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Stein–Zimmermann–Gould notation: Difference between revisions

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This article discusses about an alternative set of symbols based on current practice of microtonal music with some experimental extrapolation. The notation is, in essence, [[Ups and Downs Notation|ups and downs notation]] with a different look.  
The '''Stein–Zimmermann–Gould''' ('''SZG''') '''notation''' is a [[musical notation]] system that expands [[chain-of-fifths notation]] for [[edo]]s with higher [[sharpness]] values as well as certain [[rank-2 temperament]]s. It combines Stein–Zimmermann accidentals and Gould arrows with consistent semantics, where:
* Standard and Stein–Zimmermann accidentals represent multiples of half [[chromatic semitone]]s;
* Gould arrows represent indefinite small modifications, commonly edosteps.  


Alternative symbols may be useful for the following reasons:
This notation started as a practical way to notate edos in [[MuseScore]], first suggested by [[Flora Canou]] around 2020 and was adopted by the [https://github.com/euwbah/musescore-microtonal-edo-plugin Microtonal plugin for Musescore 3.4+]. The use of arrows to represent edosteps was inspired by [[Kite's ups and downs notation]].  
# One may prefer a more conventional look of the score;
# The up and down symbols may not be quite accessible in computer-aided score typing.
All symbols proposed in this article are available in [[MuseScore|MuseScore 3]], and is used by the [https://github.com/euwbah/musescore-microtonal-edo-plugin Microtonal plugin for Musescore 3.4+]. (Feel free to address others.)


== Symbol set ==
== Symbol set ==
The symbol set comprises traditional accidentals (sharps, flats, and naturals), Stein-Zimmermann quartertone accidentals (semisharps and semiflats), and up and down arrows in Gould arrow quartertone symbols or part of [[Helmholtz-Ellis notation|Helmholtz-Ellis just intonation accidentals]].   
The symbol set comprises standard accidentals (sharps, flats, and naturals), Stein–Zimmermann quartertone accidentals (semisharps and semiflats) and Gould arrows. When Gould arrows are not available, one may borrow the similar-looking [[Helmholtz–Ellis notation|Helmholtz–Ellis just intonation accidentals]] for prime 5.   


The traditional accidentals, as specified in ups and downs notation, modify the note by the sharpness value, which equals the steps of a chromatic semitone (apotome, [[2187/2048]]).   
The standard accidentals modify the note by multiples of a chromatic semitone (Pythagorean apotome, [[2187/2048]]).   


The Stein-Zimmermann quartertone accidentals modify the note by half the sharpness value. They apply to [[edo]]s of even sharpness value only.   
The Stein–Zimmermann quartertone accidentals modify the note by half a chromatic semitone. They apply to edos of even sharpness values only.   


The Gould arrow quartertone symbols are up and down arrows attached to sharps, flats or naturals. They modify the note by one step. The syntonic comma in Helmholtz-Ellis just intonation accidentals looks very similar, and ''can be used interchangeably''. 
The Gould arrow accidentals are arrows attached to any of above, and modify the note by an indefinite small amount, but most commonly one step of the edo.  


== Usage guide for each sharpness value ==
== Usage guide for each sharpness value ==
{{See also|Sharpness}}
A usage guide for edos of sharpness value below 8 will be provided here.  
A usage guide for edos of sharpness value below 8 will be provided here.  


=== Sharp 1 ===
=== Sharp-1 ===
Sharp-1 edos have a sharp that raises 1 step. Edos of this category include {{EDOs| 5, 12, 19, 26, and 33 }}. Standard chain-of-fifths notation can be used since an up-arrow is exactly equivalent to a sharp.
{{Sharpness-sharp1-szg}}


Sharp-1 edos have a sharp that raises 1 step. Edos of this category include {{EDOs|5, 12, 19, 26}}, etc.
=== Sharp-2 ===
 
Sharp-2 edos have a sharp that raises 2 steps. Edos of this category include {{EDOs| 10, 17, 24, 31, 38, and 45 }}. Stein–Zimmermann accidentals, Gould arrows, or a combination of both may be used.  
Traditional notation can be used since an up is exactly equivalent to a sharp.
{{Sharpness-sharp2-szg}}
 
{| class="wikitable center-all"
!|Step offset
| +2
| +1
| 0
| −1
| −2
|-
!|Symbol
|[[File:accidental doublesharp.png]]
|[[File:accidental sharp.png]]
|[[File:accidental natural.png]]
|[[File:accidental flat.png]]
|[[File:accidental doubleflat.png]]
|}
 
=== Sharp 2 ===
Sharp-2 edos have a sharp that raises 2 steps. Edos of this category include {{EDOs|10, 17, 24, 31}}, etc.
 
As are commonly seen in 24edo and 31edo, semisharps and semiflats could be used for a substitution of the tilde.  
 
{| class="wikitable center-all"
!|Step offset
| +4
| +3
| +2
| +1
| 0
| −1
| −2
| −3
| −4
|-
!|Symbol
|[[File:accidental doublesharp.png]]
|[[File:accidental sesquisharp.png]]
|[[File:accidental sharp.png]]
|[[File:accidental semisharp.png]]
|[[File:accidental natural.png]]
|[[File:accidental semiflat.png]]
|[[File:accidental flat.png]]
|[[File:accidental sesquiflat.png]]
|[[File:accidental doubleflat.png]]
|}


=== Sharp-3 ===
=== Sharp-3 ===
Sharp-3 edos have a sharp that raises 3 steps. Edos of this category include {{EDOs|8, 15, 22, 29}}, etc.  
Sharp-3 edos have a sharp that raises 3 steps. Edos of this category include {{EDOs| 8, 15, 22, 29, 36, 43, and 50 }}. This is first sharpness value where Gould arrows must be used.  
 
{{Sharpness-sharp3-szg}}
This is where you ''really'' want to use ups and downs.  


{| class="wikitable center-all"
In some cases, some notes or intervals may be best spelled with double arrows:
!|Step offset
{{Sharpness-sharp3-extended-szg}}
| 7
| '''6'''
| 5
| 4
| '''3'''
| 2
| 1
| '''0'''
|-
!|Sharp symbol
|[[File:accidental doublesharp up.png]]
|[[File:accidental doublesharp.png]]
|[[File:accidental doublesharp down.png]]
|[[File:accidental sharp up.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sharp down.png]]
|[[File:accidental natural up.png]]
| rowspan="2" |[[File:accidental natural.png]]
|-
!|Flat symbol
|[[File:accidental doubleflat down.png]]
|[[File:accidental doubleflat.png]]
|[[File:accidental doubleflat up.png]]
|[[File:accidental flat down.png]]
|[[File:accidental flat.png]]
|[[File:accidental flat up.png]]
|[[File:accidental natural down.png]]
|}


=== Sharp-4 ===
=== Sharp-4 ===
Sharp-4 edos have a sharp that raises 4 steps. Edos of this category include {{EDOs|13, 20, 27, 34}}, etc.  
Sharp-4 edos have a sharp that raises 4 steps. Edos of this category include {{EDOs| 20, 27, 34, 41, 48, 55, and 62 }}. This is first sharpness where the Stein–Zimmermann–Gould notation works in its full form.  
 
{{Sharpness-sharp4-szg}}
A full combination of semisharps, semiflats, ups and downs looks very neat.  
 
{| class="wikitable center-all"
!|Step offset
| 9
| '''8'''
| 7
| 6
| 5
| '''4'''
| 3
| 2
| 1
| '''0'''
|-
!|Sharp symbol
|[[File:accidental doublesharp up.png]]
|[[File:accidental doublesharp.png]]
|[[File:accidental doublesharp down.png]]
|[[File:accidental sesquisharp.png]]
|[[File:accidental sharp up.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sharp down.png]]
|[[File:accidental semisharp.png]]
|[[File:accidental natural up.png]]
| rowspan="2" |[[File:accidental natural.png]]
|-
!|Flat symbol
|[[File:accidental doubleflat down.png]]
|[[File:accidental doubleflat.png]]
|[[File:accidental doubleflat up.png]]
|[[File:accidental sesquiflat.png]]
|[[File:accidental flat down.png]]
|[[File:accidental flat.png]]
|[[File:accidental flat up.png]]
|[[File:accidental semiflat.png]]
|[[File:accidental natural down.png]]
|}


=== Sharp-5 ===
=== Sharp-5 ===
Sharp-5 edos have a sharp that raises 5 steps. Edos of this category include {{EDOs|18, 25, 32, 39}}, etc.  
Sharp-5 edos have a sharp that raises 5 steps. Edos of this category include {{EDOs| 32, 39, 46, 53, 60, 67, and 74 }}.
 
{{Sharpness-sharp5-szg}}
Starting with sharp-5, which includes the famous [[53edo]], you'll have to use double ups and downs.


{| class="wikitable center-all"
In some cases, some notes or intervals may be best spelled with triple arrows:
!|Step offset
{{Sharpness-sharp5-extended-szg}}
| 12
| 11
| '''10'''
| 9
| 8
| 7
| 6
| '''5'''
| 4
| 3
| 2
| 1
| '''0'''
|-
!|Sharp symbol
|[[File:accidental doublesharp up2.png]]
|[[File:accidental doublesharp up1.png]]
|[[File:accidental doublesharp.png]]
|[[File:accidental doublesharp down1.png]]
|[[File:accidental doublesharp down2.png]]
|[[File:accidental sharp up2.png]]
|[[File:accidental sharp up1.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sharp down1.png]]
|[[File:accidental sharp down2.png]]
|[[File:accidental natural up2.png]]
|[[File:accidental natural up1.png]]
| rowspan="2" |[[File:accidental natural.png]]
|-
!|Flat symbol
|[[File:accidental doubleflat down2.png]]
|[[File:accidental doubleflat down1.png]]
|[[File:accidental doubleflat.png]]
|[[File:accidental doubleflat up1.png]]
|[[File:accidental doubleflat up2.png]]
|[[File:accidental flat down2.png]]
|[[File:accidental flat down1.png]]
|[[File:accidental flat.png]]
|[[File:accidental flat up1.png]]
|[[File:accidental flat up2.png]]
|[[File:accidental natural down2.png]]
|[[File:accidental natural down1.png]]
|}


=== Sharp-6 ===
=== Sharp-6 ===
Sharp-6 edos have a sharp that raises 6 steps. Edos of this category include {{EDOs|30, 37, 44, 51}}, etc.  
Sharp-6 edos have a sharp that raises 6 steps. Edos of this category include {{EDOs| 44, 51, 58, 65, 72, 79, and 86 }}.
{{Sharpness-sharp6-szg}}


{| class="wikitable center-all"
Attaching arrows to semi- and sesquisharps and flats is also another option instead of using double arrows:
!|Step offset
{{Sharpness-sharp6-qt-szg}}
| 14
| 13
| '''12'''
| 11
| 10
| 9
| 8
| 7
| '''6'''
| 5
| 4
| 3
| 2
| 1
| '''0'''
|-
!|Sharp symbol
|[[File:accidental doublesharp up2.png]]
|[[File:accidental doublesharp up1.png]]
|[[File:accidental doublesharp.png]]
|[[File:accidental doublesharp down1.png]]
|[[File:accidental doublesharp down2.png]]
|[[File:accidental sesquisharp.png]]
|[[File:accidental sharp up2.png]]
|[[File:accidental sharp up1.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sharp down1.png]]
|[[File:accidental sharp down2.png]]
|[[File:accidental semisharp.png]]
|[[File:accidental natural up2.png]]
|[[File:accidental natural up1.png]]
| rowspan="2" |[[File:accidental natural.png]]
|-
!|Flat symbol
|[[File:accidental doubleflat down2.png]]
|[[File:accidental doubleflat down1.png]]
|[[File:accidental doubleflat.png]]
|[[File:accidental doubleflat up1.png]]
|[[File:accidental doubleflat up2.png]]
|[[File:accidental sesquiflat.png]]
|[[File:accidental flat down2.png]]
|[[File:accidental flat down1.png]]
|[[File:accidental flat.png]]
|[[File:accidental flat up1.png]]
|[[File:accidental flat up2.png]]
|[[File:accidental semiflat.png]]
|[[File:accidental natural down2.png]]
|[[File:accidental natural down1.png]]
|}


=== Sharp-7 ===
=== Sharp-7 ===
Sharp-7 edos have a sharp that raises 7 steps. Edos of this category include {{EDOs|42, 49, 56, 63}}, etc.  
Sharp-7 edos have a sharp that raises 7 steps. Edos of this category include {{EDOs| 56, 63, 70, 77, 84, 91, and 98 }}.
{{Sharpness-sharp7-szg}}
 
=== Sharp-8 ===
Sharp-8 edos have a sharp that raises 8 steps. Edos of this category include {{EDOs| 61, 68, 75, 82, 89, 96, and 103 }}.
{{Sharpness-sharp8-szg}}


{| class="wikitable center-all"
=== Higher sharpness values ===
!|Step offset
Provided with more stacks of arrows, or the ability to attach arrows to demi- and sesqui-sharps and flats, edos of higher sharpness value can be notated in the same method as above.
| 17
| 16
| 15
| '''14'''
| 13
| 12
| 11
| 10
| 9
| 8
| '''7'''
| 6
| 5
| 4
| 3
| 2
| 1
| '''0'''
|-
!|Sharp symbol
|[[File:accidental doublesharp up3.png]]
|[[File:accidental doublesharp up2.png]]
|[[File:accidental doublesharp up1.png]]
|[[File:accidental doublesharp.png]]
|[[File:accidental doublesharp down1.png]]
|[[File:accidental doublesharp down2.png]]
|[[File:accidental doublesharp down3.png]]
|[[File:accidental sharp up3.png]]
|[[File:accidental sharp up2.png]]
|[[File:accidental sharp up1.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sharp down1.png]]
|[[File:accidental sharp down2.png]]
|[[File:accidental sharp down3.png]]
|[[File:accidental natural up3.png]]
|[[File:accidental natural up2.png]]
|[[File:accidental natural up1.png]]
| rowspan="2" |[[File:accidental natural.png]]
|-
!|Flat symbol
|[[File:accidental doubleflat down3.png]]
|[[File:accidental doubleflat down2.png]]
|[[File:accidental doubleflat down1.png]]
|[[File:accidental doubleflat.png]]
|[[File:accidental doubleflat up1.png]]
|[[File:accidental doubleflat up2.png]]
|[[File:accidental doubleflat up3.png]]
|[[File:accidental flat down3.png]]
|[[File:accidental flat down2.png]]
|[[File:accidental flat down1.png]]
|[[File:accidental flat.png]]
|[[File:accidental flat up1.png]]
|[[File:accidental flat up2.png]]
|[[File:accidental flat up3.png]]
|[[File:accidental natural down3.png]]
|[[File:accidental natural down2.png]]
|[[File:accidental natural down1.png]]
|}


=== Sharp-8 ===
{{W|SMuFL}} curently supplies [https://w3c.github.io/smufl/latest/tables/extended-stein-zimmermann-accidentals.html extended Stein-Zimmermann accidentals (U+E290–U+E29F)], which contains demi- and sesqui-sharps and flats with a single arrow.  
Sharp-8 edos have a sharp that raises 8 steps. Edos of this category include {{EDOs|54, 61, 68}}, etc.  


{| class="wikitable center-all"
Here is an example of a notation scheme for sharp-10 edos.
!|Step offset
{{Sharpness-sharp10-qt1-szg}}
| 19
| 18
| 17
| '''16'''
| 15
| 14
| 13
| 12
| 11
| 10
| 9
| '''8'''
| 7
| 6
| 5
| 4
| 3
| 2
| 1
| '''0'''
|-
!|Sharp symbol
|[[File:accidental doublesharp up3.png]]
|[[File:accidental doublesharp up2.png]]
|[[File:accidental doublesharp up1.png]]
|[[File:accidental doublesharp.png]]
|[[File:accidental doublesharp down1.png]]
|[[File:accidental doublesharp down2.png]]
|[[File:accidental doublesharp down3.png]]
|[[File:accidental sesquisharp.png]]
|[[File:accidental sharp up3.png]]
|[[File:accidental sharp up2.png]]
|[[File:accidental sharp up1.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sharp down1.png]]
|[[File:accidental sharp down2.png]]
|[[File:accidental sharp down3.png]]
|[[File:accidental semisharp.png]]
|[[File:accidental natural up3.png]]
|[[File:accidental natural up2.png]]
|[[File:accidental natural up1.png]]
| rowspan="2" |[[File:accidental natural.png]]
|-
!|Flat symbol
|[[File:accidental doubleflat down3.png]]
|[[File:accidental doubleflat down2.png]]
|[[File:accidental doubleflat down1.png]]
|[[File:accidental doubleflat.png]]
|[[File:accidental doubleflat up1.png]]
|[[File:accidental doubleflat up2.png]]
|[[File:accidental doubleflat up3.png]]
|[[File:accidental sesquiflat.png]]
|[[File:accidental flat down3.png]]
|[[File:accidental flat down2.png]]
|[[File:accidental flat down1.png]]
|[[File:accidental flat.png]]
|[[File:accidental flat up1.png]]
|[[File:accidental flat up2.png]]
|[[File:accidental flat up3.png]]
|[[File:accidental semiflat.png]]
|[[File:accidental natural down3.png]]
|[[File:accidental natural down2.png]]
|[[File:accidental natural down1.png]]
|}


Provided with more stacks of arrows, edos of higher sharpness value can be notated in the same method as above.
Here is an example for sharp-12.  
{{Sharpness-sharp12-qt1-szg}}


The rest will be discussed below.  
And here is an example for sharp-14.  
{{Sharpness-sharp14-qt1-szg}}


=== Flat-1 ===
=== Flat-1 ===
For edos such as {{EDOs| 9, 16, 23, and 30 }}, if you notate them as if their native antidiatonic scales were diatonic, you would find that the sharp actually ''lowers'' by one step. If one wishes to "translate" diatonic songs into these edos, this is useful.
{{Sharpness-flat1-szg}}


Flat-1 edos have a sharp that ''lowers'' 1 step. Edos of this category include {{EDOs|9, 16, and 23}}. To have a sharp that actually lowers the tone can be counter-intuitive, yet reasonable for the system. Regardless, you could just flip it around.
However, a much more intuitive solution is to swap the meaning of sharps and flats in regards to fifthspan (so that sharp still raises and flat still lowers), allowing the accidentals to more naturally notate these edos' native antidiatonic (in this case, the normal set of sharp-1 accidentals would be used).
 
{| class="wikitable center-all"
!|Step offset
| +2
| +1
| 0
| −1
| −2
|-
!|Symbol
|[[File:accidental doubleflat.png]]
|[[File:accidental flat.png]]
|[[File:accidental natural.png]]
|[[File:accidental sharp.png]]
|[[File:accidental doublesharp.png]]
|}


=== Flat-2 ===
=== Flat-2 ===
Flat-2 edos (virtually [[11edo]] only) have a sharp that ''lowers'' 2 steps. So besides the special flavor of the sharps and flats, there are also semisharps and semiflats to fill up the spaces between.  
Flat-2 edos (virtually [[11edo]] only), if you pretend their native antidiatonic scales are diatonic, have a sharp that ''lowers'' 2 steps. So besides the special flavor of the sharps and flats, there are also semisharps and semiflats to fill up the spaces between. It makes the most sense to notate them as subsets.  
 
{{Sharpness-flat2-szg}}
{| class="wikitable center-all"
!|Step offset
| +4
| +3
| +2
| +1
| 0
| −1
| −2
| −3
| −4
|-
!|Symbol
|[[File:accidental doubleflat.png]]
|[[File:accidental sesquiflat.png]]
|[[File:accidental flat.png]]
|[[File:accidental semiflat.png]]
|[[File:accidental natural.png]]
|[[File:accidental semisharp.png]]
|[[File:accidental sharp.png]]
|[[File:accidental sesquisharp.png]]
|[[File:accidental doublesharp.png]]
|}


=== Zero ===
=== Zero ===
Perfect edos (sharp-0 edos) are even more special in that the traditional accidentals cannot raise or lower the pitch, so the notes can only be modified by arrows. Edos of this category include {{EDOs|7, 14, 21, 28, and 35}.
The lower three multiples of 7 ({{EDOs| 7, 14, and 21 }}) are known as "perfect" or sharp-0 edos, since, by tempering out the Pythagorean apotome of [[2187/2048]], the traditional sharps and flats are redundant and cannot raise or lower the pitch. Here, the notes can only be modified by arrows. [[28edo]] and [[35edo]] also fall into this category using their native fifths, but they are better notated as subsets.
{{Sharpness-0-szg}}


{| class="wikitable center-all"
== Limitations ==
!|Step offset
Some edos have odd-numbered sharpness values 9 and above, and are difficult to notate due to the lack of support for the stacks of arrows required. Edos below 72 known to be impractical for this reason are {{EDOs| 59 and 66 }}. Therefore, such edos are capped from their full strength.  
| +3
| +2
| +1
| 0
| −1
| −2
| −3
|-
!|Symbol
|[[File:accidental natural up3.png]]
|[[File:accidental natural up2.png]]
|[[File:accidental natural up1.png]]
|[[File:accidental natural.png]]
|[[File:accidental natural down1.png]]
|[[File:accidental natural down2.png]]
|[[File:accidental natural down3.png]]
|}


== Limitations ==
Conclusively, this set of symbols still covers most edos up to 72.  
Some notes cannot be reached in edos of sharpness value higher than 9 due to MuseScore not providing the stacks of arrows required. Edos below 72 known to be impractical for this reason: 59, 66, 71. Therefore, such edos are capped from its full strength.  


Conclusively, this set of symbols still covers most edos up to 72. 
{{Navbox notation}}


[[Category:Notation]]
[[Category:Notation]]
[[Category:Ups and Downs Notation]]
[[Category:Ups and downs notation]]
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