Anders Thidell and True Temperament have designed guitars with curved frets rather than the usual parallel line segments. Some of the designs were based on a method that J.S. Bach allegedly used to tune his harpsichord. Bach’s method highlights the keys featured in the famous Prelude in C Major found in the Well-Tempered Clavier, Book I. However, attempts to put this in a mathematical framework have led to strange musical results which do not agree with Bach. We offer a new mathematical framework using the natural action of the dihedral group D₁₂ on a regular dodecagon. The framework allows us to view Bach’s method as optimal. We then use the framework to create our own curved fret guitar which is optimal for 16 common guitar chords. Our mathematically optimal guitar design resembles one already produced by Thidell.