Sharpness
The sharpness of an edo is the number of edosteps to which it maps the chromatic semitone aka 3-limit augmented unison aka apotome (2187/2048). 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, thus 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 edo, such as 16edo, have fifths flat enough that the apotome is mapped to a negative number of steps. Since 16edo maps the apotome to −1 step, it is a flat-1 edo. 11edo is a flat-2 edo.
A sharp-0 edo (7, 14, 21, etc.) 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.
The penta-sharpness or limmanosity of an edo is the number of steps to which it maps the diatonic semitone aka 3-limit minor 2nd aka limma (256/243). In other words, it's three octaves minus five of its best approximation of 3/2. If one's notation were pentatonic instead of heptatonic, the concept of sharpness would be applied to the limma not the apotome, hence the first name. The second name is used in documenting the Sagittal limma-fraction notation.
For example, 12, 17, and 22 are all pentasharp-1 edos, and 19 and 24 are both pentasharp-2 edos. A pentasharp-0 edo (5, 10, 15, etc.) is also known as a "pentatonic edo".
Using heptatonic fifth-generated notation with a penta-flat edo (e.g. 8, 13, or 18) has counter-intuitive results. The minor 2nd is descending, the major 2nd is wider than the minor 3rd, the 4th is narrower than the major 3rd, etc. One solution is to use the second best 5th, e.g. 13b or 18b.
Below is a table showing each edo up to 72, with sharpness increasing top to bottom and penta-sharpness increasing left to right. The sharp-0 edos and the pentasharp-0 edos are bolded. Dual-fifth edos fit for subset notation are in italic.
| −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| −3 | 6b | 13b | |||||||||
| −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 | … |
Further generalizations
The concept of sharpness can be generalized further to dodeca-sharpness, which is the number of edosteps that the pythagorean comma maps to. For example, 17, 29, and 41 are dodeca-sharp-1 edos, while 19, 31, and 43 are dodeca-flat-1 edos.
The concept can be generalized even further to 17fold-sharpness, 19fold-sharpness, etc.
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 or limmanosity (EF = )
