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In general, you are rarely made conscious of just how arbitrary calendar time is. Technically, since the base year for the Indian calendar is 79AD, which is CE+78, I was born on 1, Kartik, 1896, and am already 45y old in lunar years, but still 44 by solar years.
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Temporal collisions also reveal history. The reason most ancient cultures used lunar or lunisolar calendars is that it's actually hard to estimate solar year directly. You have to precisely track solstices ("sun standing still") which means keeping daily noontime elevation logs
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The other alternative is to measure day length and know when the longest or shortest day passes. This requires minute-level precision in sub-day timekeeping, which is actually quite hard even with high quality hourglasses etc.
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But solar calendars despite being harder to technologically track, are more useful, since harvest cycles etc. follow solar patterns, and not many practically important things track lunar time (night-time safety?)
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Lunisolar calendars try to triage the problem, by using lunar for daily needs, and a complex human-powered priesthood for the rare, expensive solar synchronization. If you think about it, it's like solving a small Y2K type problem every couple of years, by adding a month.
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There are no calendars that do not require such janky adjustments. Only the scale differs. Regular leap years (4 year, 1 day), modern leap years, every 4th year, except every 100th year not divisible by 400. Modern atomic clocks need to add leap seconds here and there.
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A calendar is more real based on the amount of temporal phenomenology it captures by default. If we picked most absolute T=0, which would be 13.2 billion years ago, and captured ALL the dynamics (including relativistic) between the Big Bang and us, how would the calendar look?
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Science fiction stories with alien species gloss over calendar tech too easily. A space-faring species would have to construct meaningful, galactic-scale, relativity-accommodating calendar systems. It would also need to reverse engineer the local society's calendar.
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Back on earth, lunar and solar dissonance also keep astrologers employed, since the moon is very significant, but its movement against the stars is determined by both its own movement and the solar year.
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