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I have a question (or two) about weight interpolation schemes.
My question regards, in part, Luc(as) de Groot's method. It seems to be, more or less, Palladio's 'geometric mean', or the maths that govern equal temperament of fixed-pitch musical instruments. So I'm wondering if the compulsion to use that particular mathematical series is based on something more than just convenience or mathematical ease. Is there something about it that I'm missing?
I'm certainly not the most prolific or accomplished type designer, but my own personal work-flow doesn't mesh with the geometric mean method. That method seems to demand that you draw 2 weights, and let the geometric mean determine the third (and the weights amongst them). On the few occasions I draw fonts, I draw the 'medium' weight, then the lightest and darkest weights. And I'd like to interpolate intermediate weights from there.
It seems that if you have three stem weights already drawn, and know how many 'steps' you'd like between them, you can (fairly) easily determine a quadratic equation that defines the intermediate weights.
We have values for x and y (see the graph below), x being the 'weight' or 'step', y being the stem width. Given that the form of a quadratic equation is y = Ax^2 + Bx + C--and using our three predetermined stem widths and steps within the weight scheme--we can easily figure out what A, B, and C equal. Then it's merely a matter of plugging in the 'steps' we desire (as x in our generic quadratic equation), and the values for A, B, and C we just sorted, and we get our stem weights.
The graph above shows a hypothetical situation in which you want your lightest weight to have stems of 25 units, medium weight 100 units, heaviest weight 255 units, and 3 weights in between.
The GM L+M curve shows the interpolated weights using the geometric mean system when you start with the light and medium weights. In order to get the amount of heavy weights desired, you need to draw a font with stems 435 units wide. If we give up the heaviest weight, we still need to draw a font with 301 unit stems.
The GM M+D curve assumes that you start with medium and darkest weights. To get to a 25 unit stem, you need to draw two extra light weights.
But the quadratic curve closely matches the top and bottom 'halves' of these two curves.
Is this a case of 'six of one, half-a-dozen of the other'? Or is there some fundamental aspect of the geometric mean that I'm not understanding, that makes it desirable? Other that it being one of the classical mathematical series that we find 'aesthetically pleasing'.