Mean Tone Scale

The next evolution from the Just Scale was an attempt to solve the problem of uneven perfect fifths, while maintaining the just major thirds.  This was accomplished at the expense of the Pythagorean perfect fifth and is called the Mean Tone Scale.  The idea is to keep all three major thirds: C to E, F to A and G to B at a 5:4 ratio, make all of the major seconds equal and then make the minor seconds so the math works out (i.e. everything fits in one octave.)


To split the major third from C to E down the middle, we take the square root of 5/4:

Now, there are 5 major seconds in the major scale: C -- D, D--E, F--G, G--A and A--B.  This leaves two minor seconds left to fill in the octave.  So, what we do is take the whole octave, or 2:1 and divide by 5 of these new major seconds (which is the ratio we came up raised to the fifth power).  What's left needs to be divided in half (by taking the square root), and put in for the half steps between E and F and between B and C.  Got it?

Now we combine all of the above and create the mean tone scale: