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What I found annoying about "ian misali's" video is the implied criticisms and subtle lampooning of dozenal and its usage by all the past generations of dozenalists. Lots of strawman and reductio ad absurdum arguments, nested in satirical snark. I hear echoes of that from you here. I'd prefer it if you and he would lay off the indirectness and just lay out the critique forthrightly. Then see what parts those of us present here, and now actually defend or disavow. I personally don't think every past idea of the DSA necessarily has merit, but somehow the critics seem to think we now have to answer for all of them.SenaryThe12th wrote: *chuckle* I confess, one of my favorite aspects of dozenalism is that we need two new cool symbols!! Which opens up all kinds of neat things to think about: which two symbols should we pick? Or should we just chuck them all and come up with 12 new ones!! Endless opportunities for fun.
Kodegadulo wrote: I hear echoes of that from you here. I'd prefer it if you and he would lay off the indirectness and just lay out the critique forthrightly.
Well that's just the thing; I haven't critiqued anything, *except* senary. My request was explicitly for a list of critiques of senary.I'd prefer it if you and he would lay off the indirectness and just lay out the critique forthrightly.
Well, if anybody has call to complain about being strawmaned, I think that would be me. And I didn't do any reductio's, ad absurdum or ad anywhere else :-) BTW, what *is* your beef with reductio ad absurdum? Its a valid form of argument.Lots of strawman and reductio ad absurdum arguments
Indeed, you helped me out with nomenclature in this very thread. And I religiously read all your stuff on the Hexican thread. You're a good egg man. Can't wait to see what you and the rest of the folks on this forum come up with next.Kodegadulo wrote: Like I said, I'm generally supportive of thought experiments like senary.
THAT is what the world needs now!!!Is there an emoticon that means "plain-speaking"?
I didn't mean that 5 ought to be a top priority; it's just the treatment of 1/5 in senary is not that much better than its treatment in dozenal. It still doesn't terminate. Yes, it certainly looks nicer, and you get a nicer divisibility test for it, but you're only getting half of the possible improvement - and not the half that lets you use fifths in contexts with a limited number of significant figures to play with, such as measurement.SenaryThe12th wrote:Well of course, if 5's are particularly important to you, a base which has a factor of 5 will beat anything. Kind of like how my tribe (which finds powers of 2 important) uses binary, octal, and hexidecimal.
But really, this isn't really an objection against senary per se, as a generally used base, is it? I mean, senary *is* better at 5's than dozenal. And your original point, that 2 is waaaaay more important than 3 is, and 3 is waaaay more important than 5 is--that was a good point, IMHO.
Double sharp wrote:Compounded on that, there is still no good way to use an auxiliary base containing 5 in senary, because the fifths always get absorbed into the mantissa instead of the trailing zeroes and end up outshining the more important fractions.
We use numbers like 60 or 360 in decimal as common groupings instead of 10, 100, and 1000: on the forum, we've been calling such things auxiliary bases. The mantissa of a number is the significant part, before the trailing zeroes start.SenaryThe12th wrote:I don't understand this; can you unpack it a bit for me?
Indeed, one need look no further than the fact that we can co-opt the standard Latin 1 alphabet for transdecimal digits supporting up to base 6^{2}. American Sign Language has long mastered the use of one hand to express all the decimal digits plus all the letters of the alphabet (and more):Double sharp wrote:On the other hand (sorry), this multitude of possible conventions easily suggests to us that it would be quite easy to make reasonable schemes for any reasonable base, as you have just done for dozenal.
I agree with this: if powers of 2 are important to you, switch to binary, octal, or hexidecimal. If powers of 5 are important to you, switch to decimal. Use the right tool for the right job. Switching from dozenal to senary would absolutely make division by 4 and higher powers of two more prolix, even though it would make powers of 5 marginally easier.Double sharp wrote: Senary then sits at an awkward compromise which fits neither situation very well: the switch from dozenal to senary has allocated valuable resources that could have been spent on greater concision and better powers of two onto the much less important 5.
This I still don't quite understand. 5ths are unusable in senary? No disrespect to dozenal, but 5ths are even worse in that base, but I suspect that dozenalists wouldn't have any more trouble with 5ths than the whole metric-using world has with 3rds. If you want 1/3rd of a metre, there's not even a line on the meter stick for that. Yet they use thirds all the time. Honestly, the success of the metric system is proof to me that there's no problem that any base has w.r.t. measurement which can't be overcome *chuckle*.the fifths are not good enough to actually use,
Is there *any* human scale base which both makes 2,3, and 5 easy to work with, and still keeps them in the right priority? I guess base 2*2*2*3*3*5 would, but that's hardly a human-scale base.Another way of looking at it is that auxiliary bases have to give up a factor of the base in order to get one that isn't there already. With decimal, this is a no-brainer; you give up 5 to buy a better 3 and maybe 4. With dozenal, this is harder; the first thing you can buy is 5, and the only thing you can give up for it to keep 2 better than or equal to 3 better than or equal to 5 is 4. With senary, there's only 2 and 3 to give up, and neither can be given up for 5 if you want to keep {2, 3, 5} with the right priority.
delightful! Kodo, how did you make the drawings of the hands?Kodegadulo wrote: jan Misali makes the point that each of our hands can naturally represent a single senary digit, and using them together we can actually count to 55_{6} = 35_{d} = 2Ɛ_{z}. This is essentially the same point I demonstrated in the Xohox Numbers thread, with this image of how my fictional "Xanadunni" supposedly finger-count:
Theft. Ahem, strategic borrowing. I found the ventral drawings on some site about American Sign Language somewhere, and then generated the dorsal views by copying, rotating, and erasing interior lines.SenaryThe12th wrote:delightful! Kodo, how did you make the drawings of the hands?
They're "unusable" in the sense that you can't use them directly. If you want to use exact 5ths in senary, then you have to start with something that is already divisible by 5, so that you can divide it into 5ths and still get an integer - which is the same situation if you want exact 3rds in decimal, or exact 5ths in dozenal. So it doesn't matter that "0.111..." is a nicer recurrence than "0.24972497..."; neither terminates, and so in measurement you have to use an auxiliary if you want to deal with 1/5 in a senary or a dozenal context, just like how we deal with 1/3 in a decimal context.SenaryThe12th wrote:This I still don't quite understand. 5ths are unusable in senary? No disrespect to dozenal, but 5ths are even worse in that base, but I suspect that dozenalists wouldn't have any more trouble with 5ths than the whole metric-using world has with 3rds. If you want 1/3rd of a metre, there's not even a line on the meter stick for that. Yet they use thirds all the time. Honestly, the success of the metric system is proof to me that there's no problem that any base has w.r.t. measurement which can't be overcome *chuckle*.
You can't possibly have 2, 3, and 5 all as factors, because then your base is at least 2*3*5=30, and that's certainly not human-scale; so you have to give up one of them, and it may as well be 5 as the least important among them. This is similar to Jean Essig's argument for dozenal: you'd like to have all of {1, 2, 3, 4, 5, 6} as factors, but that way lies pure sexagesimal, which is too big, so you have to jettison at least one of those factors. If you want to do so, but retreat as little as possible from sexagesimal, then you ought to be giving up the least important factor among these, 5 (since having 2 and 3 guarantees 6 already), and that leavs us with lcm(1,2,3,4,6) = 12. Choosing decimal instead would be giving up {3,4,6} for {5} instead of the other way round; although the way decimal gives up {3,4,6} is significantly better than the way dozenal gives up 5, you're still giving up three factors for one. Choosing senary is essentially giving up 4 as well as 5 when you don't actually need to; as stated above, the shift from dozenal to senary means that 5ths, while better, still aren't fully usable without provisions being made for them, and this doesn't feel to me like sufficient justification for worsening quarters and concision.SenaryThe12th wrote:Is there *any* human scale base which both makes 2,3, and 5 easy to work with, and still keeps them in the right priority? I guess base 2*2*2*3*3*5 would, but that's hardly a human-scale base.
Wouldn't the fact that the sumerians used a 6-on-10 base be an existence proof that senary works just fine with auxilliary bases? I mean, that actually was a civilizational base. One of the few non-decimal civilizational bases, for that matter.
Sorry to hear that, and I hope you're feeling better now!SenaryThe12th wrote:P.S. sorry it took so long to get back to you; I've been feeling under the weather.
Well, its true that you can't use them *directly*, but:They're "unusable" in the sense that you can't use them directly.
actually, that makes all the difference. Just *because* 1/5 = 0.1 repeating, dividing by 5 is is as close as you can get to being directly usable. Take a look at how easy this is in senary: To find 1/5th of a number, all you have to do is multiply it by 0.1 repeating.So it doesn't matter that "0.111..." is a nicer recurrence than "0.24972497...";
Code: Select all
1.2
+ 1.2
+ 1.2
+ 1.......
----------
1.3 3 3 ......
I guess I see where you are coming from: would it be correct to say that (in twistaff notation) the upper staff is the auxiliary base and the lower staff is the ....uh, non-auxilliary base?6-on-10 is not really a senary auxiliary base but a decimal one. You can see this from the fact that it looks like decimal if you start counting with it: it goes 1, 2, 3, 4, 5, 6, 7, 8, 9, and then overflows to 10. This easily harmonises with decimal as you don't have to relearn how to count. If you wanted to use sexagesimal as a senary auxiliary, it would have to be 10-on-6 instead, and that is not as good as 6-on-10 because the top subdigit is decimal and hence 1/5 is represented more nicely ("20") than 1/3 ("32")
How about a 5-on-6 twistaff notation then? But perhaps you are right. I mean, given how easy 5's really are in senary, using an auxilliary base to make it even easier does seem like gilding the lily.(It is also not as good as 6-on-10 because the top subdigit uses a larger subbase than the bottom, so senary users seeking to use 10-on-6 have to agree on and learn to use extra figures beyond the standard range.)
We should see if we can persuade Icarus to expand his tour-des-bases to the ballenced bases as well :-)pcyrus wrote: Just a thought on the senary vs. dozenal debate: a reverse (balanced) dozenal system (such as Shwa, discussed elsewhere) has some characteristics of both.
Wow! That is a very cool trick. I totally didn't know about that. Thanks!Double sharp wrote: Actually, dividing by 5 in dozenal is also pretty easy, precisely because the period length is the maximum it could be. You just need to remember 1/5 = 0.2497..., and then all the other fifths are just a frameshift: 2/5 = 0.4972..., 3/5 = 0.7249..., 4/5 = 0.9724.... Again, there is no need to do any real multiplication.
Follow your bliss, man. I mean, the name of this board isn't "senaryonline"; It would be pretty reasonable to expect that the default position here would be dozenal :) I'm just glad ya'll tolerate a few of us wingnuts.{c} default dozenal