Tolkien's use of Septmeber and other months was simply a modernization of the ancient Shire Calendar, which of course would not have had the same month names as us. In fact, they had a different system for dividing their year, which is described quite nicely in Appendice D of the Lord of the Rings.
As Squatter said, our modern 1 Jan would match up with the 9th day of the Shire's first month. Calculating through the year, you get 14 September on regular years and 13 September on Leap Years. (Note: I still had this year set as a leap year until recently, so the site mistaken reported 13 September as the birthday day this year - Sorry. BW)
To somplify things, when JRRT said their birthdays fell on 22 September, he really was saying their birthdays were on the 22nd day of thier 9th month - the 257th day of their year. The 257th day of our year was 14 September.
I hope that clears it up a bit. JRRT took the time to publish Appendice D, so I'm sure he would like us all to understand it.
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The Barrow-Wight
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