The next simplest possibility for a calendar (although not necessarily for the people using it) is a cycle of days of arbitrary length. The calendar can proceed by simply counting days, over and over, ad infinitum. We have just such an arbitrary cycle in our week. Notice that the cycle of week days runs independently of our other cycles of month and year. Another example is the nundinae, an 8-day interval used in the Roman republic. An arbitrary cycle need not be quite so simple (the Maya have a 260-day count), but its advantage is that we need not bother adjusting our calendar to keep it in step with the irregular periods of the sun and the moon.

If you are a farmer, an arbitrary period has its problems. Of particular interest is the best time to plant crops—an activity tied to the seasons, which are in turn linked to the earth's revolution around the sun. If all you have is an arbitrary cycle, it's much more difficult to tell just when you should plant. Some agricultural societies, therefore, have tried to link their calendar to the length of the time it takes the earth to rotate around the sun. Of course most agricultural societies have not known that the earth orbits around the sun. From the perspective of an earth-bound observer, this motion (along with the tilt of the earth's axis relative to its orbit around the sun) translates to the changes in position relative to the horizon where the sun rises (or sets) each day. Over the course of a year, this position will change, day by day. If you look East at sunrise, the point at which the sun appears to rise each day will move over a year between a maximum north and south position (the solstices). The midpoint of this motion, through which it passes twice each year, is the equinox. If you live within the tropics, at some point in the year the sun will pass directly overhead (i.e., at noon a straight stick will cast no shadow). The length of one complete cycle is known as the mean tropical year. In 1900 CE this period was 365.24219878 days, and the length of the tropical year has been shown to decrease by 0.0053 seconds per year (a value which becomes important when we start talking about 1000-year intervals).

So precise a measurement of the year, of course, would have been irrelevant to an early farmer, even if the astronomers could have measured it. A shift of five or ten days won't really matter too much, since annual variations in the weather will be greater than that. Once the calendar is out by a month or more, however, problems will arise if the farmer depends upon the calendar. It doesn't necessarily follow, however, that an agricultural society needs to keep its calendar tied, even loosely, to the seasons. Even if we assume that farmers really need someone to tell them when spring is coming, an alternate procedure would be for a specialist, for example a calendar priest, to announce when the proper planting time fell in the calendar.

An alternative to measuring the passage of the sun through a fixed point (equinox or solstice) would be to measure the appearance of a particularly prominent star or constellation, either just before sunrise or just after sunset. These events are called heliacal rising and setting and will yield a period that is fairly close to the mean tropical year. Such a measurement might seem like more work, but it can be handy, particularly if you live in someplace like the desert or a small island where the horizon is too flat to make a reliable measurement of the changes in the sun's rising position. Old Hawaiian religious ceremonies, for example, began each year with the heliacal rising of the Pleiades. Over a long time, however, a calendar based on heliacal rise and set times will slip with respect to the solar equinox—this is the phenomenon known as the precession of the equinox and is the result of the top-like wobble of the Earth's axis, which takes about 25,800 years to complete one full wobble. A year based on this kind of measurement is known as a sidereal year (sidus = Latin for "star"), and is about 51.15 seconds longer than the tropical year.

Just as a year can be defined relative to the stars or the sun, so can a moon's orbital period. The length of time it takes the moon to complete one orbit around the Earth is 27.32166 days. From an Earth-bound observer's point of view, this is the time it takes the moon to return to the same place relative to the stars, and is called the sidereal month. But because earth and moon are both moving around the sun, it takes the moon a bit longer to get back to the same position relative to the sun, and hence show the same phase (i.e., the amount of the moon's disk illuminated by the sun). This period, the synodic month, is 29.53058773 days.

Lunar cycles were the most common basis for early calendars. These calendars were often not based on any mathematical determination of the synodic month at all, but relied, particularly in their earliest formulations, upon direct observation to determine the new month. So when do you define the month as beginning? The two obvious starting points are the new or the full moon. Since the astronomical new moon occurs appears so close to the sun as to be invisible, this gives us three observational possibilities: the full moon, the first crescent of the ascending moon after the new moon, visible at sunset, or the last crescent of the descending moon, visible just before sunrise. There is a correlation between when the next day is considered to begin and which observational choice is made. For a society that begins the new day at dawn, the first day the waning moon is invisible just before sunrise is a good time to start the new month. If the next day begins at sunset, e.g., the Jewish Sabbath, the first observation of the new moon makes more sense. A full moon observation would seem to imply a midnight start. Such observations can be used in either a lunar or a lunisolar calendar. It will result in months that generally alternate between 29 and 30 days, but because the synodic month is not exactly 29.5 days, the alternation will not be completely regular.

Direct observation does have its drawbacks. Clouds might obscure observation, and in a large civilization, there is a problem with ensuring each location stays in sync with the rest. The historical development of lunar and lunisolar calendars is largely a matter of replacing direct observation by mathematical formulas and tables that allow the prediction of the month's start without the need for observation.

Lunisolar calendars have the additional problem of keeping the lunar months roughly aligned with the solar year. They generally do this by the insertion of a 13th, intercalary month at periodic intervals. Once again, in a calendar's earliest stages, this intercalation was generally accomplished through decree from a central source. In later times, regular rules were developed.

In principle, any astronomical cycle, e.g., the orbit of Venus or Mars, could be used to construct a calendar. In practice, very few societies bothered. These events were carefully recorded by astronomers, but only where these planets played an important role in religious observances (once again, see the Maya) were these events incorporated into the regular calendar.

Imagine yourself as an early astronomer. Your local ruler has been funding your research for several years, and now he wants some practical results. Your job: make a calendar. Which cycles are you going to include? What approximations are you going to make?

Obviously some of your decisions will be based on just how precise your measurements have been, but even more on the needs of your society (and your patron, who is footing the bill, after all). You will need to account for any preexisting traditions. If your people have been counting seven-day cycles for hundreds of years, resting on the seventh day, they will probably not accept a change to a ten-day cycle, which is what was attempted in revolutionary France. Religious aspects and cultural conservatism aside, neither inconsequential, this change would mean fewer days of rest for people. You also might want to consider your own needs. A complex calendar that is constantly changing with respect to the solar year, for instance, might seem like a bad choice. But if you can get this calendar accepted, you are guaranteed lifetime employment (i.e., tenure). After all, someone has to interpret this thing to tell people when to plant each year, when to celebrate the religious festivals, etc.

Once we've decided on what cycles, we have to decide how to count them. Do we cycle through a small number of names (e.g., our months)? Do we make a fixed numerical count of cycles from some predetermined epoch (e.g., our years)?

Since most calendars have been either solar (based on the tropical year), lunar (based on the synodic month), or lunisolar (a combination of the two), let's look at the problems we face making these cycles mesh.

Assume we take the day as the basic period (we don't have to, but it makes sense, since it's the light/dark rhythms that people base their activities upon). This means we need to fit the year and/or month cycles into some whole number scheme that approximates the observed values. As an approximation, we could round off the tropical year to 365.25 days and the synodic month to 29.5 days. These fractional days we can represent fairly easily, for the year by intercalating (inserting) one day every four years and for the months by making our lunar months alternate between 29 and 30 days.

With these approximations, we are only off the true period of the tropical year by about 1 day every 128.5 years, but with the lunar cycle it takes less than 3 years to slip a day (hence we?ll need to add some leap days here as well). Moreover, whether we use the true values or their approximations, there is no obvious way to get a whole number of lunar cycles into a whole number of solar cycles (in case we want a lunisolar calendar). The nearest rough approximation, and one that seems to have been a common early choice, is 365 days and 12 1/3 lunar cycles (also 365 days if we alternate 29 and 30 day months) per solar cycle.

A much better approximation, and one that we will see cropping up repeatedly, is the Metonic cycle, which observes that 19 tropical years are equal to 235 synodic months. Plugging in the actual lengths year and month to these numbers, we find 19 years = 6939.6018 days 235 months = 6939.68865 days which means this approximation has an error of only about 1 day every 219 years. This error value will change somewhat, of course, when we use the 19:235 ratio in a civil calendar, which contains its own approximations to the year length (see below, particularly the Julian calendar).

These considerations should also make it clear that just because a calendar doesn't match up to the solar or lunar cycles exactly doesn't necessarily imply that astronomers in that culture didn't have a much better idea of what the true values were. In fact, there's direct evidence to the contrary. A calendar must balance accuracy against simplicity, and once a calendar is well established in a society, changing it can be very difficult.