The concept of a 13-month lunar calendar dates back thousands of years and was used by many ancient cultures, including the Babylonians, Egyptians, and Mayans. However, the transition to a 12-month solar calendar was a gradual process that took place over many centuries.

The reason for the transition from a lunar to a solar calendar is primarily due to the fact that the lunar calendar is based on the cycles of the moon, which lasts approximately 29.5 days. This means that a lunar year is approximately 354 days, which is about 11 days shorter than a solar year. As a result, the lunar calendar did not accurately reflect the changing seasons, which caused problems for agricultural societies that relied on the seasons for planting and harvesting.
The transition to a solar calendar was a gradual process that began in ancient Rome. In 45 BCE, Julius Caesar introduced the Julian calendar, which was a solar calendar that consisted of 12 months, with each month having either 30 or 31 days, except for February, which had 28 days (and 29 days in a leap year). The Julian calendar was based on the movements of the sun and was more accurate than the lunar calendar, which helped to improve agriculture and commerce.
However, the Julian calendar was not perfect and had some inaccuracies. Specifically, the length of a solar year is actually slightly less than 365.25 days, which meant that the Julian calendar was slowly becoming out of sync with the seasons. This led to the development of the Gregorian calendar in 1582, which was a reform of the Julian calendar introduced by Pope Gregory XIII.
The Gregorian calendar also consisted of 12 months, but it made some changes to the length of the year to correct the inaccuracies of the Julian calendar. Specifically, the Gregorian calendar added a leap year every 4 years, except for years that are divisible by 100 but not divisible by 400. This change helped to bring the calendar more in line with the seasons, which improved agricultural practices and made it easier to plan for seasonal changes.
Overall, the transition from a 13-month lunar calendar to a 12-month solar calendar was a gradual process that took place over many centuries. The development of the Julian and Gregorian calendars helped to improve the accuracy of the calendar and made it easier to plan for seasonal changes. While the 13-month lunar calendar is no longer used for practical purposes, it is still used in some cultures for religious or cultural events.