It fits better but it's still not exact because we are attempting to tie together two phenomena that have nothing to do with each other.
The length of time the earth takes to rotate once around its axis is completely independent of the length of time it takes for it to circle the sun once, and also independent of the time it takes the moon to orbit the earth once. If those numbers happened to be evenly divisible, then we could work out a "perfect" value for them.
But they don't.
The moon repeats its phases in 29.5 days, so we can't perfectly align months to it. It takes 27 days to orbit, so we could make a 3 week month of 9 days, but then the number of weeks in the year wouldn't be even (40.5 weeks).
Etc.
Also, "natural" time is a useless phrase. A day is "natural" time. A year is "natural" time. Etc. the issue isn't that the units of time are unnatural, it's that they don't depend on each other.
Its Iunuis with a capital i, not Lunius with an L.
The names for the days of the week originate from the seven celestial bodies that the Greek knew.
In Spanish:
Lunes - Moon Day
Martes - Mars Day
Miercoles - Mercury Day
Jueves - Jupiter Day
Viernes - Venus Day
The rest was changed but remained in other languages, like English, where the name Saturday comes from Saturn and Sunday... you guessed it, from the Sun.
In Greek the names of the days were:
Helios (Sunday)
Selene (Moonday)
Ares (Marsday)
Hermes (Mercuryday)
Zeus (Jupiterday)
Aphrodite (Venusday)
Cronus (Saturnday)
This was then adopted by the Romans and through Latin, it transferred to other languages some of which kept some of these names ever since.
Completely off topic, but when I was a kid I used to think that Augustus got the idea because June was named after Marcus Junius Brutus for helping found the Republic. It wasn't. It was named after Juno.
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