12=1.05946

so each fret increases the frequency by that factor. But wavelength is the reciprocal of this. or w1=w*.94387

Well, to get this, one has to have adjacent frets .94387 length of the before length.

original length 65cm
next length 65*.94387
next lenght=65*(.94387)^2 and so on.

thats a very good observation., i tried to look up the formula and couldn't find it.. the formula "2^[1/12]" that you posted.. there was no mention. so i started using the computer to determine trhe correct number and i came up with .94387 after writing a script using trial and error,, that yielded that number which i searched for and found this blog. thanks .. yet disturbingly while i was searching ,, i got a lot of search results about Pythagoras using the fundamental frequency and the some vague inference that he used the 3rd and 5th harmonics to make chords. but they seemed to divert from the fact that, that made it obvious that he would have built a mini harp and tuned the strings so that he could play the fundamental key for the 3rd and 5th harmonics of the third and 5th harmonics of every chord. which could have only in the end have perfectly yielded every single note in a 12 note even temperament .. but they want to credit this to some German in the 1800's. someone is trying to plagiarize the work of Pythagoras but that formula is really the one i needed , thanks

To determine the positions of the first six frets, you need to use the formula that relates the length of the vibrating portion of the string to the frequency and the note difference. The formula is:

length = (1 / (2^(1/12)))^n * unfingered length

Where:
- length is the length of the vibrating portion of the string.
- n is the number of frets above the unfingered position.
- unfingered length is the original length of the string.

We can solve for the positions of the six frets one by one.

1. For the first fret (n=1):
length = (1 / (2^(1/12))) * unfingered length
= (1 / (2^(1/12))) * 65 cm

2. For the second fret (n=2):
length = (1 / (2^(1/12)))^2 * unfingered length
= (1 / (2^(1/12)))^2 * 65 cm

3. For the third fret (n=3):
length = (1 / (2^(1/12)))^3 * unfingered length
= (1 / (2^(1/12)))^3 * 65 cm

4. For the fourth fret (n=4):
length = (1 / (2^(1/12)))^4 * unfingered length
= (1 / (2^(1/12)))^4 * 65 cm

5. For the fifth fret (n=5):
length = (1 / (2^(1/12)))^5 * unfingered length
= (1 / (2^(1/12)))^5 * 65 cm

6. For the sixth fret (n=6):
length = (1 / (2^(1/12)))^6 * unfingered length
= (1 / (2^(1/12)))^6 * 65 cm

By substituting the appropriate values of n, you can calculate the positions x for each of the six frets.

To determine the positions of the first six frets on a guitar string, we need to calculate the length of each fret.

Let's start with the first fret:
From the question, we know that the length of the unfingered string (l) is 65 cm.
When we press the first fret, we create a shorter vibrating portion of the string. To increase the fundamental frequency by one musical note, we need to reduce the length of the string by half of the current wavelength.

Using the formula for the fundamental frequency of a vibrating string:

f = (1/2L) * sqrt(T/μ)

Where:
f is the fundamental frequency,
L is the length of the vibrating portion of the string,
T is the tension in the string, and
μ is the linear mass density of the string.

Given that the tension and mass per unit length don't change, we can ignore T and μ and focus on the change in length (ΔL) needed to raise the pitch by one musical note.

Using the formula for the ratio of frequencies in the equally tempered chromatic scale:

ratio = 2^(1/12)

Since we want to raise the pitch of the string by one musical note, we can use this ratio to find the difference in length.
Let's call the difference in length for the first fret ΔL1.

ratio = (L - ΔL1) / L

Simplifying and solving for ΔL1:

1 + (ΔL1 / L) = 2^(1/12)

ΔL1 / L = 2^(1/12) - 1

To find the length of the first fret, we can substitute the given unfingered length (l = 65 cm) into the equation:

ΔL1 = (2^(1/12) - 1) * l

Now that we have ΔL1, we can find the length of the first fret (L1):

L1 = l - ΔL1

Using the same process, we can calculate the length of the second fret (L2) using the difference in length (ΔL2) and the length of the first fret (L1):

ΔL2 = (2^(1/12) - 1) * L1
L2 = L1 - ΔL2

Repeating this process, we can calculate the lengths of the third, fourth, fifth, and sixth frets (L3, L4, L5, and L6).

To summarize, you can calculate the positions of the first six frets using the following steps:

1. Calculate ΔL1: ΔL1 = (2^(1/12) - 1) * l
2. Calculate L1: L1 = l - ΔL1
3. Calculate ΔL2: ΔL2 = (2^(1/12) - 1) * L1
4. Calculate L2: L2 = L1 - ΔL2
5. Repeat steps 3 and 4 for L3, L4, L5, and L6 to calculate the positions of the remaining frets.

Note that these calculations give approximate values since the equally tempered chromatic scale is an approximation. The actual positions of the frets on a guitar may vary slightly depending on the specific instrument and other factors.