If you could learn to play any musical instrument

[Pythagorean]

What I'd like to do is place struts on my guitar in between the manufactured ones to create quarter tones. Sorry for a math question in a general discussion thread, but I was wondering: if I do this, how far between the manufactured struts do I place them? I think I set it in such a way that the ratio of the distance between the lower manufactured strut and the quarter tone strut over the distance between the higher manufactured strut and the quarter tone strut is equal to 2^(1/24) units (since there are 24 quarter tones). Is this right?

To make the 12 fret octave on stringed instruments, master luthiers used the 1/18th rule. Each successive fret is 1/18th the distance remaining on the unfretted portion of the board. Not sure how the derivation comes out - probably the relation between frequency, length, and tension:

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

That's probably about right, but the pitch might not be as accurate as the maths. For a start, pressing down the guitar string changes the tension and holding it down closer to or further away behind the fret therefore also changes the pitch. (This effect can mean that acoustic guitar strings made from different materials can result in different pitches for the same fret position, especially for higher frets). That means that when you add a new fret, you will now need to hold the string down closer to the existing fret, which will actually raise the pitch of the existing note slightly. Perhaps you need a fretless instrument!

The 1/18th rule leads to an error that's slightly flat, so the increase in tension ideally cancels out the error. Of course, if you use light vs. heavy strings it probably affects the amount of compensation, but we're probably talking on the order of ~10 cents here.