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Pianists sometimes do have to play white keys with their fingers between the black keys. For example, if you want to play E flat major (Eb-G-Bb), you might put your thumb on Eb, your little finger on Bb, and then your middle finger on G, right up the back between the black keys. As the keys are parallel, the width of the white keys at the back is the same as the width all the way down to the front of the black keys. Apart from the need to fit the hand to the keyboard, a piano key is a lever and it gives more resistance further back, which may be useful to the pianist in controlling the attack and dynamics.

If you place the black keys right in the middle, as you suggest, the space between them is too narrow for a finger, while there is wasted space for those white keys that only have one adjacent black key. So piano makers push the C# and D# keys and the F# and A# keys further apart.

The mathematical problem discussed in the article is that there is no way to distribute the space equally between the keys, so various compromises are considered. The "B/12 solution" is practical and widely used. The suggested "optimum arrangement" is amusing to consider but unlikely to be worth the trouble.

As the article says, it is impossible to make the back ends of the white keys equal in width so piano makers have to compromise. The difference is about a millimetre so many pianists may never have noticed.

The naive approach of placing each black key at the midpoint of its adjacent white keys makes B, C, E and F quite wide at the base, but it is harder to fit a finger between the black keys to play D, G or A, which can be necessary when the hand has to stretch or play both black and white keys. On a real piano, therefore, the C# and D# keys and the F# and A# keys are offset a little from the midpoints of their adjacent white keys. G# is the only black key that is actually at the midpoint of its adjacent white keys.

In the photograph of the Yamaha piano, you can see that the cutouts on the D key are symmetrical but less than half the width of a black key because the C# and D# keys are offset. Looking at the G key, the right cutout, at half the width of a black key, is deeper than the cutouts on the D key, so to compensate the left cutout is less deep than the cutouts on the D key. As a consequence, the cutout on the F key is deeper than the cutout on the E key, so the E key is wider at the back than the F key. Similarly, the C key is wider at the back than the B key.

As described in the article, on some keyboards just the C and E keys are wider at the back, with D the same width at the back as F, G, A and B. More often, however, the C# and D# keys are placed a little further out to spread the extra width equally between the C, D and E keys.