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 The Higher Natural Scale, The Awkward Stairway
Double sharp
Posted: Jan 15 2018, 10:42 AM

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In fact, I am increasingly quite convinced that the bulk of this "higher natural scale" is just the mid scale. I might even provocatively claim that the higher natural scale does not actually exist, because it lacks defining properties. Instead, there is simply a "human-scale approach" and a "mid-scale approach" to both addition and multiplication: in the former you don't break things up, while in the latter you do to avoid manipulating some numbers directly. The bits of the "higher natural scale" that could most justifiably be claimed to be distinct - {16, 18, 20} - could more reasonably simply be considered a reflection of the break not happening at the same place for each operation, not to mention for each person, which creates a transition where addition is single-step and multiplication isn't.

Consider bases 28 and 84 as {2:14 or 4:7, 6:14}. Now it seems to me that in both bases there simply isn't any way to memorise a full addition table: it's just a lot more ridiculous to try in 84 than in 28. Rather, we use breakpoints at every seven or fourteen, and split the problem into chunks, breaking the base down into factors. (I would've used 20 and 60 for examples, but you can actually almost finish the addition table in vigesimal with Sauter's strategies.)

Now consider multiplication. In both of these bases, every factor has a line that is simply too long to be memorised, so we end up splitting problems multiplicatively. To do 2*m in octovigesimal, we'd simply see that m is 8 over e, so the answer is 1g. The same goes for 2*23 in tetraoctogesimal: you'd surely see first that 2*20=40, then add 2*3=6 to give 46. (In fact, not even the whole of icarus' abbreviated table seems to be memorised.)

Similarly, whenever complementary divisors arise, they are the obvious and pretty much the only way to see where you are in a line. 7*j is pretty obviously dividing j0 in four to give 4l, since l is 3/4 of twenty-eight. And in base eighty-four, 17*42 = 42'00 / 4 = 10'30.

Lastly, when we do numbers that are "off the grid", we simply choose from a number of strategies, like multiplicative breaking (like hexadecimal regrouping to bunch up powers of two), additive breaking (into factors), and the quarter-square identity if the numbers happen to be close. We just see the last a lot more often in a small base, because then the digits can't be too far apart, and the method works in a higher proportion of all cases. Even DD-shifting human-scale products from the smaller base finds a parallel in how we just tweak the upper end of the multiplication table slightly to do things in {6:10}, writing 8*9=1'12 instead of 72; similarly, in vigesimal, we might think "seventy-two" and write "3c". (Surely if the French can think "soixante-douze" and write "72" it should be no different the other way around.)

It therefore seems to me that the difference is purely cosmetic, in the sense that a small mid-scale base like {4:6, 4:7} may be felt as though each place had one digit, rather than two subdigits. So, for example, the Babylonian use of {6:10} has equally spaced senary and decimal figures, while the Mayan use of {4:5} has each vigesimal digit as a separate block containing both the fives and the units. In fact, the latter rather remind me of the Chinese rod numerals: despite the fact that this and the abacus are constructed as {2:5}, not only is there no evidence of any biquinary alternation in arithmetic at all, there is instead copious evidence of the use of pure decimal and a multiplication table.

This would then make {(20), 24, 28, 30, 32, 36} look like human-scale bases even if the tables needed to make them run as such are enormous. However, the clear omnipresence of their factors should facilitate decay once fast multiplication stops being the preserve of a few learned scribes, and the algorithms these scribes would have been using beforehand would surely have been thoroughly mid-scale.

(I am considering 20 a special case because it is small enough that you can largely proceed with addition much as you would in a smaller base, so that you have enough of a foundation to start before abacus-like fiddling with fives and ones fills in the leftovers, which are mostly adding {3, h} and {4, g}. This being said, its multiplication table for me is definitely quite mid-scale, or at least it would be if I wasn't just DD shifting from decimal for pretty much all of it. I think that might be a significant barrier for us to find out how vigesimal feels like intuitively as a base in itself, assuming of course that it can be thought of that way: I somehow don't do this to get hexadecimal from octal, so we cannot rule out the hypothesis that this isn't only because we're all immersed in decimal.)
Double sharp
Posted: Jan 15 2018, 03:26 PM

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Come to think of it, this suggests that you might be able to save vigesimal by a sort of numeric diglossia with decimal. We could consider adding with the "teens" digits (that go down to the feet) to be an exercise that is covered only after adding with single digits (on the hands) is completely understood, so that the gradation of addition facts would be:

1. The trivial facts (adding 0 and 1);
2. The facts whose answer doesn't surpass a;
3. The facts whose addends don't surpass a;
4. The facts where one addend surpasses a;
5. The facts where both addends surpass a.

The beginning, in fact, would look exactly like decimal, except that once we get to group 3, we're writing "8 + 6 = e" instead of "8 + 6 = 14". Of course we'd still say "fourteen" or something like that, but we'd write it with its own transdecimal digit. Starting group 4 would presumably be facts like "plus a, without carries", "plus 5, without carries", and "plus f, without carries", as Oschkar's exposť has them. (I may write a revised version going through this route.) Group 5 can either round off from 10 using subtractive strategies or simply group the "teens" together into a carry and then use group 3 strategies for the remainders.

In this sort of vigesimal, you'd learn the tables up to ten times ten, but in base twenty, and you could have a lot of code-switching involved with a decimal nomenclature. So you could say "eight nines make seventy-two", but you'd always write it as "8*9=3c". Once again we have:

1. The trivial facts (multiplying by 0 and 1);
2. The facts whose multiplicands don't surpass a;
3. The facts where one multiplicand surpasses a;
4. The facts where both multiplicands surpass a.

I imagine that the effect of multiplying by a (halving a number) has to be taught early, somewhat apart from these groups. Then we can learn the tables up to a*a=50 using something close to the decimal strategies (although I confess I haven't really thought much about the alterations needed because the base is double the size). To do the facts in group 3, you break up the larger multiplicand into a and some leftovers; so g*7 = a*7 + 6*7 = 3a + 22 = 5c. (I think the facts with a need to be thoroughly drilled in, so that they feel almost as round as the true scores. Perhaps there needs to be finger-only counting as well as finger-and-toe counting?) To do the facts in group 4, you use the quarter-squares identity, just as you can for the high facts in group 2 if the decimal nine-times strategies aren't obvious enough. For example, d*e = (a+3)(a+4) = (h*a) + (3*4) = 92. Of course you can in some cases use 10 instead, like in g*h = (10-4)(10-3) = d0 + c = dc. And if you do them enough, it might well be possible to memorise them all.

I'm not sure if this can really be taken past twenty, because of the need to learn lots of different numerals. But if it can, it seems to be one of the best hopes available for such a double-sized base.

P.S. I think I've forgotten the even bigger problem that this doesn't really reduce the number of facts you have to learn, but only it makes it possible to learn the ones that are already there, so that I think it's massively unlikely that this would actually work to get people to memorise the facts for {24, 28}. This being said, these bases are larger, so it takes less efficiency to get them to work as long as the bottom of the range is well-understood. I think I'll have to try them all out again. This is honestly still using mid-scale strategies with the exception that removing the alternation perhaps removes the biggest hurdle, but they might not feel that way.

This post has been edited by Double sharp on Jan 15 2018, 11:42 PM
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