Month

A month is a unit of time, used with calendars, that is approximately as long as a natural phase cycle of the Moon; the words month and Moon are cognates. The traditional concept of months arose with the cycle of Moon phases; such lunar months are synodic months and last approximately 29.53 days, making for roughly 12.37 such months in one Earth year. From excavated tally sticks, researchers have deduced that people counted days in relation to the Moon's phases as early as the Paleolithic age. Synodic months, based on the Moon's orbital period with respect to the Earth–Sun line, are still the basis of many calendars today and are used to divide the year.
Calendars that developed from the Roman calendar system, such as the internationally used Gregorian calendar, divide the year into 12 months, each of which lasts between 28 and 31 days. The names of the months were Anglicized from various Latin names and events important to Rome, except for the months 9–12, which are named after the Latin numerals 7–10 because they were originally the seventh through tenth months in the Roman calendar. In the modern Gregorian calendar, the only month with a variable number of days is the second month, February, which has 29 days during a leap year and 28 days otherwise.

Types of months in astronomy

The following types of months are mainly of significance in astronomy. Most of them were first recognized in Babylonian lunar astronomy.

The sidereal month is defined as the Moon's orbital period in a non-rotating frame of reference. It is about 27.32166 days. It is closely equal to the time it takes the Moon to twice pass a "fixed" star.
A synodic month is the most familiar lunar cycle, defined as the time interval between two consecutive occurrences of a particular phase as seen by an observer on Earth. The mean length of the synodic month is 29.53059 days. Due to the eccentricity of the lunar orbit around Earth, the length of a synodic month can vary by up to seven hours.
The tropical month is the average time for the Moon to pass twice through the same equinox point of the sky. It is 27.32158 days, very slightly shorter than the sidereal month days, because of precession of the equinoxes.
An anomalistic month is the average time the Moon takes to go from perigee to perigee—the point in the Moon's orbit when it is closest to Earth. An anomalistic month is about 27.55455 days on average.
The draconic month, draconitic month, or nodal month is the period in which the Moon returns to the same node of its orbit; the nodes are the two points where the Moon's orbit crosses the plane of the Earth's orbit. Its duration is about 27.21222 days on average.

A synodic month is longer than a sidereal month because the Earth-Moon system is orbiting the Sun in the same direction as the Moon is orbiting the Earth. The Sun moves eastward with respect to the stars and it takes about 2.2 days longer for the Moon to return to the same apparent position with respect to the Sun.
An anomalistic month is longer than a sidereal month because the perigee moves in the same direction as the Moon is orbiting the Earth, one revolution in nine years. Therefore, the Moon takes a little longer to return to perigee than to return to the same star.
A draconic month is shorter than a sidereal month because the nodes move in the opposite direction as the Moon is orbiting the Earth, one revolution in 18.6 years. Therefore, the Moon returns to the same node slightly earlier than it returns to the same star.

Calendrical consequences

At the simplest level, most well-known lunar calendars are based on the initial approximation that 2 lunations last 59 solar days: a 30-day full month followed by a 29-day hollow month — but this is only roughly accurate and regularly needs intercalation by a leap day.
Additionally, the synodic month does not fit easily into the solar year, which makes accurate, rule-based lunisolar calendars that combine the two cycles complicated. The most common solution to this problem is the Metonic cycle, which takes advantage of the fact that 235 lunations are approximately 19 tropical years : 12 years have 12 lunar months, and 7 years are 13 lunar months long. However, a Metonic calendar based year will drift against the seasons by about one day every 2 centuries. Metonic calendars include the calendar used in the Antikythera Mechanism about 21 centuries ago, and the Hebrew calendar.
Alternatively in a pure lunar calendar, years are defined as having always 12 lunations, so a year is 354 or 355 days long: the Islamic calendar is the prime example. Consequently, an Islamic year is about 11 days shorter than a solar year and cycles through the seasons in about 33 solar = 34 lunar years: the Islamic New Year has a different Gregorian calendar date in each year.
Purely solar calendars often have months which no longer relate to the phase of the Moon, but are based only on the motion of the Sun relative to the equinoxes and solstices, or are purely conventional like in the widely used Gregorian calendar.
The complexity required in an accurate lunisolar calendar may explain why solar calendars have generally replaced lunisolar and lunar calendars for civil use in most societies.

Months in various calendars

Beginning of the lunar month

The Hellenic calendars, the Hebrew Lunisolar calendar and the Islamic Lunar calendar started the month with the first appearance of the thin crescent of the new moon.
However, the motion of the Moon in its orbit is very complicated and its period is not constant. The date and time of this actual observation depends on the exact geographical longitude as well as latitude, atmospheric conditions, the visual acuity of the observers, etc. Therefore, the beginning and lengths of months defined by observation cannot be accurately predicted.
While some like orthodox Islam and the Jewish Karaites still rely on actual moon observations, reliance on astronomical calculations and tabular methods is increasingly common in practice.

Ahom calendar

There are 12 months and an additional leap year month in the Ahom sexagenary calendar known as Lak-ni. The first month is Duin Shing.

No.	Name	Ahom script	Gregorian month
1	Duin-Shing	????? ????	November-December
2	Duin-Kam	????? ??	December-January
3	Duin-Tsam	????? ??	January-February
4	Duin-Shi	????? ??	February-March
5	Duin-Ha	????? ??	March-April
6	Duin-Rok	????? ????	April-May
7	Duin-Shit	????? ????	May-June
8	Duin-paet	????? ?????	June-July
9	Duin-kauo	????? ???	July-August
10	Duin-sip	????? ????	August-September
11	Duin-tsip-it	????? ???? ????	September-October
12	Duin-sip-song	????? ???? ?	October-November

Roman calendar

The Roman calendar was reformed several times, the last three enduring reforms during historical times. The last three reformed Roman calendars are called the Julian, Augustan, and Gregorian; all had the same number of days in their months. Despite other attempts, the names of the months after the Augustan calendar reform have persisted, and the number of days in each month have remained constant since before the Julian reform. The Gregorian calendar, like the Roman calendars before it, has twelve months, whose Anglicized names are:
The famous mnemonic Thirty days hath September is a common way of teaching the lengths of the months in the English-speaking world. The knuckles of the four fingers of one's hand and the spaces between them can be used to remember the lengths of the months. By making a fist, each month will be listed as one proceeds across the hand. All months landing on a knuckle are 31 days long and those landing between them are 30 days long, with variable February being the remembered exception. When the knuckle of the index finger is reached, go over to the first knuckle on the other fist, held next to the first and continue with August. This physical mnemonic has been taught to primary school students for many decades, if not centuries.
This cyclical pattern of month lengths matches the musical keyboard alternation of wide white keys and narrow black keys. The note F corresponds to January, the note F corresponds to February, the exceptional 28–29 day month, and so on.

Numerical relations

The mean month-length in the Gregorian calendar is 30.436875 days.
Any five consecutive months that do not include February contain 153 days.

Calends, nones, and ides

Months in the pre-Julian Roman calendar included:

Intercalaris an intercalary month occasionally embedded into February, to realign the calendar.
Quintilis, later renamed to Julius in honour of Julius Caesar.
Sextilis, later renamed to Augustus in honour of Augustus.

The Romans divided their months into three parts, which they called the calends, the nones, and the ides. Their system is somewhat intricate.
The ides occur on the thirteenth day in eight of the months, but in March, May, July, and October, they occur on the fifteenth. The nones always occur 8 days before the ides, i.e., on the fifth or the seventh. The calends are always the first day of the month, and before Julius Caesar's reform fell sixteen days after the ides.