Sharpness - Revision history

FloraC: Undo (see talk)

2026-04-14T19:17:49Z

Undo (see talk)

Haidilao: Added sharp 10-12 edos larger than 72edo compared with Kite's book because some edos larger than 72 (such as 87,94,99) are also useful

2026-04-14T17:17:57Z

Added sharp 10-12 edos larger than 72edo compared with Kite's book because some edos larger than 72 (such as 87,94,99) are also useful

Show changes

FloraC: FloraC moved page Kite's thoughts on sharpness to Sharpness over a redirect without leaving a redirect: Public article not of anyone's own thought

2025-08-20T10:24:18Z

FloraC moved page Kite's thoughts on sharpness to Sharpness over a redirect without leaving a redirect: Public article not of anyone's own thought

← Older revision	Revision as of 10:24, 20 August 2025
(No difference)

TallKite: TallKite moved page Sharpness to Kite's thoughts on sharpness: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

2025-08-20T01:33:30Z

TallKite moved page Sharpness to Kite's thoughts on sharpness: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

← Older revision	Revision as of 01:33, 20 August 2025
(No difference)

Dave Keenan: /* External links */ Repaired broken link to Sagittal Periodic Table

2024-12-28T05:02:56Z

External links: Repaired broken link to Sagittal Periodic Table

← Older revision		Revision as of 05:02, 28 December 2024
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	* [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide to EDOs 5-72]: (paper by [[Kite Giedraitis]] introducing the concept)		* [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide to EDOs 5-72]: (paper by [[Kite Giedraitis]] introducing the concept)
	* [https://github.com/euwbah/musescore-microtonal-edo-plugin n-EDO Retuner plugin for Musescore 3.4+]: uses sharpness to categorize EDOs for retuning		* [https://github.com/euwbah/musescore-microtonal-edo-plugin 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 ({{nbhsp}}{{sagittal\| # }} = ) and penta-sharpness or limmanosity (EF = )		* [https://sagittal.org/#periodic-table Sagittal notation's Periodic Table of EDOs]: arranges EDOs by their sharpness ({{nbhsp}}{{sagittal\| # }} = ) and penta-sharpness or limmanosity (EF = )

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

ArrowHead294: /* Further generalizations */

2024-12-20T06:23:32Z

Further generalizations

← Older revision		Revision as of 06:23, 20 December 2024
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	=== Further generalizations ===		=== 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 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.		The concept can be generalized even further to 17fold-sharpness, 19fold-sharpness, etc.

ArrowHead294 at 06:23, 20 December 2024

2024-12-20T06:23:00Z

← Older revision		Revision as of 06:23, 20 December 2024
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	The '''penta-sharpness''' or <span id="limmanosity";>'''limmanosity'''</span> 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 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation\|Sagittal limma-fraction notation]].		The '''penta-sharpness''' or <span id="limmanosity";>'''limmanosity'''</span> 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 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation\|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".		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.		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''.		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''.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+Sharpness value \ penta-sharpness value		\|+ style="font-size: 105%;" \| Sharpness value \ penta-sharpness value
	~~! \|\| -2 \|\| -1 \|\| 0 \|\| 1 \|\| 2 \|\| 3 \|\| 4 \|\| 5 \|\| 6 \|\| 7 \|\| 8~~
	\|-		\|-
	! -3		! \|\| −2 \|\| −1 \|\| 0 \|\| 1 \|\| 2 \|\| 3 \|\| 4 \|\| 5 \|\| 6 \|\| 7 \|\| 8
			\|-
			! −3
	\| \|\| \|\| \|\| \|\| \|\| ''6b'' \|\| ''13b'' \|\| \|\| \|\| \|\|		\| \|\| \|\| \|\| \|\| \|\| ''6b'' \|\| ''13b'' \|\| \|\| \|\| \|\|
	\|-		\|-
	! -2		! −2
	\| \|\| \|\| \|\| \|\| ''4'' \|\| ''11'' \|\| ''18b'' \|\| \|\| \|\| \|\|		\| \|\| \|\| \|\| \|\| ''4'' \|\| ''11'' \|\| ''18b'' \|\| \|\| \|\| \|\|
	\|-		\|-
	! -1		! −1
	\| \|\| \|\| \|\| 2 \|\| 9 \|\| ''16'' \|\| ''23'' \|\| ''30b'' \|\| \|\| \|\|		\| \|\| \|\| \|\| 2 \|\| 9 \|\| ''16'' \|\| ''23'' \|\| ''30b'' \|\| \|\| \|\|
	\|-		\|-

BudjarnLambeth: /* See also */ Add scale tree

2024-12-20T04:29:34Z

TallKite: added dodeca-sharpness

2024-12-18T23:20:38Z

added dodeca-sharpness

← Older revision		Revision as of 23:20, 18 December 2024
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	The '''sharpness''' of an [[EDO\|edo]] is the number of ~~steps~~ 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.		The '''sharpness''' of an [[EDO\|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.		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.
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	\| \|\| \|\| \|\| \|\| ''64b'' \|\| ''71'' \|\| … \|\| \|\| \|\| \|\|		\| \|\| \|\| \|\| \|\| ''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 ==		== See also ==

Dave Keenan: Changed "eefness" to "limmanosity" as it changed in the Sagittal limma-fraction documentation, which is now up on the wiki.

2024-10-20T21:56:42Z

Changed "eefness" to "limmanosity" as it changed in the Sagittal limma-fraction documentation, which is now up on the wiki.

← Older revision		Revision as of 21:56, 20 October 2024
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	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 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 <span id="~~eefness~~";>'''~~eefness~~'''</span> 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 ~~derives from the limma being notated as the interval E-F and~~ is used by the [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation\|Sagittal limma-fraction notation]].		The '''penta-sharpness''' or <span id="limmanosity";>'''limmanosity'''</span> 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 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation\|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".		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".
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	* [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide to EDOs 5-72]: (paper by [[Kite Giedraitis]] introducing the concept)		* [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide to EDOs 5-72]: (paper by [[Kite Giedraitis]] introducing the concept)
	* [https://github.com/euwbah/musescore-microtonal-edo-plugin n-EDO Retuner plugin for Musescore 3.4+]: uses sharpness to categorize EDOs for retuning		* [https://github.com/euwbah/musescore-microtonal-edo-plugin 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 and penta-sharpness		* [https://sagittal.org/Periodic%20table%20of%20small%20EDOs%20large.png Sagittal notation's Periodic Table of EDOs]: arranges EDOs by their sharpness ({{nbhsp}}{{sagittal\| # }} = ) and penta-sharpness or limmanosity (EF = )

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

← Older revision		Revision as of 04:29, 20 December 2024
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	* [[Alternative symbols for ups and downs notation]]		* [[Alternative symbols for ups and downs notation]]
	* [[User:Xenllium/Xenllium's microtonal notation\|Xenllium's microtonal notation]]		* [[User:Xenllium/Xenllium's microtonal notation\|Xenllium's microtonal notation]]
			* [[Scale tree]]

	== External links ==		== External links ==