Sharpness
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.
Table
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.
| -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| -3 | 6b | ||||||||||
| -2 | 4 | 11 | 18b | ||||||||
| -1 | 2 | 9 | 16 | 23 | 30b | ||||||
| 0 | 7 | 14 | 21 | 28 | 35 | 42b | |||||
| 1 | 5 | 12 | 19 | 26 | 33 | 40 | 47 | 54b | |||
| 2 | 3 | 10 | 17 | 24 | 31 | 38 | 45 | 52 | 59b | ||
| 3 | 1 | 8 | 15 | 22 | 29 | 36 | 43 | 50 | 57 | 64 | 71b |
| 4 | 6 | 13 | 20 | 27 | 34 | 41 | 48 | 55 | 62 | 69 | … |
| 5 | 11b | 18 | 25 | 32 | 39 | 46 | 53 | 60 | 67 | … | |
| 6 | 23b | 30 | 37 | 44 | 51 | 58 | 65 | 72 | … | ||
| 7 | 35b | 42 | 49 | 56 | 63 | 70 | … | ||||
| 8 | 47b | 54 | 61 | 68 | … | ||||||
| 9 | 52b | 59 | 66 | … | |||||||
| 10 | 64b | 71 | … |
See also
External links
- Notation Guide to EDOs 5-72: (paper by Kite Giedraitis introducing the concept)
- n-EDO Retuner plugin for Musescore 3.4+: uses sharpness to categorize EDOs for retuning
- Sagittal notation's Periodic Table of EDOs: arranges EDOs by their sharpness and penta-sharpness
