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
For years after 3,500 A.D., in looking up dates on the Gregorian Calendar, one simply has to subtract 2,000 years as often as required, since the Gregorian calendar repeats itself every 400 years.
In addition to the Gregorian and Julian calendars, the Revised Julian calendar proposed by Milutin Milanković, of Milanković Cycle fame, with a more accurate cycle of 900 years indicating the century leap years to omit, is shown. This calendar has been adopted by some branches of the Orthodox Church, but even where Orthodox Christianity is the main religion in a given country, and the branch of the church there has adopted this calendar, the governments use the Gregorian calendar instead as the civil calendar. Recently, in the Ukraine, some of the Orthodox faithful have shifted to celebrating Christmas on December 25 to distance themselves from Russia, but their chuches have not yet formally adopted a different calendar, and, from all indications, were they to do so, the calendar they would adopt would likely be the Gregorian calendar instead of the Revised Julian calendar.
How do these calendars compare? Incidentally, because of the residual errors in the Gregorian Calendar, it has also been proposed to correct it by having the year 4,000 and other years that are multiples of 4,000 no longer be leap years. Let us call the adjusted Gregorian Calendar the "Reformed Gregorian Calendar", and then the average lengths of the year in all of them are as shown in the following table:
Actual mean tropical year: 365.242199 days Julian calendar: 365.25 days Gregorian calendar: 365.2425 days Reformed Julian calendar: 365.24222222222... days Reformed Gregorian calendar: 365.24225 days
This table makes it obvious what is really needed. Instead of a cycle of 900 years, a cycle of 1,000 years is needed. While in the past, the year 1600 as well as the year 2000 were restored as leap years, however, with a 1,000 year period it would seem more logical to restore the leap year in years that are multiples of 500.
That would give the year a length of 365.2422 days, with an error of one day in a million years. While that could be dealt with by making 1,000,000 AD and other years that are multiples of a million no longer leap years, no doubt by then the length of the tropical year will have changed.
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