The three necessary primes to represent on the organized lattice are 2, 3, and 5. Given a key like 16, each of these primes should be represented in its adjacent buttons, hopefully in an axis-like manner, because these are the consonant notes to it. Left of 16 is 5/4, right is 3/4, and up-down is 2 and 1/2. But why this way?
Consonant pitches stay within a range - so large leaps in pitch are the least common. That means next to any number, the adjacent keys must have close by numbers. That's why the three primes 2, 3, 5, are converted to 2, 3/4, 5/4 - because these are the closest numbers to 1, after multiplying by a power of 2.
Your hand has a harder time reaching vertically than horizontally. 2 is the largest of these three numbers, which is why 2x is on one of the 4 vertical directions, and 3/4 and 5/4 are on the horizontal directions. Furthermore, it means that 2x must be on the 5/4 side, not the 3/4 side - because this keeps pitch distances the smallest.
There's only one remaining decision - why right to left? Your hand has an easier time turning inward than outward. And, whichever way it turns, the horizontal notes should have pitches as close as possible. Thus, 5/4 must be to the left direction.
Luckily, divisors don't exist, so the fact that the 3/4 axis and 5/4 axis are on the same line - rather than 3/4 turning to 4/3 on the opposite side - is perfectly fine.
So thanks to the above, the keyboard is forced to be this way, and every note is in its logical and rightful place, with no subjective choices. Except not, because there's not enough space. And because 7 and 15 can't be on the lattice, so they are just placed wherever there's room. 2 should really be below 1, and 3 should be where 2 is. 30 should be 40, etc. I tried to maintain the structure as much as possible but had to make things fit.