I suspect there is no such fussing about "back of white key widths" in actual piano design. What's actually going on, when you look at any piano keyboard, is t

I suspect there is no such fussing about "back of white key widths" in actual piano design. What's actually going on, when you look at any piano keyboard, is that the groups of black keys are given a substantially wider spread: they are not exactly centered on the dividing lines between the white keys. What most likely matters to playability is the width of the black keys and this spread amount. Now if you cover the front of the keys (e.g. put your fallboard felt over it) you can visualize the back of all the keys as a kind of barcode: alternating black and white strips, sometimes with two adjacent white strips. It so happens that these semitone strips do look about equal width, thanks to the sizing of the black keys and their spread. It might not be exact though. A good starting configuration might be to start with a keyboard in which all the semitones are strips of equal width. Then we identify the C major keys, and paint them white, making the others black. Next, we shorten the black keys, and adjust the frontage of the white keys to be of equal width. Say, by keeping the division between every B and C exactly where it is and interpolating the others divisions. You will find that the E-F division does not fall exactly halfway between the surrounding black keys, Eb and F#. (That's what's observed on real piano keyboards!)