Harmo­ni­zation

Tutorial
by Rainer Erich Scheichelbauer

3 January 2022

Harmony is a good thing. The term harmonization can refer to various things.

Outline dilemma

PostScript and TrueType paths have one thing in common: they are outline formats. That means that if you want build any shape with such paths, you describe the edges (the ‘outline’) of the shape with on-curve points (nodes) and off-curve points (handles). That has an advantage and a disadvantage.

The good thing is that we have an incredibly large degree of freedom. We can pretty much describe any shape with an outline. Yes, there are a few edge cases that are a bit more difficult to master, but generally, outlines are a fantastically flexible way to get any job done.

The downside is a bit more tricky. No matter which script you look at, the shapes we need most in type design are strokes. Each script has its own history, but each has one or more typical writing tools that influenced its letter shapes: all kinds of pens, brushes, pencils, chalks, needles, you name it. The good thing about these tools is that by a single stroke of your arm, you automatically create all edges of the shape in one go. And, what’s even better, they all fit to each other consistently. Or, as we say, they ‘harmonize’ with each other. Because, well, they stem from the same stroke with the same tool, how could they not harmonize?

The trouble is now that we do not have this luxury of automatic harmonization when we describe the outline. In our outline formats, each part of the edge, each segment, is completely independent from any other part. So it is our tough job as type designers to establish the proper relationship between all the outline segments, the way they would have come out automatically if the shape had been drawn with the stroke of a tool. And establishing these relationships is called ‘harmonization’.

The takeaway is that harmonization is only an issue because we work with outlines.

Three steps of harmonization

Theoretically, there are many ways to get your outlines to be in harmony. Here is one method that continues from what we discussed about good path-drawing practices, and will work for most scenarios.

Establishing harmony across all segments of a shape has three steps, and each step builds upon the previous step:

  1. Point placement: opposing on-curves nodes relate to each other, opposing off-curve handles point more or less in the same direction
  2. Inside vs. outside curve fitting: inside curve segments have a higher relative handle length than opposing segments on the outside
  3. Segment-to-segment harmonization: transition from a curve segment to an adjacent curve or line segment

Let me explain in detail.

Step 1: point placement

Let’s take a look at a stroke, e.g., the shoulder of a Latin n or h:

Node pairings

The placement of the green and blue on-curve points, a.k.a. ‘nodes’, adheres to the logic of the writing tool you are trying to immitate. You can see that they always come in opposing pairs on either side of an imaginary center stroke. If you connect them, the pairs appear to be more or less perpendicular to the center line.

However, they are not exactly perpendicular: the top pair (extremum points) is almost vertical, but with a slight tilt from bottom left to top right; and the pair between the straight and curvy parts of the shape (transition points) is almost horizontal, but with a slight tilt from bottom left to top right. This deviation is due to the writing tool we are immitating. For the classic Latin broad-nib pen held in the right hand at an angle of 20 to 40 degrees, however abstracted your design may be, you will get these tilts between pairs of extremum points or pairs of transition points. Because we assume continutity of tool usage, it is always the same tilt we can expect: in this case we always get a bottom left to top right tilt, no matter whether it is a vertical or horizontal node pair. For other tools, the tilt may be the other way around.

Of course, type design is not calligraphy. That means that we have a certain degree of freedom, so take this with a grain of salt. Some pairings will be normalized more than others. Take for instance the bottom two nodes on the baseline: since the style of this typeface demands a horizontal ending on the baseline, the two nodes will have to be completely horizontal. On the other hand, you will have pairings with a more severe tilt, perhaps due to an optical correction. And some pairs will even tilt the opposite way in order to make a curve fit. These are little tricks we take advantage of in order to make the appearance of the typeface more consistent in its intended size.

One detail I want you to pay attention to is how I placed the leftmost point pair. Since they will be in the overlap with the vertical stem, you might argue that their exact position will not matter that much. And you’d be right, at first sight. However, placing them roughly perpendicular to the center line will make the following steps easier to manage. Read on.

Handle directions

Now let’s focus on the path segments, the black lines between the nodes. You can see that you have corresponding segments on either side: each inside segment has an opposing outside segment of the same type. So, a line segment on one side will typically demand a line segment on the otherside, a curve segment will have a corresponding curve segment on the opposing side.

What’s more, the angles will be (roughly) the same. Ornamentats and calligraphic effects like tapering left aside, opposing line segments will face into the same direction, i.e., both vertical or both horizontal, or both at roughly the same angle. Again, some deviation will be caused by the logic of the tool or by optical corrections.

This is also true for curve segments, of course, just that in their case, we can focus on the angle of the grey off-curve points (a.k.a. ‘handles’, ‘Bézier control points’, ‘BCPs’). A horizontal or vertical handle will typically have an opposing horizontal or vertical handle as well, assuming we have a classic arrangement with extremum points. The angles of the diagonal handles depend on the logic of the tool, that is why the leftmost handles in our example spread apart a bit.

But still, roughly, you can say, they point in the same direction. You will not see one vertical and the other horizontal, unless you are doing an experimental design with a design principle that would demand such a placement. But that, clearly, goes beyond the scope of this tutorial.

Step 2: curve fitting

So much for the placement of on-curves and the direction of off-curves, but what about the handle length? Relative handle length is sometimes (imprecisely) referred to as the curvature of a curve segment. To recapitulate, 100% is the maximum length of a handle, because that is where the handles intersect. If you exceed that length, the curve segment will inflect, meaning it will bend into different orientations, causing cusps and false extremums, and usually you do not want that. But actually, 100% is more than enough in real-life type design practice. You never need that much unless you want to do a revival of Eurostile. In the very most designs, average handle lengths stay well below 80%.

On the other end of the spectrum, completely retracted handles (0% length) make no sense, because then you might just as well use a line segment instead. The next neuralgic percentage is 55%, which is the handle length we need for an ellipse or circle. Typically, in type design, we need something longer than that. Even extremely geometric designs start counting at 58%. Why? Good question, let’s have a look:

The colorful curvature visualisation is provided by the Speed Punk plug-in, available for free in Window > Plugin Manager. Looking, there are a couple of things to note:

  • Below 55%: curvatures cause a ‘hang-through’, and the curvature peaks move to the edges of the segment. Peaks mean more turning, ‘more steering wheel’, if you allow the comparison.
  • 55%: the curvature comb is pretty much the same amplitude everywhere, this is the arrangement that is closest to a circle.
  • Above 55%: the curvature peak moves into the middle of the segment. The longer the handles are, the flatter the segment becomes at its edges, and the curvier it will be in its middle.
  • 100%: at maximum handle lengths, note that the curvature flattens out completely at the ends of the segment.

A quick measurement of (normalized) balanced handle lengths in modern multi-script designs shows that almost all handle lengths on 90° curve segments end up between 58% and 76–78%. In other words, this is typically the range you want to aim for. So it is a good idea to set these extremes as our values in the Fit Curve palette:

Adjust for your designs as needed, of course. These values will not fit for every design, but from experience, they will fit surprisingly often. I remember how disappointed I was when I first found out that most curvatures in type design are located in such a small range. Sigh. But then again, it goes to show how much influence the handle length has.

You will ask, but what has this to do with harmonization? Easy. Since, at least in this method, each curve segment will have a corresponding opposing curve segment, we need to establish a relationship between the curvatures of the segment pairs. In other words we need to harmonize the curvatures of opposing curve segments. And here is the deal if you want to match inner and outer curves: the smaller inside curve needs higher relative handle lengths; the corresponding larger outside curve needs smaller relative handle lengths. Let that sink in.

This presupposes that point placement and handle orientation have been chosen as laid out in the previous step. But if you have done that and set your Fit Curve values in a sensible way, you can use the keyboard short cuts for the eight steps of Fit Curve: Ctrl-Opt-1 through 8. There are three scenarios:

  • In very thin weights ranging from Hairline to Ultralight, the inside and outside curves are practically the same. So you can get away with using the same Fit Curve value for both segments. So, you could select handles from both segments, and press Ctrl-Opt-5 to set the curvature for both inner and outer curve at the same time.
  • In normal weights from Light to Bold, you will have a small discrepancy between inner and outer curves. It depends on your precise Fit Curve settings of course, but in typical ranges (e.g., 58% and 76%), you will find that one step difference will typically suffice. So, select the inner curve, press Ctrl-Opt-5, then select the outer curve and press Ctrl-Opt-4.
  • In very bold weights like Extrabold, Black and Extrablack, the discrepancy in curvature between inner and outer curves tends to be bigger, and usually, you will not need the lowest values anymore. You can accommodate for that either by raising the Fit Curve values accordingly, e.g., to 64% and 78%. Or, you make sure you have two or even three steps of difference between inner and outer curves, e.g., Ctrl-Opt-8 on the inside and Ctrl-Opt-6 on the outside.

That’s it, you have harmonized inside and outside curves with each other.

Step 3: segment-to-segment transition

So far, I went on about harmonizing opposing items, i.e., points and segments on either side of the imaginary center stroke. However, when type designers use the term ‘harmonization’, they usually mean visual continuity of curvature from one curve segment to the next.

Technically, each curve segment is a world in itself and knows nothing of the adjacent segments. So, it is not unlikely that we end up with a situation like this:

Now, you can achieve a smooth transition in three ways:

  1. Move the on-curve: hold down the Option key and move the green node between the two handles towards the ‘flat’ segment. This method is applied by the Green Harmony filter.

  2. Move the off-curves: select both grey handles and move them towards the ‘curvy’ segment. In essence, the opposite of the on-curve moving described above. The advantage is that you can keep the position of the extremum point if that matters to you. The downside is that the shape is changed quite a lot. This harmonization mechanism is applied by SuperTool’s Edit > Harmonize function and the Grey Harmony plug-in.

  3. A little bit of both: if you do a bit of both wisely, you can keep the deterioration of the shape to an absolute minimum.

G0, G1, G2 and G3 continuity

The harmonization I have shown you above is often referred to as ‘G2 continuity’. G2 stands for geometric continuity by means of equating the 2nd derivatives of both curves in their connecting point. Whoa, sounds too techy? The second derivative is the amplitude (or color) of the Speed Punk visualisation. And making them the same on either side of the green point means you are establishing G2 harmonization.

But wait, if there is G2, there are probably other G’s as well, right?

Actually, yes. Here is a quick rundown:

  • G0 a.k.a. positional or point continuity: the two (curve or line) segments meet in the same position. The zero means there is no derivative, so all that counts is the same point coordinate. Any corner will do. In other words, any blue node between two segments creates G0:

  • G1 a.k.a. tangency, tangent or angular continuity: the two segments have the same first derivative, i.e., the same tangent in their meeting point. They have the same direction in their transition. Any green node will establish G1 continuity:

  • G2 a.k.a. curvature or radial continuity: the two segments have the same radius (of an imaginary inscribed circle) in their meeting point. Not only do they point in the same direction, but their curvature is the same. The curvature is the rate of change of direction. You could also say, the same steering wheel position on either side of the green point. Making the Speed Punk combs the same size on either side of the green node establishes G2 continuity:

  • G3 a.k.a. torsion or acceleration continuity: the third derivative is the same, meaning the change of curvature is continuous across the green node. If the Speed Punk combs develop continuously, you have G3 continuity. Sometimes, this is also referred to as ‘supersmooth’ or ‘extrasmooth’ transition among type designers. But honestly, there are only a few designs where stepping up to G3 continuity made much of a difference. So, I recommend G3 only for the most OC among you. Best way to achieve it is to get rid of the green point altogether, and perhaps reintroduce it by (Shift) clicking on the segment with the Draw tool (shortcut P). But if you can rebuild the curve without an on-curve point, it is guaranteed G3:

It goes on, of course. Mathematically you can also equate the 4th, the 5th or whatever derivative. But rest assured, the significance of G4 and above for type design is, ahem, limited. What you want between curves will be G2 continuity. And in the very most cases, approximate G2 continuity will suffice just nicely.

Special case: curve-to-line harmonization

Can we have a smooth transition between curve and line segments?

Maximized handles

Two problems: TT conversion and circumstance
Paths > Harmonise Curve to Line in the mekkablue scripts

Keep your sanity: consider this if it makes a difference in the intended font size. Not every curve-to-line transition needs to be 100% smooth. Ask yourself if it is smooth enough, at least for the type size you are designing the font for. If you have a high curvature (a high Fit Curve button), chances are that the curvatures towards the edges of the curve segment are pretty flat anyway, and maximizing the handle structure will not make much of a visible difference anymore.

Thus, maximizing makes sense if you make a (display) typeface for big sizes, and in those cases where the bumps are too visible

Useful scripts and plug-ins

Speed Punk by Yanone

SuperTool by Simon Cozens

Green Harmony and Grey Harmony by Alex Slobzheninov

Remix Tools by Tim Ahrens

The mekkablue scripts offer a few Paths > Tunnify and Paths > Harmonise Curve to Line. The Tunnify script balances handles, but tries to keep the overall curvature of the segment. And Harmonise Curve to Line does the opposite: it maximizes the curve handle that opposes the following line segment, thus harmonizing the transition from curve to line.