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Double sharp 
Posted: Jan 3 2017, 01:42 PM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Base 504 Base 504 is a “grand base” that lies well beyond our current ability to wield as a number base of general human arithmetic. Its multiplication table is astronomical (127,260 unique products, over 2310 times the size of the decimal table) so any notion of memorizing it is beyond the ability of the average kid in school. Why consider such a gigantic number as a base? The number 504 is similar to the superior highly composite number 360, but swaps a factor of 5 for a factor of 7. Hence it may come in handy as the tetradecimal analogue of the usage of decimal 360, being written in tetradecimal as "280". It has the same number of factors as 360. This said, it is somewhat bigger and 360's "sphere of influence" is so great that we have to go all the way to its double (720) to outdo its number of factors. Indirect relationships with neighboring integers helps; while the number 504 does not have the sort of convenient neighbors on both sides like 120 does, the prime 5 which it skips is included in its neighbour upstairs, 505 = 5 * 101. Despite the greater number of divisors, 504 has a burgeoning set of 144 totatives, which conspire to resist human use of a base as a tool of arithmetic. Like base 360, over half (59.9%) of the digits of base 504 are semicoprime to the base. Base 504 serves us better as an auxiliary base, its arithmetic tamed by that of tetradecimal, but its divisibility minimising our need to turn to fractions. Let’s take a look at 504 as a number base. Note that this examination is not as thorough as those for smaller bases, for obvious reasons. (Please refer to “Icarus’s Standard Nomenclature for Number Bases” for the legend of the digit map below and any terminology. References to elementary number theory books are given in that post and thread to support what is written in this post.) This post concerns base 504, but all of its information is also represented here. Base 504 has the following properties: Digit Map


Double sharp 
Posted: Jan 3 2017, 01:44 PM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Intuitive Divisibility Rules
As is evident, there are relatively few intuitive divisibility tests in base 504 considering the large scale of the base. This said, most common 7smooth numbers (in fact, all that have at most a single factor of 5 included) have intuitive divisibility tests. Some of the regular tests for “remote” regular numbers like digits 128, 243, and 256 would prove highly impractical. 

jaycee 
Posted: Jan 4 2017, 03:00 AM

Unregistered 
Incidentally, I was thinking about base 2200* yesterday myself! I rather like how its prime decomposition is very balanced (i.e. twocubed, threesquared, and seven are all approximately equal), similar to that of base 'Sixty'. However, also like base 'Sixty', encoding base 2200* in a way that removes the largest prime divisor from the significant digits while keeping the values of the significant digits relatively high or low is awkward. Essentially, similar to the use of ten for encoding base 'Sixty', the best encoding for base 2200* uses 22*, expressing 2200* as '280'. Unfortunately, this value is neither high nor low enough to make its divisions comparable to those of base 22*. (Ironically, the commonly used decimal auxiliaries 60; and 360; experience the same issue, while 120; is all but unused despite its divisions being very similar to those of decimal). As such, I'd prefer to use a 22* encoding of 15400* as a base, which also has the further advantages of being a SHCN and possessing five as a divisor.


Double sharp 
Posted: Jan 4 2017, 05:53 AM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
I think the fact that 60 and 360 are stranded between powers of 10 actually accounts for their survival, because 120 can be replaced by 100 all too easily. One has to take into account that while choosing an SHCN has merits, choosing a power of one's base also has merits.
I think a tetradecimal world would be strangely familiar with regards to auxiliary bases. When a coarse one is desired, we'd use "c" or its double "1a". Finer ones might very well be "60" or "280", swapping out a quinary factor for a septenary factor. So instead of 23:59:59, the last second of a tetradecimal day would be "19:5d:5d", except when we add a leap second. The tetradecimal seconds would be almost exactly half of our seconds. We probably wouldn't see "c0" since "100" is too close, which would be used as "percent". {e} (default tetradecimal) So tetradecimal users would probably have c months of just over 20 days each (maybe 22 and 23, as in the Gregorian calendar) in a year, which would seem to cry out for 7day weeks even more than in decimal. Then each day would have 1a hours of 60 minutes of 60 seconds, which would be subdivided tetradecimally into "milliseconds" and "microseconds" (we need a Systematic Tetradecimal Nomenclature now). While most of the adhoc numbers used in the old units would be the same, we'd see 10 being used in the tetradecimal world everywhere a is used in ours. Thus the metre would have been defined as 10^6 instead of a^7 of a meridian (so it would be somewhat longer, as seems to befit a slightly longer base), and we'd have had all the prefixes for powers of fourteen rather than ten. Meanwhile, we'd similarly have Dozenal Societies, with the advantage of not needing to invent new digits (but instead trying to take out c and d), complete with a forum where, among the most "humanscale" bases {8, a, c, 10} and maybe {6, 12}, fans of high divisibility would argue for {6, c}, fans of powers of two would argue for {8, 12}, and fans of pseudo5smoothness would argue for {a} and construct a strikingly similar pentadactyl conworld. 
Oschkar 
Posted: Jan 4 2017, 07:44 AM


Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
A long time ago, I proposed the roots "zen", "cist", "frat", "gim" for the first few transdozenal IUPAC roots. At least "zen" has a rational idea behind it: it is the second syllable of "dozen", already proposed by Pendlebury. I don’t remember what I derived the others from, but what I am aware of is that their initial consonants fill in the gaps left by the previous roots in alphabetical order. After Kodegadulo’s "des", a decimal base marker that replaces the coda of the SDN root "dec" with an "s", "os" for octal was derived from "oct", first by you, and then independently by me. Doing the same process to "frat" would make the tetradecimal base marker "fras". So "trifrasciaseconds" and "hexfrasciaseconds", maybe? The tetradecimal Primel analogue would divide the day into 14^{5} timels of 160.647 milliseconds, or a sixteenth note at 93 BPM. The velocitel would come out to about 1.57396 m/s, 5.66624 km/h or 3.52084 mi/h, a comfortable walking speed. The lengthel would be about 252.852 millimetres or 9.95480 inches, something that could pass for both a longish Continental palm (the length of the hand), and a shortish foot. The trifrasqualengthel is 693.825 metres, or 0.431123 miles. The bifrascialengthel is 1.29006 millimetres. The areanel is a square lengthel: 63934 mm² or 99.0980 in². The volumel is 16.1658 litres or 4.27056 US gallons, a little smaller than the 5gallon jugs that water is commonly sold in. A trifrasciavolumel is 5.89134 millilitres, barely larger than a teaspoon. The massel would be about 16.1654 kilograms or 35.6386 pounds. This is a bit large, but not entirely inconvenient. It’s about the same as the mass of an average 4yearold child, or a quarter of the average adult’s mass. An unfrasciamassel is 1.15467 kilograms (2.54561 pounds), and a bifrasciamassel is 82.476 grams (2.90927 ounces). The forcel is the weight of 1 massel subject to Earth’s gravity: 158.382 newtons. The pressurel is 2477.27 pascals, or 0.359297 psi. Atmospheric pressure in tetradecimal Primel would be 2c.c8_{{e}} pressurels. The energel is 40.0471 joules, and the powerel is 249.485 watts or 0.334298 horsepower. It all seems usable, I think. The size of the base units reminds me of TGM. Now for Kodegadulo to furnish this system with a good brandsymbol... 

Double sharp 
Posted: Jan 4 2017, 01:26 PM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
I find the timel a little short: one would get a little out of breath counting them one by one. But counting only the even ones ("two, four, six, eight, dess, zen, zeff") works fine.
I would love to have more transparent particles for at least thirteen and fourteen as transdecimals, to make tetradecimal usable, but I think this is the best we could do. After all, "cist" does remind me a bit of argam "thise" and thus very vaguely "thirteen", even if "frat" is completely opaque to me. Oh, and congratulations for 360 posts! 
Oschkar 
Posted: Jan 4 2017, 11:03 PM


Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
I would treat them as eighth note septuplets in double meter, a rhythm that, though not at all common, seems to be fairly easy for me to feel. 

Double sharp 
Posted: Jan 5 2017, 02:30 AM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Lucky you! I find even quintuplets reasonably easy, but even septuplets tend to turn into a quadruplet plus a triplet for me unless I'm really concentrating. A real septuplet feels strangely rushed to me. It took too long for me to get it right in Schumann's Eusebius (seven septuplet eighth notes filling a 2/4 bar in the right hand, with the left hand in normal 2/4, so you have to make sure that they don't coincide and both halves of the bar sound at the same speed because later he fills the first half with a sixteenthnote quintuplet and the second half with an eighthnote triplet).
I wonder if any of this increased difficulty has to do with 7 being a little more beyond the subitisation limit for most people than 5, so that the span gets split. If so, the heptadactyls are not going to have this problem. 