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# A Simplified Calendar

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 calendar that would achieve this would be the following:

```       January                February                March
====================   ====================   ====================
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  4  5  6
3  4  5  6  7  8  9    7  8  9 10 11 12 13    7  8  9 10 11 12 13
10 11 12 13 14 15 16   14 15 16 17 18 19 20   14 15 16 17 18 19 20
17 18 19 20 21 22 23   21 22 23 24 25 26 27   21 22 23 24 25 26 27
24 25 26 27 28 29 30   28                     28 29 30 31
31

April                   May                    June
====================   ====================   ====================
SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
--------------------   --------------------   --------------------
1  2  3                      1          1  2  3  4  5
4  5  6  7  8  9 10    2  3  4  5  6  7  8    6  7  8  9 10 11 12
11 12 13 14 15 16 17    9 10 11 12 13 14 15   13 14 15 16 17 18 19
18 19 20 21 22 23 24   16 17 18 19 20 21 22   20 21 22 23 24 25 26
25 26 27 28 29 30      23 24 25 26 27 28 29   27 28 29 30
30 31

July                  August               September
====================   ====================   ====================
SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
--------------------   --------------------   --------------------
1  2  3                      1       1  2  3  4  5  6
4  5  6  7  8  9 10    2  3  4  5  6  7  8    7  8  9 10 11 12 13
11 12 13 14 15 16 17    9 10 11 12 13 14 15   14 15 16 17 18 19 20
18 19 20 21 22 23 24   16 17 18 19 20 21 22   21 22 23 24 25 26 27
25 26 27 28 29 30      23 24 25 26 27 28 29   28 29 30
30

October                November               December
====================   ====================   ====================
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
```

This keeps December as a 31-day month, so that the distance between Christmas and New Year remains a week. Also, October is kept as a 31-day month. Here, the emphasis is placed on ensuring that the 13th never falls on a Friday, instead of on avoiding "perverse months" which span six weeks instead of five.

In a leap year, I now envisage lengthening February by a week, instead of adding days to several different months.

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 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.

For example, one possible approximation is to have eleven leap years every sixty-two years. Thus, seven of those leap years are one year in six, and four of them are one year in five.

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, one possibility would be to have 11 leap weeks distributed as evenly as possible over a cycle of 62 years.

This would consist of three cycles of 17 years with three leap weeks (each composed in turn of two cycles of six years with one leap week and one cycle of five years with one leap week) and one cycle of 11 years with two leap weeks (composed, of course, of one cycle of six years with one leap week and one cycle of five years with one leap week).

This would be 0.016338 days longer than 62 tropical years, for an error of about one day in 3,795 years.

### Another Compromise

If it is held that the irregular way in which years begin on different days of the week makes our calendar too complicated, but adding a leap week so that every year might begin on the same day of the week leads us too far out of step with the seasons, there is another choice.

A year of 365 days causes the next year to begin one day later in the week.

Thus, if we had a span of six years instead of four with one leap year, each such span would start on the same day of the week.

Of course, that would not keep pace with the seasons by itself, as there would not be enough leap years. But such spans could alternate with spans of five years in which there were two leap years.

As the sequence of leap years would now be less regular, the calendar would fall out of step with the seasons to a greater extent than our present one, but this would not be too severe.

Now, a perpetual calendar could be built up out of one sequence of six calendars and another very similar sequence of five calendars, instead of requiring a sequence of twenty-eight calendars.

Whether or not this would be even noticed by anyone as a useful improvement is, of course, open to question.

Six years with one leap year would fall short of six tropical years by 0.453194 days; five years with two leap years would be longer than five tropical years by .789005 days, and so the appropriate ratio of six-year spans to five-year spans would be 1.740987; thus, the first obvious approximation would be to have seven six-year spans to four five-year spans, for a cycle of 62 years.

The structure of such a cycle would be something like this:

```nnnnnLnnnnnLnnLnL
nnnnnLnnnnnLnnLnL
nnnnnLnnnnnLnnLnL
nnnnnLnnLnL
```

and if we denote the days of the week by the letters SMTWHFR, showing normal years by small letters and leap years by capital letters, the regularity, however inconsequential, that we gain from this scheme can now be exhibited, by representing each year by the letter of its first day:

```smtwhFsmtwhFsmThF
smtwhFsmtwhFsmThF
smtwhFsmtwhFsmThF
smtwhFsmThF
```

Here, leap years only begin on Tuesdays and Fridays, normal years begin on any of the five days from Sunday to Thursday, and no year ever begins on a Saturday.

Since this uses a cycle much longer than the four years of the Julian calendar, it might be hoped that the approximation to the tropical year is a close one. In this, the scheme does not disappoint; the average length of the year is 365.241935 days, compared to 365.2425 days for the Gregorian calendar - and 365.242199 days for the actual tropical year. The difference, at about .0003 days, is nearly the same for both, although this scheme actually is very slightly more accurate; .000264 days short as against .000301 days long.

It is not, by any means, the most accurate cycle of more than four years that could be used; 31 leap years in every 128 years was the next obvious step, giving an average year of 365.2421875 days, for an error of one day in almost 87,000 years; however, what I intended to note, but did not initially remember, was that the simpler cycle of 8 leap years in 33 years, discovered by Omar Khayyam, is already more accurate than the Gregorian calendar, giving a length of 365.242424 days to the year.

As a cycle of 62 years of this type is short by .016338 days, the next step would be to alternate such cycles with short cycles of 45 years, following the pattern

```smtwhFsmtwhFsmThF
smtwhFsmtwhFsmThF
smtwhFsmThF
```

thus containing 11 leap years in 45 years, and thus being .101045 days too long. Six cycles of 62 years followed by one cycle of 45 years would be the closest simple approximation to balancing out:

```      1
-------------    6: 1
1
6 + ---------   31: 5
1
5 + -----   68:11
1
2 + -  167:27
2
```

although one can take the continued fraction approximation further, 101 leap years in 417 years gives an average length of 365.242206235 days to the year, an error of one day in 138,000 years, which should be adequate for the time being.

Incidentally, the maximum variance of the calendar from the seasons could be decreased by putting the leap years at the beginning and the end of the five-year cycles, and in the middle of the six-year cycles, like this:

```smtWfr smtWfr StwhF
smtWfr smtWfr StwhF
smtWfr smtWfr StwhF
smtWfr StwhF
```

but by making the five year cycles so much more different from the six year cycles, the unformity of each cycle beginning on the same day of the week is less visible.

>The reason for the improvement is that now two leap years have at least two normal years between them (instead of one) on the one hand, and on the other, the case where two leap years have five normal years between them no longer occurs twice in a row.

Six years where five have 365 days and one 366 days, and five years where three years have 365 days and two 366 days, both have seven days more than the same number of years of 364 days, so the alternation of five and six year cycles in this proposal is equivalent to the distribution of leap years in the preceding type of calendar proposal.

### The ISO calendar

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.

### When Months Are Vague

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.

### An Impossible Calendar

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
====================   ====================   ====================
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  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
====================   ====================   ====================
SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
--------------------   --------------------   --------------------
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
====================   ====================   ====================
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                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
====================   ====================   ====================
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  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
====================   ====================
SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
--------------------   --------------------
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.

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