## Le Tour Des Bases

This forum examines bases other than twelve and less than sixty.
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
I don't think the colors will ever be used in the same work. We have the colors you are familiar with, the red divisors, orange rich regulars, yellow semicoprimes, and gray coprimes. The pale blue and aqua omega and alpha factors become full-bore blue and aqua in divisibility test maps.

Essentially we have several kinds of regular number. All regular numbers are products restricted to the prime divisors of the base (or simply the number 1, which we call a unit on account of its special status as both a divisor and coprime to the base, and neither prime nor composite). We can divide the regular numbers four ways. Regulars that divide the base evenly are divisors, those that are non divisors are called rich regulars. Regulars less than or equal to the base are digits; the others are non digits and they are all rich regulars. Rich regular digits, i.e., nondivisor regular digits are called semidivisors. If we look at regular numbers as a product of a regular root gamma and a power of the base, then we have three possible classes: regular roots (regular numbers that are not multiples of the base, e.g., "2" vs. the family {20, 200, 2,000, …} or "125" vs. {1,250, 12,500, 125,000, …}) regular compounds (like the numbers in families just mentioned as counterexamples of regular roots), and pure powers of the base (by this I mean nonnegative integer powers of the base, I am not being as rigorous as I normally am in writing this, so as to be brief). All the regular digits that are not the base itself are regular roots (I called regular roots "regular figures", what Wendy calls "units"). The trivial divisors and any nonnegative integer power of the base are obviously pure powers. The concept of regular root helps to make the range of practical regular numbers finite in some cases (not in the case of prime power bases). There are practical and impractical regular roots and powers of bases. So this produces many flavors of regular numbers, not all are of interest generally.

There are kinds of coprimes, we all know and love them. The types have to do with dividing the neighbors evenly. So there are alphas, omegas, and each have factors, alpha factors and omega factors. In base ten, we have the alpha 11 and the omega 9; the only neighbor factor of interest is the nontrivial omega factor 3. Then there are products of alpha and omega factors called alpha-omega factors, thus in base ten we have {33, 99}. The number 2 in odd bases is a factor of both alpha and omega, but it is more efficient simply to consider it omega factor. A coprime that is not a factor of the neighbors is opaque. A coprime that is a digit is called a totative. All neighbor factors have practical intuitive divisibility tests; they are never impractical. Neighbor factors have practical reptends but the multiples of these unit reptends may be impractical to keep in mind.

We then have semicoprimes, composite numbers that have at least one prime divisor of the base and at least one prime coprime to the base as factors. Generally any semicoprime is the product of a regular component and a coprime component. Practicality is a concern of the regular component, while neighbor-relatedness is a concern of the coprime portion. Concerning the coprime portion, we have alpha semicoprimes (aka alpha inheritors), omega semicoprimes, and alpha-omega-semicoprimes. In odd bases, we have even-semicoprimes that are always neighbor-related. Each of these is practical or impractical dependent on the regular component. A semicoprime digit is called a semitotative.

Since embarking on the fundamental portion of the website, the regular numbers have received a lot of attention, solidly integrating the concept of regular roots and the compound nature of regular numbers (when these are not compound, i.e., prime, one of the factors , the regular root or the power of the base, equals 1. When the regular number is 1, then both are 1). This affected what we'd called inheritors, which really are semicoprimes via the notion of practicality.

Practicality itself is a weird thing to define because it isn't fully arithmetic in nature. It's cognitive, so what we deem practical really is provisional until we have scientific research. But we can make a consistent computation of it provided these stand-in assumptions and then compare apples to apples. Practicality in divisibility tests is governed by the multiplicative complement to regular root, in the number of combinations less than the power of the base whose exponent is the regular root's richness (what I called remoteness). With some regular IDTs, we can reduce this range through "range folding", exhibited by the decimal test for divisibility by 8. Range folding can make some regular IDTs practical, but it is only applicable to powers or certain products of {2, 3}, maybe 5. Curiosly, in bases that are semiprimes like ten, the upper divisor has a larger range of practical powers. In base ten, the fourth power of 5 has a practical regular IDT (there are only 16 combinations; as these are the multiples of the decimally expanded 16th, they are relatively well known) but the fourth power of two is opaque (625 combinations, range folding is not practical, since it is easier to actually divide continually by 2 to determine divisibility by 16).

So all this new stuff is going in to the work. As I wrote, fortunately not all of these distinctions are useful at any one time.

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
Sounds very comprehensive, Icarus. And I like the new, distinctive names like 'rich regular' to distinguish from 'divisor' and 'semidivisor', which are exclusively digits of the base.

One expansion of the coprimes category has cropped up on the boards in the context of the dozenal divisibility test for 5: the power-neighbour. It generalises the rules of the neighbours to multiple digits, though its practicality depends, much like rich regulars, on remoteness. We know that for every prime p that is not a divisor or neighbour of a certain number n, the number n to the power (pâˆ’1)/2 will have a neighbour that divides p. So, if a number does not have 7 as its divisor or neighbour, we can be sure its 3rd power (ie cube) will have a neighbour divisible by 7.

All power-neighbours allow compaction of the number to prepare it for divisibility testing. For example, any number not near enough to being a multiple of E will have a relation to E through its 5th power (see above rule), which will allow us to sum groups of 5 digits or subtract the sums of alternating 5-digit places according to the power-neighbour relationship. However, this is of little use practically speaking, because a number would have to be more than 5 digits long to benefit from compaction, and even then the remaining 5-digit number would need a universal test (long division, placeholder et cetera).

In decimal, the potentially helpful power-neighbours are the cube-Ï‰ 999 = 3â†‘3Â·37 and the cube-Î± 1001 = 7Â·11Â·13. We can use the former relationship to test for the third power of 3 by summing groups of three digits, then testing the compacted number, but this would be of little help; it would be far simpler to divide the tested number by 3, then test the result for divisibility by 9 using the digit-sum test. Likewise, a number to be tested for divisibility by 7 or 13 can be compacted down to three digits by successively subtracting the last three digits from all the rest, or by subtracting the sums of alternating 3-digit places. Again, this is only a slight aid.

It looks as if dozenal really hits the jackpot with a complete divisibility test using a power-neighbour relationship. Viewing again the rule above, it is inevitable that the square of base dozen should have a neighbour divisible by 5, but it is not inevitable that there should be few enough 2-digit multiples below the gross to be memorised. SPD is like the practical tests for rich regular numbers in that it is operative only below a certain memorisation threshold; the main difference is SPD tends to reach it much faster than the regular tests do, hence its rarity (beside dozenal, octal is probably the only other usable base that benefits from SPD). Still, given the way SPD brings the hitherto totally opaque totative 5 into the dozenal fold (at least as far as divisibility tests go, not so for fractions unfortunately), I think the category of power-neighbours is worth a mention in your work.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Treisaran, indeed these are important. I think they are not as immediately apparent as omega and alpha. They would be mentioned in an article about dozenal simply because they ameliorate the five problem twelve has. I think through the Dozenal test for fivefoldness, base twelve has an appropriately evident and easily taught test for the third prime.

Indeed I do not think alpha is immediately apparent in decimal. I don't know if most folks know the test for elevenfoldness but the "stuttering" in the multiplication table is evident. Systemically, alpha is evident across many bases.

But you are right, power neighbors merit attention. It's funny, it's taken a month to get to basic things, and I don't know if I've fully explained the basic things. The basic things look pretty complex so they must be made basic, but in order to get to that point, we have to take it to the top first like a painter does his art before we can pare it down to essentials.

 Posts 773
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
There is an interesting pattern with bases 8, 13 and 18, which have relatively factorable betas of 65=5Ã—13, 170=2Ã—5Ã—17 and 325=5Â²Ã—13. Unfortunately, the pattern isnâ€™t continued into bases 23 and 28, but coincidentally, bases 8, 13 and 18 also have decent psis: 63=3Â²Ã—7, 168=2Â³Ã—3Ã—7. (Yes, I know that the psi is the product of the alpha and omega, but still...) In fact, tridecimal has memorable tests for the first seven primes except 11. I wouldnâ€™t use this as a treizenalist argument, only an interesting curiosity, because base 13 fails at everything else (no useful divisors, all fractions repeating except powers of 13...).
Actually I think this strengthens the case for the 10-12-14 base triangle. SPD is useful, but it is suboptimal, especially considering that three of every five bases have a simpler test for five. Seven, on the other hand, is only remotely reachable by SPD, and eleven even further (this really complicates divisibility up to eleven).
In decimal, for example, to test for seven through the regular SPD test, one would have to promote groups of three digits to a multiple of 7. By coincidence, decimal 301 is divisible by 7, so you could promote groups of two digits and correct the rest by triple the amount. Dozenal is the same, only instead of triple, correct it by the double in the opposite direction, and tetradecimal has seven as a factor. For eleven, decimal has it as its alpha, dozenal as its omega, and tetradecimal has it as a factor of the terribly inconvent 601 (1177 decimal).
However, casting off 98â€™s in decimal and 102â€™s (decimal 198) in tetradecimal is a quite intuitive way of grabbing hold of seven and eleven.

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
Icarus, from the standpoint of number theory, I might venture to make two sweeping generalisations:
1. The regular relationship is the father of all regular-type tests (divisor, semidivisor, rich regular whether through memorisation or range folding).
2. The trim-right test achieved by virtue of modular arithmetic is the father of all neighbour-type tests (digit-sum, alternating-digits and the multiple-digit versions of them; successive subtraction of some least-significant digits from all the rest; and SPD).
We could call the first large grouping 'modulo-0 based tests' and the second large grouping 'modulo-1 based tests'. What makes one test special and another test unremarkable is its ease of application, which in turn is usually a function of its closeness to a number we consider 'round' in the base. So the power-neighbour tests for 5 and 7 in dozenal, based on *101 and *1001 being multiples of 5 and 7 respectively, are more special than the trim-right tests for them, based on *21 and 2E (for 5) and on *41 and 2E (for 7), because the powers *100 and *1000 are rounder than the multiples *20, *30 and *40.
Oschkar wrote:In fact, tridecimal has memorable tests for the first seven primes except 11.
Only 9 (which has an unhelpful cube-Ï‰ relation to the base) and B (dozenal E, decimal 11) are opaque in ununimal. 2, 3, 4 and 6 benefit from the proximity to dozen, getting them the digit-sum test; 7 gets the alternating digit sum difference test from the neighbour upstairs; 5 uses SPD as in dozenal; and 8, like the one additional binary power in all odd bases, is a square-Ï‰ inheritor and can therefore be tested for after compacting the tested number down to two digits by summing up pairs of digits.

I find the two neighbours of dozenal interesting to compare for their divisibility tests, because they're slightly different even though they both leech off their highly composite neighbour. In levimal, 3, 4 and 6 are tested with the alternating digit sum difference test; 2 and 5 get the digit-sum test; 7 and 9 are opaque (they're cube-Î± and cube-Ï‰ inheritors, respectively); 8 gets the paired digit-sum compaction option. To summarise, the dozenal factors are easier to test for in ununimal than in levimal, except 2, which in odd bases is always both Ï‰ and Î± and therefore of the same difficulty; 5 is significantly harder in ununimal, while 7 is out of reach in levimal; 8 is of the same difficulty in both, and 9 is opaque in both.

Naturally, there's no import to this comparison beyond its value as a curio. A base rich in divisors is always preferable, and dozenal gives us divisors in spades while leaving only the prime 7 opaque, which is no big loss.

The comparison also brings to mind another comparison, between octal and unquadral, which highlights the difference between theoretical and practical benefits. Except for eleven, opaque in both, the first six primes and 9 differ greatly, with octal having tests for 3 (alternating), 5 (SPD), 7 (digit-sum), 9 (alternating) and *11 (SPD), while unquadral has tests for 3 and 5 (both digit-sum) but leaves 7, 9 and *11 (0xD) opaque (they're all cube-Ï‰ inheritors). Does this make octal better than unquadral for divisibility testing? I don't think so. For practical purposes we need no more than the first three primes to be covered; since octal and unquadral both have that coverage, they now compete on ease of application, and here unquadral wins, offering the easy digit-sum test while octal needs the harder alternating-digits test for 3 and even harder SPD test for 5.

Of course, for coverage of the first three primes, one might opt for the divisor relationship for all of them, which brings us to 30 and its multiples.

wendy.krieger
wendy.krieger
What is generally useful is a 'preferred interval', rather than short periods. A preferred interval means that the periods divide a common small number (like 4 or 6), and that the various products of these primes thus divide a short period.

In this list, i include the factors of the base to give completeness. These contribute to the pre-period. For example, in (note * should read **)
• 10, the omega6 cluster is (2).3Â³.(5.).7.11.13.19.
• 21 gives for a 4-place cluster 2*4 (3) 5 (7) 11 13 17, which is all but one prime.
• 30 gives on 6 places, (2) (3) (5) 7Â² 13 29 31 67
• 68 gives on 6 places (2*2) 3*2 7*2 13 19*2 23 31 67.
• 80 gives on the inverted-3 place (ie 1.0.0.1) (2*4) 3*5 (5) 7*2 43
• 99 gives on four places 2*4 (3) 5*2 7*2 (11) 13*2 29 or 1820Â² * 29
• 120 gives on two places (2*3) (3) (5) 7 11*2 17.
Another interesting thing is that when one uses omega-2 sums in bases like 68 and 200, the possible number of sums that a regular number comes to is much smaller. Even in 20, the sums come to one of these eighteen 0.1, 0.2, 0.4, 0.5, 0.8, 0.10, 0.16, 1.5, 1.12, 2.10, 2.13, 3.4, 5.6, 5.13, 6.8, 6.11, 10.12, 12.16, and their reversals (eg 5.1 or 16.0). For base 200, a similar process gives 198 possibilities.

Even in base 120, the single place sum of the powers comes down to just 48 possibilities out of the 96 available (less 16 mulitoples of 7 and 6 multiples of 17, and one of both).

One can not really tell that twelve's relative usefulness comes from its particular factors. 12 has a pretty tight 'opposition', in that the recriprocals of regulars are about the same size as the regulars themselves. But then 56 and 72 have this. It's also pretty small, and the power of 2 is the larger of the two prime-powers to make it.

 Posts 2,143
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
icarus wrote:Again the designation â€œintuitiveâ€ is a construction; in the new work weâ€™ll go beyond intuitive, to â€œpracticalâ€.
Both are subjective terms. I only objected to 'intuitive' because it implies a elementary schooler can discover the test all on his own, which I think holds true only for the divisor, semidivisor and placeholder* tests. Even the digit-sum wouldn't occur to a young child except as a fluke.

'Practical' sounds better, but it's still subjective. The problem here is deciding what the upper threshold of practicality is. I consider the number of two-digit multiples to be memorised for the SPD test for 5 in base 18 to be over the threshold, but others like Oschkar have disputed this. Who's right? Since it's subjective, we may both be. We can safely call a test 'impractical' only when the consensus as to its impracticality is overwhelming.
I think techniques like SPD and the higher mod-1 tests wonâ€™t be â€œfirst-tierâ€ considerations but do need to be discussed when we delve into the divisibility test portion of any base.
SPD is an involved test; perhaps easy enough once you get the required multiples memorised, but it still hasn't got the same out-of-hand usability to it as the digit-sum and alternating-digits tests. The only thing that makes it worthy of note is that, finally, finally, after a long time trying to get the prime 5 into the dozenal fold in various ways, we've got a complete test for 5 in dozenal. I might say SPD is remarkable in a 'world's tallest midget' kind of way.

â”€â”€â”€â”€â”€â”€â”€â”€â”€
* The placeholder test is the one where you test divisibility by seeing if the tested number is the sum of two known multiples. For example, decimal 238 divides by 7 because it is the sum of 210 and 28, both known to be multiples of 7 from the basic multiplication table.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
In the work, I attempt to measure the quantity of practical tests, etc. and resort to defining "intuitive" as including only the regular, alpha, omega, and their compounds as such. Practicality is similarly defined by a definition. We indeed would need scientific research, the messy, iterative, empirical type that involves the resources of a multitude to conduct. The "intuitive" tests would be so to educated young adults, so that these would be used in their work. Many adults use the rules of 3 and 9 and the regular tests in a way that they are not necessarily conscious of.

Practicality would concern the feasibility of teaching children how to use the tests. This sets the bar rather low! I am not saying that kids have to understand it coming into the subject, but that the kids would need to be able to be taught and to retain the knowledge so that further lessons can build to new topics in math. The whole subject of elementary arithmetic would have to "fit" into the 4 or so year long period in grade school. This is another reason why bases larger than 16 (maybe even including 16!) are not "human scale", and why it is important to have a base in the human scale.

It is the very argument or constraint I've done a very good job of ignoring when we consider bases like 20, 24, or 60, etc. There is a way to sneak around the barrier, i.e., using decimal coded sexagesimal, etc. We'd simply teach a different set of algorithms and that might fit into the school box. If we teach arithmetic for too long, we will delay subsequent topics and "set back" society because we haven't addressed algebra till age 16. This is why bases between about 7 and at most (and I am trying hard to be nice to the hexadecimalists) 16, possibly certain decimal coded bases and quinary coded vigesimal are likely the only practical bases to teach.

In recent weeks I am thinking that the practical bases to teach civilizationally really are very limited: {7, 8, (9), 10, (11), 12, 14, 20 via 4:5, 60 via 6:10, maybe 120 via 12:10}, that they might even be as limited as {8, 10, 12, 14, 4:5, 6:10} (I have to say 4:5, 6:10 because these were civilizational bases, but I think these are alien to the way we do things now). Clearly out of this range the dozen stands out as offering the most support for human arithmetic, decimal a decent second, followed by 8 and then 14, the two rather close to one another, rather much less practical than decimal. With octal we have problems with compression but they are not unmanageable, we have alpha-dominance and SPD for 5. With tetradecimal we have more static with opaque totatives but the scale is not yet unmanageable. Hexadecimal simply seems too laden with opaque totatives and a massive magnitude; I think we're in 5th grade learning our digit-{7, 9, 11, 13} multiplication facts while the decimal and dozenal worlds are applying these in geometry and early algebra. Every other number base seems impractical in education or application, with too little compression, insufficient pattern-recognition support for arithmetic, or massive arithmetic tables.

I do think the dozenal SPD test would be taught in grade school in a duodecimal society. It is rather simple, and the multiples of 5 would be common knowledge as the multiples of 7 are in decimal. Thankfully, base 12 is human-scale and fully within the grasp of human cognitive capacity. So I think that one can grasp the dozenal multiples of 5 less than one gross through a sort of range folding:

00 05 0a 13 18 21 26 2b 34 39 42 47
50 55 5a 63 68 71 76 7b 84 89 92 97
a0 a5 aa b3 b8

in effect all we need is what we would have learned in 2nd grade: the multiples of 5 in the multiplication table, then recognizing that the pattern repeats every five dozen. With this we can construct the required range until it stands on its own.

I don't think SPD for 5 would prove as prominent as the decimal rules for 3 and 9 (the omega IDT), but I do think it would find enough use to be common knowledge among young adults moving into their professions in a duodecimal civilization. It would be the "rule of 5".

wendy.krieger
wendy.krieger
A preferred interval comes when you get a large omega that is still managable, and has lots of useful primes in it. What you get is something like ;abcd or ;abcdef with lots of useful primes in it.

In decimal, one has 1/7 as 0.142857 1/11 = 0.090909, and 1/77 as 0.012987.

The six-place period has a lot of more useful factors than other shorter or longer periods, and it's pretty much about the only decent choice.

In base 16, (which is an eighth period), the three-place and five-place periods vye for attention, eg 3*3*5*7*13, and 3*5*5*11*31*41, but it's hard to pick from one or the other. Base 18 (3: 7*7*7*17 vs 4: 5*5*13*17*19) vye for attention so it doesn't have a preferred interval either.

Very few bases as sucg do have preferred intervals.

 Posts 773
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
Just a weird anecdote I had. I was looking for properties of base 18 and then for the rest of the day I found myself thinking in octodecimal completely. I hadn't realized it until I found myself counting "five, ten, trick, twenteen, seventeen, twelfteen,..." (yes, that's what I called them). Just makes me realize how much time I spend with other bases.

wendy.krieger
wendy.krieger
I was looking for properties of base 18 and then for the rest of the day I found myself thinking in octodecimal completely.
Lucky you. I usually doodle in bases 14 and 22, and sometimes 70.

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
This is very interesting, Icarus, the issue of teachability. Decimal is so ingrained that it's strange yet enlightening to think how elementary arithmetic would be taught in another base.

Again I can only appreciate how relatively fortunate we are to have settled on decimal as a non-dozenal base. In the entire human-scale range, putting odd bases out of consideration, ten is truly the second best. The alpha relationship you have to 3 in octal and unbinal makes things too hard. The importance of 3, the major reason for the existence of dozenalism, makes the digit-sum test a minimum requirement. Having an alternating-digits test is acceptable for 5 (as in base *20), but not for 3. For fractions, even the single-digit thirds of decimal aren't good enough, so the two-digit thirds of bases 8 and *12 would be simply intolerable.

Regarding the tables for SPD:
icarus wrote:in effect all we need is what we would have learned in 2nd grade: the multiples of 5 in the multiplication table, then recognizing that the pattern repeats every five dozen. With this we can construct the required range until it stands on its own.
I'm for such range folding as a scaffold, to be used until you've memorised the table straight out. There must, however, be some upper limit beyond which the scaffold becomes permanent. I think I could memorise the basic multiplication table row of 5 in base *28 (2â†‘5), but never the entire extended block of a bit over two hundred 2-digit multiples.

Hmm, looks like I'm set to agree with Oschkar that base *16 can use the SPD test for 5 after all. But I'm not interested in memorising it, just as I haven't memorised the small (smaller than dozenal) SPD table for 5 in octal. I value SPD for being, as you said, the 'rule of 5' in dozenal.

Apropos, your Dozenal FAQs contain the note that the inability to test for divisibility by 5 'bugs' you. That section could probably do with an update. When you have the time and inclination, of course.

Another note: I decided on the name 'SPD' to stand for 'Split, Promote, Discard', the three operations involved, as a descriptive initialism that, as it were, includes the manual in the tool itself. Formally the particular SPD test for dozenal could be called the Abbreviated Square-Alpha Test, and SPD in general could be called the Abbreviated Power-Alpha Test. As I have noted elsewhere, there exist SPD tests based on power-omega relationships, but they make things harder rather than easier; the power-Ï‰ invites you to sum groups of digits, nice and easy, so no SPD needed.

 Posts 2,143
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Treisaran,

Indeed I think that it wouldn't be long, after a period of application, that someone might memorize the multiples of five in a dozenal civilization. I used to work as a custodian aide in college; I grew up in an industrial town. In the biomechanical department at the hospital and the maintenance shop during my high school and college summers, I noticed the guys simply knew their 16ths, 32nds, and 64ths. Especially in the machine shop. These are usually the same guys that tell you they didn't want to be in school after high school, completely disinclined to the classroom, yet they knew 64 multiples of .015625! (To be sure, though, the shops had a big chart with these multiples printed out). These guys would tell you they weren't "book smart" but knew their multiples because it was a matter of course in their work; fractions of an inch, something practical for them. We wouldn't say that 64 is a practical regular number; I don't think we'd teach young kids the multiples of 1/64. I think we could rely on them to know the eighths after a few years in school, maybe have a feeling for the 16ths. I think 8 is the limit of practicality for powers of 2 in decimal. This doesn't mean people simply don't get and can't reach larger regulars like 64. It simply means that a young kid in school wouldn't be able to handle memorization. In the course of a profession, some things are handy to know, thus "impractical" regulars like 64 fall into use.

Surely the multiples of 5 less than 144 (only 28 such, a figure that falls within the range of practicality in my estimation) is not beyond acquisition by memory through use. I think the dozenal SPD for 5 is practical. I don't know if the 64 multiples of 5 less than 324 in base 18 is practical, but we can range-fold the 5 table to reach that figure (it's just that I think that base 18 is beyond practicality in memorizing the arithmetic tables).

Truly, no work introducing dozenal should stand without this rule mentioned, because it fills a key gap in the utility of base twelve. With the SPD for 5, or the "rule of 5", we can't get a compound rule for {10, 15, 20, 30, 40, 45, 55, 60, etc.}. The rule of five simply renders dozenal even more powerful. So if it doesn't really appear anywhere in the FAQs, they deserve to be updated.

Most of my pondering happens in very large bases, but I usually use dozenal and sexagesimal for things like counting laps, etc. while swimming. I use dozenal massively nearly daily in my work, as it is a handy way of manipulating (US Customary) dimensions, which I use daily. I use sexagesimal also in the computation of fees and other things. I am fond of tetradecimal because it is awkward and serves as a sort of analogy for decimal. The "pondering" normally is restricted to integers, but sometimes involves fractions especially of regular numbers.

wendy.krieger
wendy.krieger
One can learn a great list of useful numbers, and retain them some time. For example, i did a lot of calculations on four-function calculators, so i set the equations up to use the short-chord (the chord of a triangle, bounded by two edges), of the regular polygons. The shortchords for 3-7, 10, and 12 are usually given to 12 digits.

One of the bits of magic one can weave with an SWS system is to take something like "Btu-inch / sq foot hr degF", and reduce it to the sws-coherent measure with little fuss. Each of these units equate to a number in sws, so that unit is simply an equation in numbers, eg 6d6. 3 / 9d2 1d5 1d4 = 2d-5. d = exponent duodecimal. I use the tiof in twelfty to do this. This is the sort of magic that an sws in a high-factor scheme does. The current duodecimal version lines up the units in a pretty tight scale, while still allowing things like 1/16, so it nearly gets the advantages of 16 (where the regulars are actually powers of the base).

There are also useful conversion points, and ratios, which means that you can look at metric or imperial figures, and simply convert the number at duodecimal, although not always at the radix. For example, the pressure given in millibars convert into dbo (my sws scheme), by supposing that 1013 mb is 101.3, gives 85;4 add 1/20; to get 89; gives 890; mbars (DD). Minutes are converted at 10 min = 10; min DD, Because bdo passes through metric at nearly the dm, kg values, many of the handy values convert at that ratio (eg 183 cm = 18.3 = 16;4 gives 164 bc or 6 ft 1 inche.)

This sort of trick allows you to read regular instruments and get conversions.

My previous conversion tables were based on giving decimal expansions of metric or imperial units to many places. Now it takes this form

Metrics are rounded to the

c ton : 1 us 25/28 m 100 t = 125/126
ton 6/5 us 15/14 m 1 t = 25/21
cwt ditto 50c = 25/21
stone 9/10 7c = 1
pound 9/7 500 g = 10/7
ounce 28/27 25 g = 6/7
dram 3/5 * 1.25 g = 3/7

exact 539/540 * 385/384 m = 203/205

troy:
ounce = 16/15 m 30 g = 1
drachm = 4/3 m 3 g = 1
dwt = 16/15 m 1.5 g = 1
scruple = 4/3 m = 1
ounce / c = 32/25 m = 6/5 mg
carat = 64/75 m = 4/5 200 mg
grain = 4/3 m = 1 50 mg
mite = 8/5 m = 24/25 mg

exact 99/100 metric = 7308/7175

c = 100 in the unit, either 100 units, or unit divided into 100. so c ton is a hundred tons (120 TW, 100 ME or BI).

This is read as a BI stone is 9/10 of the twelfty stone, but if you want a more exact value, you have to multiply it by 539/540. The conversions happen from the nearest unit to the nearest unit, avoiding any preferential base unit. So weight say of 108 kgs (or c chogs), is converted into 15 st 6 lbs ME, which gives 15 st 8 lb TW.

 Posts 615
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Paul Rapoport
Dozens Disciple
Joined: Dec 26 2012, 01:59 AM
Le tour des bases est vraiment la tour . . .

If only I had the time to go through it all. The setup of the relatively low bases after 12 reminds me of music written in equal temperaments other than 12, a subject I've had a fair amount of involvement with over the past decades. Easley Blackwood wrote music in 13 to 24 and wrote about those tunings, demolishing received opinion about most of them.

I have no idea whether we'd have a rapprochement. Although I've devised principles for the best notation for a very large number of equal temperaments, I doubt that they'd benefit from a numbers-heavy approach, especially if one tries to maintain centric pitch organization (tonal/modal widely interpreted), which implies a clear first principle that necessarily biases the notation.

I've written music in 19-equal, 31-, and 25-, with a few bars in 13- (and 12-, of course). Although equal temperaments are by no means the only way to go outside 12, they provide many surprises in their basic setup and their harmonic relations, partly because of their completely transposable enharmonicities.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Paul,

Thanks! The tour is intended to be a resource against which we can test our statements regarding duodecimal or any other base. If most of our argument is based on number theory (arithmetic), we have logic on our side. Some of the argument has to do with human cognition, which requires scientific research. Some of the human cognition argument is evident through observation. If we stick with the arithmetic algorithms and positional notation we are used to, then the bases up to about 20 are "wieldable". The "word length problem" restricts us to bases above 6 or 7, but the loss of compression below decimal makes us reluctant to settle for any base smaller than ten. The implicit duration of elementary instruction in arithmetic seems to restrict us to bases 16 or less (and I think the bases between 7 and 14 represent the range of tolerable duration of instruction: 16 seems to have us learning multiplication for two and a half years, so a total of 5 years, meaning all one's primary school, we're only achieving basic arithmetic). Within the range of bases 7 to 14, dozenal is evidently the most richly patterned base, working with human pattern recognition faculties, facilitating the acquisition and practice of arithmetic.

With base 16, we might say that we could use duplation and mediation to calculate, as it is "natural" in a base that is 2^2^2. I am unsure whether this approach to arithmetic at the scale of our current civilizational level of development would be as efficient as our mnemonic multiplication. If D&M is as efficient or nearly so, then hexadecimal is part of the range {7-14, 16}. my guess is that D&M is akin to the complementary divisor method of multiplication, dividing a single problem into parts, this division seems inefficient. Also hexadecimal might be buoyed if we redesigned numerals to resemble their binary values. Then the numerals aid arithmetic. This is why I have not written off base 16 as a "human scale base". So I consider the range 7-16 "human scale". Hexadecimal is also buoyed by its neighbor 15 = 3 * 5, giving it transparency for {3, 5, f} to multiplicity = 1, and products of powers of two with these neighbor factors. But transparency is no substitute for having those small primes as factors. Hexadecimal is depressed by "putting all its eggs in one basket", i.e., being a prime power. It suffers a paucity of regular numbers, which are the font of a richly patterned base. Any way you look at it, hexadecimal is an interesting base.

With decimal, it is not optimum but it is buoyed by the neighbor relationship with 9, which grants decimal good transparency for the "skipped" prime 3, its commonly encountered product 6, and its square 9. Because of this, and the "native" properties (compact size, relatively small prime factors) I think it is second best.

Duodecimal represents the convergence of several small cycles, leaving it in a "void", i.e., with prime neighbors. So its strength is concentrated its "native" properties. The patterns and brief representations of product cycles and expansions of unit fractions of small integers overcomes the relatively poor handling of small primes that do not divide twelve evenly. The patterning is a great aid to human arithmetic, making it easy to acquire and exercise.

I think it's neat to take a look at different scales, even very large ones. If we admit mixed radix representation, we might see that decimal coded sexagesimal and base 120, or quinary coded vigesimal are only about as difficult as hexadecimal, maybe more meritorious to use than base 16. The studies, like your own studies of music in scales other than 12 and this of others, only contribute to one's understanding of how scales work/affect music or arithmetic. At the least, they act as a foil for work in a favorite base of choice, especially if that base is optimized.

The tour is intended for reference, so if you're interested, read it now and then. No one has to read every post or read the range like a book. This way it's more like a dictionary or encyclopedia.

 Posts 773
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
Censeo Icarum occupatum esse, quoniam multas dies in foro non postavit.

 Posts 773
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
I was going to try to write a comment in Hebrew, but I really couldn't find any grammar references to guide me...

Dozens Disciple
Dan
Dozens Disciple
Joined: Aug 8 2005, 02:45 PM
Treisaran @ Mar 27 2013, 06:43 AM wrote: Censeo Icarum occupatum esse, quoniam multas dies in foro non postavit.
Quidquid latine dictum sit, altum viditur.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Folks we are coming out of the darkness. Have had to do some business development, then have had a surge of work that commanded all my time. Still about a week from when I can return. There is a lot to read in the forum.

Hope all are well.

 Posts 773
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
icarus @ Jun 19 2013, 12:18 PM wrote: Folks we are coming out of the darkness. Have had to do some business development, then have had a surge of work that commanded all my time. Still about a week from when I can return. There is a lot to read in the forum.

Hope all are well.
I was about to write a suitably Icarian overview of bases 17, 19, 34, 55, 64, 99 and 2520, thinking that you wouldnâ€™t return to the Forum. At least Iâ€™m glad to see that youâ€™re back. Also, my Argam Kinsevoctove extension is done.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Oschkar,

There is no prohibition for you to do so, actually! I am very interested in your argam extension!

I have one big job moving out and a less intensive job starting, then it seems quiet. So next month will resume the production of numberbases.com. I'd spent about a month of cut-up time on polychora. It seems whenever I am rebooting my affection for number bases, I go through polychora first. There, I see Wendy's name and contributions.