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 ?

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