Sharpness: Difference between revisions

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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]].

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''.

{| 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'' || || || ||

|-

! -2

! −2

| || || || || ''4'' || ''11'' || ''18b'' || || || ||

|-

! -1

! −1

| || || || 2 || 9 || ''16'' || ''23'' || ''30b'' || || ||

|-

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=== 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.

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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)

* [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]]

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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]].
-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''.
 {| 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
+! || &minus;2 || &minus;1 || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8
+|-
+! &minus;3
 | || || || || || ''6b'' || ''13b'' || || || ||
 |-
-! -2
+! &minus;2
 | || || || || ''4'' || ''11'' || ''18b'' || || || ||
 |-
-! -1
+! &minus;1
 | || || || 2 || 9 || ''16'' || ''23'' || ''30b'' || || ||
 |-
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 === 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.
@@ Line 77: / Line 78: @@
 * [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://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]]