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The Bible does not directly mention the addition of "embolismic" or intercalary months.
The Hebrew calendar is lunisolar with an embolismic month.
The 8-year cycle (99 synodic months, including 3 embolismic months) was used in the ancient Athenian calendar.
The Chinese calendar is lunisolar, so a leap year has an extra month, often called an embolismic month after the Greek word for it.
However, without the insertion of embolismic months, Jewish festivals would gradually shift outside of the seasons required by the Torah.
For example, in the Hebrew calendar seven years out of every nineteen (37%) have the "embolismic month" Adar II.
In the Hindu calendar, which is a lunisolar calendar, the embolismic month is called adhika maasa (extra month).
Regular years have 12 months, but embolismic years insert a 13th "intercalary" or "embolismic" month every second or third year.
This has been ruled as implying a requirement for the insertion of embolismic months to reconcile the lunar cycles to the seasons, which are integral to solar yearly cycles.
Whenever the epact reaches or exceeds 30, an extra (embolismic or intercalary) month is inserted into the lunar calendar, and the epact is reduced by 30.
The lunar year consists of 30-day and 29-day lunar months, generally alternating, with an embolismic month added periodically to bring the lunar cycle into line with the solar cycle.
To determine when an embolismic month needs to be inserted, some calendars rely on direct observations of the state of vegetation, while others compare the ecliptic longitude of the sun and the Moon phase.
Therefore, for a lunisolar calendar, every 2 to 3 years there is a difference of more than a full lunar month between the lunar and solar years, and an extra (embolismic) month needs to be inserted (intercalation).
He concluded that the ancient Rapanui used a lunisolar calendar with kotuti its embolismic month ( "leap month"), and that Thomson chanced to land on Easter Island in a year with a leap month.
Therefore, whenever this excess accumulates to about 30 days, or a little more or less, one month is added and the particular year is made to consist of 13 months, and this is the so-called embolismic (intercalated) year.
When the observational form of the calendar was in use, whether or not an embolismic month was announced after the "last month" (Adar) depended on 'aviv [i.e.the ripeness of barley], fruits of trees, and the equinox.
They had knowledge of the Denkard's proposal: at some point between 1125 and 1129, the Parsi-Zoroastrians of the Indian subcontinent inserted such an embolismic month, named Aspandarmad vahizak (the month of Aspandarmad but with the suffix vahizak).