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Yugadi (New Year) today in Karnataka and many other places of India.

ಯುಗಾದಿಯ ಶುಭಾಶಯಗಳು

Interestingly, we use the lunar calendar.
In a lunar calendar, what is important is the tithi. Unlike the seven day week which we now follow (introduced much later in India), tithi is based on the fifteen days between full moon and new moon (and vice versa).
So a lunar month has two phases/paksha, depending on the waning (krishna paksha) and the waxing (shukla paksha) moon. Each of the pakshas has 15 days in them as tithi.
So what determines a tithi? Since we are looking at a rotation of 360 degrees and 30 days, the tithi starts when the angle between the sun and the moon (with respect to earth, of course) is 12 (=360/30) degrees. This tithi/day goes on until this angle changes by 12 degrees.
The problem with this day is that it is not fixed to be 24 hours. This is because we have to both consider the earth and the moon's orbit. The elliptic orbits make the lunar days non-uniform in length. From 20 to 26hrs.
So yesterday at 2:20om, the moon made an appropriate angle to the sun after the new moon day, signifying the first day of the month; and therefore the first day of the year. A new year!

But we celebrate it today as we push the tithi to sunrise.
What the the tithis called? Very practically, it is just called by a number! Prathama/dwithya/tritiya which is just 1/2/3. No names of gods, nothing; just numbers!
Because of the presence of the sun in measuring angles etc, this calendar is not strictly lunar. It is something called lunisolar calendar. The islamic calendar is lunar, with no reference to the sun.
So why does the new year differ between the (say) Kannada and Tamil people? It is because while both use the same tithis, the marking of the first day of the month differ.

The Tamils mark the month based on the angle of the sun to the first point of Aries (astronomical term).
So as a tamil-kannadiga, I get to celebrate both new years (along with the one on Jan 1st! Indians are not just multi-lingual, we are multi-celebratory as well!).
But I am mildly partial to Yugadi because I get to eat Obbattu!

Happy Yugadi. Happy New Year. (Until next month!)
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