Music Spiral

Hello.

My name is Clark. I build musical instruments as a hobby and am building a stringed instrument that requires a spiral shaped gear. To generate this gear I need the geometry for the spiral (I can add the teeth). I have attempted to express the problem in the simplest way that I can. I have also attached a spreadsheet, which has the setup work completed. What I need is a formula that generates the polar coordinates for the spiral described in step 4. Here are the steps to follow:

1. The spacing of guitar frets is calculated using the formula: L-(L/(2^(1/12))) = first fret of L, where L is the remaining string length. Starting with L=32", the total length of 24 fret spacings will equal 24". (See the "From Bridge", "From Nut" and "From Last Fret" columns in the attached spreadsheet).

2. Subtract a constant 0.375" from each fret-to-fret spacing. The total length of the fret spacings will now equal 15". (See the "Shifted" column)

3. Draw a circle with a 15" circumference, mark the spacings resulting from step 2 around the circumference, and draw radii from these points to the center. (See the "Arc Angle" column)

4. There exists a 360-degree spiral curve that is exactly divisible by the above defined radii into 0.375" long segments. Define points on this curve by providing a function that, given the angle of a radius, returns the distance along that radius a point lies from the center.

My guess is that the function will actually generate a closed curve bounded at 0 & 360 degrees rather than an infinitely expanding spiral. If you want to modify the spreadsheet so that it calculates the 24 polar coordinates for any L that would be awesome! Otherwise, the formula and a decent explanation should be enough for me to work with. Thank you and good luck!