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 limna not the apotome to get penta-sharpness. In the table below, the sharp-0 edos and the pentasharp-0 edos are bolded.

Sharpness value \ penta-sharpness value

	-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

• Alternative symbols for ups and downs notation

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