Xhovian calendar

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The Xhovian Calendar is the most widely used civil calendar of South Baredina. The reformed version used today was devised in 1480 by the Haldian mathematician and astronomer Ajkkën Veuttinec.

It is a solar calendar beginning on the Gregorian date of the 21st of June, in accordance with the occurrence of the winter solstice. An additional day is inserted between the Gregorian dates of the 20th and 21st of June every leap year, which in terms of the Xhovian calendar is defined as being every year divisible four, excluding any year that is also divisible by 100 and yields a remainder of 200 or 600 when divided by 900.

The Xhovian calendar does not specify divisions of weeks, although 73 weeks of 5 days are often used, and the year is not split up into months, so the date is typically written in the formats of DDD/YY, DDD/YYYY, or more rarely, YYYY/DDD.

Arithmetic

The following are Gregorian minus Xhovian date differences, calculated for the beginning of March in each century year, which is where differences arise or disappear, until 10000 AD. These are exact arithmetic calculations, not depending on any astronomy. A negative difference means that the proleptic Xhovian calendar was behind the proleptic Gregorian calendar. The Xhovian calendar is synchronised with Gregorian calendar from 1 March 1600 to 28 February 2800. A positive difference means that the Xhovian calendar will be ahead relative to the Gregorian calendar, which will first occur on 1 March 2800 (excluding the basic difference of ~240 years between the two calendars):

Gregorian minus Xhovian date differences
Century
Difference
100
0
200
−1
300
−1
400
0
500
0
600
−1
700
−1
800
0
900
0
1000
0
Century
Difference
1100
−1
1200
0
1300
0
1400
0
1500
−1
1600
0
1700
0
1800
0
1900
0
2000
0
Century
Difference
2100
0
2200
0
2300
0
2400
0
2500
0
2600
0
2700
0
2800
+1
2900
0
3000
0
Century
Difference
3100
0
3200
+1
3300
0
3400
0
3500
0
3600
+1
3700
+1
3800
0
3900
0
4000
+1
Century
Difference
4100
+1
4200
0
4300
0
4400
+1
4500
+1
4600
+1
4700
0
4800
+1
4900
+1
5000
+1
Century
Difference
5100
0
5200
+1
5300
+1
5400
+1
5500
+1
5600
+1
5700
+1
5800
+1
5900
+1
6000
+1
Century
Difference
6100
+1
6200
+1
6300
+1
6400
+2
6500
+1
6600
+1
6700
+1
6800
+2
6900
+1
7000
+1
Century
Difference
7100
+1
7200
+2
7300
+2
7400
+1
7500
+1
7600
+2
7700
+2
7800
+1
7900
+1
8000
+2
Century
Difference
8100
+2
8200
+2
8300
+1
8400
+2
8500
+2
8600
+2
8700
+1
8800
+2
8900
+2
9000
+2
Century
Difference
9100
+2
9200
+2
9300
+2
9400
+2
9500
+2
9600
+2
9700
+2
9800
+2
9900
+2
10000
+3

The Xhovian leap rule omits seven of nine century leap years, leaving 225−7 = 218 leap days per 900-year cycle. Thus the calendar mean year is 365+218900 days, but this is actually a double-cycle that reduces to 365+109450 = ~365.242 days, or exactly 365 days 5 hours 48 minutes 48 seconds, which is exactly 24 seconds shorter than the Gregorian mean year of 365.2425 days, so in the long term on average the Xhovian calendar pulls ahead relative to the Gregorian calendar by one day in 3600 years.

The number of days per Xhovian calendar cycle = 900 × 365 + 218 = 328,718 days. Taking mod 7 leaves a remainder of 5, and taking mod 5 leaves a remainder of 3, so the Xhovian calendar cycle does not contains neither a whole number of Gregorian 7 day weeks nor optional Xhovian 5 day weeks. Therefore, a full repetition of the Xhovian leap cycle with respect to the five-day weekly cycle is seven times the cycle length = 5 × 900 = 4500 years.

Conversion

The epoch of the Xhovian calendar is June 21st, 240 BCE, so in order to convert a date from Gregorian to Xhovian, observe the following process:

  1. Add 240 to the Gregorian year, or if it is a date before the 21st of June in the Gregorian calendar, add 239 (value y)
  2. Find the month then day of the Gregorian date in the table below, and find the value in the cell on its right (value d)
  3. Write the date in the format [d]/[y]

Example - conversion of Gregorian date 1st September 2017 to Xhovian calendar:

  • September 1st is after June 21st in a Gregorian year, so add 240 to the Gregorian year to give the Xhovian year of 2257
  • '73' is the value right of September 1st in the table, so the day is the 73rd
  • The full Xhovian date is therefore 73/2257, also commonly written as 73/57

The following is a table listing the conversion of all dates of the year from the Gregorian to the Xhovian calendar, excluding during leap years.
It is read from left to right in pairs of columns, so for example, the Gregorian date January 1st is the Xhovian date '195th'.

January Xhovian February Xhovian March Xhovian April Xhovian May Xhovian June Xhovian July Xhovian August Xhovian September Xhovian October Xhovian November Xhovian December Xhovian
1 195 1 226 1 254 1 287 1 317 1 346 1 11 1 42 1 73 1 103 1 134 1 164
2 196 2 227 2 255 2 286 2 316 2 347 2 12 2 43 2 74 2 104 2 135 2 165
3 197 3 228 3 256 3 287 3 317 3 348 3 13 3 44 3 75 3 105 3 136 3 166
4 198 4 229 4 257 4 288 4 318 4 349 4 14 4 45 4 76 4 106 4 137 4 167
5 199 5 230 5 258 5 289 5 319 5 350 5 15 5 46 5 77 5 107 5 138 5 168
6 200 6 231 6 259 6 290 6 320 6 351 6 16 6 47 6 78 6 108 6 139 6 169
7 201 7 232 7 260 7 291 7 321 7 352 7 17 7 48 7 79 7 109 7 140 7 170
8 202 8 233 8 261 8 292 8 322 8 353 8 18 8 49 8 80 8 110 8 141 8 171
9 203 9 234 9 262 9 293 9 323 9 354 9 19 9 50 9 81 9 111 9 142 9 172
10 204 10 235 10 263 10 294 10 324 10 355 10 20 10 51 10 82 10 112 10 143 10 173
11 205 11 236 11 264 11 295 11 325 11 356 11 21 11 52 11 83 11 113 11 144 11 174
12 206 12 237 12 265 12 296 12 326 12 357 12 22 12 53 12 84 12 114 12 145 12 175
13 207 13 238 13 266 13 297 13 327 13 358 13 23 13 54 13 85 13 115 13 146 13 176
14 208 14 239 14 267 14 298 14 328 14 359 14 24 14 55 14 86 14 116 14 147 14 177
15 209 15 240 15 268 15 299 15 329 15 360 15 25 15 56 15 87 15 117 15 148 15 178
16 210 16 241 16 269 16 300 16 330 16 361 16 26 16 57 16 88 16 118 16 149 16 179
17 211 17 242 17 270 17 301 17 331 17 362 17 27 17 58 17 89 17 119 17 150 17 180
18 212 18 243 18 271 18 302 18 332 18 363 18 28 18 59 18 90 18 120 18 151 18 181
19 213 19 244 19 272 19 303 19 333 19 364 19 29 19 60 19 91 19 121 19 152 19 182
20 214 20 245 20 273 20 304 20 334 20 365 20 30 20 61 20 92 20 122 20 153 20 183
21 215 21 246 21 274 21 305 21 335 21 1 21 31 21 62 21 93 21 123 21 154 21 184
22 216 22 247 22 275 22 306 22 336 22 2 22 32 22 63 22 94 22 124 22 155 22 185
23 217 23 248 23 276 23 307 23 337 23 3 23 33 23 64 23 95 23 125 23 156 23 186
24 218 24 249 24 277 24 308 24 338 24 4 24 34 24 65 24 96 24 126 24 157 24 187
25 219 25 250 25 278 25 309 25 339 25 5 25 35 25 66 25 97 25 127 25 158 25 188
26 220 26 251 26 279 26 310 26 340 26 6 26 36 26 67 26 98 26 128 26 159 26 189
27 221 27 252 27 280 27 311 27 341 27 7 27 37 27 68 27 99 27 129 27 160 27 190
28 222 28 253 28 281 28 312 28 342 28 8 28 38 28 69 28 100 28 130 28 161 28 191
29 223 --- --- 29 281 29 313 29 343 29 9 29 39 29 70 29 101 29 131 29 162 29 192
30 224 --- --- 30 283 30 314 30 344 30 10 30 40 30 71 30 102 30 132 30 163 30 193
31 225 --- --- 31 284 --- --- 31 345 --- --- 31 41 31 72 --- --- 31 133 --- --- 31 194

Xhovian calendrical calculations

The calendrical arithmetic discussed here is adapted from Gregorian and Julian calendar arithmetic published by Dershowitz and Reingold.[1] They define the MOD operator as x MOD y = x − y × floor(x / y), because that expression is valid for negative and floating point operands, returning the remainder from dividing x by y while discarding the quotient.[2] Expressions like floor(x / y) return the quotient from dividing x by y while discarding the remainder.

Leap rule

isLeapYear = (year MOD 4 = 0)

IF isLeapYear THEN

IF year MOD 100 = 0 THEN
Century = (year / 100) MOD 9
isLeapYear = (Century=2) OR (Century=6)
END IF

END IF

References

  1. Dershowitz, Nachum; Reingold, Edward M. (2008). Calendrical Calculations (3rd ed.). Cambridge University Press. p. 47, footnote 3. 
  2. Dershowitz, Nachum; Reingold, Edward M. (2008). Calendrical Calculations (3rd ed.). Cambridge University Press. p. 18, equation 1.15.