Diesis (scale theory): Difference between revisions

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The '''diesis''' ~~(precisely, the~~ '''~~mosdiesis~~'''~~) of a [[MOS scale]]~~ is ~~the interval that is the difference between two generalized enharmonic notes, i.e.~~ the diminished ~~mosstep~~ or inverse diminished ~~mosstep~~, whichever is positive~~. In~~ [[~~soft~~]] ~~{{nowrap|(L:s < 2:1)}} scales, it~~ is the ~~diminished mosstep~~. In [[~~hard~~]] ~~{{nowrap|(L:s > 2:1)}} scales, it~~ is the ~~inverse diminished mosstep, since the diminished mosstep itself is negative. In basic {{nowrap|(L:s {{=}} 2:1)}} scales, it~~ [[~~vanish~~]]es.

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.

~~The diesis or '''enharmonic diesis''' in the [[diatonic scale]] is the diminished second or inverse diminished second, whichever is positive, such as the interval between C♯ and D♭.~~ If the fifth represents the just interval [[3/2]], the diesis or inverse diesis represents the [[Pythagorean comma]]. In [[meantone]], it 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

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]].

* [[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 {{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 ==

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-The '''diesis''' (precisely, the '''mosdiesis''') of a [[MOS scale]] is the interval that is the difference between two generalized enharmonic notes, i.e. the diminished mosstep or inverse diminished mosstep, whichever is positive. In [[soft]] {{nowrap|(L:s &lt; 2:1)}} scales, it is the diminished mosstep. In [[hard]] {{nowrap|(L:s &gt; 2:1)}} scales, it is the inverse diminished mosstep, since the diminished mosstep itself is negative. In basic {{nowrap|(L:s {{=}} 2:1)}} scales, it [[vanish]]es.
+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.
-The diesis or '''enharmonic diesis''' in the [[diatonic scale]] is the diminished second or inverse diminished second, whichever is positive, such as the interval between C♯ and D♭. If the fifth represents the just interval [[3/2]], the diesis or inverse diesis represents the [[Pythagorean comma]]. In [[meantone]], it 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
+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]].
 * [[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 {{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 ==