Pythagorean Tuning produces an almost perfect set of fifths. The diagram below shows all of these 5ths at 702 cents with the exception of one, which has to be at 678 cents to close the circle of fifths. Looking at this relative to the Just Intonation this wolf note is 24 cents flat, a gap known as the Pythagorean comma. The other key feature of Pythagorean Tuning, is the sharpness of the 3rds. Again, relative to the natural ratio of 5/4, the Pythagorean 3rds are 22 cents sharp. This 22 cent difference is known as the syntonic comma (or the comma of Didymus)

The syntonic comma was seen as beneficial in medieval music where 3rds were not meant to be consonant. However, the baroque style demanded 3rds in their natural ratio, so the syntonic comma had to disappear. In the intervening centuries, the two basic methods have been to detune all of the 5ths equally, creating regular temperaments or to share the comma amongst a few of them, creating the irregular temperaments with 5ths of two different sizes.

The regular temperaments are usually specified by the fraction of the syntonic comma that is used to reduce each fifth. Thus, Meantone 1/4 comma reduces each 5th by 5.5 cents (22/4). It may be seen in the diagram that this creates a set of perfect 3rds, in the near keys, but renders the far keys useless. It may also be seen that wolf note 5th present in Pythagorean Tuning at 24 cents too small is now worse at 36 cents too large.

Is it possible to get the benefits of the perfect thirds without the drawback of the wolf 5th? The answer is yes, and the method is to flatten some of the 5th, but not all. Temperaments which use this solution ar known as 'irregular' since they lead to two different sizes of 5ths. One example, Kirnberger III uses the 1/4 comma of the previous example, but only the 5.5 cent flattening to 4 of the 5ths. The diagram shows that the wolf has disappeared, but at the expense of the 3rds which are still relatively consonant in the near keys, but drift back to the syntonic comma in the far keys.

As a last example, if the syntonic comma is spread evenly across all of the 5ths, we have Equal Temperament. The 5ths are now all 2 cents flat, indistinguishable from the natural 5th to the human ear. However, the 3rds are all 14 cents sharp; an acquired taste, but given that this has been the norm for all keyboard instruments since the early 20th century, probably one which we have now acquired.

In conclusion, the Pythagorean comma is an unwanted, but natural outcome of musical geometry. Its partner, the syntonic comma, started as a 'feature' but quickly became a 'bug' (as the computer programmers would have it) and the history of tunings is very much about the reduction of the syntonic comma to acceptable levels, with the removal of the Pythagorean comma being a useful by-product, if possible,