There’s a stairway at my university that goes uphill towards the management and acounting faculties. Its tread length is about 1.1 metres (3.5 feet) per step, too long for me to comfortably step once on each stair, but too short to take two strides. I usually take a running start and try to climb them faster than normal, or alternate between one and two strides, but they’re still awkward, and I’ve tripped multiple times on them. The higher natural scale reminds me of this stairway: these bases are too large to use a single digit per place, but too small to use two.

The limits of the higher natural scale must be somewhat fuzzy. Its lower boundary should probably be set at {16}, the first base that requires help from something other than Stevinian arithmetic. The upper boundary is somewhat debatable, but I’ll set it at {36}, because it’s the square of 6, the largest number without any opaque totatives.

The numbers between 16 and 36 inclusive can be sorted by their prime signature and number of divisors:

- The primes, {17, 19, 23, 29, 31}, have 2 divisors, 1 and themselves.
- The only square of a prime in this range is {25}. It has 3 divisors.
- The semiprimes, {21, 22, 26, 33, 34, 35}, and the only cube of a prime, {27}, have 4 divisors.
- {16} is the only fourth power of a prime at this scale, with 5 divisors.
- The products of a prime by the square of another prime, {18, 20, 28}, and the only fifth power in this range, {32}, have 6 divisors.
- {24} is the product of a prime by the cube of a different prime, and {30} is the product of three distinct primes. They both have 8 divisors.
- {36} is the product of the squares of two primes, and as such, has 9 divisors.

We are left with {16, 18, 20, 24, 28, 30, 32, 36} to try to wrangle. These bases are all multiples of either 4 or 6. On the one hand, except for {16}, they are clearly not in the human scale. On the other hand, they don’t seem too alien for human thinking, especially because one of them, {20}, was continuously employed by the civilizations of Mesoamerica for centuries. So, in this thread, I’ll try to determine not if these bases could be treated as if they were in the human scale, but what methods a modern society that ended up with these bases could use to calculate, and whether these methods could ever be as efficient as Stevinian arithmetic.

At first glance, it looks like there’s no one single method that can be used for all of these bases. The human scale bases can use Stevinian arithmetic, the lower mid-scale bases can use reciprocal divisors, and the higher mid-scale have alternating arithmetic to help them out. But the bases in the higher natural scale don’t seem to share methods of computation, and it seems that each base here has its own personality:

- {16, 32} are binary powers, and should best pretend to be binary,
- {18} is unbalanced and distant, though still 3-smooth and divisible enough to make it on the list,
- {24} tries to solve the main issue with dozenal by putting two powers of 5 in the alpha,
- {36} is the senary “hundred”, and as such, inherits all of senary’s strengths at a larger scale,
- {20} is naturally found on the human body as the total number of a person’s fingers and toes,
- {28} is its septenary analogue, the number of phalanges on both hands,
- {30} is three-dimensional and full of regulars, though not very deep.