Why are a piano’s black keys arranged in groups of three and two?

Question

Why are a piano’s black keys arranged in groups of three and two?

In: 44

Answer 1

As yet, no-one has given the complete answer, despite giving the correct background about overtones and temperament etc, Namely, **the white notes in fact are equally spaced** in a particular rigorous sense (and it is this nearly mathematical symmetry which defined the role of the diatonic scale compared to other modes in the German/Austrian tradition, which the Russian Tchaikovsky was criticised for seeming to ignore in his early career, and was later consciously rejected as a necessary ingredient of composition by composers like Shonberg.)

 

so let’s start over at the beginning with a very uconfusing experiment. Start at

 

**F, a white note**

 

Now go up 7 semitones, you reach

 

**C, a white note**

 

Now go up 7 semtones, you reach

 

**G, a white note**

 

Now go up 7 semitones, you reach

 

**D, a white note**

 

Now go up 7 semitones, you reach

 

**A, a white note**

 

Now go up 7 semitones, you reach

 

**E, a white note**

 

Now go up 7 semitones, you reach

 

**B, a white note**

 

In that sense, the white notes *are* equally spaced. If you continue then you go through the 5 black notes, also equally spaced. What I am going to say next is *less* important than just that observation.

 

Not only are the white notes equally spaced in this sense, they are all a 3-adic ‘open ball neighbourhood’ of D in the sense that if you tuned your piano to perfect fifths going up and down 3 steps from D, the frequency component of 3 in the frequency ratios define a notion of “3-adic distance” https://en.wikipedia.org/wiki/P-adic_valuation and you are llooking at D and the three closest points to each side.

 

I have ignored a relation with octaves (that an unnecessary complication can be removed if you replace 7 with 19 in the exercise above) and the relation with the fact that 3^{12} is close to 2^{19} but it was frustrating to see people describing the backgroud for this notion and not actually mentioning the precise fact.

 

One could remove F and B and have a pentatonic scale equivalent to the set of black notes, or include Bb and F# and have an extended scale both centered around D.

 

Also, there is the combinatorial fact that if you replace F with F# it is the same as deleting the first term in the sequence and including one extra term, 7 semitones higher than the last term of B. Thus two transformations of the set which seem different — one changing one note by a semitone, the other shifting all notes by 7 semitones — are in fact the same transformation of the note classes mod 12. This is related to how 7 mod 12 is invertible.