
Probably the tithi was invented to avoid the complications of having to deal with the extra half day in the 29.5 solar day month. It however created other complications. The tithi could begin at any time of a solar day so it was decided that the tithi that prevailed at sunrise would be the tithi for the whole current day. The year normally consisted of twelve months, of 30 tithi each and was 360 tithi long. It usually began with the Caitra (March-April) month, but in some systems the it began with the Karttika (October-November). Three hundred sixty tithi is only 354 solar days and the solar year is 365.25 solar days thus every second or third year an extra month was added. It is interesting to note that a 360 tithi long year makes use of the very convenient number 360. Ancient Babylonians and ancient Maya also used a 360 day year for their calendars, but they used solar days rather than lunar days.
The solar calendar was imported to India in the time of Gupta (300 A.D.) with months named for the signs of the zodiac. The names of the months are a close translation of the Greek originals. Thus the months are Mesa (Aries), Vrsagha (Taurus). Mithuna (Gemini), Karaata (Cancer), Simha (Leo), Kanya (Virgo), Tula (Libra), Vrscika (Scorpio), Dhanus (Sagittarius), Makara (Capricornus), Kumbha (Aquarius) and Mina (Pisces.) The seven day week was also introduced with the name of the day being the name of the planet which precides over the days as in the Greco-Roman system. In fact the order of the planets is the same as in the Greco-Roman system, a vestige of the ordering by ancient Babylonians.
Today in India many religious festivals are determined by their lunar calendar date. Diwali, the new year festival, celebrated all over India, occurs on the New Moon which begins the lunar month Karttika (usually occurring in October or November).