Gregorian Accumulator

Victor Engel has introduced a class of calendar where intercalation is regulated by an accumulator. It ensures that intercalation is as evenly spread as possible and hence calendar jitter in minimised. Victor uses a monthly accumulator in 28/293 calendar and the 43/450 calendar.

The idea can be applied to many types of calendars, but to keep the note simple, I'll only consider solar calendars with a leap day. The accumulator idea can be applied annually to a solar calendar with a Y-year cycle of L leap years. Each year has an accumulator number, which is incremented by L each year until the limit of Y is reached then Y is subtracted and the year becomes a leap year.

The accumulator of year y1 less the accumulator of y2 is the number of 1/Y day units that year y1 ends earlier than year y2 relative to the mean calendar year and so measures the jitter of the calendar from year y2 to year y1.

This can be expressed by the following rules:

  1. The accumulator of a common year is L greater than the accumulator of the previous year.
  2. The accumulator of a leap year is Y- L less than the accumulator of the previous year.
  3. The minimum accumulator is 0
  4. The maximum accumulator is Y-1
Only rules 1 and 2 are necessary for the accumulator to be a measure of calendar jitter from year y2 to year y1.

Furthermore, rules 1 to 3 can be applied to any leap year solar calendar. This includes the Gregorian calendar.

In the case of the Gregorian calendar, the year 2096 has the minimum accumulator by rules 1 and 2 and so by rule 3, 2096 has an accumulator of 0. This gives the accumulators for the 1600s, 2000s, 2400s, etc.. centuries as 

      0   1   2   3   4    5   6   7   8   9

00s  288 385 482 579 276  373 470 567 264 361
10s  458 555 252 349 446  543 240 337 434 531
20s  228 325 422 519 216  313 410 507 204 301
30s  398 495 192 289 386  483 180 277 374 471
40s  168 265 362 459 156  253 350 447 144 241
50s  338 435 132 229 326  423 120 217 314 411
60s  108 205 302 399 096  193 290 387 084 181
70s  278 375 072 169 266  363 060 157 254 351
80s  048 145 242 339 036  133 230 327 024 121
90s  218 315 012 109 206  303 000 097 194 291

For the 1700s, 2100s, etc. centuries you add 100 to these figures
for the 1800s, 2200s, etc. centuries you add 200 to these figures and 
for the 1900s, 2300s, etc. centuries you add 300 to these figures.

As expected the year 1903 has the maximum accumulator (579+300 = 879), so the maximum jitter is 879/400 days = 2.1975 days.

The remainder of the Gregorian accumulator divided by seven equals the number of days that December 31 comes before a Monday. For example, the accumulator for 2004 is 276, which has a remainder of 3 when divided by 7. Hence 31 December 2004 is 3 days before a Monday, hence is a Friday.

Karl Palmen June 2004