Finding the relationship between the length of the string and the frequency it produces

It is clear that when someone plays guitar, the instrument produces different "notes" - frequencies.
The question is? how the luthier (the person that build the musical isntruments) knew where to place the frets to produce the desired frequencies?

Lets study a guitar for a while. The guitar has different strings some more robust than other. This means that the different strings are made from different materials, and therefore they have different "DENSITY"

Then the player changes the tension of the "OPEN STRING" to match a desired "OPEN STRING FREQUENCY"

And then, the player gives energy to the string by plucking it (moving the string out of its point of equilibrium), and the strings starts to vibrate, releasing sound.

Pressing the string to one fret makes the frequency changes

Now lets investigate how the guitar (and for all matters any string instrument) works. There are Three possible independent variables we can change to change the sound

1- Density of the String - D
2- Tension of the String - T
3- Length of the String - L

It is obvious that the player is not changing strings between notes, neither change the tension when playing. Therefore the Density D and the Tension T will be PARAMETERS that will be adjusted and remind constand during the execution of the play.

Now is time to explore the string in a scientific way - Let implement an experiment that you will design. You can find the experiment design worksheet in the link here .

We added a movie containing the samples of the guitar string sound with the specific string lengths you can use for your experiment. To access to this movie please click here .

With the worksheet and the movie, you will be able to collect information about the frequency the string produces - the dependent variable (the fundamental) - and the Length L of the string - the independent variable - when the Density D and the Tension T remind constant - the Parameters of the experiment.
Please implement the experiment and share with everybody your findings!!!