Audio: tritone (0:04)
tritone plays two tritones in Pythagorean tuning. The tritone figure shows that the score contains a tritone in the first bar, the augmented fourth DG#, and a tritone in the second bar, the diminished fifth DAb.
A tritone is an interval of an augmented fourth or a diminished fifth. An augmented interval is an interval that has been slightly sharpened, a diminished interval is an interval that has been slightly flattened.
A Pythagorean augmented fourth has a frequency ratio of 729:512 and a Pythagorean diminished fifth has a frequency ratio of 1024:729. The difference between the frequency ratios is called the Pythagorean comma, or comma for short. It is miniscule, a ratio of 531441:524288 to be precise, However, the human ear is very sensitive to tiny differences in frequency and you might be able to hear the comma in tritone.
Pythagorean tuning could continue indefinitely, but now is the time to stop. For one thing, the frequency ratios numbers have become awfully big numbers. The real problem, however, is that Pythagorean tuning will never produce an octave. It does not matter how many times you continue to multiply the number three by itself, it will never be divisible by two to get an octave with a ratio of 2:1. There will always be a gap.
Pythagorean tuning does not extend beyond 12 notes. In practice, the augmented fourth and diminished fifth are treated as the same interval, a tritone, with a frequency ratio of 729:512 (the 1024:729 ratio is quietly dropped).
The direct consequence of treating two Pythagorean tritones as the same interval is that there will always be an interval of a fifth in Pythagorean tuning that is not a ratio of 3:2. This unfortunate beast is called a wolf fifth because it sounds like a howling wolf.
Pythagorean tuning has an elegant simplicity and a long pedigree. The end result is a tuning system containing 12 notes. This system has lasted for centuries and is a tuning option that is still available to you today.
Pythagorean tuning is sometimes cited as the reason there are 12 notes in an octave. Seems plausible enough, although, in truth, nobody really knows why. What we do know is that Pythagorean tuning set the precedent for the design of future tuning systems by focussing on frequency ratios within an octave.