# Is there a ''mathematical'' reason for the distinction of black/white keys on a piano or is there only a cultural reason?

This question can be found in different forms on the internet and this website. Unfortunately, I am not satisfied with the given answers: They are partly even different from another. So please let me ask again and point out more specifically, where my problem is.

I have a (for me fairly satisfying) reason, why there are 12 notes in an octave on the piano

C, C#, D, D#, E, F, F#, G, G#, A, A#, H, (C)

(and it is not important for this reason if they are tuned in equal temperament or whatsoever): The Pythagorean method of stacking perfect fifth from the root note C ends up (after rescaling by 1/2^n in order to land again in the given octave) at the top C (i.e. a multiple of 2) approximately with 12 iterations. The 'natural choice' of the perfect fifth here is from the lowest non-trivial position in the harmonic series. Alright.

But what about the 7 white keys

C, D, E, F, G, A, H, (C)

in an octave on the piano? Is there a non-cultural reason for taking these notes of the above-mentioned 12-tone scale?

• For example, it could be reasonable to take the first notes of the Pythagorean stacking-fifth-system from above but they are C, G, D, A, E, H and then comes F# which is not of the wanted form. For the last step, one could go a perfect fifth down from the root note C to obtain F. This is the same a starting to stack fifths from F on. But what is a reason to start with F instead of C? Why not start at B for example? Is there a non-cultural reason?
• The first notes of the harmonic series from C are (approximately) C, G, E, ~B, D, ~F, but there is no A in sight, only much later. So this does not seem convincing for me either.
• The harmonic series from C contains the C-major chord C-G-E very early, and in particular the major third E of C. So is there a way to stack major thirds to obtain the C major scale? Well: C, E, G#. No, this does not work. Even a combination of fifth and thirds does not yield the desired scale.
• As already mentioned, the harmonic series contains a major chord on the fundamental very early. In fact all the desired seven notes are in the major chords on C (C, E, G) on G (G, H, D) and F (F, A, C). Starting from C, G is a fifth above and F a fifth below (or a fifth from F to C). This is probably very natural. Could this be the reason?

Is there conceptual (non-cultural) reason for the white keys on the piano?

Please let me clarify that I do not want to diminish a 'cultural reason' (i.e. something like: it is like this because it developed by chance through music history) below a conceptual one.

• Search the site for some of the posts on why the keyboard layout is as it is. Combination of math and history. Apr 16 at 20:46
• “it could be reasonable to take the first notes of the Pythagorean stacking-fifth-system from above but they are C, G, D, A, E, H and then comes F#”. Not if you start on F. Starting on F gives you F C G D A E B(H in the German system). The white keys are the first seven pitch classes derived from going up in fifths from F, not from C. Apr 16 at 22:00
• @ToddWilcox, yes, this is what I've wrote as well but why then start with F and not with B (the international notation is Bb), further down a fifth? Apr 17 at 6:21
• @mrpotato Seems like a separate question, but there has to be a starting note. Any note is a good as any other to start on. Why not start on F? It doesn’t really matter. As far as I can see, your first bullet point is incorrect. The white keys do form a cycle of stacked 5ths. Apr 17 at 10:58
• C was not the default home note when the natural notes of western music were developed, so there’s no reason to expect the notes of the whites keys were derived from C in any way. That’s the point I was trying to make. The idea that F is not easily derived from C makes sense because C was not as important for most of the history of western music. Apr 17 at 13:51

It is a musical reason.

The white keys form the tones of the diatonic gamut of pitches, lettered `A-G`.

The black keys are the half steps between the white keys separated by whole steps.

You can think of the interaction/relationship of the white and black keys in two important ways:

• The black keys allow chromatic inflection both melodic and harmonic to the diatonic tones.
• With the combination of white and black keys the diatonic gamut can be transposed to any of the 12 chromatic steps.

That begs the question "why a diatonic gamut?"

That isn't the OP's main question, but it might as well be addressed.

The question to that is mostly cultural, but depending on how you look at it, a wide range of cultures need to be included. Also, it is partly acoustical. In a nutshell the overtone series supports the acoustical notion that a root position major triad is resonant and therefore preferred by people. By extension the octave and the perfect fifth a preferred intervals. From a starting pitch jumping up four perfect fifths yields tones for a pentatonic scale and two more jumps yields the diatonic gamut of seven tones. Those scales can be viewed as a kind of manifestation of an acoustic fundamental (the perfect fifth) which is the reason for their cultural significance.

• I do not understand your last paragraph about the diatonic scale derived from perfect fifths. As I pointed out above, this yields only F# as the 7th tone, not F. Apr 16 at 21:05
• @mrpotato So you derive seven pitch classes from C and get an F sharp and then play G to G and you’ve still got a major scale. Or, put another way, we can guess the pitches were derived from F leading to B natural (H) a half step below C and therefore the major scale is C to C and F to F is Lydian. Apr 16 at 21:55
• @ToddWilcox As I've written, this is clear to me, but I don't understand why one starts at F and not at C itself. Apr 17 at 6:24
• @mrpotato because the choice of white keys was not made as a result of stacking fifths. Also note that the overtone series does not contain E very early as the result of stacking Pythagorean fifths; that E isn't the fifth harmonic; it is the eighty-first. Apr 17 at 6:41
• @phoog I am confused about yout comment on E: I define it as the major third from C. Then it is the 5th harmonic. Or am I wrong? Apr 17 at 12:20