I agree. I do not think I said he discovered the table. He discovered the "law of octaves". And, in a way, he did not really discover that. Musicians had already learned that the first note of a scale and its ending note are companiable. ("companiable"? Is that a word?) Anyway, pleasant to the ear. Not all notes can be paired and have a pleasing sound. In the case of music, the sound is the property. What is it in the elements? Can you select one pair of first and eighth elements that does have specific properties to be seen and named? I am not challenging the idea on that basis. I just want to know what I am looking for.
And - "7 note scale"?
Now, regarding the similar properties in the first two short periods, here are some examples:
Li, Na: very reactive metals, forming ions with a single +ve charge
Be, Mg: reactive metals forming ions with a double +ve charge
B, Al: B is a semimetal forming 3 bonds, Al is a metal forming ions with a triple +ve charge
C,Si: C is a non-metal forming 4 bonds, Si is a semimetal forming 4 bonds
N, P: N is a gas forming 3 bonds, P non-metal forming 3 or 5 bonds
O, S: O is a gas forming 2 bonds or ions with double -ve charge, S non-metal forming 2,4 or 6 bonds or ion with double -ve charge
F, Cl: both pungent reactive gases with yellow-green colour forming one bond or ion with single -ve charge
Ne, Ar: "noble"or "inert" gases, extremely unreactive, with almost no chemical reactions at all.
As you can see there are plenty of big differences as well as similarities, but the similarities are sufficiently marked to be useful as an organising pattern or principle. It was only with the much later advent of quantum theory that the basis of these similarities could be properly understood, as being due to the numbers of electrons in the atoms of each element and which orbitals they occupy.