The concept of a 13th month has existed in various cultures and calendars throughout history, but the name of the 13th month varies depending on the calendar system in question. In this response, we will explore some examples of calendars with a 13th month and their respective names.

One of the most well-known calendars with a 13th month is the Chinese calendar. This calendar system, which has been used for thousands of years, is based on lunar cycles and consists of 12 or 13 months in a year. The 13th month is added to the calendar in order to align it with the solar year and ensure that the start of each lunar year is close to the start of the solar year. The name of the 13th month in the Chinese calendar is “闰月” (rùn yuè), which means “intercalary month” or “leap month”. This is because the 13th month is added as an extra month to the calendar in order to “leap” over the discrepancy between the lunar and solar years.
Another calendar system that includes a 13th month is the Ethiopian calendar, which is also known as the Ge’ez calendar. This calendar is based on the Coptic calendar, which in turn is based on the ancient Egyptian calendar. The Ethiopian calendar consists of 12 months of 30 days each, followed by a 13th month of 5 or 6 days, depending on whether or not it is a leap year. The 13th month in the Ethiopian calendar is called “Pagume”, which is derived from the Greek word “epagomene”, meaning “added” or “intercalary”.
In the Islamic calendar, which is based on lunar cycles, a 13th month is not included. Instead, the calendar consists of 12 months of either 29 or 30 days, depending on the sighting of the new moon. Because the Islamic calendar is shorter than the solar year, the start of each Islamic year moves forward by approximately 11 days each year relative to the Gregorian calendar. This means that the Islamic holidays, such as Ramadan and Eid al-Fitr, occur at different times each year. However, there is a tradition of adding an extra month to the Islamic calendar in order to align it with the solar year, but this is done infrequently and is not a regular part of the calendar system.
In the Julian calendar, which was introduced by Julius Caesar in 45 BCE and was used throughout much of Europe until the 16th century, a 13th month was not included. Instead, the calendar consisted of 12 months of varying lengths, with a total of 365 days in a year. However, because the Julian calendar did not accurately account for the length of the solar year, it gradually fell out of sync with the seasons. This led to the introduction of the Gregorian calendar, which is the calendar system that is used in most of the world today.
In summary, the name of the 13th month varies depending on the calendar system in question. In the Chinese calendar, the 13th month is called “闰月” (rùn yuè), in the Ethiopian calendar it is called “Pagume”, and in the Islamic calendar, a 13th month is not included. Other calendar systems, such as the Julian calendar, did not include a 13th month.