I guess I must be dumb or something, but I'm simply not seeing the problem.
Imagine the piano had only white keys, no problem right? Now just place the black keys at the back, between some of the white keys, right in the middle, such that each black key takes like a quarter of the width of the sandwiching white keys.
Now what's the problem with this again? Can someone explain in clearer terms?
If the issue is that we are trying to make the white key all have the same width at the back, well, why should that matter? Pianists don't press the white keys all the way at the back, do they?
Pianists don't press the white keys all the way at the back, do they?
The good ones do it all the time because moving the entire hand forward and back can be significantly more fluid than contorting to play another way…the keyboard is three dimensional.
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.
Yes, you do need to press white keys further back sometimes. Imagine trying to play on black keys with your thumb and pinky finger while playing a white key with your middle finger. You won't be pressing all the way at the back, but your finger will have to press between the black keys.
That design you describe is what is pictured at the top of the article.
Problem is that then the keys are not equally spaced chromatically (e.g. larger spacing between B and C than between C and C#).
You could probably get used to play like that, but it would be ineficient in terms of space for both the fingers and the mechanics of the piano (hammers, strings).
So what you do, in reality, is move some of the black keys down a bit (C#, F#) and some up (Eb, Bb) so that the spacing between the center of the keys is regular.
I don't think that's what's described in the article though?
> Pianists don't press the white keys all the way at the back, do they?
I do, and I'm not even a good pianist. Many chords will need it, just pick any chord that required the thumb or the pinky (or both) in black keys.