A full calendar system has a different calendar date for every day. Thus the week cycle is by itself not a full calendar system; neither is a system to name the days within a year without a system for identifying the years.
The simplest calendar system just counts time periods from a reference date. This applies for the Julian day or Unix Time. Virtually the only possible variation is using a different reference date, in particular one less distant in the past to make the numbers smaller. Computations in these systems are just a matter of addition and subtraction.
Other calendars have one (or multiple) larger units of time.
Calendars that contain one level of cycles:
week and weekday – this system (without year, the week number keeps on increasing) is not very common
year and ordinal date within the year, e.g. the ISO 8601 ordinal date system
Calendars with two levels of cycles:
year, month, and day – most systems, including the Gregorian calendar (and its very similar predecessor, the Julian calendar), the Islamic calendar, and the Hebrew calendar
year, week, and weekday – e.g. the ISO week date
Cycles can be synchronized with periodic phenomena:
A lunar calendar is synchronized to the motion of the Moon (lunar phases); an example is the Islamic calendar.
A solar calendar is based on perceived seasonal changes synchronized to the apparent motion of the Sun; an example is the Persian calendar.
A “luni-solar calendar” is based on a combination of both solar and lunar reckonings; examples are the traditional calendar of China, the Hindu Calendar in India or the Hebrew calendar.
There are some calendars that appear to be synchronized to the motion of Venus, such as some of the ancient Egyptian calendars; synchronization to Venus appears to occur primarily in civilizations near the Equator.
The week cycle is an example of one that is not synchronized to any external phenomenon (although it may have been derived from lunar phases, beginning anew every month).
Very commonly a calendar includes more than one type of cycle, or has both cyclic and acyclic elements.
Many calendars incorporate simpler calendars as elements. For example, the rules of the Hebrew calendar depend on the seven-day week cycle (a very simple calendar), so the week is one of the cycles of the Hebrew calendar. It is also common to operate two calendars simultaneously, usually providing unrelated cycles, and the result may also be considered a more complex calendar. For example, the Gregorian calendar has no inherent dependence on the seven-day week, but in Western society the two are used together, and calendar tools indicate both the Gregorian date and the day of week.
The week cycle is shared by various calendar systems (although the significance of special days such as Friday, Saturday, and Sunday varies). Systems of leap days usually do not affect the week cycle. The week cycle was not even interrupted when 10, 11, 12, or 13 dates were skipped when the Julian calendar was replaced by the Gregorian calendar by various countries.
Main article: Solar calendar
Days used by solar calendars
Solar calendars assign a date to each solar day. A day may consist of the period between sunrise and sunset, with a following period of night, or it may be a period between successive events such as two sunsets. The length of the interval between two such successive events may be allowed to vary slightly during the year, or it may be averaged into a mean solar day. Other types of calendar may also use a solar day.
Main article: Calendar reform
There have been a number of proposals for reform of the calendar, such as the World Calendar, International Fixed Calendar, Holocene calendar, and, recently, the Hanke-Henry Permanent Calendar. The United Nations considered adopting such a reformed calendar for a while in the 1950s, but these proposals have lost most of their popularity.
Main article: Lunar calendar
Not all calendars use the solar year as a unit. A lunar calendar is one in which days are numbered within each lunar phase cycle. Because the length of the lunar month is not an even fraction of the length of the tropical year, a purely lunar calendar quickly drifts against the seasons, which don’t vary much near the equator. It does, however, stay constant with respect to other phenomena, notably tides. An example is the Islamic calendar. Alexander Marshack, in a controversial reading, believed that marks on a bone baton (c. 25,000 BC) represented a lunar calendar. Other marked bones may also represent lunar calendars. Similarly, Michael Rappenglueck believes that marks on a 15,000-year old cave painting represent a lunar calendar.
A lunisolar calendar is a lunar calendar that compensates by adding an extra month as needed to realign the months with the seasons. An example is the Hebrew calendar which uses a 19-year cycle.