Diesis (scale theory): Difference between revisions
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The '''diesis''' | 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 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. | ||
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 [[comma]]s 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]]. | * [[648/625]], the major diesis, tuned pure in [[1/3-comma meantone]]. | ||
* [[2048/2025]], the diaschisma, tuned pure in [[1/6-comma meantone]]. | * [[2048/2025]], the diaschisma, tuned pure in [[1/6-comma meantone]]. | ||
* [[32805/32768]], the schisma, tuned pure in 1/12-comma meantone. | * [[32805/32768]], the schisma, tuned pure in 1/12-comma meantone. | ||
== Generalization == | |||
The diesis can be generalized to any [[mos scale]] as the '''mosdiesis''', defined as {{nowrap| {{!}}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 ''n''L 1s scales, it is the double-diminished mosstep or inverse thereof since the small step itself is diminished. Except for ''n''L 1s scales, it is the diminished mosstep in [[soft]] {{nowrap|(L:s < 2:1)}} scales and the inverse diminished mosstep in [[hard]] {{nowrap|(L:s > 2:1)}} scales. It vanishes in basic {{nowrap|(L:s {{=}} 2:1)}} scales. | |||
== See also == | == See also == | ||
Revision as of 17:22, 28 April 2025
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 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.
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.
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.