Before we get started, I'm trying out a new website background technique! I made this in blender. I couldn't quite figure out how to get the formatting how I wanted, but let me know if you like it or not.
This is an in order list from worst to best bases, from the range 1 to 4*4! (base neutral way to write dozen four). There will be the name of the base, followed by the base number in seximal, decimal, and then binary. In that order. If there's less than three, it means some of them have been merged for brevity.
Ah, unary. Tally marks. Every integer is expressed as a series of 1s equal in length to the integer it's representing. This does have some advantages, that math is really easy, and that it's very visual, but that's about it. It's not actually totally unusable if you cheat a little bit and use math. For example, you can write the largest known prime as millions of decimal digits, or you can write it as 2136,279,841 - 1DEC. So we can write numbers like the number of degrees in a circle as 11111 * 11111 * 11111. Better than ten nif ones, but still worse than 1400SEX, or 360DEC.
This is basically prime factorizations trying to be a positional numbering system. Though you can write the number in other ways that use composites, but I think they're all less efficient. Maybe I'll write a program to find the best unary representation of a number. Still by far the worst base in this range.
This base is prime and near decimal. It truly has nothing going for it, other than it has a kind of decent size.
Prime base. Super small, and it doesn't get away with being prime and small like binary does. Binary gets away with it by being the smallest and even. Trinary is neither. At least it's good with thirds, the second simplest fraction.
Prime base. Decent size. Doesn't do too badly with thirds since five squared minus one is divisible by three. One third is simply <0.13>. But that's two digits repeating for the second smallest prime. Easy to derive on your own but god damn. Quinary is one more than 4 so it's not actually too bad with fours, but still no excuse for fumbling halves and thirds.
This is a prime base. A bit big, but still in the usable range. This prime base piggy backs off of dozenal, making it almost usable. Because of its size, I can't put it any higher.
The last prime base on this list. Now I usually say that dozenal's fourths are a gimmick, and they are, but because septimal and baker's dozenal both do fifths with 4 digits, they don't matter. Like baker's dozenal, septimal piggy backs off of a multiple of six base to become sorta usable. But in this case, baker's dozenal wins in terms of fractions because of its quarter being ever so slightly better. But because baker's dozenal is so large, I have to give this to septimal.
Base 3 * 5. This one isn't a terrible odd base, as it's composite. There's actually only two odd composites on this list, and this is the slightly worse one. Mostly because it's not a square base and it doesn't get much from its predecessor. Triquinary is good with thirds, fifths, and sevenths. That's about it. Not great at quarters or eighths, and kind of big. The main thing it has over nonary is that it's really good with fifths and sevenths. Honestly it's really hard to say whether this is better or worse than nonary. You can call this a tie if you want.
Base three squared! Nonary is good with thirds and nineths, and is sorta decent with powers of two because it succeeds 1000. One eighth, fourth, and half are all one digit repeating. Pretty good for an odd base. Also decent with fifths, just being <0.17>. This base is also a good size. Sevenths aren't terrible either, being just <0.125>. This is because square bases usually have pretty good fractions. They have finite or no cyclic numbers. This is also a pretty good sized base.
The worst even base on this list. Base 2 * 7. As the prime factorization suggests, it's only good with twos and sevens. Its predecessor is thirteen so it gets really screwed over here.
By far the worst power of two base in this range. Its neighbor gives it sevenths, which aren't terribly useful. It's terrible at thirds and fifths, and isn't a meta power of two. Over all just kind of awful.
Ah, base three four. Great with twos, threes, and fours, at the cost of everything else. It's large too. Like seximal but bad.
Yeah, this one. I put it in fifth place, which is actually pretty high, and that's because this base is genuinely pretty good. I hate on it a lot but that's just because it's what I'm used to so it's extremely dull. In truth, it's a pretty good base. Good with twos and fives, but also good with threes and nines using nonary. Just barely covers eleven by being before it and- oops. It forgot sevenths. Yeah, it's not a great base, but it is one of the best in the usable range. A bit big for my tastes, but still a decent size.
Coming in at fourth place, these last few bases are the ones I actually like. Tessimal is very similar to decimal, in that it's not great on its own but becomes great using its predecessor. Tessimal piggy backs off of triquinary to become usable. It has good thirds and fifths. And obviously, is very good with the powers of two. One half is <0.8>, one quarter is <0.4>, one eighth is <0.2>, and of course one tesseth is <0.1>. Because it's a square base, it has few cyclic numbers. I think it has none but I wasn't able to find info.
I can't not address the size of this base though. Being the biggest on the list, it is quite big. But this base cheats by being a power of two. You can think of every digit as four bits, and then it doesn't feel as large anymore. Adding to this, it does do one seventh as <0.249>, but once you realize that it's just 0.'001 in binary it becomes really easy to calculate. Great base.
Base four! This is like tessimal but smaller. It has no cyclic numbers, and has relatively simple fractions. It has a great size too, so it gets to go in front of tessimal.
Binary! One of my favorites, I put it in second place. (Isn't that funny? I put decimal in fifth place, binary in second place, and tessimal in fourth place?) For a few months, I was torn on whether I liked binary or seximal better. Of course there was this amazing video arguing for binary, but something didn't sit right. Yes, judging it purely on how many strokes it takes to write small numbers is shallow, but it's not pointless. Numbers do get long fast, even if it is maybe worth it.
There's all the cool stuff about it being small making it maneuverable, and yes that's true, but it still has to do that, while seximal just works. It's really cool that I can test for divisibility by 37DEC really easy, but do I ever need to? No, not really. Having to add up pairs of bits for three and quads of bits for fives is just a bit cumbersome.
In conclusion, binary is no doubt a great base, but it is still binary. It kinda takes effort to use, and usually repeating fractions are just kind of a deal breaker sometimes. I don't want to type in "0.01010101" for one third. Though I do think it's unfair that this base is thought of as having so many cyclic numbers while quaternary has none. It's just a slightly more efficient form of quaternary, so they should be treated the same. A digit length of 4 in binary should be thought of as the same worth as a digit length of 2 in quaternary.
Binary is one of those things that is fun to play around with, but ultimately, whenever I don't think in decimal, I find myself thinking in seximal. It's conventional and easy to use. Binary is okay at everything, while seximal is really good at the first few primes. And honestly that's just better. Still really cool how informationally efficient it is. If you like binary you may want to check out my binary systemplex. Oh, and lastly, binary finger counting is by no doubt my favorite finger counting method. Seximal's is nice, but binary is just wayy better. If you can't move all of your fingers individually I totally get it though.
Uh, I kind of spoiled all of this in the previous base's entry. Anyway, seximal is the most convienent general purpose base. It's the perfect size, I wouldn't want a digit more or a digit less (though I may be biased), and it prioritizes the most important two primes first. Then it prioritizes the third most important prime, and then it prioritizes the fourth most important prime last. It does things in the correct order. Decimal could learn from this, putting fifths foolishly above thirds. Like binary, seximal being small is its biggest strength.
In conclusion, we should all count in decimal. It's what everyone uses so it's the easiest to "switch" to, and it's already pretty good. The only thing better than perfect is standardized. But seximal comes in close second.