Pythagorean Scale

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 The Pythagorean scale is one of the most primitive musical systems.  It dates back to Pythagoras, who lived from 579 to 520 B.C.  The Pythagorean system is an attempt to build a complete chromatic scale from only two of the pure tones.  Pythagoras used the octave and the perfect fifth (3:2) to attempt this.

He started with an arbitrary frequency represented by 1, since it has a 1:1 ratio with itself.  When this frequency is ascended by a perfect fifth, the ratio between the two is 3:2.  Then if the perfect fifth is augmented by another perfect fifth, the ratio between this tone and the root is (3/2)2 or 9/4.  This process is repeated:

 
or
 

The goal was to close what is known as the Circle of Fifths, meaning Pythagoras eventually wanted to end up with the same note he started on.  In order to work with the new notes generated, it was necessary to reduce the notes down into the octave between the the original note and its first octave, the ratio is halved until it is between one and two (as a fraction).

Think of a keyboard if you want to look at this from a different perspective.  In a C major key, the perfect fifth of C is G.  The perfect fifth above G is D (on the octave of C'), D to A, A to E (on C" octave), and E to B, and so on. You can see that D in the figure below does not fall in the octave between C and C', so in order to get it there we divide by two (remember octaves have an interval ratio of 2:1).  All the other notes that fall outside the initial octave are brought back down in the same way.  The table below may help.

Suppose our starting note is C.  Then we have the ratios:

Octaves Above root Pitch Ratio Number to Divide By.  2(octaves above) Ratio Reduced to 1 Octave
0 C 1:1 1 1:1
0 G 3:2 1 3:2
1 D 9:4 2 9:8
1 A 27:8 2 27:16
2 E 81:16 4 81:64
2 B 243:32 4 243:128
3 F# 729:64 8 729:512
4 C# 2,187:128 16 2,187:2,048
4 G# 6,561:256 16 6,561:4,096
5 D# 19,683:512 32 19,683:16,384
5 A# 59,049:1,024 32 59,049:32,768
6 E# 177,147:2,048 64 177,147:131,072
7 B# 531,441:4,096 128 531,441:524,288

But if B# = C (as it does on the keyboard), then there is a problem.  The problem is that when the pitch C is arrived at again, it should have a 1:1 ratio when brought back into a single octave, but it doesn't.  In fact, if one were to continue this process, they will find that there will never be a pitch that reduces down to the original tone.  This is not hard to figure out why.

Since the perfect fifth has integer coefficients in its ratio (3:2) that are relatively prime, then using simple number theory and the Fundamental Theorem of Arithmetic allows us to conclude that

for all m and n that are non-zero integers.  In other words, 

will never be reducible into an integer (which is needed to return to the root).  Therefore, the circle of fifths is left open, and can never be closed with all pure tones.  In fact, the octave cannot be completely built out of any of the pure tones because they are all relatively prime.  So what did the Pythagorean Scale end up like?

Instead of tuning only above the root, the Pythagorean Chromatic Scale tunes both above and below the root, and then adjusts into one octave.  Finding a fifth below the root gives a ratio of:

From this, we can determine that two consecutive perfect fifths below the root will be (2/3)2. The process continues.  When considering C as the root, the perfect fifth below C is a F.  From the above, we know the ratio of F to C is 2:3.  However, this F is not within the octave we want it, so finding the octave of F, we have

Notice that this is the same ratio for the pure perfect fourth (this is good to have in the scale)  The Chromatic Scale (with 12 tones) finishes as follows:

or

Reducing the ratios to arrange them in a single octave, we arrive at:

We run into the same problem as before (as we suspected we would); there are two different interval ratios for the same note.  By choosing one or the other, there will be a loss of the perfect fifth ratio within the scale at some point (even though the difference will be small, and perhaps undistinguishable.

Another problem, as we will see with the whole steps of the pure tones, is that there are two different half step intervals.  The first half step is clearly 256:243.  The second half step (from the major second to the minor second) is

The Pythagorean Scale will sound quite good in the key it is initially developed from.  However, if one were to try and modulate (or change keys), the sound would become