You may have remembered seeing, years ago, the "World Calendar", a proposed calendar which would be the same every year. This would simplify planning for businesses, as well as saving a few trees. However, it had one problem.
The uniformity of the years was achieved by placing one day each year, plus a second day in leap years, outside of the week. Thus, every now and then, there would be an extra day sitting between a Saturday and a Sunday.
The trouble is that, even if the more "advanced" and "progressive" denominations go along with this idea, there are going to be a great many individuals, and churches, that will say that if God commanded them to go to church (or the synagogue, or the mosque) every Sunday (or Saturday, or Friday) He meant on the day that really is Sunday, et cetera, not whatever day we choose to call Sunday.
To me, it is quite surprising that the inventors of this scheme did not recognize that any such proposal would inevitably be a complete non-starter. (Of course, I also felt that the novel Dayworld would have been much more plausible if people were let out of the suspended animation chambers on the basis of an eight-day rotation, since eight is equal to one modulo 7. That way, everyone who wished, in any of the eight cohorts, could go to church on a subjective Sunday which is also an objective Sunday, and so on. That would, however, have messed up the title of the short story on which it was based.) However, that does not mean that it is impossible for a simplified calendar to be designed that is more respectful of religious tradition, belief, and practice.
The simple way to do so is this: let a year normally be 364 days, or exactly 52 weeks, long, and have leap years with an extra week which are 371 days long. Then, every year can start on a Sunday.
One set of calendars for a normal year and a leap year that would achieve this with a minimum of change would be the following:
January February March
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
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1 2 3 4 5 6 7 1 2 3 4 1 2 3
8 9 10 11 12 13 14 5 6 7 8 9 10 11 4 5 6 7 8 9 10
15 16 17 18 19 20 21 12 13 14 15 16 17 18 11 12 13 14 15 16 17
22 23 24 25 26 27 28 19 20 21 22 23 24 25 18 19 20 21 22 23 24
29 30 31 26 27 28 29 25 26 27 28 29 30 31
April May June
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 3 4 5 6 7 1 2 3 4 5 1 2
8 9 10 11 12 13 14 6 7 8 9 10 11 12 3 4 5 6 7 8 9
15 16 17 18 19 20 21 13 14 15 16 17 18 19 10 11 12 13 14 15 16
22 23 24 25 26 27 28 20 21 22 23 24 25 26 17 18 19 20 21 22 23
29 30 27 28 29 30 31 24 25 26 27 28 29 30
July August September
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 3 4 5 6 7 1 2 3 4 5 1 2
8 9 10 11 12 13 14 6 7 8 9 10 11 12 3 4 5 6 7 8 9
15 16 17 18 19 20 21 13 14 15 16 17 18 19 10 11 12 13 14 15 16
22 23 24 25 26 27 28 20 21 22 23 24 25 26 17 18 19 20 21 22 23
29 30 27 28 29 30 31 24 25 26 27 28 29 30
October November December
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 3 4 5 6 7 1 2 3 4 1 2 3
8 9 10 11 12 13 14 5 6 7 8 9 10 11 4 5 6 7 8 9 10
15 16 17 18 19 20 21 12 13 14 15 16 17 18 11 12 13 14 15 16 17
22 23 24 25 26 27 28 19 20 21 22 23 24 25 18 19 20 21 22 23 24
29 30 31 26 27 28 29 25 26 27 28 29 30 31
This keeps the amount of change low, but also divides the year into four quarters of equal length, by having both February and November 29 days long, January, March, May, August, October, and December 31 days long. Thus, November and July are made one day shorter, and February is made one day longer as it would be in a leap year. As shown, the 13th of the month falls on a Friday four times each year, but since the 1st of the month is never a Monday nor a Saturday, one could simply start the year with January 1st falling on a Monday instead, and never have a Friday the 13th.
In my original proposal for creating a 364-day year, July and August were each shortened to 30 days, but February is lengthened to 29 days. Thus, the changes said to have been made to the calendar to feed the vanity of ancient Roman emperors are undone. (One might even consider renaming those months Quintilis and Sextilis once again!)
It might also be felt preferable, however, to keep July at 31 days, and leave February at 28 days, which would reduce the amount of change needed still further.
Or, to make a more symmetrical year, the layout of which would be easier to remember, only January, March, October and December could be the 31-day months, and all other months, including February, would be 30 days long.
A leap year is simple enough to construct; every month will be 31 days long, except February, which will be 30 days long.
January February March
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 3 4 5 6 7 1 2 3 4 1 2
8 9 10 11 12 13 14 5 6 7 8 9 10 11 3 4 5 6 7 8 9
15 16 17 18 19 20 21 12 13 14 15 16 17 18 10 11 12 13 14 15 16
22 23 24 25 26 27 28 19 20 21 22 23 24 25 17 18 19 20 21 22 23
29 30 31 26 27 28 29 30 24 25 26 27 28 29 30
31
April May June
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 3 4 5 6 1 2 3 1 2 3 4 5 6 7
7 8 9 10 11 12 13 4 5 6 7 8 9 10 8 9 10 11 12 13 14
14 15 16 17 18 19 20 11 12 13 14 15 16 17 15 16 17 18 19 20 21
21 22 23 24 25 26 27 18 19 20 21 22 23 24 22 23 24 25 26 27 28
28 29 30 31 25 26 27 28 29 30 31 29 30 31
July August September
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 3 4 1 1 2 3 4 5
5 6 7 8 9 10 11 2 3 4 5 6 7 8 6 7 8 9 10 11 12
12 13 14 15 16 17 18 9 10 11 12 13 14 15 13 14 15 16 17 18 19
19 20 21 22 23 24 25 16 17 18 19 20 21 22 20 21 22 23 24 25 26
26 27 28 29 30 31 23 24 25 26 27 28 29 27 28 29 30 31
30 31
October November December
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
-------------------- -------------------- --------------------
1 2 1 2 3 4 5 6 1 2 3
3 4 5 6 7 8 9 7 8 9 10 11 12 13 4 5 6 7 8 9 10
10 11 12 13 14 15 16 14 15 16 17 18 19 20 11 12 13 14 15 16 17
17 18 19 20 21 22 23 21 22 23 24 25 26 27 18 19 20 21 22 23 24
24 25 26 27 28 29 30 28 29 30 31 25 26 27 28 29 30 31
31
There are two problems with this calendar, however. The first problem is that with the existing calendar, some people would have birthdays on February 29th in a leap year, a day not found in other years. Here, in a leap year, February 30th, November 30th and 31st, and the 31st of April, June, July, and September as well, would all have the same problem, seven days instead of just one.
A year that is 365.242199 days long is quite closely approximated by a 365 day year with an extra day every four years. But such a year is also 52.177457 weeks long. Thus, the second problem is that one leap year every five years is too many, and one leap year every six years is too few, so instead of a simple rule, one leap year every four years with very few exceptions, the necessary rule would have to be more complicated.
Thus, while our current Gregorian Calendar can make exceptions to the rule of one leap year every four years only once a century (and then in only three out of every four centuries), a calendar based on adding whole weeks would have to have a rule the complications of which would be more obtrusive.
One way to handle leap years, perhaps the simplest, would be to proceed like this:
one leap year every five years 52.2 omitting a leap year every 40 years 52.175 but putting the leap year back every 400 years 52.1775
To keep the year in closer correspondence with the seasons, a gradual scheme like the following could be used:
A CYCLE is 17 years long, and consists of:
five normal years, one leap year, four normal years, one leap year, five normal years, one leap year
and is .016769 weeks short of exactly 17 years.
A NORMAL SPAN consists of seven cycles and five years as follows:
seven cycles, four normal years, one leap year
and is .004668 weeks short of exactly 124 years.
A SHORT SPAN consists of six cycles and five years as follows:
six cycles, four normal years, one leap year
and is .012101 weeks in excess of exactly 107 years.
The calendar then consists of successive rounds, where a ROUND consists of:
three normal spans, one short span, two normal spans, one short span, three normal spans, one short span, two normal spans, one short span, three normal spans, one short span
for an error of only .000179 of a week, or .001253 of a day, in 2,147 years.
Beginning the year on the same day of the week is sufficiently convenient that various companies will use fiscal years that are made up of an even number of weeks. An international standard exists for a calendar of this type.
In the ISO calendar, the year always begins with a week, starting on Monday and ending on the following Sunday, that includes January 4 of the Gregorian calendar. This makes the rule for relating dates in the two calendars simple, and lets the leap years in the Gregorian calendar take care of deciding when to insert leap weeks in the ISO calendar.
Since, conventionally, Sunday is considered to be the first day of the week, a calendar of conventional weeks, if it were to be synchronized with the ISO calendar, would have its first conventional week be the one including January 3 of the Gregorian calendar.
The 19-year cycle relating the phases of the Moon to the solar year is called the Metonic cycle, after Meton of Alexandria, although this cycle had been known previously. The ancient Greek calendar had been based on an 8-year cycle which approximated the Metonic cycle, called the Octaeteris. At least one source claims that before the Octaeteris was adopted, it had been a solar calendar which was equivalent to the Julian calendar. The calendar had months which approximated a lunar month in length, but one month was added every two years, alternating between 22 and 23 days in length, so the months did not have a fixed relation to the phases of the Moon.
Since an extra 22-day month would be very awkward for current economic purposes, it is quite unlikely that we would go back to that calendar, except for making Deutero-Poseidonus 22 instead of 23 days long once every 400 years, for the sake of having a calendar the names of whose months are "neutral".
If Deutero-Poseidonus were usually 20 days long, then the calendar would be composed of two years whose form, compared to the days of the week, did not change, and one could always increase the length of Deutero-Poseidonus to 27 days to keep the year in step with the seasons, so a uniform calendar could be constructed on this basis as well.
Also, if Deutero-Poseidonus were added once every three years, it could be about the same length as a normal month.
The Wikipedia article on calendar reform notes the fact that the month does not correspond to a lunar month as one of the things that some calendar reformers would like to correct.
As the article correctly notes, addressing this would conflict with making the kind of uniform solar calendar where every year is very nearly the same as every other that addresses other objects of calendar reform.
But how close could one get?
The lunar month is 29.530588853 days long.
A purely lunar calendar that always starts on the same day of the week could be built on the basis of exactly two kinds of periods.
One period consists of fourteen lunar months, alternating between 30 and 29 days. In such a period, the month would have an average length of 29.5 days, with 413 days for 14 months.
The other period would consist of nine lunar months, five of 30 days and four of 29 days. This works out to an even number of weeks because 30 days is two more than 28, a multiple of 7, and 29 days is one more than 28, and 5*2+4 equals 14, a multiple of 7. In this period, the average length of a month would be 29.555... days, with 266 days for 9 months.
So one could have two calendars, one 14 months long, and one 9 months long, but with the names of the months according to the solar year assigned by the Metonic cycle. This would come as close as possible to the impossible goal of total regularity.
If one has a group of two 9 month periods and one 14 month period repeated eight or nine times, alternating with one group of three 9 month periods and two 14 month periods, one can very closely approximate the true length of the lunar month.
Thus, here is Calendar A:
I II III
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
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1 2 3 4 5 6 7 1 2 3 4 5 1 2 3 4
8 9 10 11 12 13 14 6 7 8 9 10 11 12 5 6 7 8 9 10 11
15 16 17 18 19 20 21 13 14 15 16 17 18 19 12 13 14 15 16 17 18
22 23 24 25 26 27 28 20 21 22 23 24 25 26 19 20 21 22 23 24 25
29 30 27 28 29 26 27 28 29 30
IV V VI
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
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1 2 1 1 2 3 4 5 6
3 4 5 6 7 8 9 2 3 4 5 6 7 8 7 8 9 10 11 12 13
10 11 12 13 14 15 16 9 10 11 12 13 14 15 14 15 16 17 18 19 20
17 18 19 20 21 22 23 16 17 18 19 20 21 22 21 22 23 24 25 26 27
24 25 26 27 28 29 23 24 25 26 27 28 29 28 29
30
VII VIII IX
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
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1 2 3 4 5 1 2 3 1 2
6 7 8 9 10 11 12 4 5 6 7 8 9 10 3 4 5 6 7 8 9
13 14 15 16 17 18 19 11 12 13 14 15 16 17 10 11 12 13 14 15 16
20 21 22 23 24 25 26 18 19 20 21 22 23 24 17 18 19 20 21 22 23
27 28 29 30 25 26 27 28 29 24 25 26 27 28 29 30
X XI XII
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SU M TU W TH F SA SU M TU W TH F SA SU M TU W TH F SA
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1 2 3 4 5 6 7 1 2 3 4 5 6 1 2 3 4
8 9 10 11 12 13 14 7 8 9 10 11 12 13 5 6 7 8 9 10 11
15 16 17 18 19 20 21 14 15 16 17 18 19 20 12 13 14 15 16 17 18
22 23 24 25 26 27 28 21 22 23 24 25 26 27 19 20 21 22 23 24 25
29 28 29 30 26 27 28 29
XIII XIV
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SU M TU W TH F SA SU M TU W TH F SA
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1 2 3 1
4 5 6 7 8 9 10 2 3 4 5 6 7 8
11 12 13 14 15 16 17 9 10 11 12 13 14 15
18 19 20 21 22 23 24 16 17 18 19 20 21 22
25 26 27 28 29 30 23 24 25 26 27 28 29
and Calendar B is just the first 9 months of Calendar A.
And the sequence of using the calendars would go like this:
BAB BAB BAB BAB BAB BAB BAB BAB BABAB BAB BAB BAB BAB BAB BAB BAB BAB BAB BABAB
which involves a regular cycle of 654 months for a close approximation to the lunar month.
Combined with the Metonic cycle for the solar element in the calendar, one would have a table of months that would look like this, the columns corresponding to successive years:
Jan I Jan IV Jan III Jan VI Jan IX
Feb II Feb V Feb IV Feb VII Feb X
Mar III Mar VI Mar V Mar VIII Mar XI
Apr IV Apr VII Apr VI Apr IX Apr XII
May V May VIII May VII May I May XIII
Jun VI Jun IX Jun VIII Jun II Jun XIV
Mid X Mid I
Jul VII Jul XI Jul IX Jul III Jul II
Aug VIII Aug XII Aug I Aug IV Aug III
Sep IX Sep XIII Sep II Sep V Sep IV
Oct I Oct XIV Oct III Oct VI Oct V
Nov II Nov I Nov IV Nov VII Nov VI
Dec III Dec II Dec V Dec VIII Dec VII
so one could basically use one fourteen-month calendar for year after year, although after ten years or so, one would need to get a new calendar in order to get a new table on the bottom of which months of the calendar to use.
Or, of course, each month could have a little slot above it, through which the name of the month shows through from a little card in the back, which is gradually lifted up, being moved up one step as a nine or fourteen month cycle completes.
A luni-solar calendar that is almost as uniform as the World Calendar, at least for a suitable value of almost.
However, the objection that this is cheating, because the real year is twelve or thirteen months long, and can start on any day of the week, is legitimate too. If we allow the start of the lunar month to coincide less well with the new moon, just as a calendar of 364 or 371 days coincides only imperfectly with the tropical year, what would a uniform yet luni-solar calendar look like?
Such a calendar could consist of three quarters of months with lengths 30, 30, and 29, with the fourth quarter having months 30, 30, and 30 days in length; 357 days is divisible by 7. This would, of course, approximate the 29 1/2 day lunar month by one that is 29 3/4 days long.
Having the intercalary month be only 28 days long, thus 1 1/2 days too short, while the year is normally 3 days too long, doesn't make up for the discrepancy, and so occasionally the intercalary month would have to be only 21 days long.
Thus, one could indeed make a calendar which tries to be like the World Calendar, and yet which is lunisolar, should one feel like doing so.