There are four basic moves available on the lattice of fifths and thirds. They are:
Up a fifth (x3)
Down a fifth (÷3)
Up a major third (x5)
Down a major third (÷5)
Each of these moves has its own harmonic flavor, and they can be combined to create new flavors.
The major scale only uses the first three building blocks. What about the fourth one?
The land of reciprocal thirds is where most of the black keys reside. It’s the world of minor tonality. Here is the sound of a pure reciprocal third:
The new note is a mirror of the 3, an upside-down 3. Its ratio is 1/5, which can be octave-shifted to 8/5. That ratio puts it a little over halfway up the scale, between the 5 and the 6. It’s called the minor sixth or flatted sixth; I use the symbol b6.
There is a beautiful shift of feeling when you move from overtonal energy to reciprocal and back again. To me it feels like breathing in and out. Maybe that’s because I play harmonica. When you blow on a few holes of a Marine Band, you get the 1 chord. When you draw, you get the 4. Breathing in and out takes you back and forth between reciprocal and overtonal territory.
The same action can happen on the 5-axis, with a more exotic flavor:
Hear the shift? Overtonal, reciprocal, back again. Every note on the lattice except the 1 has its mirror twin.