My Accidental Glyphs

[Fluffeh]

After a bit of procrastination I got around to finishing these.

I created these from scratch through defining smaller components by mathematical and/or geometrical principles and combining these simpler shapes into the complex shapes seen here. A good deal of the proportions revolve around √2, one of my favourite numbers in terms of mathematical aesthetics, but that's probably getting a little too detailed.

I've also attached a .pdf for high-resolution viewing.

All thoughts/comments/feedback/suggestions appreciated.

[tisimst]

Very nicely done, fluffeh! I'm interested to learn exactly how you formed them based on mathematical principles. Hope you can take a post or two to show how math helped you arrive at the final designs.


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[Fluffeh]

My design philosophy is quite simple: each component of the shape should be able to be defined by simple ratios or formulae.

To explain how I arrived at the quarter-tone, standard and three-quarter-tone sharp, we will first need to define a few constants.

Let 1 stave space = 250

We will now decrease this by one order of magnitude (10^−1 or 0.1):

250 × 10^−1 = 25 is the width of a staff line.

Now we bring my favourite number in: √2 (If you analyse the rest of the glyphs you'll find that it appears literally everywhere).

As the width of a stem should be thicker than that of a staff line, we will multiply it by √2:

25 × √2 = 25√2 ≈ 35.3554 is the width of a stem.

Now to mimic the wear of manual engraving tools, we will round the corners of the shapes.

25 ÷ 2 ÷ √2 = 6.25√2 ≈ 8.8388 is the radius of the rounding circle.

Let us consider that a standard sharp is 2× a quarter-tone sharp, and a three-quarter-tone sharp is 3× a quarter-tone sharp. Thus we can focus on one ‡ component knowing that these can be combined later to derive the other glyphs.

A standard sharp is defined as having the width of one stave space i.e. 250. Therefore one ‡ component is ½ of this.

250 ÷ 2 = 125 is the width of a ‡ component.

As a ‡ component is vertically symmetrical, we can find the space on either side of the vertical stem:

(125 − 25√2) ÷ 2 ≈ 44.8223

As for the width of the horizontal bar, it is said to be under the width of a beam, which is defined as half of a stave space i.e. 125.

125 ÷ √2 ≈ 88.3883

But because a staff line has width, if we align the horizontal bar with the centre of a staff line, there will be 25 ÷ 2 = 12.5 that we need to compensate for.

88.3883 + 12.5 = 100.8883 is the width of the horizontal bar.

From these figures we can now simply align the shapes together to create the sharp glyphs.

The slant is created by shearing the glyph such that the top-left and bottom-right of the horizontal bars align with the top and bottom of a staff line.

This also explains a phenomenon discussed in a previous thread, where the slants of the natural and sharp symbols sometimes appeared to be different. As a glyph gets narrower, the angle of the shear must increase to compensate for the distance. This effect can most clearly be seen in my various sharp symbols.

I hope you can understand this wall of text. The natural symbol pretty much follows the exact same principles, but the flat symbol is slightly trickier, so I'll get to that another time.

[erelievonen]

I like the look of your accidentals a lot. What you show and say about your design principles makes sense to me.

As someone who has to read microtonal notation regularly, I have some suggestions about the microtonal accidentals.

To me, the 1/4-sharp is too narrow, it would be easy to miss it inbetween other accidentals on a chord, or maybe even if standing on its own. To have enough visual impact, it should be made weighty (=wide) enough, definitely more than half of the regular sharp. (Myself, I regularly use the Maestro glyphs, and even there the 1/4-sharp feels a bit flimsy, even though is it wider than yours.) You say that the width proportion of the 1/4-sharp and the regular sharp is now 1:2. How would it look if the proportion was 1:√2 instead? Would you please try how that would look? Or maybe √2:2?

Your 3/4-sharp, on the other hand, is unnecessarily wide. In many contexts, such as a chord with many accidentals, it would be good for it to take less horizontal space. Could the widths of the regular sharp and the 3/4-sharp also be in a 1:√2 (or √2:2) proportion? Consequently, the vertical lines of the 3/4-sharp would be closer to each other in that same proportion.

With the varying slopes of the crossbars on the different sharps, I have no problem with.

About the flat signs, first a little digression: I thank you from the bottom of my heart for choosing this shape for the 3/4-flat. This is, in my opinion, the best shape for 3/4-flat, but it's impossible to find in any of the most popular music fonts. This situation may have led to the relative popularity of the "db" shape (which combines a 1/4-flat and a regular flat side-by-side, but with two separate stems). The combined shape with only one stem is superior in every way, takes less horizontal space, and cannot be misinterpreted as two separate accidentals under any circumstances. So thank you for using it!

My only other remark is an aesthetic one on the 3/4-flat sign. At the bottom tip of your 3/4-flat, there is a rectangular "bump" which interrupts the diagonal lines from both sides before they reach the bottom of the glyph. It would look better when the diagonal lines are allowed to continue uninterrupted until the lowest point of the glyph (be it rounded or not). In case my explanation is not clear enough, I attach an example.

I also like your double flat sign, where the two flats are joined and thus the whole glyph is less than twice as wide as a single flat.

[Fluffeh]

Thank you erelievonen for your thoughts! As I myself am not so well acquainted with microtonal music, receiving your feedback is invaluable!

I absolutely agree that the combined three-quarter-tone flat symbol is in every way superior to the separated one! It's worth noting that as I do intend to make this SMuFL-compliant, both variants (listed as the Couper variant and Zimmermann variant respectively) are included in the lookup.

I have made amendments to my glyphs as per your suggestions. The quarter-tone and three-quarter-tone sharps have been resized to a 1:√2 ratio in reference to the standard sharp, and the three-quarter-tone flat's lowermost vertex has had its rectangular protrusion removed. I was however, unable to make a separate set for √2:2 proportions, because 1:√2 and √2:2 in fact represent the same ratio! (1 × √2 = √2 : √2 × √2 = 2)



A .pdf of this image is once again attached for high-resolution viewing.

I would like to ask for your opinion on whether the quarter-tone flat should be made narrower (as found in the Sagittal-2 font) to decrease lateral confusion, or whether it should be kept at full width as to remain aesthetically consistent.

I will have several more microtonal accidentals posted soon! As a side note, I am also working on the Sagittal accidental glyphs here on the off chance there may be some microtonalists here interested in that specific notation.

[erelievonen]

Sorry for my late reply. We're on holidays at the moment...

I was however, unable to make a separate set for √2:2 proportions, because 1:√2 and √2:2 in fact represent the same ratio!

Of course, silly me. No wonder I wasn't sure which ratio would work better... I should have spent another 2 seconds thinking about it!

Your new 1/4-sharp looks very good! I totally like it.

The new 3/4-sharp is now less wide, but its internal proportions seem a bit strange. When I look closely, it looks like the distance between the left extremity of the glyph and the left edge of the first vertical line in the 3/4-sharp is greater than the corresponding distance in the regular sharp (and similarly on the left sides of the glyphs). This is odd; one would expect that distance in the 3/4-sharp to be the same or less compared to that in the regular sharp. I'm not sure exactly what the best proportions are. Maybe, keep this width but make the 3 vertical lines less close to each other, to satisfy the above-mentioned requirement?

The 3/4-flat symbol looks very good now. Also, I'm happy to see that this "Couper" 3/4-flat symbol is now included in SMuFL. It wasn't there when I last checked (a long time ago).

I would like to ask for your opinion on whether the quarter-tone flat should be made narrower (as found in the Sagittal-2 font) to decrease lateral confusion, or whether it should be kept at full width as to remain aesthetically consistent.

I think you should keep the 1/4-flat at full width. I don't think there's any serious lateral confusion issue here within such a small set of symbols as this. It's quite a different issue within the Sagittal system, which is a much more complex system and less familiar. The narrower 1/4-flat there reflects the aesthetics and needs of the Sagittal system, but in my opinion that should not be mixed in with other, already established systems.
(I've been following the development of Sagittal with interest, but haven't actively used it myself yet.)

I would also be curious to hear how you arrived at the flat signs and their proportions, if you have time to elaborate.

[Knut]

Those accidentals are very nice, Fluffeh!

This discussion is also very interesting, as deciding on accidental proportions is difficult and involves a lot of different considerations. The main ones, at least to me, being size, weight and proportion. I do not deal much with quarter tone accidentals in a live setting, I have to say. Nevertheless, here is my take on the subject:

Given the width of the double- and 3/4-flat symbols, I don't see why the 3/4-sharp symbol needs to be any narrower than a pair of merging conventional sharps. In most cases, the width of the latter will be less than the first two anyway. Nor do I see any need to shorten the diagonal strokes of the 1/4-sharp too much, for the same reason.

My own basic and quarter tone accidentals look as follows (Pardon the separated 3/4-flat. It was included here for the sake of convenience):

Skjermbilde 2016-07-24 kl. 01.33.15.png (49.89 KiB) Viewed 15603 times

In the case of the 3/4-sharp, making it narrower will inevitably decrease the distance between the vertical strokes, thereby altering the weight of the glyph. The 1/4-flat needs to be narrowed down a bit to make it proportionally pleasant, but too much will compromise readability.

For me, balancing the weight between all accidentals (and the font in general) is of the utmost importance to a pleasant reading experience, even if this means compromising on horizontal space efficiency in some cases. This has nothing to do with mathematics, and everything to do with perception.

[OCTO]

For me, balancing the weight between all accidentals (and the font in general) is of the utmost importance to a pleasant reading experience, even if this means compromising on horizontal space efficiency in some cases. This has nothing to do with mathematics, and everything to do with perception.

Agreed!

[Fluffeh]

Forgive me for the delay, but it seems I missed the part where it said OpenType only accepted integer values for coordinates... meaning I have to redo a bunch of glyphs as the rounding from the Adobe Illustrator importation hasn't worked out quite as nicely as I'd have liked it.

The new 3/4-sharp is now less wide, but its internal proportions seem a bit strange. When I look closely, it looks like the distance between the left extremity of the glyph and the left edge of the first vertical line in the 3/4-sharp is greater than the corresponding distance in the regular sharp (and similarly on the left sides of the glyphs). This is odd; one would expect that distance in the 3/4-sharp to be the same or less compared to that in the regular sharp. I'm not sure exactly what the best proportions are. Maybe, keep this width but make the 3 vertical lines less close to each other, to satisfy the above-mentioned requirement?

I suspect this may be due to an optical illusion where if the rounding "opens" instead of "closes" their perpendicular edges, the area enclosed is greater and thus the height appears bigger than it actually is, if you can understand that. I will make amendments as soon as I get my font repaired...

My own basic and quarter tone accidentals look as follows (Pardon the separated 3/4-flat. It was included here for the sake of convenience):

Your accidentals are very nice too!

A note on the three-quarter-tone sharp (which I've found out can be shortened to sesquisharp, from sesqui- (Latin prefix for "one-and-a-half") + sharp): as the horizontal bar is slanted at the same angle as the others, the top-left and bottom-right hover just below and above the staff line, creating a wedge that could be potentially undesirable, especially considering the conventions around avoiding such wedges with beams. This is one of the reasons I opted for variable horizontal beam slants.

I also find it quite interesting to see your take on the double sharp — this must be at least the seventh unique design I've seen so far! Whilst some people consider the treble clef the distinguishing mark of a font, perhaps it should actually be the double sharp. It could do with a slight weight increase though, as it seems a tad light next to the other accidentals.

[tisimst]

I also find it quite interesting to see your take on the double sharp — this must be at least the seventh unique design I've seen so far! Whilst some people consider the treble clef the distinguishing mark of a font, perhaps it should actually be the double sharp.

That's a very interesting assessment, though I'm not sure I feel the same way. I mean, there really are just as many unique designs of virtually any other glyph and I think it would be a stretch to say that the double sharp falls in the "most commonly used/seen" category. It's also very small compared to most other glyphs and the general design is not that complex. The treble clef, however, is not only much larger and more complex in design, but the clef is virtually the first thing seen when reading music. Hence, it is often looked at as the most distinguishing mark of a font since it is most likely to be noticed more than anything else (font-related, of course).

I must say, though, that I love seeing all these designs. You have crafted your accidentals very nicely and methodically. Next step is to put them out on the playing field to see how they do in practice.