Pythagorean Tuning Part 2

We left off with the scale consisting of {C, G, C'} and containing frequencies with ratios of {1:1, 3/2:1, 2:1} with the fundamental pitch.  Notice that we get a second interval for free: the prefect fourth between G and the octave C.  To determine the ratio of these pitches (Pitches be crazy), we divide the higher ratio by the lower ratio.  If you remember back to grade school when you learned how to divide fractions, we simply flip the bottom and multiply straight across:

the upshot being that the ratio of a perfect fourth is 4:3.  

So now, if we start with C and multiply by 4:3, we get a new note in our scale, F.

{C, F, G, C'} = {1:1, 4:3, 3:2, 2:1}

This leads us to the interval between F and G, a major second.  Applying the same technique of dividing 3/2 by 4/3 gives us 9/8.  Start at C again and now we have

{C, D, F, G, C'} = {1:1, 9:8, 4:3, 3:2, 2:1}

A major second up from D gives us E, so multiply 9/8 by 9/8 and get 81/64, which is a Pythagorean major third and do the same for A by multiplying G's ratio by 9/8:  3/2 by 9/8 gives us 27/16 which is a Pythagorean Major sixth and finally, a major second up from A (which has a ratio of 27/16) gives us B: 27/16 by 9/8 = 243/128, completing the Pythagorean Major Scale:

{C, D, E, F, G, A, B, C'} = {1:1, 9:8, 81:64, 4:3, 3:2, 27:16, 243:128, 2:1}

Notice the minor seconds:

The note E to the note F is a minor second, also called a half step.  Unfortunately, that name doesn't apply to this tuning, which presents problems.

Calculating the ratio of a minor second involves either dividing F's ratio by E's or equivalently dividing C' by B.  We'll do the former: 

To find half of a Pythagorean Major second, we need to take the square root, because these ratios are multiplicative:

Notice that the Pythagorean Minor Second is slightly less than half of a Pythagorean Major Second.