The horizontal spines in the lattice are formed using the fifth partial of the harmonic series. If we start with C and keep adding the fifth above, conventional music theory tells us that after 12 fifths we will end up back at C. We will have gone around the circle of fifths.

There is a problem here but in order to see it we need a to use a little math. On the previous page we said that a perfect fifth is the the interval between the 2nd and 3rd partials in the harmonic series. So the frequency of G (which is a perfect fifth above C) is 3/2 times the frequency of C. Moving around the circle of fifths, it follows that the frequency of D is 3/2 times the frequency of G. That D has a frequency that is 2.25 times the frequency of C.

Notice that this D is actually the D that lies an octave plus a whole step above the original C. This makes sense since stacking two perfect fifths produces a 9th. We can confirm that this D is in fact greater than an octave by observing that it's frequency is greater than 2 times the frequency of the original C.

If we repeat this process 12 times the circle of fifths say that we should end up back at C (actually it would be a C seven octaves above the original note).

But the note seven octaves above middle C has frequency that is 128 times the frequency of middle C.

128 is definitely not equal to 129.746. So what is going on here?!?

In order to construct an instrument like a piano that has fixed notes and can also be played equally well in any key some compromises have to be made in the way the notes are tuned. This is called "tempering" and it was a hot topic around the time of J.S. Bach. But what is "temperament" and what