When I collected data for the new final chapter of "The Moon in 'The Hobbit'", I realised that there are not two but three different modes of counting time used in different regions of Middle-earth. I believe that Tolkien was inspired to use them by the Venerable Bede, from whom he also took the names of the Shire months. Briefly, those three modes are:
- The Arnorian mode, used in the Shire, Bree and by the Dúnedain (Aragorn alludes to it). Like in our modern system, a full day is divided into 24 hours of equal lengths, and clocks are made to count them. The Venerable Bede called this mode natural hours or equinoctial hours because the counting represents the natural division of day and night at the exact moment of an equinox, i. e. 2 x 12.
- The Gondorian mode, used in Minas Tirith and probably the rest of Gondor. Here, the period from sunrise to sunset is divided into 12 hours and counted from 1 to 12. This count is the inevitable result of observing sundials. It is correspondingly ancient and alluded to already in the Silmarillion (Maedhros coming "at the third hour of morning"!). The Venerable Bede called this mode artificial hours because "artifice" was his word for a sundial.
- The Northmannish mode, used in Rohan and possibly in Rhovanion. It uses artificial hours, too, but counts them before or after noon, not from sunrise to sunset. (Tolkien yet applied it once in "Cirion and Eorl" though his source was allegedly Gondorian.) Strangely, this mode pops up among the Fellowship all of a sudden soon after Gandalf is lost. If this is not simply a negligible oversight by Tolkien it might suggest that someone else assumed the task of observing the sky then - Legolas, possibly, for Boromir and Aragorn would have used the Gondorian or Arnorian mode, respectively. (Legolas also makes it his habit to count "two days ago" as "thrice twelve hours". Maybe that's his personal twirk for it never occurs anywhere else, as fas as I am aware.)
Artificial hours represent true solar time at your place, not an averaged time. Their drawback is that these hours are not constant over the year, hence they are unsuitable for mechanical clockworks. Instead, they will be longer in summer and shorter in winter as the daytime divided into twelve equal parts expands and shrinks; hence the Roman distinction of summer hours and winter hours. The difference also is the harder felt the further north you are, hence, it may have suggested to Elendil to fall back on equinoctial hours instead.
The following diagram shows the lengths of artificial hours over the year at about the latitude of Pelargir (or Rome, in our modern world). You will see that Pippin makes a pretty good guess when he equates the 3rd hour of Minas Tirith to 9 o:clock in the Shire at this season.
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