Abstract
I argue that the chronology of the Priestly (P) Flood narrative encodes the replacement of an "old" 30-day-month-based calendar with years of 360, 390 or (every 60 years) 405 days and mean year length 365.25 days by a "new" week-based calendar with years of 364 or 371 days and mean year length 365.24 days, both calendars having been designed by the author within the P source who composed the Creation account of Gen ch. 1, the genealogy of Gen ch. 5, and the P Flood narrative, whom I call P_chron. The encoded calendar replacement reflected the proposal, by P_chron, to enact the week-based sabbatical calendar in Persian Judah in replacement of the lunisolar calendar, in order to favor the replacement of the observation of the full-moon holiday with that of the weekly Sabbath. The sabbatical calendar - in which the year and each quarter begin on Sunday, in line with Genesis ch. 1 - is kept in sync with the solar year by intercalating an extra week in two independent cycles of 7 years (sabbatical year) and 25 years (mid-Jubilee and Jubilee years), resulting in an intercalation cycle of 7 · 25 = 175 years (same as Abraham's lifespan), with the last year having only one extra week. The P Flood narrative assumes that the "old" calendar has been used by the descendants of Seth since Creation up to, and including, the year of the Flood (1656 AM & Noah's 600th year), and that the enactment of the new calendar at the beginning of Noah's 601st year, consistently with P_chron's view of the Flood as a new creation, required a 10-day date advancement, just as the enactment of the Gregorian calendar did in 1582. Collaterally, the 30-day-month-based calendar is the key to understand the 1290 and 1335 days of Dan 12:11-12, and the reason why Lamechs' fathering age was changed to 182 in the proto-masoretic text.
Introduction
The Astronomical Book, chapters 72-82 of 1 Enoch (dated to the 3rd century BC or earlier), the Book of Jubilees (dated to the middle of the 2nd century BC), and about 20 non-biblical scrolls found at Qumran are witnesses to a 364-day week-based calendar that the authors of those texts regarded as normative, holding also, in the cases of 1 Enoch and Jubilees, that it faithfully reflected the God-given law of the movement on heavenly luminaries. Moreover, 1 Enoch ch. 80 implied that any discrepency between the 364-day calendar and observation was due to sin, either of man or of the angels to whom God had given the task of guiding the heavenly luminaries. Consistently with the belief in the divinely-decreed nature of the 364-day year, none of those texts mentioned an intercalation system to keep the calendar aligned with the seasons. In that calendar, the year was divided in quarters of 91 days each, in turn comprised of three months of 30, 30 and 31 days. In the Book of Jubilees implicitely [1] and in the Qumran scrolls explicitely - but not in 1 Enoch - the first day of the year and of each quarter was Wednesday.
The hypothesis that a 364-day week-based calendar was used in Judaism since long before the time of composition of 1 Enoch was put forward first by Annie Jaubert in 1953 and defended by James C. Vanderkam in 1979 [2]. Since then a number of scholars have adopted that position, of whom a list can be found in a 2009 paper by Ron H Feldman [3] where he argues that both the weekly Sabbath and the 364-day calendar were introduced simultaneously and sinergistically during the early Persion period. Among those scholars, Philippe Guillaume stands out by his study on the chronology of the Flood narrative in the context of, and as support to, the introduction of the 364-day calendar, and by his inference of its intercalation system from biblical passages [4]. In this study I propose a thesis that was inspired by the work of Guillaume but differs from it in several key points.
Summary of the thesis
I will refer as P_chron to the author within the Priestly (P) source who composed the Creation account in Gen ch. 1, the genealogy from Adam to Noah in Gen ch. 5, and the P Flood narrative. He writes primarily for returnees from the Babylonian Exile and in his theology the Flood, the Exodus, and the return from Exile are new creations of progressively smaller demographic and geographical scope, with the narratives of Creation, the Flood and the Exodus aimed to illuminate the life of the returnees:
- Flood: new creation for mankind: all the peoples on the whole Earth;
- Exodus: new creation for Israel: all 12 tribes on the whole Promised Land;
- Return from Exile: new creation for the remnant of Israel: Judeans, priests and Levites in Judah.
1. P_chron worked around 450 BC in the time and circle of Ezra and Nehemiah, and might have been Ezra himself. He was familiar with the current state of knowledge of Babylonian mathematical astronomy, in particular with the length of the solar year as it was computed at that time. He held that the actual solar year length of 365 days and a fraction, as opposed to 364 days, was due exclusively to a divine decree and not to some human or angelic sin, an opinion that might have already been held by some in his circle, long before it was recorded in 1 Enoch. He stated that position symbolically by writing twice that Enoch, who had lived 365 years, "walked with God".
2. P_chron devised a 364/371-day week-based calendar that, if enacted in the Persian province of Judah (Yehud Medinata) in replacement of the predominant lunisolar calendar, would facilitate the observance of the weekly Sabbath and the Sabbatical and Jubilee years and would be fit for agriculture. For the latter purpose, the intercalation system was devised so that only one extra week would be added to any year.
3. The previous objective was achieved by adding an extra week according to two cycles that run independently from each other, i.e. every M and every N years counting from the year of Creation, with the last year of each composite cycle of M · N years having only one extra week.
4. In the 364-day calendar devised by P_chron, the first day of the year and of each quarter was Sunday, consistently with the first day in the Creation narrative in Genesis ch. 1, also composed by P_chron. I show that this scheme is consistent not only with observance of Sabbath rest in all biblical events, but also with P_chron's view of the Exodus as a new Creation, thus refuting Annie Jaubert's defense of the Qumranites' adoption of Wednesday as New Year day.
5. Since P_chron viewed the Flood as a type of the return from the Babylonian Exile, and since he wanted a calendar replacement to take place in the province of Judah, he encoded a calendar replacement in the Flood narrative, both in the dates of events and in the partition of Noah's lifespan by the Flood. The "old" calendar to be replaced in the Flood narrative was not the 354/384-day lunisolar calendar to be replaced in the real world, a calendar which P_chron might have considered unworthy of having been followed by the Sethian line, as it was both linked to idolatry and a direct obstacle to P_chron's intention of replacing the observation of the full-moon holiday, named from the Accadian word sapattu [5] [6], with that of the weekly Sabbath. Rather, it was a theoretical 30-day-month-based calendar with a month-based intercalation system (also devised by P_chron) to keep it in line with the solar year (system whose knowledge survived P_chron and resurfaced in Dan 12:11-12). The chronology of the P Flood narrative assumes that the old calendar had been observed by the Sethian line since Creation up to, and including, the year of the Flood (1656 AM, Noah's 600th), and that the new calendar became effective at the beginning of Noah's 601st year, consistently with P_chron's view of the Flood as a new creation. Since 1656 years since Creation computed according to the old calendar comprised 10 more days than 1656 years computed according to the new calendar, a 10-day date advancement occurred when the new calendar was enacted at the beginning of Noah's 601st year, just as a 10-day date advancement occurred when the Gregorian calendar was enacted in 1582. Thus, whereas the date Noah.600.2.17 of the day when the Flood began was acccording to the old calendar, the date Noah.601.2.27 of the day when the earth was dry was according to the new calendar, so that it was Noah.601.2.17 in the old calendar.
Each of the above 5 points will be developed below. To note, we do not know whether P_chron's circle had any degree of success in their endeavor to have the 364-day calendar enacted in the province of Judah. Even if the calendar was enacted then, the province might have reverted to the lunisolar calendar under the Ptolemaic administration in the III century BC, so that knowledge of the intercalation system devised by P_chron faded over time to the point that it had already disappeared by the time of composition of 1 Enoch.
1. Year length reckoning in Babylonian astronomy ca. 460 BC
The 19-year cycle of leap-year intercalations of an additional lunar month at fixed intervals was implemented in year 10 of the reign of Xerxes I (486–465 BC), i.e in 476/5 BC [7], implying that by that time Babylonian astronomy had already discovered the "metonic" cycle according to which 235 mean synodic months = 19 mean solar years. (To note, Babylonan astronomy did not distinguish between sidereal and tropical solar years.) Therefore the implicit mean length of the solar year in Babylonian astronomy was 235 · mean synodic month length / 19.
The best source on the evolution of the epistemic state of Babylonian astronomy regarding the length of the mean synodic month is Britton (2007) [8], section "5. Mean Synodic Month" in pp. 116ff. It lists, in apparent chronological order, five values for the mean synodic month, m, which are attested or implied in Babylonian lunar texts.
- The second value, 29;31,50,18,..., results from the approximation to the length of the Saros cycle known by the middle of the 6th century.
- The third value, 29;31,50,12,..., results from the corrected length of the Saros cycle according to the procedure in tablet BM 45861 studied in Steele (2002) [9].
- The fourth value, 29;31,50,6, is that of System A. Although the computational system itself was developed ca. 400 BC, the value it uses for the length of the mean synodic month could have arisen earlier from taking the average of the maximum and minimum lengths of the Saros cycle (pp. 117-118), which had been bounded by respective pairs of eclipses occurring in −594 and −576 with an interval of six months (p. 111). From the expression of the Saros cycle using this average length, 223 m = 6585;19,10 d, we obtain a mean synodic month length of 29.53058 = 29;31,50,5,23.
Taking either 29;31,50,5,23 or 29;31,50,6 and using the relation of the metonic cycle we obtain a mean solar year length of 365.2467 = 6,5;14,48, which is the length computed by Hipparchus.
Thus, according to our current state of knowledge of the state of knowledge of Babylonian astronomy ca. 460 BC, the most probable value of the reckoned length of the mean solar year at that time is 365.2467 = 6,5;14,48, whose difference to 365.25 is half its difference to 364.24. Therefore if someone living at that time whose only available knowledge was that of then-current Babylonian astronomy had to choose between 365.25 and 365.24 as the mean year length of a new calendar which should track the equinox, he should choose 365.25.
2. Real equinoctial solar year ca. 500 BC
The situation changes completely if we move from the epistemic to the physical plane, i.e. if we take as reference the real mean length of the equinoctial year ca. 500 BC and compare it with both 365.24 and the length which is effectively assumed by using the lunisolar calendar of metonic cycle and the real length of the mean synodic month ca. 500 BC.
Mean length of the equinoctial year |
Value in decimal and sexagesimal notations |
assumed by the 364-/371-day "sabbatical" calendar developed ca. 460 BC by the author of the chronologies of the P source |
365.24 = 6,5;14,24 |
Difference real - sabbatical = |
0.002252 = 0,0;0,8,6,26 |
real ca. 500 BC [10] |
365.242252 = 6,5;14,32,6,26 |
Difference lunisolar - real = |
0.004593 = 0;0,16,32,5 |
effectively assumed by using the lunisolar calendar of metonic cycle (235 m = 19 a) and the real length of the mean synodic month ca. 500 BC [11]: a = 235 · 29.530596 / 19 |
365.246845 = 6,5;14,48,38,31 |
Now the difference between the real mean length of the equinoctial year ca. 500 BC and 365.24 is half the difference between the real mean length and that effectively assumed by using the lunisolar calendar of metonic cycle ca. 500 BC. Therefore, if someone living at that time who had accurate knowledge or the real mean length of the equinoctial year and whose sole criterion for evaluating calendars were the closeness of its tracking of the equinox had to choose between a calendar with mean year length of 365.24 days and the lunisolar calendar, he should choose the former. (Not to mention if the choice were between the sabbatical and a calendar with mean year length 365.25, whose difference to the real mean year length is 3.5 times that of 365.24.)
If in the second half of the first millennium BC the count of years in the sabbatical calendar described below begins with a year whose first day coincides with the day of the equinox, such as 459 BC, the deviation of the first day of any year of the sabbatical calendar with respect to the equinox within a 350-year interval is within the range (-10, 6), with the first case occurring in years 20, 195, and 223 [12]. This means that the equinox always occurs before or on 1.11, thus fulfilling the "rule of the equinox" according to which the equinox must occur before or on the 14th of the first month. Notably, if day 1 of year 1 (or 3501) was the equinox of 459 BC, Sunday (Julian) March 27 in Babylon and also in Jerusalem in Jewish day reckoning, Julian Day 1553858.5, then day 1 of year 351 (or 3851) was 109 BC, Sunday (Julian) March 23, Julian Day 1681692.5 (1553858 + 350 * 364 + 62 * 7 = 1681692), on which the equinox occurred at 17:39 Jerusalem time, just half an hour before sunset [13] [14].
3. Study of intercalation in each calendar
For each calendar, we determine first the period of a basic intercalation cycle that makes its mean year length 365 days (except in the case of the Zoroastrian calendar where it is already that) and then the period of an additional independently-running intercalation cycle that takes its mean year length to a value close to that of the solar year, defining in the sabbatical calendar that only one extra week is added to the last year of the composite cycle.
3.a. 360-/390-/405-day month-intercalating calendar (A)
As shown in Brack-Bernsen (2007) [15], a 360-day calendar comprised of 12 30-day months was used in Mesopotamia for administrative purposes since the early dynastic time ca. 2600 BC until Ur III times ca. 2100 BC. Then it was used in the training of scribes in an unbroken tradition until it appeared in astronomical tables in the astronomical compendium MUL.APIN, composed sometime before 700 BC, as an "ideal" year. Its use as an "ideal" year for astronomical purposes is documented until ca. 300 BC. Of course, the intercalation system described below was devised either by P_chron or by a Babylonian astronomer ca. 450 BC at his request for the purpose of its use in the biblical text. As shown in Appendix A, this calendar with this intercalation system is also the calendar assumed in Daniel's prophecy of the 70 weeks of years (Dan 9:27 & 12:11).
Basic intercalation: add 1 month every M years to make mean year length (myl) = 365.
myl = [360 · (M - 1) + 390] / M = 360 + 30 / M = 365
30 / M = 5
M = 30 / 5 = 6
Additional intercalation: since 365.25 is 6,5;15 in sexagesimal notation, it is immediately evident for anyone using a base-60 numerical system that a mean year length of 6,5;15 days can be achieved by simply adding 15 additional days every 60 years. Making that choice, the 60-year composite intercalation cycle comprises:
- 50 regular 360-day years, totalling 18000 days,
- 9 leap 390-day years, totalling 3510 days, and
- 1 extra-leap 405-day year (the 60th)
for a total of 21915 days and mean year length = 21915 / 60 = 365.25 days.
3.b. Intercalated calendars based on the 365-day calendar
These calendars are based on the Zoroastrian aka Young Avestan calendar adopted for religious purposes in the Persian empire during the reign of Xerxes I (486-465 BC) [16] [17], which was in turn probably based on the Egyptian calendar. Since the year length of this calendar is already 365 days, the intercalation system to be added needs only to increase the mean year length by 0.25 days, which in principle can be done in any of the following 4 ways:
- adding an additional 30-day month every 120 years (Zoroastrian calendar since ca. 900 CE),
- adding 15 additional days every 60 years,
- adding 5 additional days every 20 years, or
- adding an additional day every 4 years (Julian-like calendar with 30-day months).
3.b.1. 365-/395-day month-intercalating calendar (B)
This intercalation system was implemented ca. 900 CE at the request of the Caliph by the Zoroastrian clergy, who invented the legend that such intercalation had actually been carried out since the beginning of the Zoroastrian calendar. To note, the earliest extant and datable reference to an intercalated 13th month in that calendar is by the historian al-Masudi in his book "Meadows of Gold" (Murūj aḏ-Ḏahab), a first version of which was completed in 947 CE [16]. The 120-year intercalation cycle comprises:
- 119 regular 365-day years, totalling 43435 days, plus
- 1 leap 395-day year (the 120th)
for a total of 43830 days and mean year length = 43830 / 120 = 365.25 days.
3.b.2. 365-/380-day 15-day-intercalating calendar (C)
Of the 4 intercalated calendars based on a 365-day year, this has the highest probability of being the "old" calendar to be replaced in the Flood narrative, since the length of its intercalation cycle, 60 years, is the one hinted by the partition of Noah's lifespan by the Flood. The 60-year intercalation cycle comprises:
- 59 regular 365-day years, totalling 21535 days, plus
- 1 leap 380-day year (the 60th)
for a total of 21915 days and mean year length = 21915 / 60 = 365.25 days.
3.b.3. 365-/370-day 5-day-intercalating calendar (D)
The 20-year intercalation cycle comprises:
- 19 regular 365-day years, totalling 6935 days, plus
- 1 leap 370-day year (the 20th)
for a total of 7305 days and mean year length = 7305 / 20 = 365.25 days.
3.b.4. 365-/366-day Julian-like calendar with 30-day months (J)
The 4-year intercalation cycle comprises:
- 3 regular 365-day years, totalling 1095 days, plus
- 1 leap 366-day year (the 4th)
for a total of 1461 days and mean year length = 1461 / 4 = 365.25 days.
3.c. 364-/371-day week-intercalating calendar (W)
Basic intercalation: add 1 week every M years to make mean year length (myl) = 365.
myl = [364 · (M - 1) + 371] / M = 364 + 7 / M = 365
7 / M = 1
M = 7 (sabbath year (shmita))
Additional intercalation: add 1 week every N years, with only 1 week added to year 7N.
In each cycle of 7N years:
No of 371-day years = 7N / 7 + 7N / N - 1 = N + 7 - 1 = N + 6
myl = [364 · 7N + 7 · (N + 6)] / (7N)
myl = [365 · 7N + 42] / (7N)
myl = 365 + 6 / N
N - 6 / N
24 - 0.25 * Accuracy of Julian calendar. Composite cycle of 24 · 7 = 168 years.
25 - 0.24 * Close to 0.2423. Composite cycle of 25 · 7 = 175 years.
Clearly a sabbatical calendar with a 25-year second intercalation cycle is the one most compatible with the biblical text, since its leap years are the sabbatical, Jubilee and mid-Jubilee years. Note that the Jubilee year does not reset the sabbatical year cycle. This is in line with Leviticus 25:8-13, which states only the time relationship between the first seventh sabbatical year and the first Jubilee year, after which both sabbatical and Jubilee years are assumed to keep their respective 7-year and 50-year cycles.
The 175-year composite intercalation cycle comprises:
- 144 regular 52-week years, totalling 7488 weeks and 52416 days, and
- 31 leap 53-week years, totalling 1643 weeks and 11501 days
for a total of 63917 days and mean year length = 63917 / 175 = 365.24 days.
Recalling that patriarchal lifespans come from P_chron, I will show that he provided clear clues pointing to calendars A and C in two very important lifespans, Noah's and Abraham's.
Regarding Noah, the Flood partitions his lifespan in two intervals whose durations point clearly to the calendar in effect during the respective interval, i.e.:
- 600 years is the product of the intercalation cycle length of 60 years of Calendar A times 10, the number of generations up to, and including, Noah's that had been on Earth up to the Flood.
- 350 years is the product of the intercalation cycle length of 175 years of Calendar C times 2, the number of generations from, and including, Noah's that were on Earth at the end of the Flood.
Regarding Abraham, his lifespan of 175 years is the intercalation cycle length of Calendar C, while the number of regular years in that intercalation cycle is 144, the previous item in the sequence of integers of form (m · m · n) where (m + m + n = 17) that includes the lifespans of Abraham, Isaac and Jacob in that order:
4 · 4 · 9 = 144. Number of regular years in a composite intercalation cycle.
5 · 5 · 7 = 175. Number of years in a composite intercalation cycle and Abraham's lifespan.
6 · 6 · 5 = 180. Isaac's lifespan.
7 · 7 · 3 = 147. Jacob's lifespan.
4. Consistency of a 364-day, Sunday-beginning calendar year with the biblical text
As noted in the introduction, in the Book of Jubilees implicitely [1] and in the Qumran scrolls explicitely - but not in 1 Enoch - the first day of the year and of each quarter in the 364-day calendar was Wednesday.
This feature is usually attempted to be justified by arguing that, since the year is determined by the movement of the sun (as the moon plays no role in the 364-day calendar) and the sun was created on the 4th day (Gen 1:14-19), then that was the first day of the year. Thus stated, this argument misses the critical point that the first day of the first year in the 364-day calendar is not determined arbitrarily but is the day of the spring equinox (Ex 12:2). (The first day of subsequent years will deviate from that coincidence, deviation that would be corrected periodically by intercalating an additional week.) Therefore, the correct version of this argument is that God created the sun on the 4th day in the position corresponding to the first day of the year, i.e. the position of the spring equinox. Thus stated, it is clear that this is a wholly arbitrary assumption, as God could have created the sun in the position corresponding to any day of the year.
An alternative attempt at justifying this feature is by noting that day counting by man became possible only when the heavenly lights were created, i.e. on the 4th day. This is simply idiotic because day counting by man became possible only when man was created, i.e. on the 6th day. Which is in addition to the fact that the day when day counting became possible has no relevance whatsoever on the issue of the first day of the year, which is determined exclusively by the position of the sun.
Keeping in mind that the first day of the first year in the 364-day calendar is not determined arbitrarily but is the day of the spring equinox, the scenarios implied by each flavor of the 364-day calendar, namely the "year starts on a Sunday" and the "year starts on a Wednesday" flavor, are, respectively:
A. God created the sun on the 4th day in the position corresponding to the 4th day of the year, i.e. as if the spring equinox had occurred on the 1st day. Then, when Adam and Eve were created on the 6th day, they learned that that day was the 6th day of creation, the 6th day of the week, and the 6th day of the year.
B. God created the sun on the 4th day in the position corresponding to the 1st day of the year, i.e. in the position of the spring equinox. Then, when Adam and Eve were created on the 6th day, they learned that that day was the 6th day of creation, the 6th day of the week, and the 3rd day of the year.
Now, why on heaven would God have acted according to scenario B? Perhaps creating the sun in the position corresponding to the 4th day of the year was too complicated for Him? Perhaps He just wanted to make calendrics more complicated for Adam, Eve, and their descendants? Clearly scenario B does not fit God's infinite wisdom and goodness.
4.1. Verification of compliance with Sabbath rest in all bibilical events dated on Sabbath
The only valid motive for choosing a particular day of the week as New Year day is to achieve by this choice the compliance with Sabbath rest in biblical events dated on Sabbath. In order to check that, I will use the list of events in Beckwith 2005 pp. 57-65 [18]. Since the days in that list are based on a scheme where the year starts on Wednesday, which we will call We.1, we need to convert it to a scheme where the year starts on Sunday, which we will call Su.1.
Day #: 1 - 2 - 3 - 4 - 5 - 6 - 7
We.1: We, Th, Fr, Sa, Su, Mo, Tu
Su.1: Su, Mo, Tu, We, Th, Fr, Sa
Therefore, in order to study the compliance with Sabbath rest in scheme Su.1, we need to study the events falling on Tuesday in scheme We.1, keeping in mind that labors forbidden in Sabbath do NOT include those commanded by the LORD, either by his word or by fact such as birth labor. Dates are denoted as month.day, m.d, or as day number since the start of the year, d.n.
A. Sabbath 1.14: Slaughtering of the Passover lamb at twilight
Ex 12:6 You shall keep it [the lamb] until the fourteenth day of the same month, then the whole assembly of the congregation of Israel is to slaughter it at twilight (“between the two evenings”).
Lev 23:5 In the first month, on the fourteenth day of the month at twilight is the LORD’S Passover.
Num 9:2-5 “Now the sons of Israel are to celebrate the Passover at its appointed time. On the fourteenth day of this month, at twilight, you shall celebrate it at its appointed time; you shall celebrate it in accordance with all its statutes and all its ordinances.” So Moses told the sons of Israel to celebrate the Passover. And they celebrated the Passover in the first month, on the fourteenth day of the month, at twilight, in the wilderness of Sinai; in accordance with everything that the LORD had commanded Moses, so the sons of Israel did.
Num 28:16 “On the fourteenth day of the first month is the LORD’s Passover,
Josh 5:10 While the sons of Israel camped at Gilgal they celebrated the Passover on the evening of the fourteenth day of the month on the desert plains of Jericho.
2 Chro 35:1 Then Josiah celebrated the Passover to the LORD in Jerusalem, and they slaughtered the Passover animals on the fourteenth day of the first month.
Eze 45:21 “In the first month, on the fourteenth day of the month, you shall have the Passover, a feast of seven days; unleavened bread shall be eaten.
Ezra 6:19 The exiles held the Passover on the fourteenth of the first month.
OK, as the task was commanded directly by the LORD.
B. Sabbath 1.21: Holy assembly on the seventh day of the Feast of Unleavened Bread
Ex 12:16 And on the first day [1.15, Sunday] you shall have a holy assembly, and another holy assembly on the seventh day [1.21, Sabbath]; no work at all shall be done on them, except for what must be eaten by every person—that alone may be prepared by you.
Ex 12:18 In the first month, on the fourteenth day of the month at evening, you shall eat unleavened bread, until the twenty-first day of the month at evening.
Ex 13:6 For seven days you shall eat unleavened bread, and on the seventh day there shall be a feast to the LORD.
Lev 23:8 But for seven days you shall present an offering by fire to the LORD. On the seventh day is a holy convocation; you shall not do any laborious work.
Num 28:25 On the seventh day you shall have a holy assembly; you shall do no laborious work.
2 Chro 8:13
OK, as the holy assembly was commanded directly by the LORD, and moreover its compliance with Sabbath rest is explicit in Lev 23:8 and Num 28:25.
C. The word of the LORD comes to a prophet on a Sabbath.
Eze 30:20 In the eleventh year, in the first month, on the seventh of the month, the word of the LORD came to me, saying,
Hag 2.1 On the twenty-first of the seventh month, the word of the LORD came by Haggai the prophet, saying,
Hag 2.10 On the twenty-fourth of the ninth month, in the second year of Darius, the word of the LORD came to Haggai the prophet, saying,
Hag 2.18 ‘Do consider from this day onward, from the twenty-fourth day of the ninth month; from the day when the temple of the LORD was founded, consider:
Hag 2.20 Then the word of the LORD came a second time to Haggai on the twenty-fourth day of the month, saying
OK, as listening to the word of the LORD is most fitting on a Sabbath. Moreover, if the 364-day calendar year starts on a Sunday then the Israelites listened to God in Sinai on a Sabbath.
"In the third month after the Israelites went out from the land of Egypt, on the very day, they came to the desert of Sinai" (Ex 19:1)
=> 3.01 = d.61, Thursday.
The Lord said to Moses, “Go to the people and sanctify them today and tomorrow, and make them wash their clothes and be ready for the third day, for on the third day the Lord will come down on Mount Sinai in the sight of all the people." (Ex 19:10-11)
=> "today" 3.01 = d.61, Thursday and "tomorrow" 3.02 = d.62, Friday the people wash their clothes, and "on the third day" 3.03 = d.63, Saturday the people listen to God.
D. A sacrifice is commanded to be offered on a Sabbath.
Eze 45:20 And you shall do this on the seventh day of the month for everyone who does wrong inadvertently or is naive; so you shall make atonement for the house.
OK, as offering a sacrifice to the LORD (describe in vv. 18-19) is most fitting on a Sabbath.
E. Work on the construction of the Second Temple is performed on a Sabbath.
Hag 1:14-15 So the LORD stirred up the spirit of Zerubbabel the son of Shealtiel, governor of Judah, and the spirit of Joshua the son of Jehozadak, the high priest, and the spirit of all the remnant of the people; and they came and worked on the house of the LORD of armies, their God, on the twenty-fourth day of the sixth month in the second year of Darius the king.
Ezra 6:15 Now this temple was completed on the third day of the month Adar; it was the sixth year of the reign of King Darius.
OK because of either or both of the following:
1. The work forbidden in Sabbath is that performed for economic profit, not that performed for the LORD. If offering sacrifices to the LORD is fit on a Sabbath, so is working on the house of LORD.
2. The narrative and dates of the books of Haggai and Ezra are fully historical, not allegorical, and at the times of Haggai's prophecy (520 BC) and of the completion of the second Temple (515 BC) the Israelites were still using the lunisolar calendar, as the sabbatical 364-day calendar was devised in the time of Ezra and Nehemiah.
We have thus verified that the 364-day calendar with Sunday as New Year day (scheme Su.1) is consistent with the observance of the Sabbath rest in all biblical events listed in [18].
4.2. Verification of compliance with Sabbath rest in the books of Esther and Tobit
Since according to [18] p. 65 no event in the book of Esther occurs on Tuesday in scheme We.1, no event occurs on Saturday in scheme Su.1. Thus there is no need to remove that book from the biblical canon for violating the Sabbath rest, as the Qumranites apparently did.
Finally, the day when Tobit's wife Anna worked is not a Sabbath in scheme Su.1.
Tobit 2:12 When she delivered the material to her employers, they would pay her a wage. On the seventh day of the month of Dystrus, she finished the woven cloth and delivered it to her employers. They paid her the full salary and also gave her a young goat for a meal.
Given that 91 · 3 + 30 = 303
If Dystrus is the 11th month, then 11.7 = d.(303+7) = d.310, which is not a Sabbath.
If Dystrus is the 12th month, then 12.7 = d.(333+7) = d.340, which is not a Sabbath.
4.3. Bonus: the water of the Jordan was divided on Tuesday
An additional consequence of having the 364-day calendar year start on Sunday is that the day when God divided the waters of the Jordan (Josh 3:17), turning its bottom into dry land in order to allow the Israelites to cross it on foot, namely 1.10 (Josh 4:19), was Tuesday, the same day of the Creation week when God gathered the waters to let dry land appear (Gen 1:9-10).
This week day coincidence is consistent with P_chron's view of the Flood, the Exodus, and the return from the Exile as successive new creations, and it occurs also with the day when the tops of the mountains became visible after the Flood, as we will see in section 5.4.
5. Interpretation of the chronology of the P Flood narrative as encoding a calendar replacement with date adjustment
5.1. Conventions and recalling of a basic notion
Dates will be stated as year.month.day, where year is AM if only the year number is stated or the nth year in the life of a person if the year is stated as name.n. Thus, the Flood began on 1656.2.17 = Noah.600.2.17. Also, month.day may be replaced by d.nnn, with nnn being the day number within the year, so the previous date is also 1656.d.047 = Noah.600.d.047. The first year after Creation is year 1. People are 1 year old (yo) in the 2nd year of their lives and in general N years old in the (N+1)th year of their lives.
5.2. Basic chronological framework
1.1.6: Adam was created.
1 AM (after d.006) = Adam's 1st year, when he was 0 yo.
2.1.6: Adam turned 1 yo.
2 AM (after d.006) = Adam's 2nd year, when he was 1 yo.
...
131.1.6: Adam turned 130 yo. At this moment, he had lived 130 years.
When Adam was 130 yo he begot Seth. So Seth was born in the year 131 AM.
Let us assume that it was precisely on Adam's birthday, 131.1.6. Thus:
131 AM (after d.006) = Seth's 1st year, when he was 0 yo.
132.1.6: Seth turned 1 yo.
Thus, if the sum of the begetting ages of the previous patriarchs is n, a patriarch was born, at the earliest, in the year (n + 1) AM. Thus, assuming that all patriarchs up to Lamech begot their first sons on their birthdays, Noah was born in 1057.1.6, so that 1057 AM is the 1st year of Noah's life. Summarizing the chronology of Noah and Shem that is relevant for our purposes:
1057, Noah.1, Noah was 0 yo.
...
1557, Noah.501, Noah was 500 yo.
- 1557.sm.sd: Noah begot Shem (Gen 5:32), with sm.sd > 2.17.
...
1656, Noah.600, Noah was 599 yo.
- 1656.2.17: Flood began.
1657, Noah.601, Noah was 600 yo.
- 1657.sm.sd: Shem turned 100 yo, with sm.sd > 2.17.
1658, Noah.602, Noah was 601 yo.
- 1658.2.17: Shem begot Arpachshad, exactly 2 years after the Flood began (Gen 11:10).
- 1658.sm.sd: Shem turned 101 yo, with sm.sd > 2.17.
Thus, while the statement by P_chron that "Shem was a hundred years old, and begot Arpachshad two years after the flood." (Gen 11:10) does not mean necessarily that Arpachshad was born exactly two years after the Flood began, it is clear that the statement was made by P_chron to ensure that the Flood chronology was understood correctly as above.
5.3. Number of days in the first 1656 years from Creation in each calendar
I will denote the number of days in an interval according to calendar x as x.days(interval).
5.3.a. 360-/390-/405-day month-intercalating calendar (A)
We first divide 1656 by the length of the composite intercalation cycle: 1656 / 60 = 27.6
The number of days in the first 27 · 60 = 1620 years is:
A.days(1..1620) = 1620 · 365.25 = 591705
The 36-year interval (1621..1656) has:
- 6 leap 390-day years, and
- 30 regular 360-day years.
Therefore the number of days in that interval is:
A.days(1621..1656) = 30 · 360 + 6 · 390 = 13140
and the total number of days is:
A.days(1..1656) = 591705 + 13140 = 604845
5.3.b.1. 365-/395-day month-intercalating calendar (B)
We first divide 1656 by the length of the intercalation cycle: 1656 / 120 = 13.8
The number of days in the first 13 · 120 = 1560 years is:
B.days(1..1560) = 1560 · 365.25 = 569790
The 96-year interval (1561..1656) does not have any leap year, so that:
B.days(1561..1656) = 96 · 365 = 35040
giving a total number of days of:
B.days(1..1656) = 569790 + 35040 = 604830
5.3.b.2. 365-/380-day 15-day-intercalating calendar (C)
We first divide 1656 by the length of the composite intercalation cycle: 1656 / 60 = 27.6
The number of days in the first 27 · 60 = 1620 years is:
C.days(1..1620) = 1620 · 365.25 = 591705
The 36-year interval (1621..1656) has only regular years, so that:
C.days(1621..1656) = 36 · 365 = 13140
and the total number of days is:
C.days(1..1656) = 591705 + 13140 = 604845
5.3.b.3. 365-/370-day 5-day-intercalating calendar (D)
We first divide 1656 by the length of the composite intercalation cycle: 1656 / 20 = 82,8
The number of days in the first 82 · 20 = 1640 years is:
D.days(1..1640) = 1640 · 365.25 = 599010
The 16-year interval (1641..1656) has only regular years, so that:
D.days(1641..1656) = 16 · 365 = 5840
and the total number of days is:
D.days(1..1656) = 599010 + 5840 = 604850
5.3.b.4. 365-/366-day Julian-like calendar with 30-day months (J)
Since 1656 is divisible by 4, the total number of days is:
J.days(1..1656) = 1656 · 365.25 = 604854
5.3.c. 364-/371-day week-intercalating calendar (W)
We first divide 1656 by the length of the composite intercalation cycle: 1656 / 175 = 9.4628...
The number of days in the first 9 · 175 = 1575 years is:
W.days(1..1575) = 1575 · 365.24 = 575253
The 81-year interval (1576..1656) has the following leap years:
- 11 sabbatical years, and
- 3 jubilee or mid-jubilee years: 1600, 1625 and 1650.
Therefore the number of days in that interval is:
W.days(1576..1656) = 81 ·364 + 14 · 7 = 29582
and the total number of days is:
W.days(1..1656) = 575253 + 29582 = 604835
5.4. Discernment of the calendar in effect up to and including 1656, Noah.600
It is evident that the calendar used to compute time during the year of the Flood, 1656 or Noah.600, is either A or B, since the passing of 150 days from Noah.600.2.17 to Noah.600.7.17 is possible only in those calendars, because the length of the months in the lunisolar calendar alternates between 29 and 30 days while the length of the months in each quarter of the sabbatical calendar is 30, 30 and 31 days.
To discern which calendar fits as the one in effect during Noah.600, we require that:
- Noah.600.2.10, when they entered the ark (7:7-11), must not be Saturday.
- Noah.600.7.17, when the ark rested upon the mountains of Ararat (8:4), must be Friday.
- Noah.600.10.1, when the tops of the mountains became visible (8:5), must be Tuesday.
The third requirement is based on P_chron's view of the Flood as a new creation and the obvious analogy between the appearance of the tops of the mountains after "the water decreased steadily until the tenth month" (8:5) and the appearance of dry land after "the waters below the heavens" were "gathered into one place" (1:9) on the 3rd day of Creation.
For the last two requirements to be mutually consistent, the difference between the modulo 7 values of the day numbers within the year must be equal to the difference between the ordinal numbers of the respective weekdays. To verify this mutual consistency is satisfied, we first compute the modulo 7 values:
- Noah.600.2.10 = Noah.600.d.40; 40 mod 7 = 5
- Noah.600.7.17 = Noah.600.d.197; 197 mod 7 = 1
- Noah.600.10.1 = Noah.600.d.271; 271 mod 7 = 5
Then we compute the differences between the modulo 7 values and between the ordinal numbers of the respective weekdays:
271 mod 7 - 197 mod 7 = 5 - 1 = 4
Tuesday - Friday = 3 - 6 + 7 = 4
Thus the last two requirements are mutually consistent and their fulfillment (or lack thereof) will be simultaneous. Also, their fulfillment will imply the fulfillment of the first requirement, as the equality of the modulo 7 of the day numbers within the year of Noah.600.2.10 and Noah.600.10.1 implies the equality of their weekdays. Either of the last two requirements leads in turn to the required weekday of the last day of Noah.599:
Friday - 1 = Thursday
Tuesday - 5 = Thursday
which means that the modulo 7 of the number of days since Creation up to, and including, the last day of Noah.599, i.e. x.days(1..1655) where x may be A or B, must be 5.
5.4.a. 360-/390-/405-day, month-intercalating calendar (A)
Since 1656 was a leap year, as it is divisible by 6, it consisted of 390 days, so that:
A.days(1..1655) = A.days(1..1656) - 390 = 604845 - 390 = 604455
604455 mod 7 = 5, as required.
5.4.b.1. 365-/395-day month-intercalating calendar (B)
Since 1656 was a regular year, it consisted of 365 days, so that:
B.days(1..1655) = B.days(1..1656) - 365 = 604830 - 365 = 604465
604465 mod 7 = 1
5.4.b.2. 365-/380-day 15-day-intercalating calendar (C)
Since 1656 was a regular year, it consisted of 365 days, so that:
C.days(1..1655) = C.days(1..1656) - 365 = 604845 - 365 = 604480
604480 mod 7 = 2
5.4.b.3. 365-/370-day 5-day-intercalating calendar (D)
Since 1656 was a regular year, it consisted of 365 days, so that:
D.days(1..1655) = D.days(1..1656) - 365 = 604850 - 365 = 604485
604485 mod 7 = 0
5.4.b.4. 365-/366-day Julian-like calendar with 30-day months (J)
Since 1656 was a leap year, as it is divisible by 4, it consisted of 366 days, so that:
J.days(1..1655) = J.days(1..1656) - 366 = 604854 - 366 = 604488
604488 mod 7 = 3
We have thus verified that calendar A is the "old" calendar in the Flood narrative.
5.5. Date adjustment and time elapsed since Flood beginning till the earth was dry
As this section deals with the transition from the old calendar to the sabbatical calendar, I will prefix dates with the calendar according to which the date is expressed.
Since A.days(1..1656) = 604845 while W.days(1..1656) = 604835, the enactment of the sabbatical calendar required advancing the date by 10 days. This was achieved by designing that the last day when calendar A was in effect, A.1657.1.6 = A.Noah.601.1.6 - which was Noah's 600th birthday (*), as all patriarchs since Adam were born on 1.6 - was succeeded by W.1657.1.17. This particular day was chosen so that calendar A was in effect throughout the entire span of Noah's first 600 years of life (**). Therefore the day when the earth was dry, W.1657.2.27 in the new calendar, would have been A.1657.2.17 in the old calendar, the same date as that of the beginning of the Flood. Thus, since 1656 was a leap year in calendar A, the time elapsed since the beginning of the Flood till the earth was dry was 390 days.
This allows for a straightforward explanation of the differences in the dating of Flood events between the masoretic text (MT), the Septuagint (LXX), the book of Jubilees (Jub) and the Qumran scroll 4Q252, summarized in p. 286 of [19]. Noting that, with only one exception, the four sources have the same day, either 17 or 27, in the dates of three events:
E1: the day when the Flood began (Gen 7:11),
E2: the day when the ark came to rest on the mountains of Ararat (Gen 8:4),
E3: the day when the earth was dry (Gen 8:14),
the only exception being the 3rd date in the MT, which is 27 instead of 17 as the previous two dates for the reason explained above, we can explain the differences in the dates by postulating that the authors/editors of all four texts were in agreement about the following points:
a. The calendar in effect during the Flood had months of 30 days.
b. According to Gen 8:3, 150 days elapsed between E1 and E2.
c. E1 and E3 occurred on the same calendar date according to the calendar in effect during the Flood.
Points a and b imply that the day of the month of E1 and E2 must be the same, and it is so in all four texts.
Point c implies that, unless a new calendar was enacted at the beginning of Noah.601 with a date adjustment, the dates of E1 and E3 must be the same. Since the author/editors of the LXX vorlage, Jub and 4Q252 were not aware of the notion of calendar replacement (or, in the case of the LXX, rejected that notion as the date adjustment required by his longer time since Creation to Flood would have been larger), they had to change the dates in the MT text in either of two ways:
- LXX: change the day of the month of E1 (and consequently also of E2) to be the same as that of E3 (27).
- Jub and 4Q252: change the day of the month of E3 to be the same as that of E1 (17).
Notably, the date advancement involved in the 1657 AM transition to the sabbatical calendar was exactly the same as that involved in the 1582 AD transition to the Gregorian calendar, in both magnitude and sign.
(*) And also Methuselah's 969th birthday and death day, as I explain in Appendix C.
(**) And also throughout the entire span of Methuselah's life.
Appendix A: Explanation of the 1290 and 1335 days in Dan 12:11-12
Clearly referring to this passage of the prophecy of the 70 weeks previously revealed to Daniel by the angel Gabriel:
"And he will confirm a covenant with many for one week, but in the middle of the week he will bring an end to sacrifice and offering; and on the wing of abominations will be one who makes desolate, even until the consummation which is determined is poured out on the desolator." (Dan 9:27)
another angel said then to Daniel:
"From the time that the daily sacrifice is taken away and the abomination of desolation is set up there will be 1290 days. Blessed is he who waits and comes to the 1335 days." (Dan 12:11-12).
I will show that 1290 and 1335 days are respectively the minimum (and most probable) and the maximum (and least probable) possible lengths of the second half of a week of years in the 30-day-month-based calendar assumed in the P Flood narrative, from the subjective viewpoint of someone who, not knowing the position of that week of years within the calendar, has reckoned the length of the first half of that week of years on the basis of regular 360-day years, i.e. as consisting of 360 · 3.5 = 1,260 days.
Adopting the notation *leap = {leap | extra-leap}, in that calendar every 6th year is *leap and every 10th *leap year is extra-leap, i.e. has 405 days.
Numbering the years in a 6-year interval 1 to 6, with 6 being the *leap year, a week of years can start in any year of that interval, so that each of the 6 possible cases of a week of years has a probability of 1/6:
1234561, 2345612, 3456123, 4561234, 5612345, 6123456
Therefore, the probability that an arbitrary week of years has only one *leap year is 5/6. Since extra-leap years come every 60 years, the probability that an arbitrary *leap year is extra-leap is 1/10. Thus we have the following probabilities:
P(6 regular years + 1 leap year) = (5/6) (9/10) = 3 / (2 · 2) = 3/4
P(6 regular years + 1 extra-leap year) = (5/6) (1/10) = 1/12
For the case of a week of years of the form 6123456, we need to calculate the probability that one of its *leap years is extra-leap. Numbering I to X the *leap years in a sequence of 10 consecutive *leap years, with X being the extra-leap year, a week of years of the form 6123456 can have as its first year any of the 10 *leap years of the sequence, so that each of the 10 possible configurations of weeks of that form has the same probability of 1/10:
(I..II), (II..III), (III..IV), (IV..V), (V..VI), (VI..VII), (VII..VIII), (VIII..IX), (IX..X), (X..I)
Therefore, the probability that an arbitrary week of years of the form 6123456 has one extra-leap year is 2/10 = 1/5, and the probability that it has two leap years is 8/10 = 4/5. Thus we have the following probabilities:
P(5 regular years + 2 leap years) = (1/6) (4/5) = 2/15
P(5 regular years + 1 leap year + 1 extra-leap year) = (1/6) (1/5) = 1/30
Returning to Dan 12:11-12, the faithful who must wait to the end of the 70th week of years do not know to which of the 4 possible cases that particular week belongs. Therefore, if they have reckoned the middle of that week on the basis of 3.5 regular years, i.e. as comprising 1260 days, there are 4 possibilities for the remaining days till the end of the week, depending on the week's case:
6 regular + 1 leap: 6 · 360 + 390 = 2550 = 1260 + 1290 (Probability 3/4)
6 reg + 1 extra-leap: 6 · 360 + 405 = 2565 = 1260 + 1305 (Probability 1/12)
5 regular + 2 leap: 5 · 360 + 2 · 390 = 2580 = 1260 + 1320 (Probability 2/15)
5 reg + 1 leap + 1 extra-leap: 5 · 360 + 390 + 405 = 2595 = 1260 + 1335 (Probability 1/30)
Appendix B: Solution to the problem of Lamech's original fathering age
As we saw in section 5.4, the mutually consistent requirements that Noah.600.7.17 must be Friday and Noah.600.10.1 must be Tuesday lead in turn to the requirement that the last day of the previous year, Noah.599,must be Thursday, meaning that the modulo 7 of the number of days since Creation up to, and including, the last day of Noah.599 must be 5.
At that point in the redaction process, P_chron had already increased Jared's, Methuselah's and Lamech's fathering ages (FA's) from their values in the first draft to avoid those patriarchs outliving the Flood, which in the first draft of the chronology was on 1342. Specifically, he had increased Jared's FA from 62 to 162, Methuselah's FA from 67 to 187, and Lamech's FA from 88 to 188, so that Methuselah died on 1657.1.6, Lamech died on 1652.1.6, and the Flood began on 1662.2.17. The result was quite neat: a 5-year margin for Methuselah, a 10-year margin for Lamech, and a leap year of Flood as 1662 is multiple of 6.
The only problem was that A.days(1..1661) mod 7 was 4, not 5. Since P_chron wanted the year of the Flood to be leap, i.e. multiple of 6, he went back 6 years, computed A.days(1..1655) mod 7, and found that it was 5 as desired. Therefore he made 1656 the year of the Flood by subtracting 6 from Lamech's FA, which became 182. The Flood now began on 1656.2.17, giving a 4-year margin for Lamech and...
Appendix C: A usually unnoticed consequence on Methuselah's last year
Since the first year after Creation is 1, Adam was created on 1.1.6 and became 130 years old 130 years after that, i.e. on 131.1.6, which is Seth's earliest day of birth. Therefore, the earliest day of birth of any patriarch is 1.6 of the AM year which is the sum of the previous patriarch's fathering ages + 1. Adding 969 years to Methuselah's day of birth gives an earliest day of death of 1657.1.6, if he died on the very day when he turned 969 years old.
Since 1657.1.6 is Noah.601.1.6, this means that Methuselah died on the ark. This is not inconsistent with the biblical text, because what matters is not that God did not include Methuselah in the list of people who were commanded to enter the ark (Gen 6:18 and 7:1), or that Methuselah is not included in the list of people who boarded the ark before the Flood began, either 7 days before in the J narrative (Gen 7:7-10) or on the very same day in the P narrative (Gen 7:13), but only that God did not command Noah to get Methuselah off the ark, because he was already there! Seriously, with Grandpa 968 years old and barely able to walk, and probably with significant parts of his house and furniture having been used to finish the ark, isn't it logical to assume that Noah had moved Grandpa onto the ark as soon as it was habitable?
Appendix D: Comparison with the 364-/371-day calendar system actually exact ca. 460 BC
The actual best fitting 364-/371-day calendar system in the interval (1300 BC - 400 CE) is 63/355, meaning 63 leap years in an intercalation period of 355 years. This system, whose mean year length is exactly equal to the mean equinoctial year length ca. 460 BC, can be built by concatenating two 31/175 intercalation periods plus one short 1/5 intercalation period:
31 + 31 + 1 = 63
175 + 175 + 5 = 355
Calling this system E, from "Exact", let's calculate the number of days in the first 1656 years since creation according to this system, E.days(1..1656), and compare it with the corresponding number of days using the 31/175 364-/371-day calendar, W.days(1..1656).
1656 / 355 = 4,66
355 x 4 = 1420
1656 - 1420 = 236
236 - 175 = 61
In the first 61 years of a 31/175 period there are:
- 8 sabbatical years
- 2 mid-jubilar or jubilar years (25 and 50).
Thus:
E.days(1..1656) = 4 [355 · 364 + 63 · 7] + [175 · 364 + 31 · 7] + [61 · 364 + 10 · 7] =
= 4 · 129661 + 63917 + 22274 = 604835
which is exactly equal to W.days(1..1656), which was calculated in section 5.3.c.
References
[1] "And in the fifth year of the fourth week of this jubilee, in the third month, in the middle of the month, Abram celebrated the feast of the first-fruits of the grain harvest. And he offered new offerings on the altar, the first-fruits of the produce, unto the Lord,…" (Jub 15:1-2). This date is consistent with the calendar year starting on Wednesday, the Shabbat in Lev 23:11,15 being a Saturday, and the Sunday after it being the Sunday after the Matzot festival has ended, i.e. 1.26, as shown next:
Calendar days of Matzot Festival: 15 16 17 18 19 20 21 -- 22 23 24 25 26
Weekdays in Jub 364-day calendar: We Th Fr Sa Su Mo Tu -- We Th Fr Sa Su
Thus the Sunday after the Matzot festival ended is 1.26 and the Sunday 49 days later is 3.15 (4 + 30 + 15 = 49).
[2] VanderKam, James C., “The Origin, Character, and Early History of the 364-Day Calendar: A Reassessment of Jaubert's Hypotheses”, The Catholic Biblical Quarterly, vol. 41, no. 3 (1979), pp. 390–411. JSTOR, www.jstor.org/stable/43714717.
[3] Feldman, Ron H., “The 364-Day “Qumran” Calendar and the Biblical Seventh-Day Sabbath: A Hypothesis Suggesting Their Simultaneus Institutionalization by Nehemiah”, «Henoch», vol. 31 (2009), pp. 342–365. Online at http://www.ronhfeldman.com/uploads/2/2/1/9/22191114/364-day_calendar_and_sabbath_-_henoch.pdf
[4] Guillaume, Philippe, “Sifting the Debris: Calendars and Chronologies of the Flood Narrative”, in Silverman, Jason M. (ed.), Opening Heaven's Floodgates. The Genesis Flood Narrative, its Context, and Reception, Gorgias Press, 2013, pp. 57–84.
[5] Wright, Jacob L., “Shabbat of the Full Moon”, TheTorah.com (2015), https://thetorah.com/article/shabbat-of-the-full-moon
[6] Wright, Jacob L., “How and When the Seventh Day Became Shabbat", TheTorah.com (2015), https://thetorah.com/article/how-and-when-the-seventh-day-became-shabbat
[7] Ossendrijver, Mathieu, “Babylonian Scholarship and the Calendar During the Reign of Xerxes”, in: C. Waerzeggers & M. Seire (eds.), Xerxes and Babylonia: The Cuneiform Evidence (Louvain [etc]: Peeters, 2018 [= Orientalia Lovaniensia Analecta, nr. 277]), pp. 135-163.
[8] Britton, John P., “Studies in Babylonian Lunar Theory: Part I. Empirical Elements for Modeling Lunar and Solar Anomalies.” Archive for History of Exact Sciences, vol. 61, no. 2, 2007, pp. 83–145. https://doi.org/10.1007/s00407-006-0121-9
[9] Steele, John M., "A Simple Function for the Length of the Saros in Babylonian Astronomy". In John M. Steele and Annette Imhausen (Eds.), Under One Sky: Astronomy and Mathematics in the Ancient Near East, Münster: Ugarit-Verlag, 2002, pp. 405–420.
[10] Irv Bromberg's site, 6 "The Lengths of the Seasons", topic 7 "Graphical Analyses of the Length of the Solar Year", image # 6, "Landscape Layout, With Calendars".
http://individual.utoronto.ca/kalendis/seasons.htm#years
[11] Irv Bromberg's site, 7 "The Length of the Lunar Cycle", topic 17 "The Mean Synodic Month". http://individual.utoronto.ca/kalendis/lunar/index.htm#msm
[12] This can be checked using a spreadsheet with the following columns:
A: cumulative day number of the equinox - 364 * (year number - 1),
B: cumul. day number of the 1st day of the year of the sabbatical calendar - 364 * (year number - 1),
C: B - A.
A1 & B1 are seeded with 1. The formulas of A2, B2 & C2, then copied & pasted into the ranges A3:A350, B3:B350 & C3:C350 respectively, are as follows ("," may have to be changed to ";"):
A2: =A1+(107331/86400)
B2: =IF(OR(MOD(ROW(),7)=0,MOD(ROW(),25)=0),B1+7,B1)
C2: =B2-A2
The minimum and maximum values of the difference are:
D1: =MIN(C1:C350)
D2: =MAX(C1:C350)
[13] A list of solstices and equinoxes, with times in GMT, is downloadable from the 2nd link in
https://astronomy.stackexchange.com/a/13009
Jerusalem Mean Time (JMT) = GMT + 2 h 21′
Babylon Mean Time (BMT) = GMT + 2 h 58′
[14] https://www.fourmilab.ch/documents/calendar/
[15] Brack-Bernsen, Lis, “The 360-Day Year in Mesopotamia”, in Steele, John M. (ed.) Calendars and Years: Astronomy and Time in the Ancient Near East, Oxbow Books, 2007, pp. 83-100.
[16] De Blois, François, “The Persian Calendar”, Iran, Vol. 34 (1996), pp. 39-54.
[17] Boyce, Mary, “Further on the calendar of Zoroastrian feasts”, Iran, Vol. 43 (2005), pp. 1-38.
[18] Beckwith, Roger T., “Calendar, Chronology And Worship: Studies in Ancient Judaism And Early Christianity”, Brill, 2005, pp. 57-65.
[19] Jacobus, Helen R., “Noah's flood calendar (Gen 7:10-8:19) in the Septuagint”, «Henoch», vol. 36 (2014), num.2, pp. 283-296. Online at https://www.academia.edu/10494119/
