View source for Stein–Zimmermann–Gould notation

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. 

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]]. 

== Symbol set ==
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 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 a chromatic semitone. They apply to edos of even sharpness values only.  

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 ==
A usage guide for edos of sharpness value below 8 will be provided here. 

=== 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-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. 
{{Sharpness-sharp2-szg}}

=== Sharp-3 ===
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}}

In some cases, some notes or intervals may be best spelled with double arrows:
{{Sharpness-sharp3-extended-szg}}

=== Sharp-4 ===
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}}

=== Sharp-5 ===
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}}

In some cases, some notes or intervals may be best spelled with triple arrows:
{{Sharpness-sharp5-extended-szg}}

=== Sharp-6 ===
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}}

Attaching arrows to semi- and sesquisharps and flats is also another option instead of using double arrows:
{{Sharpness-sharp6-qt-szg}}

=== Sharp-7 ===
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}}

=== Higher sharpness values ===
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.

{{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. 

Here is an example of a notation scheme for sharp-10 edos. 
{{Sharpness-sharp10-qt1-szg}}

Here is an example for sharp-12. 
{{Sharpness-sharp12-qt1-szg}}

And here is an example for sharp-14. 
{{Sharpness-sharp14-qt1-szg}}

=== 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}}

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).  

=== Flat-2 ===
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}}

=== Zero ===
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}}

== Limitations ==
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.   

Conclusively, this set of symbols still covers most edos up to 72. 

{{Navbox notation}}

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