The first equation is a mathematical representation of a

scale as geometric series that sets aside the perfect fifth. Basically, this

equation represents that fact that an interval is necessary which upon trans

positioning 7 times would give the 7 semitones. Whereas, the second equation

represents the fact that, the geometric scale is for a scale of 12 notes. The

reason being that, 12 perfect fifths did not round off to precisely to an even

octave, instead it exceeded it by a small ratio. This ratio is called the Pythagorean

comma. “It is the small interval that exists between two notes with identical

pitch, but has different names and different roles in harmony.” (Halpin)

Now, when both

equations are solved for x, it is quite evident that, the two solutions will

not be equal, which then brings up the question, which to rely upon or ?