I am pretty convinced it is not just for historical reasons.
I have found a mathematically-sound explanation of the twelve-tone musical scale, thanks to this question. Does anyone know a similar, human-readable explanation for the piano key layout?
I suspect the current layout is optimal in some sense, for example it maximizes the number of consonant chords that can be played only with the white keys.
Interesting question, although my answer might be more historical than you'd like ;-)
One answer is that it gives you all the notes of the diatonic scale on the white keys, so by transposing to C major you can play any major-key melody that doesn't modulate using only the white keys.
Another way of saying this: assume that you are working in our musical system, which has twelve-tone equal temperament as the background "system", but within that the diatonic major scale is the most commonly used set of pitches. Then assume that you want to have one particular diatonic scale easy to play, and that you'll put the other pitches on harder-to-reach keys. Subtract the diatonic pitches C-...-B from the set of all twelve pitches and you are left with C#/Db, D#/Eb, F#/Gb, G#/Ab and A#/Bb. Put these "between" the diatonic keys, in the right order in the chromatic scale, and you have something very close to the standard piano keyboard. (You can't add any more "half-steps" between E and F, or between B and C, without expanding your tuning beyond 12-tone equal temperament).
Wikipedia and Grove Music online (subscribers only, unfortunately) note that the original organ keyboards (13th century) had only the pitches of the C major scale, plus B flat, because that made up more or less the entire pitch resources of the religious music sung at that time (and instruments would have been used only for accompanying sung music -- at least in church). On those keyboards B and B flat were both "white keys", with no "black keys" at all. The first surviving organ with a fully chromatic keyboard, from the late 14th century, still has B flat as a "diatonic"/"white" key.
I would guess that as keyboard music developed as its own genre, it became much more useful to be able to play fast runs in the major scale -- lots of early keyboard music is based on existing pieces of vocal music with the addition of fast, "improvisational"-sounding ornaments. At least at first, the chromatic notes would have largely been used in chords, not scales, so it would be an acceptable trade-off to have those keys harder to get to in exchange for being able to play the main scale quickly. Even quite a bit later, around 1600, there are pieces which are written in G major or F major, but where all the fast bits ignore the key signature sharps or flats and just use the diatonic keys -- it was easier to play fast scales on the "white" keys with the technique they used.
Finally, it's worth noting that people have often built keyboards with more than twelve tones to the octave. In the 16th century it was common to have the Eb/D# key "split", with the front half playing one of the two pitches (Eb) and the back half the other (D#). This was done by people who valued having perfectly-tuned chromatic notes over being able to easily navigate all scales at high speed. The extreme of this way of thinking, pre-20th century, is probably the 1555 Archicembalo, which has 36 keys to the octave! And 20th/21st century microtonal musicians have done lots of similar things. There is a nice introduction to different tuning systems both historical and modern at Kyle Gann's page.
The diatonic scale, as well as the 12-tone chromatic scale, are both by-products of overtones. If one examines the harmonic series, the first six pitches created from a fundamental (initial) tone outline a major chord. Arnold Schoenberg goes into great detail about this subject in his book The Theory of Harmony.
Pitches from a diatonic scale being played together are generally considered consonance, while chromatic notes being introduced to this scale would be considered dissonance. Schoenberg posits that all notes create consonance, referring to the chromatic notes as a more distant consonance.
If you continue to follow the intervals, by the ninth note in the series you have an outline of the major scale. The notes we don't have are the fourth, sixth and seventh scale degrees. These pitches are all somewhat sensitive pitches. The fourth and Seventh degrees played together form a tritone; which, for a large portion of musical history, was referred to as 'El Diablo" (Johann Fux's The Study of Counterpoint makes an amusing reference to this) and completely left out of composition all together. The sixth scale degree is the root of the relative minor scale, making it sensitive and also a somewhat distant consonance.
All of this to say that the diatonic scale satisfies the ear by creating consonance, as well as resolution of dissonance, in more perfect ratios that are traditionally aurally pleasing. The piano's arrangement, as well as many other musical concepts, can be attributed to the overtone series and its effect on the way we perceive music.
There isn't anything "optimal" about the keyboard. Not all chords or melodies are the easiest to play using only the white keys, even if they are purely diatonic (could be transposed to C major).
It can be regarded as a virtue of the keyboard that transposition to different keys leads to different fingerings. A chromatic keyboard would "feel" the same in any key. Thus, transposing would be intellectually and physically easy; however, it would also lack any surprises, such as bringing some figure "under the hand", making it easier.
Fundamentally, the keyboard has these parts: there is diatonic scale naively represented by a row of keys of equal dimensions. These are sufficiently wide that they accommodate human fingers of various thicknesses, so that a key can be struck confidently without striking adjacent keys, and without ridiculously precise positioning. Between these keys are inserted additional keys, of a slender construction, which recede toward the rear panel, so that they do not obstruct the diatonic keys. These keys provide access to the semitones which are omitted from the diatonic modes, allowing for chromatics and modulation. These keys are also spaced equally, but their striking surfaces are smaller. That yields to more effective spacing between them, allowing for variation in fingering.
What is optimal about this layout that musicians have stuck with it for so long?
So, in a nutshell, the piano keyboard is ingenious in a number of ways, which could explain why it resists being replaced by something else.
Optimality of the twelve-tone musical scale explains why we have (7+5) keys in an octave.
The 7 white keys form the the diatonic scale which is at least 9000 years old!
The 5 black keys form the pentatonic scale and this scale is also ubiquitous.
Maths shows that these scales do stand out if we insist on having the frequency ratios 2:1 and 3:2 in the scale.
That snippet about B and B♭ in earlier keyboards helps me to understand why German music uses the letter H. I guess that using A,B,C,D,E,F,G and H gives the option of playing in C maj and F maj.Thus one could modulate a little.
There are a lot of opinions here justifying the keyboard layout in terms of being able to find notes by touch. Two remarks on that: for any serious kind of playing, there will be no time to grope around the keyboard.
For another, things like chromatic button accordions don't offer any "find the diatonic scale" help. While some instruments use a different surface for selected notes (C, D, G on mine, but also in analogy to the bass side C, E, A♭ is common), quite a few instruments are totally uniform in their righthand side and rely on the repositioning skills of the player. And I don't think that this is all too dissimilar with how a piano keyboard is getting played by experienced players. After all, guitars, violins and other instruments don't have a patterned keyboard to get by with either.
It's pretty obvious that a keyboard could have been laid out with all white keys, but that would take away a major advantage of the black/white arrangement: Being able to concentrate on reading your music and being able to feel your location with your fingers. Ever wonder why there are so many blind piano tuners? A pianist can close his eyes and identify any individual key on the key board by where it is located in relationship to the group of 3 black keys and the 2 grouped Black keys. For instance, you have two places on the key board where there is a white key directly to the right of another white key (C and F). You are able to tell them apart because the F has a group of 3 black keys to its right, while the C has a group of 2 black keys to its right. All 88 keys can be uniquely identified by their placement. The key of C major on the white keys is the only arrangement that makes all of the keys unique so that they can be identified solely by touch. Other arrangements yield ambiguity.
The 5 black keys divide the 5 whole tones within the natural music register A B C D E F G plus Octave A, which is Diatonic A Natural Minor (5 whole-tones and 2 half-tones). Notice that from B to C and E to F have no black keys between them because they are already half-tones. This creates a keyboard that is a succession of 12 half tones which repeat.
The white keys represent the diatonic scale (5 whole steps and 2 half steps) which Western Europeans inherited from the ancient Greeks. The Greater Perfect System encompasses the natural pitches we call A2 to A4, reflecting the range singable by male baritones and tenors. When these pitches were first given letter names in the 6th century CE, the lowest was called "A," the next, "B," and so on, up to "O." Later, the principle of octave equivalence was leveraged to reduce them to a repeating sequence of A-G.
Additional half-steps were added in later centuries as Western European harmony developed and matured. They were represented by short, raised keys wedged in between the existing white keys that were a whole step apart. In this way, the spacing of the white keys was preserved so that instruments could be retrofitted and musicians didn't have to learn a completely new keyboard.
Incidentally, the major scale did not enjoy special status with the Greeks and did not form the basis of their System. The major scale happens to occur beginning with the third natural pitch, and thus, our white key major scale begins on C.
place the 1st finger of the right hand on E and the 5th finger of the same hand on the next C and you'll see why.
EDIT: Ok, I'll develop the answer. If you notice distance between the beginning of the white keys and the beginning of the black keys, it's more or less the distance between the tip of the thumb and the index finger.
This distance was not randomly chosen, it fits the hand anatomy. It's best seen as I first described, with your 1st finger on E and 5th on C, you can see that the other fingers (2, 3 and 4) are naturally placed on F#, G# and A#.
And then you ask me, but what about the other group of two black keys on the octave (C# and D#)? If fit's the same way, but now with only two fingers of your choice.
Well, then why does the white keys where chosen to be the "white keys"? It's because they provide the diatonic scale on C, which is the most "natural" scale.
If you take a look on the proposed fingering of the majority of songs, you'll see that it tries to avoid placing the thumb on a black key, because it does not fit well.
With this layout, it's possible to minimize the horizontal distance between keys, making it possible to easily reach large intervals (octaves) without compromising the comfort factor of playing smaller intervals (it also minimizes the travel time between keys without compromising the comfort factor).
This layout dates from before the 15th century, if my memory is not failing. (don't quote me on that one) In my humble opinion, it's the most logical layout, and that's why it survived through centuries. You even have some letter-printing telegraph keyboards which follows the same strategy.
I'm sorry for the lack of explanation in the first answer.
I always felt like the piano setup was weird. I'm a guitar player by trade, but I decided I would at least understand how the piano worked and become acquainted with the structure. I've determined that it would be much better to make the areas in between the black keys (i.e. the white keys but the part in between the black keys) raised up to be the same height as the black keys, but leave the other part of the white key the same (the bigger part not in between the black keys). These white areas are lowered and it's just very awkward to play.
Just think about what it would be like if there were the normal white keys and then the area with the black keys was an even row of half steps. It would be glorious, and all you'd have to do would be fill in those gaps with some material, such as layering tape and then cutting it to fit. Then all you would have to do is learn a universal shape and find your root note and that's all you need as opposed to individual shapes for each key. If you do want to use the white keys normally, however, you can because they're still there.
My school of thought on this comes from using EADGCF tuning on guitar where the strings are tuned in 4ths as opposed to the standard guitar tuning which is centered around barring your finger. I feel like standard piano is much like standard guitar tuning. It makes sense if you're stuck a few centuries ago, but nowadays both of these things are obsolete. EADGCF is much easier mentally, just takes a bit more finesse physically because the top two strings' notes are scooted back one half step so you have to have a "curved bar" to do full 6 note chords. I much prefer the easy layout mentally which is significantly easier for arpeggios and jazz chords.
