Difference between revisions of "Xhovian calendar"
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==Arithmetic==  ==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 [[wikipedia:proleptic calendarproleptic]] 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 {{nowrap1 March 2800}}:  +  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 [[wikipedia:proleptic calendarproleptic]] 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 {{nowrap1 March 2800}} (excluding the basic difference of ~240 years between the two calendars): 
{ class=wikitable  { class=wikitable 
Revision as of 11:32, 23 September 2018
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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):
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 900year cycle. Thus the calendar mean year is 365+^{218}⁄_{900} days, but this is actually a doublecycle that reduces to 365+^{109}⁄_{450} = ~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 fiveday weekly cycle is seven times the cycle length = 5 × 900 = 4500 years.
Conversion
The epoch of the Xhovian calendar is June 21^{st}, 240 BCE, so in order to convert a date from Gregorian to Xhovian, observe the following process:
 Add 240 to the Gregorian year, or if it is a date before the 21^{st} of June in the Gregorian calendar, add 239 (value y)
 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)
 Write the date in the format [d]/[y]
Example  conversion of Gregorian date 1^{st} September 2017 to Xhovian calendar:
 September 1^{st} is after June 21^{st} 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 1^{st} in the table, so the day is the 73^{rd}
 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 1^{st} is the Xhovian date '195^{th}'.
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
 ↑ Dershowitz, Nachum; Reingold, Edward M. (2008). Calendrical Calculations (3rd ed.). Cambridge University Press. p. 47, footnote 3.
 ↑ Dershowitz, Nachum; Reingold, Edward M. (2008). Calendrical Calculations (3rd ed.). Cambridge University Press. p. 18, equation 1.15.