Top 10 Lunisolar Calendars

Lunisolar calendars are calendars that calculate when the moon will achieve a phase on or near the day it occurred before. They have been known since ancient times.

While the names of many cycles and sub-cycles aren't often used in the public world, such as pop culture, politics, or entertainment, those names, based on Greek literature, define the periods which highlight when we see the moon phases.
The Top Ten
1 Metonic Cycle

The period of roughly 19 tropical years where the moon phases occur around the same day. It equals 235 synodic months, exceeding 19 tropical years by just 2 hours, 4 minutes, and 58 seconds on average. This makes it the most accurate for synchronizing the lunar calendar with the tropical, Gregorian, and Julian calendars for periods less than a century.

It adds 7 synodic months over 19 lunar years. With 110 hollow months and 125 full months, it yields a short eclipse series of up to 5 members, defining the total life expectancy of the Callippic period.

2 Triacontaoctaëteris

Two Metonic cycles, in which the moon phases occur around the same day after 38 years. It equals 470 synodic months and comes toward the same day of the week as roughly an integer number of weeks are achieved.

As an eclipse period, it's a period separating similar eclipses with opposite gamma values, as it equals half of the Callippic period (equaling 220 hollow months and 250 full months).

3 Pentacontaheptaëteris

Three Metonic cycles, in which the moon phases occur around the same day after 57 years. It equals 705 synodic months and was known by the Buddhists as the Buddhist lunar calendar.

As an eclipse period, it's an individual eclipse period as it's three-quarters of the Callippic period (equaling 331 hollow months and 374 full months).

4 Callippic Cycle

The period around 76 years where a day is subtracted from the calendars to improve synchronization with the moon phases. It equals 4 Metonic cycles. With the calendar adjustment, the offset is around 5 hours and 54 minutes.

As an eclipse period, it defines the total life expectancy of a Metonic eclipse series, which is why it's known by eclipses as the Callippic period. This doesn't regularly happen, so instead, one synodic month is usually subtracted, creating the short Callippic cycle.

5 Archetypes Lunar Calendar

It is the equivalent to 100 Saros eclipse cycles (which isn't possible in our time), so 22,300 synodic months are calculated, equaling roughly 1,806 years. This means the Moon phases also occur toward the same day of the week as roughly an integer number of weeks are achieved.

6 Gregoriana

The best lunisolar cycle for simultaneously achieving the moon phases as well as eclipses around the same day of the Gregorian calendar and week, as well as the moon's orbital node. It equals 4,601 synodic months, which approximates 372 Gregorian years, making it more accurate for Earth calendars than for solar calendars.

7 Tropicana

The most accurate period for synchronizing the lunar calendar with the tropical calendar. It equals 4,366 synodic months, which very well approximates 353 tropical years (hence the name), making it useful for calculating the phases in the timing of the equinoxes and solstices.

When it comes to eclipses, it's a period separating similar eclipses of opposite gamma values.

8 Grattan Guinness Cycle

The shortest period where the calendar dates and lunar years synchronize accurately, with the offset being just hours. It equals 4,836 synodic months, which approximates 391 Gregorian years and equals 403 lunar years.

Discovered by and named after Henry Grattan Guinness from a speculative reading of Revelation 9:15.

9 Dihectaicosaëteris
10 Octacontatetraëteris

The period over 84 tropical years where the Moon phases occur over 1 1/2 days later. It equals 1,039 synodic months, which is a Callippic cycle plus an octaëteris, making it useful for calculating significant eclipses near the same day of the common calendars, as 11 Inex and 13 Saros are achieved.

The Contenders
11 Hipparchic Cycle

As a calendar cycle, Hipparchus defined it as the period over 304 tropical years, improving the Callippic cycle (which itself improved the Metonic cycle).

It subtracts a day from 4 Callippic calendars, therefore 16 Metonic cycles, calculating 3,760 synodic months, leading to moon phases around 1 3/8 days later.

12 Old Lufkan Calendar

An old calendar used in Medieval times equaling 8,027 synodic months or approximately 649 tropical years on the same node. It subtracts 1 synodic month off of 669 lunar years, which on the other hand equals 36 Saros eclipse cycles.

13 Goddess Lunar Calendar

An arithmetic lunar calendar invented by Peter Meyer, with the months named after goddesses with names beginning with successive letters of the alphabet starting with A.

It equals 20,890 synodic months, which approximates 1,689 tropical years, ending on the same node.

14 Octaëteris

The period over 8 tropical years where the moon phases occur around 1 1/2 days later. It equals 99 synodic months or 8 1/4 lunar years and approximates 13 synodic and 5 sidereal periods of Venus, therefore nearly synchronizing with the planet.

Though it's rarely ever useful for eclipses, as they usually happen 1 synodic month later, forming a hectolunex.

15 Hendecaëteris

The period within 11 tropical years where the moon phases occur around 1 1/2 days earlier. It equals 136 synodic months or 11 1/3 lunar years and is slightly more than 12 synodic periods of Jupiter. The subtraction of 1 synodic month helps properly synchronize with Jupiter and also forms an eclipse cycle known as a Tritos.

16 Triacontaëteris

The period within 30 tropical years where the moon phases occur around 1 1/2 days earlier. It is an improvement to the triëteris, as 1 synodic month is added over 10 triëterides to adjust the moon phases with the calendar dates.

It equals 371 synodic months and is nearly a half-integer number of draconic and anomalistic months and weeks. This means the moon phases occur toward the opposite node from apogee to perigee, or from perigee to apogee, nearly halfway through the week.

17 Hyper Metonic Cycle

12 Metonic cycles, equaling 2,820 synodic months or 235 lunar years, leading to the moon phases occurring over 1 day later in the same node after 228 Earth years.

18 Icosiheptaëteris

The period over 27 tropical years where the moon phases occur around 1 1/2 days later. It equals 334 synodic months and is nearly an integer number of weeks and anomalistic months, allowing the moon phases to occur at nearly the same distance toward the same day of the week.

However, it's poor in draconic returns, which makes it useless for calculating eclipses of similar character. In fact, it is a period separating similar eclipses with opposite gamma values. It adds an octaëteris over the Metonic cycle and equals a Saros plus a hibbardina.

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