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.

Question

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]

Answer 1

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.

Answer 2

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