Problems with the Pythagoren Tuning

Let's say we wanted to change keys using the Pythagorean Tuning based on a fundamental pitch of C.  First we'll transpose up a fifth to move to the key of G.  We expect all of the pitches to remain the same, but the seventh degree f needs to move up to f#, so we maintain a leading tone between the 7th degree and octave.  Remember, to change pitches we multiply (to go up) or divide (to go down) by the appropriate ratio.  In this case we multiply by 3/2 to go up a perfect fifth.

In the figure below, the down arrows indicate multiplying by 3/2.  In some cases, the notes leave the octave, so the next round of arrows divides by 2 (where necessary), to keep all of the pitches in one octave.

We now have the major second from e to f# divided by "semi-tones".  I put that in quotes because if you examine the intervals, they are not even.

Between f# and g is the same minor second as between b and c, but the new minor second created between f and f# is slightly larger.

Now let's transpose in the other direction: down a fifth (or equivalently up a fourth).  This would correspond to multiplying each ratio by 2/3.

Again, the new minor second between b and b♭ is bigger than the minor second between a and b♭.

This creates problems with the circle of fifths (or fourths).  If we transpose 12 times, funny things happen.  In the table below, the key of C is in the middle, going up transposes by fifths until we reach F# and going down transposes by fourths until we reach G♭.  Notice the note names across the top use sharps and the enharmonic note names across the bottom use flats.

All of the enharmonic notes are off by the same amount: 3¹²:2<sup>19 

which is called a Pythagorean Comma.  It is formed by going up 12 fifths and back down 7 octaves.</sup>

Pythagorean f#


Pythagorean g♭


Both played simultaneously