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Murlo
Rivendell
Mar 25 2018, 8:49pm
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Appendix D: Tolkien Reckoned Mostly Correct and Why We Should Observe the Elves' New Year on March 25

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Happy Yestarë! If you've seen my article published in this month's Beyond Bree newsletter, titled "Why We Should Observe the Elves' New Year on March 25", then you'll know I'm referring to the Yestarë of the Calendar of Imladris, and not the Yestarë of Aragorn the King Elessar's New Reckoning (which falls on the Shire Calendar's 'March' 25). If you do not subscribe to Beyond Bree, you can still see my arguments presented in a slightly different format on my Shire Reckoning project's home page (this is a fan project and is completely adfree and free to use). Today I've also updated my Shire Reckoning project with a section summarizing all the holidays I know of in Middleearth. If anyone knows of any other Middleearth holidays I may have missed, I love to hear about them! Also in my project's updates for today are my notes on Tolkien's Deficit calculations. I've published my thoughts on my Shire Reckoning project's calendar simulation page, and I also plan to prepare these thoughts with a little more detail and submit them as another essay to Beyond Bree, but I'll reproduce them here to open it up for discussion for those who are also interested in this topic. For all my interest in Appendix D, I was never really interested in Tolkien's Deficit calculations and whether or not he made any mistakes. After all, others have already addressed these calculations in other articles, such as the article "The King's Reckoning: Did Tolkien Reckon Correct?" (Beyond Bree, November 1985) by Åke Jönsson (Bertenstam) and the article "J.R.R. Tolkien's Calendars or The Saga of Hador The Incompetent" (Mythlore 54, Summer 1988) by Darrell A. Martin. Since I'm confident I have accurately reproduced the rules of the Kings' and Stewards' Reckoning in my calendar simulations, I am now able to use those calculations to assist my own Deficit calculations and compare them with the results of Tolkien and others. Given Tolkien's static year length of 365 year, 5 hours, 48 minutes, 46 seconds, I was able to reproduce Tolkien's millennial deficit of 4 hours, 46 minutes, 40 seconds. I was also able to reproduce the calculations presented in "Table 5" of Martin's article, where the total deficit of T.A. 1 was given as "1 day 9 hr 46 min 6 sec", T.A. 2001 was given as "1 day 19 hr 19 min 26 sec", and later in the article the total deficit of T.A. 3001 was calculated as "1 day 6 minutes and 6 seconds". So this indicates that I have at least interpreted the Kings' and Stewards' Reckoning rules the same as Martin. My numbers didn't match Jönsson's calculations, however, and I think this is because Jönsson's calculations went astray starting with his deficit calculation for the end of T.A. 2059. Without Mardil's 2day addition to T.A. 2059, Jönsson gives the deficit as "2 days, 8 hours, 5 minutes, 26 seconds, consisting of 1 day, 2 hours, 16 minutes, 40 seconds (5.5 times the millennial deficit) together with 5 hours, 48 minutes, 46 seconds (this deficit was caused by making T.A. 4 instead of T.A. 3 the first leapyear in the Third Age...)". First, the value given for "5.5 times the millennial deficit" would only apply if the King's Reckoning added 1 day every 500 years to account for the millennial deficit. Since the millennial deficit is adjusted by adding 2 days only after 1000 years have elapsed, then the actual deficit by this system would be 5 times the millennial deficit, plus another deficit of 500 years, which I calculate as a total deficit of 2 days, 2 hours, 16 minutes, 40 seconds (2d:2h:16m:40s). This figure also agrees with the amount given in the Reader's Companion of "50 hours, 16 minutes, 40 seconds." Second, he should not have added the regular yearly deficit of 5h:48m:46s to that total. He may have been misled by Tolkien's statement in Appendix D that "By making T.A. 4 a leap year instead of T.A. 3 (S.A. 3444) 1 more short year of only 365 days was intruded causing a deficit of 5 hours, 48 minutes, 46 seconds." While it's correct that the regular yearly deficit of 5h:48m:46s was not offset by a leapday in T.A. 3, the leap day added in T.A. 4 made up the difference, so that by the start of T.A. 5 both systems (either T.A. 5 or S.A. 3446) would end up with the same deficit. After adding the 2 days of Mardil's correction, Jönsson's total deficit would come to 8h:5m:26s, whereas I have calculated a deficit of 2h:16m:40s by the end of T.A. 2059 instead. One might wonder if Tolkien was also misled in a similar way and came to a similar deficit in Appendix D when he described the "special addition of 2 days to 2059 (S.A. 5500), which concluded 5 1/2 millennia since the beginning of the Númenórean system. But this still left about 8 hours deficit." Another possibility was that Tolkien's "8 hours deficit" may be a remnant of calculations done for the first publication of Appendix D. In the first publication of Appendix D, according to the Reader's Companion and also quoted in Jönsson's article, Mardil added 2 days to T.A. 2060 instead. Since T.A. 2060 was already supposed to be a leapyear, which means it was already to be a year of 366 days, then if Mardil's 2day addition to T.A. 2060 persisted into the second edition, that would have meant a year with 368 days, which had never happened before in all of the 5500 year history of the Kings' Reckoning; not even in a "Millennial Leapyear" which has 367 days. This is probably why Tolkien emended Mardil's special 2day addition to T.A. 2059 instead, so that the last year of the Kings' Reckoning could be treated as another "Millennial Leapyear" (just like the one that occurred 59 years earlier in T.A. 2000), and then the first year of Stewards' Reckoning in T.A. 2060 could still be a leapyear of 366 days as already scheduled. Then the rules of Third Age leapyears (every 4 years except the last of a century) could continue as normal, just as they had for the past 2059 years. If Tolkien was using a shortcut of calculating the deficit at the end of S.A. 5501 (which would be the same year as T.A. 2060), that calculation would be 1 day off of the calculation for the end of T.A. 2060, since T.A. 2060 has a leapday but there would be no leapday in either S.A. 5500 or S.A. 5501. Aaron Chong's blog post, Tolkien's Legendarium versus Astronomical Reality, suggests that Tolkien may have forgotten that T.A. 2060 was a leapyear and that dropping the leapday in that year may explain his calculations (and the Reader's Companion seems to suggest something similar), but using a shortcut of calculating the deficit for S.A. 5501 could also explain the missing leapday in Tolkien's calculations. It just so happens that the deficit by the end of S.A. 5500 would match the deficit by the end of T.A. 2059, but accounting for Mardil's extra 2 days in T.A. 2059 and the leapday in T.A. 2060, I calculate a surplus of 15h:54m:33s by the end of T.A. 2060. Omitting the leapday due in T.A. 2060 (but keeping Mardil's extra 2 days) results in a deficit of 8h:5m:26s by the end of that year instead, and this may be the (incorrect) figure for Tolkien's "about 8 hours deficit" (and also works out to the same deficit as Jönsson's total). As an aside, I'll note that my deficit calculations match Aaron Chong's, but only up to T.A. 2059 (the rest appear to be hypothetical calculations if the Kings' Reckoning continued beyond that year). In any case, it appears Tolkien may have gotten back on track in his next statement about the deficit by T.A. 2360: "Hador to 2360 added 1 day though this deficiency had not quite reached that amount." My calculations also show a deficit of not quite 1 day, at 18h:6m:40s by the start of T.A. 2360. After Hador's extra day was added to T.A. 2360, however, I calculate another surplus of 1d:0h:4m:34s at the start of T.A. 2361, which explains the subtitle of Martin's article, "The Saga of Hador The Incompetent". As Martin points out in his article, "had he not added a day, and had Denethor II added the expected two days to 3Age 3000, the accumulated deficit to begin the fourth millennium would have been a mere 6 minutes and 6 seconds! As it actually occurred, the deficit was 1 day 6 minutes and 6 seconds" by the start of T.A. 3001, and my calculations agree. If Tolkien meant that the "deficiency had not quite reached" 1 day at the end of T.A. 2360, then once again he may have used a shortcut of calculating the deficit for S.A. 5801, and he would again be 1 day off from my calculation for T.A. 2361 and thought Hador only added a surplus of 4m:34s. Finally, I think my calculations also agree with Tolkien's statement about the deficit by the end of T.A. 3020, which was 660 years after Hador's adjustment: "By the end of the Third Age, after 660 more years, the Deficit had not yet amounted to 1 day." If the Stewards' Reckoning had continued through T.A. 3020, I calculate a deficit of 20h:21m:26s by the start of Stewards' T.A. 3021, but adding to this deficit about 1/4 of the yearly deficit (approximately 1h:27m), to get to the start of the New Reckoning's Fourth Age 1, would still be a deficit of less than 1 day. I believe I have accurately reproduced all of the Kings' and Stewards' Reckoning rules as presented in Appendix D in my calendar simulations, and with it I have verified all of Tolkien's statements about the Deficit, with the one exception of the "about 8 hours deficit" around T.A. 2060 (though his statement about the deficit around T.A. 2360 is still a little ambiguous to me). I also now believe any miscalculations Tolkien presented in Appendix D might be explained by him using a shortcut of calculating the deficit for Second Age years instead of S.A. 3441 plus Third Age years. So if Åke Jönsson (Bertenstam) asks "Did Tolkien Reckon Correct?", I can now answer with confidence: "Mostly." P.S. My interpretation of the Kings' and Stewards' Reckoning rules presented in Appendix D is also what I relied on to show how Yestarë of the Calendar of Imladris could have fallen on a hypothetical Gregorian calendar's March 25 in T.A. 3019.
(This post was edited by Murlo on Mar 25 2018, 8:54pm)



Otakusempai
Immortal
Mar 25 2018, 11:13pm
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So what are we to make of the April 6 meeting of Celeborn and Thranduil on the day of the New Year of the Elves? My impression is that this is also meant to represent the first day of Spring in the Calendar of Imladris. And, no, I have never seen the Beyond Bree newsletter.
"I may be on the side of the angels, but do not think for one second that I am one of them."  Sherlock



Murlo
Rivendell
Mar 26 2018, 3:23am
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April 6 by Shire Reckoning, but March 25 in our Gregorian calendar
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So what are we to make of the April 6 meeting of Celeborn and Thranduil on the day of the New Year of the Elves? My impression is that this is also meant to represent the first day of Spring in the Calendar of Imladris. Ah, sorry about that. I see now that the opening of my post was fairly ambiguous. First, you are correct that the Elves' season of "Spring" immediately follows their New Year's Day (Yestarë). Second, I recognize that the Appendix B listing of the meeting of Celeborn and Thranduil on the day of the New Year of the Elves on April 6 (i.e. Astron 6) is canon, but note that this date is 1419 Astron 6 by Shire Reckoning. I'll also point out that this is the only specific date given for the Elves' New Year, but because of the different leapyear rules between these calendars, the Elves' Yestarë would not always fall on Astron 6 in every year. My essay "Why We Should Observe the Elves' New Year on March 25" is referring to the March 25 in our modern Gregorian calendar. Usually we would align Shire New Year's Day with our December 21 (so that Midyear's Day falls on the usual date of the summer solstice, June 21). In that alignment Shire Astron 6 would usually fall on our March 27 (in nonleapyears), but in this essay, and in my Shire Reckoning project's notes, I explain how Astron 6 and the Elves' New Year in T.A. 3019 could have fallen on a hypothetical Gregorian calendar's March 25 (and the calculations I use to show that are the same used in my discussion of the Deficit above).



Otakusempai
Immortal
Mar 26 2018, 3:45am
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So you're stating that our March 25 would equate to 6 Astron (S.R.). That's a couple days off of my figures, but then, my calculations don't quite match Tolkien's either.
"I may be on the side of the angels, but do not think for one second that I am one of them."  Sherlock



Murlo
Rivendell
Mar 26 2018, 2:39pm
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The calendars drift throughout the millennia
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That's right, but only around the end of the Third Age, and only if we start with Tolkien's rules and Gregorian alignment in S.A. 1. Back then (proleptic) Astron 6 would have fallen on March 28~29. Since the different leapyear rules cause Shire and Gondor dates to drift relative to our Gregorian calendar, then by the end of the Third Age, Astron 6 would end up on March 25~26, then after S.R. 1800 it would align with March 24~25, etc.
(This post was edited by Murlo on Mar 26 2018, 2:41pm)



Murlo
Rivendell
Mar 26 2018, 2:59pm
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As an aside, I'll note that my deficit calculations match Aaron Chong's, but only up to T.A. 2059 (the rest appear to be hypothetical calculations if the Kings' Reckoning continued beyond that year). I was only referring to Chong's list of years with deficit figures in the statement above, but I notice now that he also discusses the surplus of "15:54:34" for the end of T.A. 2060 and the deficit of "20:21:26" for the start of T.A. 3021, which closely match my own figures.



Otakusempai
Immortal
Mar 26 2018, 3:01pm
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That's right, but only around the end of the Third Age, and only if we start with Tolkien's rules and Gregorian alignment in S.A. 1. Back then (proleptic) Astron 6 would have fallen on March 28~29. Since the different leapyear rules cause Shire and Gondor dates to drift relative to our Gregorian calendar, then by the end of the Third Age, Astron 6 would end up on March 25~26, then after S.R. 1800 it would align with March 24~25, etc. My own figures, based on Tolkien's general notes in Appendix D, ideally place 6 Astron on our March 27. However, that is without determining the margin of error caused by your drift. And my results would be different in a leap year.
"I may be on the side of the angels, but do not think for one second that I am one of them."  Sherlock



Murlo
Rivendell
Mar 26 2018, 6:10pm
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And I agree, in general, for the Shire Calendar in modern times
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As I mentioned before, I agree we should align Shire New Year's Day with our December 21 so that Midyear's Day falls on the usual date of the modern summer solstice, June 21; and in that alignment Shire Astron 6 would usually fall on our March 27 (in nonleapyears). I also try to address this in my concluding statements to this essay: I think that observing the Shire New Year's Day on December 21st is more seasonally appropriate in modern times, since it keeps Midyear's Day in sync with the modern summer solstice as it was originally intended, whereas others may consider it more traditionally appropriate to observe Shire New Year's Day on December 23, keeping Tolkien's alignment of our Gregorian January 1 with Shire Afteryule 9. Likewise, some may wish to always observe the Elves' New Year's Day (Yestarë) on Shire Astron 6 since that's the traditional date as observed by the Hobbits in the Fourth Age (as stated by Tolkien at the end of Appendix D). I think aligning Rivendell's Yestarë with our Gregorian March 25 in modern times can be considered traditionally appropriate from the perspective of the Elves at the end of the Third Age (the only point in time Tolkien made a correspondence with this calendar and any other); but in order to observe what I believe is Tolkien's "hidden" alignment of Rivendell's Yestarë = Gregorian March 25, we have to disconnect the alignment of Rivendell's Yestarë = Astron 6 in modern times. I'm OK with that since I'm doing something similar when I opt to disconnect Tolkien's alignment of January 1 = Afteryule 9 in order to observe a better seasonal fit of the Shire Calendar in modern times.



Murlo
Rivendell
Mar 27 2018, 3:48pm
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Minor update and giving credit where it's due
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FYI: I pushed a few minor updates to my Shire Reckoning project's Deficit notes today, including credit to Andreas Möhn (a.k.a. CodexRegius of "Lalaith’s Middleearth Science Pages") for inspiring this post. I was skeptical of his analysis of Tolkien's Deficit calculations, which inspired me to finally take a look at them myself.



Murlo
Rivendell
Apr 7 2018, 12:40am
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More Deficit notes updated on April 6
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If anyone is still following this post, then FYI: I just posted more updates on this topic to my Shire Reckoning project's Deficit notes, which now include some revisions I made for my "Beyond Bree" submission. For example, I now state what my conclusions are before I go into the details on how I arrived at those conclusions :)
I now think Tolkien attempted to use deficit calculations for S.A. 5501, 5801, and 6462 when discussing the deficit for T.A. 2060, 2360, and 3021.



Murlo
Rivendell
Apr 22 2018, 4:54am
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The Kings' Reckoning Rules and the Deficit
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Today I've updated my Shire Reckoning project with a page dedicated to The Kings' Reckoning Rules and the Deficit. That page is an expansion of the first post of this thread, where I still present what I think is the simplest explanation for Tolkien's Deficit figures that have confused just about everyone who has tried to figure out how he got the results he published in Appendix D, but I also provide all the details of how to work out the math of the Deficit so anyone else can check these results for themselves. It turns out that Beyond Bree would actually prefer less detail, so I'll be submitting just a summary of these ideas for publication in that newsletter Those not interested in the Rules or how to work out the math of the Deficit can jump directly to my discussion of these results, which is mostly the same as the first post of this thread, except I've made a few corrections, and I feel some of my explanations are more straight forward now. I think I've really broken down the figures and discussion presented in Jönsson's article and the Reader's Companion, so if anyone else felt a little lost in that part of Appendix D in the Reader's Companion, I hope my notes can help make more sense of them now. I was never all that interested in Tolkien's Deficit figures before, and I think part of the reason is that whenever anyone discussed these figures, they never discussed how they got their results. Usually they just throw figures at you, and tell you that one deficit value is correct and another is not. Even after I became very familiar with Appendix D for my Shire Reckoning project, the figures and discussion given in the Reader's Companion (or other sources online) for Tolkien's Deficit was not obvious to me. I am not mathaverse (I took advanced math courses for my college degree), but I never had any reason to figure out how to calculate the deficit on my own. Now that I've figured out the rules of the Kings' and Stewards' Reckoning correctly for my calendar simulations, it turned out to be pretty simple to use those rules to figure out how to calculate deficit results. So I hope my notes are interesting or useful for those who are not putoff by math, but who are also not experts in calendar systems, and they can now work out these figures for themselves, or at least more easily follow along with other articles that discuss Tolkien's Deficit figures.




