Diesis

What it means

Diesis, a term from Greek, denotes very small pitch intervals smaller than a tempered half step. In just intonation such leftover intervals arise when intervals meant to lead back to the starting pitch are stacked but, because of their differing frequency ratios, do not add up exactly. A well-known example is the lesser diesis: three pure major thirds (ratio 5:4) stacked on top of each other fall slightly short of an exact octave, by the amount called the diesis.

In modern Italian, diesis simply means the sharp sign, independent of its historical, tuning-theory meaning. Both meanings share the same root but refer to different things: the theoretical leftover interval in tuning theory, and the practical accidental in notation.

Musical use

In tuning theory, the diesis explains why pure intervals cannot simply be stacked without contradictions arising: a pure third and a pure octave cannot both be maintained exactly in every key at once. Equal temperament resolves this problem by spreading such leftover amounts evenly across all twelve half steps, which makes every third deviate slightly from a pure third.

The diesis should be distinguished from the syntonic and the Pythagorean comma: all three are small leftover intervals in tuning theory, but they differ in origin and size. The Pythagorean comma arises from a chain of twelve pure fifths, the syntonic comma from the difference between a Pythagorean and a pure major third, and the lesser diesis from a chain of three pure major thirds.

Other names

In modern Italian, diesis is also the everyday word for a sharp accidental, regardless of the term's tuning-theory meaning.