We can do some analysis within the range from 6 to 20 inclusive. For one, a base must be even to have any chance in society. This is not just a societal thing: evenness is not important solely because of custom. An odd base is forced to have a totient ratio over half, and is bloated beyond the size of an even base with the same prime decomposition. See Icarus' thread on the subject. So I think it is safe to ignore odd bases, and so the range shrinks to {6, 8, 10, 12, 14, 16, 18, 20}.

If hexadecimal's usefulness is questionable, so surely is that of octodecimal and vigesimal. Furthermore, following on Icarus' multiplication-table argument on why hexadecimal would take too much time, I am unsure if even tetradecimal is workable. To quote a post I made, following Icarus' argument about hexadecimal:

So maybe the upper limit is actually duodecimal. One of Icarus' old threads suggests that "we maximize our human computational capability when we use base twelve" (as it is the largest usable base), and that enlarging the base past twelve makes itDouble sharp @ Oct 6 2015, 10:14 AM wrote:I am not even sure if tetradecimal is human-scale, come to think of it. Its multiplication table is already twice as big: decimal has 55 unique facts (78 if you use a 12-by-12 table), while tetradecimal has 105. So at a minimum, presuming every fact is equally easy, tetradecimal would take at least 4/3, and at most twice, as much class time.icarus @ Sep 29 2015, 03:01 PM wrote:The "mountains" are an interesting high country to camp. We can see for miles up there. But it is harsh and not for the faint of heart.

Ultimately we'd fare better building our city in the valley, between octal and around tetradecimal.

But not all facts are equally easy. With two opaque totatives in tetradecimal {9, b} (as opposed to one, {7}, in decimal), and with the alpha dominance (more difficult than decimal's omega dominance), and the longer digit-sequences to memorize for divisors like {2}, I think tetradecimal would still be more difficult on average, even if its tables were the same size as those of decimal.

Combining these two factors, I would estimate that tetradecimal is between 3/2 and 9/4 times as difficult as decimal (assuming that the tetradecimal tables are about 9/8 times as difficultintrinsically, not counting their size), and the acquisition of multiplication would take that much more time. On average, we'll be spending a full year more on average to acquire multiplication. It's not as bad as hexadecimal, but we'll still be behind using tetradecimal, with only the greater concision to comfort us with.

So I am not sure if even tetradecimal can be civilizational. Since it seems that 2 is very fundamental, it may be that only {(6), 8, 10, 12} are possible civilizational bases (and I'm not even sure about senary), in which case the ordering is probably {12, (6), 10, 8} or {(6), 12, 10, 8} (I'm not really sure which it is).

*more*difficult to use. This gives the most easily memorized bases as {8, 10, 12}, with anything lower being trivially small.

The lower bound is a little more difficult to quantify...

P.S. It appears we can sort of assign the ideas of these four bases {6, 8, 10, 12} like so:

{6}, senary: pseudo-7-smoothness (or maybe the idea is pseudo-5-smoothness and 7 is a lucky afterthought), but 3 is more important than 5

{8}, octal: 2-smoothness (pure binary thinking)

{10}, decimal: pseudo-5-smoothness, but 3 can be sacrificed a little for 5

{12}, duodecimal: 3-smoothness