Are there any real calendar systems that have days that are not part of any month?

A non-real example might be the hobbitsʼ calendar where all 12 months consist of 30 days, and the remaining five or six days (the two equinoxes are 1+1, and the solstices are 2 for winter, and one (or two in leap years) for the summer.)

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The Polonkai family

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