How come every culture on the planet has a different calendar, yet follow the same system for a week?

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We have an enormous amount of evidence for the ancient Babylonian calendar, but no evidence at all for a seven-day week in ancient Babylonia.

In the ancient world there were two forms of the seven-day week. First, the Jewish week (eventually adopted by Christians and Muslims) has numbered days from one (Sunday) to six (Friday) and the Sabbath on the seventh day. Although the Sabbath is frequently mentioned in the Old Testament, the use of the numbered week for dating is not attested before the first century BC, first with Jews, and then in the New Testament and other Christian texts. Second, the planetary week, where each of the seven days is named after a planet (Sun, Moon, Mars, Mercury, Jupiter, Venus, Saturn) is based on the astrological doctrine of the Lords of the Hours and the Lords of the Days; it is alluded to by the Roman poet Tibullus in the 1st century BC, then fully developed by the astrologer Vettius Valens in the 2nd century AD.

The seven-day week spread with Christianity and (specifically in India) with the reception of Greek astrology, beginning with the Yavanajataka, where the week is explicitly described as a Greek thing.

There is currently an ongoing research project on this question:

Pingree’s translation of the Yavanajataka (see in particular section 77):

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I guess, people look for fixed cycles in nature. the cycle of the moon, the wandering of the shadow on its surface takes 28 days, divide the horizontal axis of the moon into 4 equal parts you get for each part 7 days.

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The seven day week is not necessarily universal, although it has spread through most of the world as certain cultures have dominated the globe. The Romans and then the Christians more or less pressed certain cultural elements on everyone.

Why 7 days? The easiest explanation is probably that there are roughly 28 days in a lunar cycle and so, dividing that into four equal "weeks" would yield roughly 7 days.

The earliest known 7 day "week" I believe was used in Babylon, on which a religious celebration took place every 7 days. However, this celebration started on the new moon. Hence, 7 day weeks, to make 4 celebrations per month.

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The solar year and the lunar month (and the solar day) are obvious, natural cycles of time. Since none of them is an integer multiple of another, a wide range of calendars have been created to reconcile them. They tend to still be in use because they've got a great deal of inertia (and often, religious tradition) behind them.

There are no natural sub-month cycles. The Abrahamic seven-day week is just one that's been in use; various other cultures have cycles ranging from the four-day market cycle of the Igbo to the Aztec/Maya 13-day ritual cycle. The domination of the Abrahamic week is probably simply because a third of the world's population follows it for religious reasons; none of the other sub-month cycles has anywhere near that following.

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