Stein–Zimmermann–Gould notation

The Stein–Zimmermann–Gould (SZG) notation is a musical notation system that expands chain-of-fifths notation for edos with higher sharpness values as well as certain rank-2 temperaments. It combines Stein–Zimmermann accidentals and Gould arrows with consistent semantics, where:

• Standard and Stein–Zimmermann accidentals represent multiples of half chromatic semitones;
• 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 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 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 5, 12, 19, 26, and 33. Standard chain-of-fifths notation can be used since an up-arrow is exactly equivalent to a sharp.

Step offset	−2	−1	0	+1	+2
Symbol					

Sharp-2

Sharp-2 edos have a sharp that raises 2 steps. Edos of this category include 10, 17, 24, 31, 38, and 45. Stein–Zimmermann accidentals, Gould arrows, or a combination of both may be used.

Step offset	−4	−3	−2	−1	0	+1	+2	+3	+4
Symbol									

Sharp-3

Sharp-3 edos have a sharp that raises 3 steps. Edos of this category include 8, 15, 22, 29, 36, 43, and 50. This is first sharpness value where Gould arrows must be used.

Step offset	0	1	2	3	4	5	6	7
Sharp symbol								
Flat symbol								

In some cases, some notes or intervals may be best spelled with double arrows:

Step offset	0	1	2	3	4	5	6	7	8
Sharp symbol									
						
Flat symbol									
						

Sharp-4

Sharp-4 edos have a sharp that raises 4 steps. Edos of this category include 20, 27, 34, 41, 48, 55, and 62. This is first sharpness where the Stein–Zimmermann–Gould notation works in its full form.

Step offset	0	1	2	3	4	5	6	7	8	9
Sharp symbol										
Flat symbol										

Sharp-5

Sharp-5 edos have a sharp that raises 5 steps. Edos of this category include 32, 39, 46, 53, 60, 67, and 74.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol													
Flat symbol													

In some cases, some notes or intervals may be best spelled with triple arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol														
									
Flat symbol														
									

Sharp-6

Sharp-6 edos have a sharp that raises 6 steps. Edos of this category include 44, 51, 58, 65, 72, 79, and 86.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp symbol															
Flat symbol															

Attaching arrows to semi- and sesquisharps and flats is also another option instead of using double arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol														
Flat symbol														

Sharp-7

Sharp-7 edos have a sharp that raises 7 steps. Edos of this category include 56, 63, 70, 77, 84, 91, and 98.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol																		
Flat symbol																		

Sharp-8

Sharp-8 edos have a sharp that raises 8 steps. Edos of this category include 61, 68, 75, 82, 89, 96, and 103.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Sharp symbol																				
Flat symbol																				

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.

SMuFL curently supplies 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.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol													
Flat symbol													

Step offset	13	14	15	16	17	18	19	20	21	22
Sharp symbol										
Flat symbol										

Here is an example for sharp-12.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Sharp symbol																
Flat symbol																

Step Offset	16	17	18	19	20	21	22	23	24	25	26	27
Sharp Symbol												
Flat Symbol												

And here is an example for sharp-14.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol																		
Flat symbol																		

Step Offset	18	19	20	21	22	23	24	25	26	27	28	29	30	31
Sharp Symbol														
Flat Symbol														

Flat-1

For edos such as 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.

Step offset	−2	−1	0	+1	+2
Symbol					

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.

Step offset	−4	−3	−2	−1	0	+1	+2	+3	+4
Symbol									

Zero

The lower three multiples of 7 (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.

Step offset	−3	−2	−1	0	+1	+2	+3
Symbol							

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 59 and 66. Therefore, such edos are capped from their full strength.

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