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There were many calendars and systems for dating throughout the history of Middle-Earth, including:

    Stewards Reckoning
    Kings Reckoning
    Númenorean Reckoning
    New Reckoning (After the War of the Ring)

Many of the systems had different names for the months:

Shire Calendar Bree Calendar Sindarin (Used in Gondor) Westron (Quenya Names)
Afteryule Frery Narwain Narvinye
Solmath Somath Ninui Nenime
Rethe Rethe Gwaeron Sulime
Astron Chithing Gwirith Virisse
Lothron Lotesse
Forelithe Lithe Norui Narie
Afterlithe Mede Cerveth Cermie
Wedmath - Urui Urime
Halimath Harvestmath Ivanneth Yavannie
Winterfilth Wintring Narbeleth Narquelie
Blotmath Blooting Hithui Hisime
Foreyule Yulemath Girithron Ringaire

The Eldar had a 6 day week, which was changed by the Númenoreans into a seven day week. They also changed the names of some of the days.

The Days of the Week

Shire (At the Time of the War of the Ring) Sindarin Quenya Dúnadan Translation
Sterrendei Sterday Orgilion Elenya Elenya Star-Day
Sunnedei Sunday Oranor Anarya Anarya Sun-Day
Monendei Monday Orithil Isilya Isilya Moon-Day
Trewesdei Trewsday Orgaladhad Alduya Aldea Tree Day (Númenorean) or Trees-Day
Hevenesdei Hevensday (or Hensday) Ormenel Menelya Menelya Heaven's-Day
Meresdei Mersday Orbelain (or Rodyn) Valanya (or Tarion) Valanya Valar-Day
Highdei Highday Oraearon Eärenya Eärenya Ship-Day (Added by the Númenoreans)

Our Days: Shire Calendar:
Monday Monday
Tuesday Trewsday
Wednesday Hevensday
Thursday Mersday
Friday Highday
Saturday Sterday
Sunday Sunday

Each of these calendars had a deficit of days, and each calendar dealt with it differently. The Númenorean Reckoning (or Kings Reckoning) dealt with it by adding a day every thousand years, but that left a small amount over, about 5 hours. This was corrected with the adaptation to the Steward's Reckoning. This one had such a tiny deficit, that by the War of the Ring it had not yet grown to one day. Each month had 30 days and the remaining days were holiday outside of the months.

The Shire Calendar used a similar system, save that the extra days were in two groups: Yule and Lithe.
Also, Year one in the Shire Reckoning was equivalent to 1601 of the Third Age in the Steward's Reckoning.

The New Reckoning
After the War of the Ring, the New Year began on March 25, the day that Sauron was destroyed. The month names remained the same, however they now began 5 days earlier than before.

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