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The '''sharpness''' of an EDO is the number of steps it maps the [[apotome]] to; in other words, it is the difference between seven of its best approximation of [[3/2]] and four octaves.
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.
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.
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".
A sharp-0 edo is also known as a "perfect edo".


== See also ==
== 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.


[http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Kite's theory using the nomenclature of sharpness]: (perhaps introducing?)
{| class="wikitable center-all"
|+Sharpness value \ steps of a diatonic semitone
!|
!|-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
|…
|
|
|
|
|}


[https://github.com/euwbah/musescore-n-tet-plugins n-EDO Retuner plugin for Musescore 3.4+]: uses sharpness to categorize EDOs for retuning
== See also ==
* [[Alternative symbols for ups and downs notation]]


[https://sagittal.org/Periodic%20table%20of%20small%20EDOs%20large.png Sagittal notation's Periodic Table of EDOs]: arranges EDOs by their sharpness
== External links ==
* [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Kite's theory using the nomenclature of sharpness]: (perhaps introducing?)
* [https://github.com/euwbah/musescore-n-tet-plugins n-EDO Retuner plugin for Musescore 3.4+]: uses sharpness to categorize EDOs for retuning
* [https://sagittal.org/Periodic%20table%20of%20small%20EDOs%20large.png Sagittal notation's Periodic Table of EDOs]: arranges EDOs by their sharpness


[[Category:EDO theory pages]]
[[Category:EDO theory pages]]
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