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 Base 504, The tetradecimal circle of degrees
Double sharp
Posted: Jan 3 2017, 01:42 PM


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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 semi-coprime 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

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
2 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
3 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
4 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
5 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143
6 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
7 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
8 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215
9 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239
10 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263
11 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287
12 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311
13 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335
14 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359
15 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383
16 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407
17 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431
18 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455
19 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479
20 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503
  • Of the 96 primes p less than r = 504, only {2, 3, 7} are divisors of r. The remaining 93 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • There are 24 divisor digits d | r: {0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 504).) Read this brief description of the divisor.
  • There are 144 totatives t such that gcd(t, r) = 1. Two-sevenths or 28.1% of the digits of base 504 are totatives (the totient ratio φ(r)/r = 2/7 for all bases r having the distinct prime divisors {2, 3, 7}). Read this brief definition and description of the significance of the totative.
  • Base 504 has 35 semidivisors (regular numbers gr that do not divide r evenly). This set includes any integer 0 < gr that is the product strictly of one or more of the distinct prime divisors of 504 that does not divide 360 evenly. Together with the 24 divisors, there are 59 regular digits in base 504; just over 1/9 (11.7%) of the digits of base 504 are regular. This post defines regular numbers, and their significance.
  • There are 337 neutral digits in the scale of 504, including the 35 aforementioned semidivisors, the remaining 302 are semitotatives, products pq of at least one distinct prime divisor p and at least one prime q that is coprime to 504. Over 2/3 (66.9%) of the digits of base 504 are neutral; the preponderance are semitotatives, making up about three-fifths (59.9%) of all the digits of base 504.
  • Let the integers ω = (r − 1) and α = (r + 1). The ω-totative of base 504 is 503, which is prime and does not distribute the omega divisibility rules to smaller digits. The “digit-sum” rule would be an academic curiosity in base 504 as it applies to digit-503, a large prime. The α-number of base 504 is 505 = 5 * 101. Digit-five and digit-101 enjoy the “alpha” divisibility tests, i.e., the “alternating sum” rule, along with 505. Multiples of either 5, 101, and 503 and any of the divisors of 504 enjoy intuitive compound divisibility tests. The test for 5 would see heavy use, but those for 101 and 503 would be essentially academic curiosities. See this post for maps and descriptions of the alpha divisibility rules for bases less than 30.
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Double sharp
Posted: Jan 3 2017, 01:44 PM


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Intuitive Divisibility Rules

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
2 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
3 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
4 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
5 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143
6 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
7 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
8 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215
9 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239
10 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263
11 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287
12 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311
13 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335
14 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359
15 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383
16 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407
17 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431
18 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455
19 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479
20 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503


As is evident, there are relatively few intuitive divisibility tests in base 504 considering the large scale of the base. This said, most common 7-smooth numbers (in fact, all that have at most a single factor of 5 included) have intuitive divisibility tests. Some of the regular tests for ďremoteĒ regular numbers like digits 128, 243, and 256 would prove highly impractical.
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jaycee
Posted: Jan 4 2017, 03:00 AM


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Incidentally, I was thinking about base 2200* yesterday myself! I rather like how its prime decomposition is very balanced (i.e. two-cubed, three-squared, 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.
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Double sharp
Posted: Jan 4 2017, 05:53 AM


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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 7-day 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 ad-hoc 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 "human-scale" 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 pseudo-5-smoothness would argue for {a} and construct a strikingly similar pentadactyl conworld.
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Oschkar
Posted: Jan 4 2017, 07:44 AM


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QUOTE (Double sharp @ Jan 4 2017, 05:53 AM)
{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 7-day 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).

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 donít 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 Kodegaduloís "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 145 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 mm≤ or 99.0980 in≤.

The volumel is 16.1658 litres or 4.27056 US gallons, a little smaller than the 5-gallon 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. Itís about the same as the mass of an average 4-year-old child, or a quarter of the average adultís 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 Earthís 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 brand-symbol...
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Double sharp
Posted: Jan 4 2017, 01:26 PM


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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!
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Oschkar
Posted: Jan 4 2017, 11:03 PM


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QUOTE (Double sharp @ Jan 4 2017, 01:26 PM)
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 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.
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Double sharp
Posted: Jan 5 2017, 02:30 AM


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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 sixteenth-note quintuplet and the second half with an eighth-note 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.
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