Diesis (scale theory)

The diesis or enharmonic diesis is the diminished second or inverse diminished second, whichever is positive, in the diatonic scale. An example of a diesis is the interval between C♯ and D♭. The diesis spans twelve perfect fifths, and is observed in any tuning whose perfect fifth is not the same as 12edo's. Notes related by the diesis are said to be enharmonic to each other.

Just intervals

If the fifth represents the just interval 3/2, the diesis or inverse diesis represents the Pythagorean comma. In meantone, the diesis approximates a class of commas separated by the syntonic comma (81/80), among which 128/125, the augmented comma, is notable for being tuned pure in quarter-comma meantone. Therefore diesis traditionally refers to the augmented comma by default. Other dieses according to this definition are

• 648/625, the major diesis, tuned pure in 1/3-comma meantone.
• 2048/2025, the diaschisma, tuned pure in 1/6-comma meantone.
• 32805/32768, the schisma, tuned pure in 1/12-comma meantone.

This is not to be confused with the related sense of the same term, for which a number of other intervals are named despite not being reached through twelve fifths.

Generalization

The diesis can be generalized to any mos scale as the mosdiesis, defined as |L - 2s|, i.e. the difference between a large step and two small steps. In terms of stepspan, it is usually the diminished mosstep or inverse diminished mosstep, whichever is positive. However, in nL 1s scales, it is the double-diminished mosstep or inverse thereof since the small step itself is diminished. Except for nL 1s scales, it is the diminished mosstep in soft (L:s < 2:1) scales and the inverse diminished mosstep in hard (L:s > 2:1) scales. It vanishes in basic (L:s = 2:1) scales.

See also

• Diesis (disambiguation page)
• Chroma