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:

- The accumulator of a common year is L greater than the accumulator of the previous year.
- The accumulator of a leap year is Y- L less than the accumulator of the previous year.
- The minimum accumulator is 0
*The maximum accumulator is Y-1*

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