I never suggested that the seasons were linked to the moon or resulted from a joint lunar-solar calculation; rather than, oddly, we seem to have a lunar New Year overlaid with solar (or sidereal?) seasons.
The oddity I was pointing out was that no
single system could have a New Year and the end of autumn a mere week (or 6 days) apart, which is what Thorin says. Thus they must somehow derive from different systems, or at least there's no other way to account for the published text.
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But I think you may have hit upon the key to the puzzle: the correction to the First Typescript.
“Tomorrow begins the last month [> week] of Autumn” said Thorin one day. “And winter comes after autumn” said Bifur.
As originally written, the last month of autumn began the next day, the day upon which the Durin's Day new moon rose; in other words, the first appearance of the moon which defines autumn's last (lunar) month. It all works.
Changing that to "week" throws all that off (as well as the later chronology). So why did T do it?