Who knew that rock and roll could be so mathematical. Let's investigate the lengths of guitar strings from the fret to the bridge. For this investigation I used my Dean Bass Guitar which should provide longer lengths that found in the mandolin data set. Using a ruler accurate to a millimeter I measured all the frets on my guitar resulting in the following data:
Taking the ratio going in both directions (from a higher to a lower fret and from the lower to a higher fret) we can find the ratio of the distances from one fret to another. Using an exponential decay model to predict the string length based on the fret number, we find that we can do a pretty accurate job with the equation y = 90 * (.945)^x (where .945 is the average ratio from one string to the next, x is the fret number, and 90 is the length of an open string). If the neck of the guitar were to shorten, the only change I would have to make to my equation is to change 90 to whatever length of the new neck is (assuming the ratio between frets remained the same) In addition I used excel to formulate it's own best fit line to the data. Plotting all this data results in the following graph:
The BLUE data represents the actual measured length of each string from the fret to the bridge. The RED data set represents my exponential decay prediction. The BLACK trendline is what excel used to predict the string length. Just by looking at the graph we can see that both predictions are pretty accurate, however my prediction seems to fall away from the data as the fret number increases. This could be from inaccurate data recording which could be remedied using more precise measurement devices. The BLACK trendline however holds true to the data all the way down the fretboard.