Sharpness: Difference between revisions

The '''sharpness''' of an [[edo]] is the number of steps it maps the apotome ([[2187/2048]]) to; in other words, it is the difference between seven of its best approximation of [[3/2]] and four octaves.

For example, [[12edo]] maps the apotome to one step; it has a sharpness of 1. We could say it is a sharp-1 edo. On the other hand, [[17edo]] maps the apotome to two steps, so it is a sharp-2 edo.

Some edos, such as [[16edo]], have fifths flat enough that the apotome is mapped to a negative number of steps. Since 16edo has the apotome mapped to −1 step, it is a flat-1 edo.

A sharp-0 edo is also known as a "perfect edo".

The sharpness of an edo has implications for the heptatonic fifth-generated notation of that edo. For example, all sharp-1 edos (5, 12, 19, 26...) can be notated conventionally with just 7 letters and #/b. Another example: the half-sharp and half-flat accidentals are applicable to an edo only if its sharpness is an even number.

Below is a table showing the characteristics of each edo up to 72 in the context of traditional fifth-generator heptatonic ups and downs notation. Each row represents the steps of a chromatic semitone. Each column represents the steps of a diatonic semitone (limma, [[256/243]]), located between E–F and B–C. If one's notation were pentatonic instead of heptatonic, the concept of sharpness would be applied to the limma not the apotome to get penta-sharpness. In the table below, the sharp-0 edos and the pentasharp-0 edos are bolded.