Digging deeper into this problem reveals another pitfall with this tuning system. Actually, it's a problem that goes beyond Pythagorean Tuning and affects other tuning systems afterward. The problem stems primarily from the interval of the Major 3rd. To put things into greater perspective, here is a list of ratios with corresponding note letters:
C D E F G A B
1/1 9/8 81/64 4/3 3/2 27/16 243/128
The Major 3rd, shown above, has a ratio of 81/64. This ratio is often called a Ditone I know that may have seemed pretty random, but this will make sense in a minute. Anyway, after whipping out the calculator, log 10 that ratio of the Major 3rd and multiply by the constant (If you missed this step, just go back to "The Problem with Tuning, Part 1." It gets progressively harder from here.)
What the answer should be is about 407.8200 cents. Now, let's try making an octave out of that. How do you do that??? Well, let's try adding three Ditones--, meaning. lets multiply 407.8200 by 3. Why 3? Because three Major 3rd stacked together should make an octave (leaving out the 5th)
Here' what you get:
407.8200*3= 1223.46
The end result is an octave. But, there's something really odd, here. Notice that cent value for this kind of octave adds 23. 46 more cents than the 2/1 octave. The 1223.46 cent value is actually something called the Pythagorean Octave. If we take the value of the 2/1 Octave (1200 cents) and subtract that from the Pythagorean Octave (1223.46 cents), we are left with this:
1223.46-1200.00= 23.46 cents
This is the value for the Pythagorean Comma. also known as the Ditonic Comma. Some of you might be asking right now:
WHAT DOES THIS EVEN MEAN???!!
It means that, even though Pythagorean Tuning tunes by fifths, it misses it's target by exactly 23.46 cents. That means that, because the Major 3rd is WAY off, Pythagorean Tuning doesn't make a complete Circle of Fifths. This is a HUGE problem that people have tried to fix for centuries... ways that will explored next post.