Diesis
The diesis (/ˈdaɪəsɪs/ DY-ə-sis; plural dieses) is a small interval that has several related definitions. Most commonly, it refers to 128/125, the augmented comma a.k.a. lesser diesis, though rarely and if the context is clear, it can refer to 648/625, the diminished comma a.k.a. greater diesis.
History
The earliest usage of the term diesis was due to Philolaus in ancient Greek music to refer to an interval that is known as the limma today. Another usage, also in ancient Greek theory and notably used by Aristoxenus, would refer to a number of quartertone-sized intervals. In more recent times, Marchetto da Padova used it for fifth-tones, and finally, the modern diesis centered around 1\31 is due to Adriaan Fokker.
As an interval region
As an interval region, the diesis is a small melodic unit of about an augmented comma's size. The specific range varies considerably among musicians, but is generally agreed to be roughly 30–60 cents.
Just intervals
Some just intervals have been named according to this sense of diesis.
- Porcupine comma, or maximal diesis (49.2 ¢)
- Magic comma, or small diesis (29.6 ¢)
- Tetracot comma, or minimal diesis (27.7 ¢)
- 49/48, the large septimal diesis (35.7 ¢)
- 50/49, the small septimal diesis or septimal tritonic diesis (35.0 ¢)
This is not to be confused with the related sense of the same term introduced next, for which the major diesis (648/625) was named while being way wider than the "maximal diesis".
As a diatonic interval category
In the diatonic scale, the diesis, more specifically enharmonic diesis, is a diminished second or inverse diminished second, whichever is positive in size. 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 introduced in the section above, 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.
Normal diesis
The normal diesis (~38.7 ¢) is an interval size measure defined as one step of 31edo.