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Perpetual Calendars

Which day of the week a particular day falls on can be determined by charts that operate like the following one:

 1) Sun Mon Tue Wed Thu Fri Sat
 2) Mon Tue Wed Thu Fri Sat Sun
 3) Tue Wed Thu Fri Sat Sun Mon
 4) Wed Thu Fri Sat Sun Mon Tue
 5) Thu Fri Sat Sun Mon Tue Wed
 6) Fri Sat Sun Mon Tue Wed Thu
 7) Sat Sun Mon Tue Wed Thu Fri

MAR   1   2   3   4   5   6   7
      8   9  10  11  12  13  14
     15  16  17  18  19  20  21
     22  23  24  25  26  27  28
     29  30  31
APR               1   2   3   4
      5   6   7   8   9  10  11
     12  13  14  15  16  17  18
     19  20  21  22  23  24  25
     26  27  28  29  30
MAY                       1   2
      3   4   5   6   7   8   9
     10  11  12  13  14  15  16
     17  18  19  20  21  22  23
     24  25  26  27  28  29  30
     31
JUN       1   2   3   4   5   6
      7   8   9  10  11  12  13
     14  15  16  17  18  19  20
     21  22  23  24  25  26  27
     28  29  30
JUL               1   2   3   4
      5   6   7   8   9  10  11
     12  13  14  15  16  17  18
     19  20  21  22  23  24  25
     26  27  28  29  30  31
AUG                           1
      2   3   4   5   6   7   8
      9  10  11  12  13  14  15
     16  17  18  19  20  21  22
     23  24  25  26  27  28  29
     30  31
SEP           1   2   3   4   5
      6   7   8   9  10  11  12
     13  14  15  16  17  18  19
     20  21  22  23  24  25  26
     27  28  29  30
OCT                   1   2   3
      4   5   6   7   8   9  10
     11  12  13  14  15  16  17
     18  19  20  21  22  23  24
     25  26  27  28  29  30  31
NOV   1   2   3   4   5   6   7
      8   9  10  11  12  13  14
     15  16  17  18  19  20  21
     22  23  24  25  26  27  28
     29  30
DEC           1   2   3   4   5
      6   7   8   9  10  11  12
     13  14  15  16  17  18  19
     20  21  22  23  24  25  26
     27  28  29  30  31
JAN                       1   2
      3   4   5   6   7   8   9
     10  11  12  13  14  15  16
     17  18  19  20  21  22  23
     24  25  26  27  28  29  30
     31
FEB       1   2   3   4   5   6
      7   8   9  10  11  12  13
     14  15  16  17  18  19  20
     21  22  23  24  25  26  27
     28 (29)

where the row in which to find the days of the week is found by looking up the year (because a day is added to February in a leap year, January and February are considered to be part of the previous year to make the table simpler in its form) in a table like the following:

     00 01 02 03    04 05
     06 07    08 09 10 11
        12 13 14 15    16
     17 18 19    20 21 22
     23    24 25 26 27
     28 29 30 31    32 33
     34 35    36 37 38 39
        40 41 42 43    44
     45 46 47    48 49 50
     51    52 53 54 55
     56 57 58 59    60 61
     62 63    64 65 66 67
        68 69 70 71    72
     73 74 75    76 77 78
     79    80 81 82 83
     84 85 86 87    88 89
     90 91    92 93 94 95
        96 97 98 99

                           Gregorian
      4  5  6  7  1  2  3       1600 2000 2400 2800 3200
      2  3  4  5  6  7  1       1700 2100 2500 2900 3300
      7  1  2  3  4  5  6       1800 2200 2600 3000 3400
      5  6  7  1  2  3  4  1500 1900 2300 2700 3100 3500

                           Julian
      5  6  7  1  2  3  4        400 1100
      4  5  6  7  1  2  3        500 1200
      3  4  5  6  7  1  2        600 1300
      2  3  4  5  6  7  1     0  700 1400
      1  2  3  4  5  6  7   100  800 1500
      7  1  2  3  4  5  6   200  900 1600
      6  7  1  2  3  4  5   300 1000 1700

                           Revised Julian
      6  7  1  2  3  4  5  1500                               4600
      4  5  6  7  1  2  3  1600 2000 2400
      2  3  4  5  6  7  1  1700 2100 2500 2900 3300
      7  1  2  3  4  5  6  1800 2200 2600 3000 3400 3800 4200
      5  6  7  1  2  3  4  1900 2300 2700 3100 3500 3900 4300 4700
      3  4  5  6  7  1  2            2800 3200 3600 4000 4400 4800
      1  2  3  4  5  6  7                      3700 4100 4500 4900

In addition to the Gregorian and Julian calendars, the Revised Julian calendar proposed by the Orthodox Church, with a more accurate cycle of 900 years indicating the century leap years to omit, is shown.

For years after 3,500 A.D., one simply has to subtract 2,000 years as often as required, since the Gregorian calendar repeats itself every 400 years.

While there is no year 0, the entry in the table for Julian years is still required to find the calendar for the years from 1 A.D. to 99 A.D. For years B.C., the required table is:

     99 98    97 96 95 94
        93 92 91 90    89
     88 87 86    85 84 83
     82    81 80 79 78
     77 76 75 74    73 72
     71 70    69 68 67 66
        65 64 63 62    61
     60 59 58    57 56 55
     54    53 52 51 50
     49 48 47 46    45 44
     43 42    41 40 39 38
        37 36 35 34    33
     32 31 30    29 28 27
     26    25 24 23 22
     21 20 19 18    17 16
     15 14    13 12 11 10
        09 08 07 06    05
     04 03 02    01 00

                           Julian (B.C.)
      7  1  2  3  4  5  6  5199 4499 3799 3099 2399 1699 999  299
      6  7  1  2  3  4  5  5099 4399 3699 2999 2299 1599 899  199
      5  6  7  1  2  3  4  4999 4299 3599 2899 2199 1499 799   99
      4  5  6  7  1  2  3  4899 4199 3499 2799 2099 1399 699
      3  4  5  6  7  1  2  4799 4099 3399 2699 1999 1299 599
      2  3  4  5  6  7  1  4699 3999 3299 2599 1899 1199 499
      1  2  3  4  5  6  7  4599 3899 3199 2499 1799 1099 399

The Gregorian calendar, which the top section of the bottom half of this table within this calendar reflects, was only adopted in the English-speaking countries in 1752. However, it was first put into practice in 1582 in countries where the Roman Catholic Church was generally followed.

Note that the Julian calendar repeats in a cycle of 700 years, while the Gregorian calendar repeats in a cycle of only 400 years, since the number of days in four centuries, 146,097 days, happens to be an exact multiple of seven. (20,871 times 7.)

The table at the top, showing the days of the week for a year starting with March 1st, therefore shows parts of two consecutive years. The rows in that table can be related to the Dominical Letter as follows:

Row number:                            1  2  3  4  5  6  7
Dominical Letter (March - December)    C  B  A  G  F  E  D
Dominical Letter (January - February)  B  A  G  F  E  D  C

thus, as the row number for a year usually advances by one, the two parts of a year have the same Dominical Letter, except for leap years, which consume two letters in the sequence.

The reason why the second table is so neat and tidy is that the number of days in a year is 365, which is just one more than a multiple of 7. Similar tables can be made for other kinds of "week", but they will have a somewhat different form, as we will see below. And in some of these cases, the table will have a somewhat more elaborate form.

For example, in the traditional Chinese calendar, days as well as years belong to a succession of 10 Heavenly stems and of 12 Earthly branches. Let us first consider the case of the 12 Earthly branches.

Setting up the first part of the chart is as straightforwards as for the days of the seven-day week:

 1)  Tzu Ch'ou Yin  Mao Ch'en Szu  Wu   Wei  Shen Yu   Hsu  Hai
 2) Ch'ou Yin  Mao Ch'en Szu  Wu   Wei  Shen Yu   Hsu  Hai  Tzu
 3)  Yin  Mao Ch'en Szu  Wu   Wei  Shen Yu   Hsu  Hai  Tzu Ch'ou
 4)  Mao Ch'en Szu  Wu   Wei  Shen Yu   Hsu  Hai  Tzu Ch'ou Yin
 5) Ch'en Szu  Wu   Wei  Shen Yu   Hsu  Hai  Tzu Ch'ou Yin  Mao
 6)  Szu  Wu   Wei  Shen Yu   Hsu  Hai  Tzu Ch'ou Yin  Mao Ch'en
 7)  Wu   Wei  Shen Yu   Hsu  Hai  Tzu Ch'ou Yin  Mao Ch'en Szu
 8)  Wei  Shen Yu   Hsu  Hai  Tzu Ch'ou Yin  Mao Ch'en Szu  Wu
 9)  Shen Yu   Hsu  Hai  Tzu Ch'ou Yin  Mao Ch'en Szu  Wu   Wei
10)  Yu   Hsu  Hai  Tzu Ch'ou Yin  Mao Ch'en Szu  Wu   Wei  Shen
11)  Hsu  Hai  Tzu Ch'ou Yin  Mao Ch'en Szu  Wu   Wei  Shen Yu
12)  Hai  Tzu Ch'ou Yin  Mao Ch'en Szu  Wu   Wei  Shen Yu   Hsu

MAR    1    2    3    4    5    6    7    8    9   10   11   12
      13   14   15   16   17   18   19   20   21   22   23   24
      25   26   27   28   29   30   31
APR                                       1    2    3    4    5
       6    7    8    9   10   11   12   13   14   15   16   17
      18   19   20   21   22   23   24   25   26   27   28   29
      30
MAY         1    2    3    4    5    6    7    8    9   10   11
      12   13   14   15   16   17   18   19   20   21   22   23
      24   25   26   27   28   29   30   31
JUN                                            1    2    3    4
       5    6    7    8    9   10   11   12   13   14   15   16
      17   18   19   20   21   22   23   24   25   26   27   28
      29   30
JUL              1    2    3    4    5    6    7    8    9   10
      11   12   13   14   15   16   17   18   19   20   21   22
      23   24   25   26   27   28   29   30   31
AUG                                                 1    2    3
       4    5    6    7    8    9   10   11   12   13   14   15
      16   17   18   19   20   21   22   23   24   25   26   27
      28   29   30   31
SEP                        1    2    3    4    5    6    7    8
       9   10   11   12   13   14   15   16   17   18   19   20
      21   22   23   24   25   26   27   28   29   30
OCT                                                      1    2
       3    4    5    6    7    8    9   10   11   12   13   14
      15   16   17   18   19   20   21   22   23   24   25   26
      27   28   29   30   31
NOV                             1    2    3    4    5    6    7
       8    9   10   11   12   13   14   15   16   17   18   19
      20   21   22   23   24   25   26   27   28   29   30
DEC                                                           1
       2    3    4    5    6    7    8    9   10   11   12   13
      14   15   16   17   18   19   20   21   22   23   24   25
      26   27   28   29   30   31
JAN                                  1    2    3    4    5    6
       7    8    9   10   11   12   13   14   15   16   17   18
      19   20   21   22   23   24   25   26   27   28   29   30
      31
FEB         1    2    3    4    5    6    7    8    9   10   11
      12   13   14   15   16   17   18   19   20   21   22   23
      24   25   26   27   28  (29)

But since there are five days left over in a year of 365 days, rather than just 1, the chart for the years looks somewhat different. To allow successive years, when they are not leap years, to be contiguous, the sequence below of displacements for the years is decimated in fives. But then one extra day no longer produces a gap of one space in the series of years, but a longer gap. (This would be true even if we did not also have the fact that 366, the number of days in a leap year, being divisible by 6, is not relatively prime to 12.)

     00 01 02 03                04 05 06
     07                08 09 10 11
              12 13 14 15
     16 17 18 19                20 21 22
     23                24 25 26 27
              28 29 30 31
     32 33 34 35                36 37 38
     39                40 41 42 43
              44 45 46 47
     48 49 50 51                52 53 54
     55                56 57 58 59
              60 61 62 63
     64 65 66 67                68 69 70
     71                72 73 74 75
              76 77 78 79
     80 81 82 83                84 85 86
     87                88 89 90 91
              92 93 94 95
     96 97 98 99

                                         GREG
      5 10  3  8  1  6 11  4  9  2  7 12      1600 1900
      1  6 11  4  9  2  7 12  5 10  3  8      1700
      9  2  7 12  5 10  3  8  1  6 11  4      1800
      2  7 12  5 10  3  8  1  6 11  4  9      2000 2300
     10  3  8  1  6 11  4  9  2  7 12  5      2100
      6 11  4  9  2  7 12  5 10  3  8  1      2200
     11  4  9  2  7 12  5 10  3  8  1  6      2400 2700
      7 12  5 10  3  8  1  6 11  4  9  2      2500
      3  8  1  6 11  4  9  2  7 12  5 10      2600
      8  1  6 11  4  9  2  7 12  5 10  3 1500 2800 3100
      4  9  2  7 12  5 10  3  8  1  6 11      2900
     12  5 10  3  8  1  6 11  4  9  2  7      3000

                                         JUL
      9  2  7 12  5 10  3  8  1  6 11  4  600 1000 1400
      6 11  4  9  2  7 12  5 10  3  8  1  700 1100 1500
      3  8  1  6 11  4  9  2  7 12  5 10  800 1200 1600
     12  5 10  3  8  1  6 11  4  9  2  7  900 1300 1700

In this perpetual calendar, the Gregorian calendar repeats with a cycle of 1600 years, and the Julian calendar repeats in 400 years.

Next, we have a table for the Ten Heavenly Stems. Again, the first table is straightforwards; a calendar with a longer week, with a set of scales at the top with the day names in all their possible displacements.

To make the table below smaller, it relates directly to a Western cycle of ten days, the last digit of the Julian Day number. Since the Julian Day begins, or at least began, at noon, the digits are shown between the days of the calendar; thus, the digit in the column between March 1st and 2nd refers to the number of the Julian Day beginning at noon (UT, or GMT) March 1st and ending at noon March 2nd.

 1) 0  1  2  3  4  5  6  7  8  9  0
 2) 1  2  3  4  5  6  7  8  9  0  1
 3) 2  3  4  5  6  7  8  9  0  1  2
 4) 3  4  5  6  7  8  9  0  1  2  3
 5) 4  5  6  7  8  9  0  1  2  3  4
 6) 5  6  7  8  9  0  1  2  3  4  5
 7) 6  7  8  9  0  1  2  3  4  5  6
 8) 7  8  9  0  1  2  3  4  5  6  7
 9) 8  9  0  1  2  3  4  5  6  7  8
10) 9  0  1  2  3  4  5  6  7  8  9

MAR   1  2  3  4  5  6  7  8  9 10
     11 12 13 14 15 16 17 18 19 20
     21 22 23 24 25 26 27 28 29 30
     31
APR      1  2  3  4  5  6  7  8  9
     10 11 12 13 14 15 16 17 18 19
     20 21 22 23 24 25 26 27 28 29
     30
MAY      1  2  3  4  5  6  7  8  9
     10 11 12 13 14 15 16 17 18 19
     20 21 22 23 24 25 26 27 28 29
     30 31
JUN         1  2  3  4  5  6  7  8
      9 10 11 12 13 14 15 16 17 18
     19 20 21 22 23 24 25 26 27 28
     29 30
JUL         1  2  3  4  5  6  7  8
      9 10 11 12 13 14 15 16 17 18
     19 20 21 22 23 24 25 26 27 28
     29 30 31
AUG            1  2  3  4  5  6  7
      8  9 10 11 12 13 14 15 16 17
     18 19 20 21 22 23 24 25 26 27
     28 29 30 31
SEP               1  2  3  4  5  6
      7  8  9 10 11 12 13 14 15 16
     17 18 19 20 21 22 23 24 25 26
     27 28 29 30
OCT               1  2  3  4  5  6
      7  8  9 10 11 12 13 14 15 16
     17 18 19 20 21 22 23 24 25 26
     27 28 29 30 31
NOV                  1  2  3  4  5
      6  7  8  9 10 11 12 13 14 15
     16 17 18 19 20 21 22 23 24 25
     26 27 28 29 30
DEC                  1  2  3  4  5
      6  7  8  9 10 11 12 13 14 15
     16 17 18 19 20 21 22 23 24 25
     26 27 28 29 30 31
JAN                     1  2  3  4
      5  6  7  8  9 10 11 12 13 14
     15 16 17 18 19 20 21 22 23 24
     25 26 27 28 29 30 31
FEB                        1  2  3
      4  5  6  7  8  9 10 11 12 13
     14 15 16 17 18 19 20 21 22 23
     24 25 26 27 28 29

These digits relate to the Ten Heavenly Stems as shown in the table below:

 0 Chia 1 I    2 Ping 3 Ting 4 Wu   5
 5 Chi  6 Keng 7 Hsin 8 Jen  9 Kuei 0

thus, by a coincidence, both series begin on the same days.

But because a year has 365 days normally, and 365 is not relatively prime to 10, the second chart takes a complicated form indeed. With care, it can still be made reasonably compact, however: but in this case, one cannot merely read across to see the years in order, but must bounce up and down within a group of two or three rows.

     00 01|04 05|08 09|12 13|16 17
     02 03|06 07|10 11|14 15|18 19
     -----------------------------
        20|   24|   28|   32|   36
     21 22|25 26|29 30|33 34|37 38
     23   |27   |31   |35   |39
     -----------------------------
     40 41|44 45|48 49|52 53|56 57
     42 43|46 47|50 51|54 55|58 59
     -----------------------------
        60|   64|   68|   72|   76
     61 62|65 66|69 70|73 74|77 78
     63   |67   |71   |75   |79
     -----------------------------
     80 81|84 85|88 89|92 93|96 97
     82 83|86 87|90 91|94 95|98 99

                                   GREG
      3  8  4  9  5  0  6  1  7  2      1600           3500      4200      4900
      7  2  8  3  9  4  0  5  1  6      1700 2400                4300      5000
      1  6  2  7  3  8  4  9  5  0      1800 2500 3200                     5100
      5  0  6  1  7  2  8  3  9  4      1900 2600 3300      4000
      9  4  0  5  1  6  2  7  3  8           2700 3400      4100      4800
      0  5  1  6  2  7  3  8  4  9      2000           3900      4600      5300
      4  9  5  0  6  1  7  2  8  3      2100 2800                4700      5400
      8  3  9  4  0  5  1  6  2  7 1500 2200 2900 3600                     5500
      2  7  3  8  4  9  5  0  6  1      2300 3000 3700      4400
      6  1  7  2  8  3  9  4  0  5           3100 3800      4500      5200

                                   JUL
      3  8  4  9  5  0  6  1  7  2 1000 1200 1400 1600
      8  3  9  4  0  5  1  6  2  7 1100 1300 1500 1700

Note that in this case the Julian calendar has a cycle of only 200 years, while the Gregorian calendar repeats in a cycle of 4000 years.

Thus, this illustrates the possible difficulties one might encounter producing perpetual calendars for different lengths of week. It is just fortunate that the seven-day week produces the simplest possible type of perpetual calendar.

The Tzolkin component of the Mayan calendar lends itself to treatment by perpetual calendars of this type as well.

Thus, the 20-day cycle is indicated as follows:

 1) Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha
 2) Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi
 3) Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix
 4) Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb
 5) Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan
 6) Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi
 7) Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim
 8) Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man
 9) Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam
10) Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul
11) Chu Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc
12) Eb  Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu
13) Ben Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb
14) Ix  Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben
15) Men Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix
16) Cib Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men
17) Cab Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib
18) Etz Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab
19) Cau Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz
20) Aha Imi Ix  Akb Kan Chi Cim Man Lam Mul Oc  Chu Eb  Ben Ix  Men Cib Cab Etz Cau

MAR   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20
     21  22  23  24  25  26  27  28  29  30  31

APR                                               1   2   3   4   5   6   7   8   9
     10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29
     30

MAY       1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19
     20  21  22  23  24  25  26  27  28  29  30  31

JUN                                                   1   2   3   4   5   6   7   8
      9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28
     29  30

JUL           1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
     19  20  21  22  23  24  25  26  27  28  29  30  31

AUG                                                       1   2   3   4   5   6   7
      8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27
     28  29  30  31

SEP                   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16
     17  18  19  20  21  22  23  24  25  26  27  28  29  30

OCT                                                           1   2   3   4   5   6
      7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26
     27  28  29  30  31

NOV                       1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
     16  17  18  19  20  21  22  23  24  25  26  27  28  29  30

DEC                                                               1   2   3   4   5
      6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
     26  27  28  29  30  31

JAN                           1   2   3   4   5   6   7   8   9  10  11  12  13  14
     15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31

FEB                                                                       1   2   3
      4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23
     24  25  26  27  28  29

This table shows the abbreviations of the Mayan day names used above:

Imi: Imix       Cim: Cimi     Chu: Chuen     Cib
Ix              Man: Manik    Eb             Cab: Caban
Akb: Akbal      Lam: Lamat    Ben            Etz: Etznab
Kan             Mul: Muluc    Ix             Cau: Cauac
Chi: Chicchan   Oc            Men            Aha: Ahau

The English translations of the Aztec day names corresponding to these are, in order:

Crocodile       Death         Monkey         Vulture
Wind            Deer          Grass          Quake
House           Rabbit        Reed           Flint
Lizard          Water         Jaguar         Rain
Snake           Dog           Eagle          Flower

and this calendar system was termed the tonalpohualli by the Aztecs.

For this cycle, in four years we have 21 excess days, not 20, so we don't quite have the overlap problem we did for the 10 day cycle, but we still have to put the years out of order:

     00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
     -----------------------------------------------------------
        20 21 22|   24 25 26|   28 29 30|   32 33 34|   36 37 38
     23         |27         |31         |35         |39
     -----------------------------------------------------------
           40 41|      44 45|      48 49|      52 53|      56 57
     42 43      |46 47      |50 51      |54 55      |58 59
     -----------------------------------------------------------
              60|         64|         68|         72|         76
     61 62 63   |65 66 67   |69 70 71   |73 74 75   |77 78 79
     -----------------------------------------------------------
     80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

                                                                 GREG
     12 17  2  7 13 18  3  8 14 19  4  9 15 20  5 10 16  1  6 11      1600      3500      4900           8200
     16  1  6 11 17  2  7 12 18  3  8 13 19  4  9 14 20  5 10 15      1700                5000 6400      8300
     20  5 10 15  1  6 11 16  2  7 12 17  3  8 13 18  4  9 14 19      1800 3200           5100 6500
      4  9 14 19  5 10 15 20  6 11 16  1  7 12 17  2  8 13 18  3      1900 3300                6600 8000
      8 13 18  3  9 14 19  4 10 15 20  5 11 16  1  6 12 17  2  7           3400      4800      6700 8100
      1  6 11 16  2  7 12 17  3  8 13 18  4  9 14 19  5 10 15 20      2000      3900      5300           8600
      5 10 15 20  6 11 16  1  7 12 17  2  8 13 18  3  9 14 19  4      2100                5400 6800      8700
      9 14 19  4 10 15 20  5 11 16  1  6 12 17  2  7 13 18  3  8      2200 3600           5500 6900
     13 18  3  8 14 19  4  9 15 20  5 10 16  1  6 11 17  2  7 12      2300 3700                7000 8400
     17  2  7 12 18  3  8 13 19  4  9 14 20  5 10 15  1  6 11 16           3800      5200      7100 8500
     18  3  8 13 19  4  9 14 20  5 10 15  1  6 11 16  2  7 12 17      2400      4300      5700           9000
      2  7 12 17  3  8 13 18  4  9 14 19  5 10 15 20  6 11 16  1      2500                5800 7200      9100
      6 11 16  1  7 12 17  2  8 13 18  3  9 14 19  4 10 15 20  5      2600 4000                7300
     10 15 20  5 11 16  1  6 12 17  2  7 13 18  3  8 14 19  4  9      2700 4100                7400 8800
     14 19  4  9 15 20  5 10 16  1  6 11 17  2  7 12 18  3  8 13           4200      5600      7500 8900
     15 20  5 10 16  1  6 11 17  2  7 12 18  3  8 13 19  4  9 14      2800      4700      6100           9400
     19  4  9 14 20  5 10 15  1  6 11 16  2  7 12 17  3  8 13 18 1500 2900                6200 7600      9500
      3  8 13 18  4  9 14 19  5 10 15 20  6 11 16 21  7 12 17  2      3000 4400           6300 7700
      7 12 17  2  8 13 18  3  9 14 19  4 10 15 20  5 11 16  1  6      3100 4500                7800 9200
     11 16  1  6 12 17  2  7 13 18  3  8 14 19  4  9 15 20  5 10           4600      6000      7900 9300      

                                                                 JUL
      4  9 14 19  5 10 15 20  6 11 16  1  7 12 17  2  8 13 18  3  200  600 1000 1400
      9 14 19  4 10 15 20  5 11 16  1  6 12 17  2  7 13 18  3  8  300  700 1100 1500
     14 19  4  9 15 20  5 10 16  1  6 11 17  2  7 12 18  3  8 13  400  800 1200 1600
     19  4  9 14 20  5 10 15  1  6 11 16  2  7 12 17  3  8 13 18  500  900 1300 1700

The length of the cycle for the Gregorian calender has doubled, to 8000 years. For the Julian calendar, the century contains 25 leap years, and so the cycle doubles for it as well, to 400 years, since 5 days repeat four times in 20.

For the 13-day cycle, the perpetual calendar will look like this:

 1)  1  2  3  4  5  6  7  8  9 10 11 12 13
 2)  2  3  4  5  6  7  8  9 10 11 12 13  1
 3)  3  4  5  6  7  8  9 10 11 12 13  1  2
 4)  4  5  6  7  8  9 10 11 12 13  1  2  3
 5)  5  6  7  8  9 10 11 12 13  1  2  3  4
 6)  6  7  8  9 10 11 12 13  1  2  3  4  5
 7)  7  8  9 10 11 12 13  1  2  3  4  5  6
 8)  8  9 10 11 12 13  1  2  3  4  5  6  7
 9)  9 10 11 12 13  1  2  3  4  5  6  7  8
10) 10 11 12 13  1  2  3  4  5  6  7  8  9
11) 11 12 13  1  2  3  4  5  6  7  8  9 10
12) 12 13  1  2  3  4  5  6  7  8  9 10 11
13) 13  1  2  3  4  5  6  7  8  9 10 11 12

MAR  1  2  3  4  5  6  7  8  9 10 11 12 13
    14 15 16 17 18 19 20 21 22 23 24 25 26
    27 28 29 30 31

APR                 1  2  3  4  5  6  7  8
     9 10 11 12 13 14 15 16 17 18 19 20 21
    22 23 24 25 26 27 28 29 30

MAY                             1  2  3  4
     5  6  7  8  9 10 11 12 13 14 15 16 17
    18 19 20 21 22 23 24 25 26 27 28 29 30
    31

JUN     1  2  3  4  5  6  7  8  9 10 11 12
    13 14 15 16 17 18 19 20 21 22 23 24 25
    26 27 28 29 30

JUL                 1  2  3  4  5  6  7  8
     9 10 11 12 13 14 15 16 17 18 19 20 21
    22 23 24 25 26 27 28 29 30 31

AUG                                1  2  3
     4  5  6  7  8  9 10 11 12 13 14 15 16
    17 18 19 20 21 22 23 24 25 26 27 28 29
    30 31

SEP        1  2  3  4  5  6  7  8  9 10 11
    12 13 14 15 16 17 18 19 20 21 22 23 24
    25 26 27 28 29 30

OCT                    1  2  3  4  5  6  7
     8  9 10 11 12 13 14 15 16 17 18 19 20
    21 22 23 24 25 26 27 28 29 30 31

NOV                                   1  2
     3  4  5  6  7  8  9 10 11 12 13 14 15
    16 17 18 19 20 21 22 23 24 25 26 27 28
    29 30

DEC        1  2  3  4  5  6  7  8  9 10 11
    12 13 14 15 16 17 18 19 20 21 22 23 24
    25 26 27 28 29 30 31

JAN                       1  2  3  4  5  6
     7  8  9 10 11 12 13 14 15 16 17 18 19
    20 21 22 23 24 25 26 27 28 29 30 31

FEB                                      1
     2  3  4  5  6  7  8  9 10 11 12 13 14
    15 16 17 18 19 20 21 22 23 24 25 26 27
    28 29

Because there are 52 weeks in a year, with one extra day, the cycle of years in a century takes the same simple form it did for the perpetual calendar for the 7-day week.

    00 01 02 03    04 05 06 07    08 09 10
    11    12 13 14 15    16 17 18 19    20
    21 22 23    24 25 26 27    28 29 30 31
       32 33 34 35    36 37 38 39    40 41
    42 43    44 45 46 47    48 49 50 51
    52 53 54 55    56 57 58 59    60 61 62
    63    64 65 66 67    68 69 70 71    72
    73 74 75    76 77 78 79    80 81 82 83
       84 85 86 87    88 89 90 91    92 93
    94 95    96 97 98 99

                                           GREG
     2  3  4  5  6  7  8  9 10 11 12 13  1      1600 2600
     9 10 11 12 13  1  2  3  4  5  6  7  8      1700 2700      4400
     3  4  5  6  7  8  9 10 11 12 13  1  2      1800      3600 4500
    10 11 12 13  1  2  3  4  5  6  7  8  9      1900 2800 3700 4600
     4  5  6  7  8  9 10 11 12 13  1  2  3           2900 3800 4700
    11 12 13  1  2  3  4  5  6  7  8  9 10           3000 3900
     5  6  7  8  9 10 11 12 13  1  2  3  4      2000 3100
    12 13  1  2  3  4  5  6  7  8  9 10 11      2100           4800
     6  7  8  9 10 11 12 13  1  2  3  4  5      2200      4000 4900
    13  1  2  3  4  5  6  7  8  9 10 11 12      2300 3200 4100 5000
     7  8  9 10 11 12 13  1  2  3  4  5  6 1500      3300 4200 5100
     1  2  3  4  5  6  7  8  9 10 11 12 13      2400 3400 4300
     8  9 10 11 12 13  1  2  3  4  5  6  7      2500 3500

                                           JUL
     3  4  5  6  7  8  9 10 11 12 13  1  2  500
    10 11 12 13  1  2  3  4  5  6  7  8  9  600
     4  5  6  7  8  9 10 11 12 13  1  2  3  700
    11 12 13  1  2  3  4  5  6  7  8  9 10  800
     5  6  7  8  9 10 11 12 13  1  2  3  4  900
    12 13  1  2  3  4  5  6  7  8  9 10 11 1000
     6  7  8  9 10 11 12 13  1  2  3  4  5 1100
    13  1  2  3  4  5  6  7  8  9 10 11 12 1200
     7  8  9 10 11 12 13  1  2  3  4  5  6 1300
     1  2  3  4  5  6  7  8  9 10 11 12 13 1400
     8  9 10 11 12 13  1  2  3  4  5  6  7 1500
     2  3  4  5  6  7  8  9 10 11 12 13  1 1600
     9 10 11 12 13  1  2  3  4  5  6  7  8 1700

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