Why no black note between B and C

Why is there no black note between B and C, and between E and F?
There has been much online discussion about this over several years.
Howard Goodall in his book Big Bangs answers many questions and I have learned a lot from him. Here is my contribution to this topic:

If you stretch a string and pluck it, it will create a sound. But it is not a pure sound. There are overtones and harmonics. The strongest or ‘dominant ’ harmonic is what we now know to be a fifth from the base sound. So, let us call the base sound F and the dominant  harmonic C and write them like this: F   C.

Now repeat the process but make C the base sound and the new dominant produced will be a G. Lower that G an octave, it will still be a G. We now have 3 notes FG  C. Repeat the process and we get a D: FG  CD. Repeat again and we get an A: FGA CD. We have the first 5 notes and they are of course, the pentatonic scale.

But we can keep going. Using A as the base we will get an E: FGA CDE. Repeat and we get a B: FGABCDE. We now have 7 notes. They are the notes that form the C scale. They just need to start on a C.

But we can still keep going. Using B as the base we get – F#. Until now we have always got a new note to add to the sequence but now we have a note that needs to be inserted into the pattern we have created. A note that has to go between F and G. This situation was handled by using a different kind of note and calling it F# or Gb. On a keyboard it is a black note.

We now have FGbGABCDE. Repeating the process we get Db, then Ab, then Eb and then Bb: FGbGAbABbBCDbDEbE. We have gone as far as we can go because if we repeat the process using Bb as the base note we get an F and we are back where we started. We now have 12 notes arranged as they are on a keyboard. There is not a note between B and C nor between E and F. (The pattern can be repeated and that will put E next to F.)

 As stated above, when using Bb as the base note we get another F. But this F is slightly different to the F we started with. This presented a major problem that the music world struggled with for years. Eventually an answer was found with the introduction of Equal Temperament.