Schisma

Two commas, one remainder

The Pythagorean comma of 23.46 cents arises when you stack twelve pure fifths and compare them with seven octaves. The syntonic comma (81:80, 21.51 cents) is the difference between the Pythagorean third (81:64) and the pure third (5:4). If you subtract the smaller from the larger, the schisma remains: 23.46 − 21.51 ≈ 1.95 cents. More exactly it follows from the ratios 32805:32768.

The equal-tempered fifth

The pure fifth (3:2) has 701.955 cents, the equal-tempered exactly 700. The difference is 1.955 cents, practically a schisma (1.954 cents). That is no accident: because twelve equal-tempered fifths hit seven octaves exactly, equal temperament distributes the Pythagorean comma evenly, one twelfth per fifth, and a twelfth of 23.46 cents is 1.955 cents. The page Cent gives the same number as “2 cents too low”.

Which part of which comma?

You often read that the schisma is “a twelfth of the syntonic comma”. That is not right: a twelfth of 21.51 cents would be 1.79 cents. The schisma is about a twelfth of the Pythagorean comma (23.46 : 12 = 1.955) and about an eleventh of the syntonic. The values are not equal: the schisma has exactly 1.9537 cents, a twelfth of the Pythagorean comma 1.9550 cents. If you calculate, check the ratios for yourself.

Common mistakes

• Wanting to hear the schisma: it lies far below the limit of perception.
• Confusing it with the syntonic comma: the syntonic is more than ten times larger.
• Believing “a twelfth of the syntonic comma”: correct is, approximately, a twelfth of the Pythagorean.