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Hebrew and Babylonian Calendar Intercalation

To ensure that intercalated months did not invalidate the correlation of dates and events reported by Jeremiah, Daniel, Ezra, and Nehemiah, the regnal tables show the intercalated months, and this is a brief overview of how those intercalated months were determined.

12-Month Luni-Solar Calendar

Both the Hebrews and Babylonians used similar calendar adjustments to keep their calendars synchronized with the Sun, moon and seasons. They both employed a system of adding entire months to their calendars (intercalation) during 7 particular years (embolismic years - years in which a month is intercalated) out of every 19 years. This was a repeating cycle.

The Babylonian's employed a spring calendar starting with the month of Nisanu:

In the period covered by this study the Babylonian calendar year was composed of lunar months, which began when the thin crescent of the new moon was first visible in the sky at sunset. Since the lunar year was about eleven days shorter than the solar year, it was necessary at intervals to intercalate a thirteenth month, either a second Ululu (the sixth month) or a second Addaru (the twelfth month) in order that New year's Day, Nisanu 1, should not fall much before the spring of the year (late March and early April).

Richard A. Parker and Waldo H. Dubberstein, Babylonian Chronology, 626 B.C.-A.D. 75, 3rd ed.
Providence: Brown University Press (1956) p. 1

Both the Babylonians and Hebrews employed solar-lunar calendrics of 12 months of alternating duration of 30 and 29 days:

Babylonian
Hebrew ( sacred / civil )
30-day months
29-day months
30-day months
29-day months
Nisanu 1 Aiaru 2 Nisan 1/7 Iyar 2/8
Simanu 3 Duzu 4 Sivan 3/9 Tammuz 4/10
Abu 5 Ululu 6 Av 5/11 Elul 6/12
Tashritu 7 Arahsamnu 8 Tishri 7/1 Heshvan 8/2
Kislimnu 9 Tebetu 10 Kislev 9/3 Tevet 10/4
Shabatu 11 Addaru 12 Shevat 11/5 Adar 12/6

Above, each month name is followed by its numerical sequence in the calendar year. The table reads left-to-right, then next row down.

Further, the Hebrews employed two calendars, a "civil" and a "sacred", with the sacred calendar following the civil by 6 months. Each Hebrew month's sequence in both the civil and sacred calendar is designated by the "s/c" following each month's name, where "s" is that month's number in the sacred calender and "c" is that month's number in the civil calender. So the Hebrew side of the table (reading left to right, then next row) shows Nisan, Iyar, Sivan and Tammuz as the first 4 months of the sacred calendar and Tishri, Heshvan, Kislev and Tevet as the first 4 months of the civil calendar.

Summing 6 30-day months plus 6 29-day months yields a total of 354 days, the "common regular" length year, which is short of the actual 365.24 (approximate) day solar year. This is an error rate of 11 days per year, every year or about 1 month every three years. If the calendar is not adjusted, after only two decades the actual observed seasons would be reversed relative to what the calendar declared, e.g. the season would actually be winter when the calendar reported summer months.

To fix this, the Babylonians (and seemingly the Hebrews to some extent) surmised that a 19-year Lunar cycle existed (sometimes called the Metonic cycle) and that if additional months were periodically inserted to correct the calendar, the calendar would be re-synchronized with the actual observed solar year. During that 19-year cycle the Babylonians (and presumably Hebrews with some variation) would insert at 7 different times an additional 29-day month. This insertion of extra months to correct the calendar is called "intercalation".

Ancient history is vague on precisely whom to credit with developing intercalation and when it was methodically adopted by the Hebrews. The Sumerian cultures circa 2100 B.C. seem to be the earliest in employing some form of it; then Hammurabi standardized the Babylonian lunar calendar circa 1750 B.C. resulting in intercalation being standardized by 541 B.C.; and Persian astronomers made refinements until about 380 B.C. It is believed the Israelites in exile acquired intercalation methods from their Babylonian captors.

Note very carefully: It is assumed (in the table below) for the purposes of determining 7th, 6th and 5th century B.C. dates of intercalated months in the chronolgies which underly the biblical books of Jeremiah, Daniel, Ezra, Nehemiah, etc., that those intercalation methods acquired by the Israelites during their Babylonian exile are very similar, at least in result, to the 2nd and 3rd century A.D. intercalation methods codified by the Rabbis and in use today. Hence current methods may be used, hypothetically, to extrapolate ancient B.C. intercalated dates (see Hypothetical extrapolation of Hebrew intercalated months). None of the analysis presented herein depends on extrapolated dates of historical events, rather the chronologies of the Fall of Judah and of Artaxerxes I are supplemented with hypothetically intercalated months to ascertain what impact, if any, intercalation has on those chronologies. None of the dates of actual events were adjusted, rather all dates are reported exactly as given by the Bible or as inscribed on artifacts and monuments.

Hebrew Intercalary Year Types

Intercalation results in several different length years. Common years can be "deficient", "regular", or "complete", having respectively 353, 354, or 355 days, whereas the year in which an extra month is inserted is called an "embolismic" year, and after inclusion of the extra 30 days (Adar I becomes 1-day longer than regular Adar while Adar II has the regular 29-day length) the resulting embolismic year can then likewise be "deficient", "regular", or "complete" having respectively 383, 384, or 385 days.

The Hebrews intercalated their civil calendar (Tishri through Elul) by lengthening the regular month Adar from 29 to 30 days, inserting a month "Adar II" (or "weAdar" meaning "second Adar") having the regular 29-day duration, and then moving the celebration of Purim to Adar II. The result is both their civil and sacred calendars are re-synchronized simultaneously while keeping Purim in the same proximity to Pesach (Passover) plus keeping the same duration between Nisan and Tishri on the sacred calendar. Additional "postponement rules" caused the Hebrew sacred calendar to further conform with Hebrew Scripture (Old Testament) festival date determination:

In the table below, the "new" number of days = 354 (common regular) plus/minus the values in the columns to the right.

year
Common #days changed
Embolismic #days changed
type
new reg
Kislev
Heshvan
Adar I
Adar II
 common deficient
353
354
 -1 (30 => 29)      
 common regular
354
354
       
 common complete
355
354
   +1 (29 => 30)    
 embolismic deficient
383
354
 -1 (30 => 29)    +1 (29 => 30)  +29
 embolismic regular
384
354
     +1 (29 => 30)  +29
 embolismic complete
385
354
   +1 (29 => 30)  +1 (29 => 30)  +29

19-Year Intercalary Cycles

The Babylonians and Hebrews employed different intercalation methods to re-synchronize their calendars with the solar year. The Babylonians usually inserted an additional 12th month "Addaru II" at the end of an embolismic year, but occasionally inserted an additional 6th month "Ululu II" in the middle of an embolismic year. The Hebrews always inserted in their civil calendar an additional 6th month "Adar II" in the middle of an embolismic year.

Both the Babylonians and Hebrews inserted 7 additional months throughout a 19-year cycle, in which the 7 embolismic years were for the:

While their respective intercalary sequences were identical (i.e. years 3, 6, 8, 11, 14, 17, and 19), the months, however, were intercalated differently and the resulting lengths of years varied considerably, as will be discussed.

The next table below shows 14 and 9 (respectively) 19-year cycles of embolismic years and intercalated months for the Babylonian and Hebrew calendar systems. The chosen timeframe spans the rise of Nebuchadnezzar through the fall of Judah and the transition from Xerxes to Artaxerxes, covering the range of dates used by Jeremiah (Fall of Judah timeline) and Daniel (Artaxerxes I timeline). Cycles 10 - 14 are shown only for the Babylonian calendar to demonstrate its gradual settling into a regular intercalary pattern in the 4th century B.C.

Note that the Hebrew intercalated months below are just hypothetical extrapolations using the Fourmilab Calendar Converter. The converter's math is correct when converting ancient B.C. dates, but what is unknown is, did the Hebrews in the 7th, 6th and 5th centuries B.C. use similar math when adjusting their calendars? For purposes of computing the beginning and end dates for Daniel's prophecy of 69 weeks, accurate to within 1 year over 483 years, it is not significant. But for purposes of verifying specific historical synchronisms of the intervening history, it could be significant but results suggest the ancient Hebrews did use a very similar computation. The merits of hypothetically extrapolating these intercalations is discussed further (see Hypothetical extrapolation of Hebrew intercalated months).

Highlighted in the table are: hypothetical Hebrew embolismic or intercalary years for the Hebrew 19-year cycles (right half of the table) as reported by the Fourmilab Calendar Converter1; and the actual Babylonian embolismic or intercalary years (left half of the table) excerpted from "Babylonian Chronology, 626 B.C.-A.D. 75"2 tables for Nabopolassar - Artaxerxes II:

  1. Fourmilab Calendar Converter
  2. Richard A. Parker and Waldo H. Dubberstein, Babylonian Chronology, 626 B.C.-A.D. 75, 3rd ed.; Providence: Brown University Press (1956) pp. 27-35

Regarding the starting year of 747 B.C. for Babylonian intercalation, Parker and Dubberstein note:

It may have been in the reign of Nabonassar, 747 B.C., that Babylonian astronomers began to recognize, as the result of centuries of observation of the heavens, that 235 lunar months have almost exactly the same number of days as nineteen solar years. This meant that seven lunar months must be intercalated over each nineteen-year period.1

1 Against recognition of nineteen-year cycles at that time see Kuglar, Sternkunde und Sterndienst in Babel II 362-71 and 422-30. We have followed Sidersky (Étude sur la chronologie assyro-babylonienne, p 38) in taking 747 B.C. as a conveneient starting point for our scheme in Plate I, but that is not to be interpreted as acceptance of that date as the point at which Babylonian astronomers consciously recognized the principle that seven intercalations were regularly needed in each nineteen years.

Richard A. Parker and Waldo H. Dubberstein, ibid. p. 1

Eduard Mahler, originally concluded the Babylonian intercalary series began in 747 B.C.:

The Calendar of the Babylonians was not clearly understood by scientists till Eduard Mahler, then assistant of the Geodetic Survey of Austria, unriddled its mysterious construction and revealed a system of Great symmetry and comparative simplicity. It will suffice here to say that two kinds of years were used, a common year of 354 days and a year of intercalation which had a length of 383 or 384 days divided into 13 months. … They began the day at 6 p.m. and this custom likewise prevailed among the Hebrews. [p. 279]

Kurt Laves, "New Light from Old Records",
Popular Astronomy, vol. 14, (1906) pp. 276-287

* Ptolemy in his canon of the Babylonian kings has recorded this very information, starting with the King Nabonassar of Babylon. It seems to be sufficiently certain that the Babylonian Calendar began when Nabonassar came to the throne and the Era of Nabonassar is equivalent with April 21st 747 B.C. [p. 280]

ibid. p. 280

[Mahler] found that the cycle of nineteen years began with the first year of the reign of Nabonassar in 747 B.C., of these nineteen years there were twelve lunar years of 354 days length and seven years of 384 days length. Sometimes a lunar year has 353 days and an intercalated year 383 days. [p. 281]

ibid. p. 281

Ostensibly, it would seem Nabonassar commemorated his reign with an edict to his astronomers to implement a more accurate calendar, an intercalated calendar. However, cuneiform data isn't sufficient to provide reliable details until Nabopolassar's reign, for which his 1st regnal year in 625 B.C. coincides with the 9th year of the 7th Babylonian 19-year intercalary cycle.

Analysis of Parker & Dubberstein's tables, show the Babylonians as of the 4th century B.C. were generally using a second Addaru (Addaru II) for their embolismic month, and had settled into a regular pattern of intercalating the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of every 19.

Earlier, however, a second Ululu (Ululu II) was often intercalated instead of Addaru II, and occasionally 'irregular' (outside the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years) intercalations of either Addaru II or Ululu II were done as well. These variations are highlighted in the table as follows:

 regular Addaru II :
 
 (hypothetical) Adar II:
 
 irregular Addaru II :
     
 regular Ululu II :
     
 irregular Ululu II :
     

The table below shows how the 19-year intercalary sequences of the Babylonians and Hebrews were shifted or offset:

For example, how the 19 years of Babylonian cycle 1 compares to the 19-years of Hebrew cycle 1: Year 625 B.C. (Babylonian cycle 1, 1st row, 1st column) corresponds to year 3136 A.M. (Hebrew cycle 1, 1st row, 17th column). The next row down is the next year in the same corresponding 19-year cycle 1 624 B.C. (2nd row, 1st column) corresponds to 3137 A.M. (2nd row 17th column).

The table portrays the same pairing of 19-year cycles, enumerated 1-9 in the 2nd row on the left for the Babylonians and enumerated 1-9 on the right for the Hebrews. The highlighting shows how the Babylonian intercalation was highly irregular in earlier cycles, then became stable and regular about the middle of the 4th century B.C. (Babylonian cycles 10-14), and how that corresponds (hypothetically) to the Hebrew intercalation adopted from the Babylonians and in use by 5 B.C.

The table shows pairs of corresponding columns (having the same 19-year cycle number at the top), and reads top-to-bottom, then next pair of columns (or cycles) to the right; i.e., Babylonian cycle 1 (column) is paired with Hebrew cycle 1 (column), then Babylonian cycle 2 with Hebrew cycle 2, etc.

* cycle: The center pair of columns shows how the intercalary sequences are shifted: in the third row, Babylonian intercalary cycle year 9 corresponds to Hebrew intercalary cycle year 1, next row down 10 corresponds to 2, ... etc., and in the last row Babylonian intercalary cycle year 8 corresponds to Hebrew intercalary cycle year 19. This offset alignment is elaborated in the next section. Note in these center columns, the same intercalary sequence is highlighted (cycle years 3, 6, 8, 11, 14, 17, and 19) for both Babylonians and Hebrews, but offset by 12 years - the essential point being demonstrated.

Babylonians intercalated2
Hebrews hypothetically intercalated
1
2
 3
 4
 5
 6
7
8
9
10
11
12
13
14
cycle*
 1
2
 3
 4
 5
 6
7
8
9
BC
BC
BC
BC
BC
BC
BC
BC
BC
BC
BC
BC
BC
BC
yr 
yr 
AM
AM
AM
AM
AM
AM
AM
AM
AM
 625  606  587  568  549  530  511  492  473  454  435  416  397  378
9
1  3136  3155  3174  3193  3212  3231  3250  3269  3288
 624  605  586  567  548  529  510  491  472  453  434  415  396  377
10
2  3137  3156  3175  3194  3213  3232  3251  3270  3289
 623  604  585  566  547  528  509  490  471  452  433  414  395  376
11
3  3138  3157  3176  3195  3214  3233  3252  3271  3290
 622  603  584  565  546  527  508  489  470  451  432  413  394  375
12
4  3139  3158  3177  3196  3215  3234  3253  3272  3291
 621  602  583  564  545  526  507  488  469  450  431  412  393  374
13
5  3140  3159  3178  3197  3216  3235  3254  3273  3292
 620  601  582  563  544  525  506  487  468  449  430  411  392  373
14
6  3141  3160  3179  3198  3217  3236  3255  3274  3293
 619  600  581  562  543  524  505  486  467  448  429  410  391  372
15
7  3142  3161  3180  3199  3218  3237  3256  3275  3294
 618  599  580  561  542  523  504  485  466  447  428  409  390  371
16
8  3143  3162  3181  3200  3219  3238  3257  3276  3295
 617  598  579  560  541  522  503  484  465  446  427  408  389  370
17
9  3144  3163  3182  3201  3220  3239  3258  3277  3296
 616  597  578  559  540  521  502  483  464  445  426  407  388  369
18
10  3145  3164  3183  3202  3221  3240  3259  3278  3297
 615  596  577  558  539  520  501  482  463  444  425  406  387  368
19
11  3146  3165  3184  3203  3222  3241  3260  3279  3298
 614  595  576  557  538  519  500  481  462  443  424  405  386  367
1
12  3147  3166  3185  3204  3223  3242  3261  3280  3299
 613  594  575  556  537  518  499  480  461  442  423  404  385  366
2
13  3148  3167  3186  3205  3224  3243  3262  3281  3300
 612  593  574  555  536  517  498  479  460  441  422  403  384  365
 3
14  3149  3168  3187  3206  3225  3244  3263  3282  3301
 611  592  573  554  535  516  497  478  459  440  421  402  383  364
 4
15  3150  3169  3188  3207  3226  3245  3264  3283  3302
 610  591  572  553  534  515  496  477  458  439  420  401  382  363
 5
16  3151  3170  3189  3208  3227  3246  3265  3284  3303
 609  590  571  552  533  514  495  476  457  438  419  400  381  362
 6
17  3152  3171  3190  3209  3228  3247  3266  3285  3304
 608  589  570  551  532  513  494  475  456  437  418  399  380  361
7
18  3153  3172  3191  3210  3229  3248  3267  3286  3305
 607  588  569  550  531  512  493  474  455  436  417  398  379  360
8
19  3154  3173  3192  3211  3230  3249  3268  3287  3306

A final cautionary note about interpreting or understanding the above table:

The Babylonian calendar years, instead of being designated by regnal year, e.g. "1st year of Nebuchadnezzar II" or "4th year of Artaxerxes I", etc. (which is how they are actually reported in the cuneiform tablets), were "normalized" by Parker and Dubberstein to the Julian B.C. year in which the corresponding Babylonian year began, thus shortening an otherwise unwieldy and inconsistent regnal designation but adding a source of confusion.

Because a Babylonian calendar year (like the Hebrew sacred calendar year) spans the last 9 months of the previous Julian year and the first 3 months of the next Julian year, a more accurate designation for the Julian B.C. years above would have been of the format "Jy / Jy-1"; for example, 463/462 to show that two Julian years are spanned by the Babylonian year. This is a significant caveat because when the Babylonians intercalated a 2nd Ululu (which is their 6th month), that intercalated month falls in the Julian year (i.e. "Jy") as shown above. But when the Babylonians intercalated a 2nd Addaru (which is their 12th month), that intercalated month actually falls into the next Julian year (i.e. the "Jy-1" year).

Consider as a practical example of this significance, in the above table for Babylonian and Hebrew cycle 9 are the Julian years 463 and 460 and their corresponding Hebrew A.M. years 3298 and 3301, which in turn correspond to the 1st and 4th regnal years of Artaxerxes I:

Because it was a 2nd Addaru (a 12th month) that was intercalated, that month falls into the next Julian years 462 and 459, respectively, and thus a more informative designation in the above table would have been 463/462 and 460/459, respectively. This discrepancy doesn't exist for the Hebrew years 3298 and 3301 because A.M. (Anno Mundi) years are reported as-is and not "normalized" to a Julian B.C. designation. Likewise, the discrepancy doesn't exist for Babylonian years in which a 2nd Ululu (a 6th month) is intercalated because being only the 6th Babylonian month, an intercalated one still falls in the same Julian year ("Jy") in which the Babylonian year began. This year labeling discrepancy only exists for 2nd Addaru intercalated years.

So, why not use the format "Jy / Jy-1"? Two reasons: because,

  1. The data shown above originated with Parker and Dubberstein and they likewise opted to use a "Jy" format where the single Julian year reported is when the corresponding Babylonian year began (around March/April), not the Julian year in which the intercalated Babylonian month actually fell (which is always the next year for 2nd Addaru). By following Parker & Dubberstein's convention, comparing the analysis here with their data (should the reader be so inclined) is straightforward and the reader does not have to translate years reported here to years reported there.
  2. The "Jy / Jy-1" format is more cumbersome and takes up even more space in an already cramped and difficult to read table.

Though identical, Babylonian and Hebrew cycles don't coincide.

The Babylonians actually designated their calendar years as regnal years, i.e. as the Nth year of the enthroned king, and there were many kings: from Nabonassar through to Nabonidus, followed by Cyrus and later the Medo-Persians to Alexander and into the Seleucid Era. As a convenience for purposes of this topic, 'Julian years B.C.' was adopted by Parker and Dubberstein as the dating convention which consistently spans all these various reigns.

The Hebrew calendar uses Anno Mundi years, consistently, spanning several thousand years, its epoch beginning 1st of Tishri 1 A.M. which equates to 7th of October 3761 B.C. Julian.

Because the Hebrew's 1st 19-year cycle began 3761 B.C. and the Babylonian's 1st 19-year cycle began in 747 B.C., the difference is 3,014 years between the start of their respective cycles, which difference includes 12 years of a partial cycle:

12 = 3,014 modulo 19 ( i.e. after dividing 19 into 3,014 an even number of times, the remainder is 12)

There is a 12 year partial Hebrew intercalary cycle already begun when the first Babylonian intercalary cycle starts. This means when the Babylonian cycle begins in year 1 of any 19-year cycle, the Hebrew cycle will already be in year 12 (an 11 year difference between their respective 1st years). Recall, from the table above, in the center columns, 8th row up from the bottom, Babylonian cycle 1 corresponds to Hebrew cycle 12.

Sometime between the 4th and 1st century B.C., the Hebrews adopted the Babylonian intercalary calendar, modified it, and retroactively defined the 1st Hebrew intercalary cycle to begin 3761 B.C., and 158 full 19-year cycles 'later', plus 12 years, the Babylonian king Nabonassar began theirs in 748/747 B.C.

That is why even though they both intercalate in years 3, 6, 8, 11, 14, 17, and 19 of any given cycle, when comparing the same cycle of years (such as shown in the tables above and below), the sequences won't align identically. They will be offset or shifted by 12 years.

Are Babylonian dates and Hebrew dates comparable?

No, because they didn't start their 19-year cycles on equivalent years, nor are the intercalary months always the same, nor are their common and embolismic years always the same lengths:

Unlike the mathematical precision repeated in the Hebrew calendar, this suggests the Babylonian intercalation was somewhat ad-hoc and not according to a precise formula.

The next table below shows the 12 year differing alignment of the Babylonian and Hebrew 19-year intercalary cycles from before the turn of the millennium to after Christ's crucifixion in A.D. 30 (as highlighted below):

For example, from 6 B.C. to A.D. 40, the Hebrew calender covered 16,804 days while the Babylonian calendar covered 16,802 days, for a 2-day difference over the same 46-year period. Moreover, in A.D. 29 the Babylonian year was 354 days while the Hebrew year was 383 days, and in the following year A.D. 30, the Babylonian year was 383 days while the Hebrew year was 354 days, which resulted in an additional 29-day discrepancy between A.D. 29 and A.D. 30.

Below, note that the Julian year spans approximately the:

  • last 9 months of the prior Hebrew civil year and the first 3 months of the next Hebrew civil year, or
  • last 3 months of the prior Babylonian regnal year and the first 9 months of the next Babylonian regnal year.

This is because Hebrew (Anno Mundi) civil years began in the fall (September/October) of a Julian year, while Babylonian regnal years began in the spring (March/April) of a Julian year (as did Hebrew sacred years).

Babylonian
Hebrew
 Julian
year 
 Cycle
year 
 Intercalated
month 
 Year
 Length 
 Cycle
year 
 A.M.
year 
 Intercalated
month 
 Year
 Length 
 18 BC 
             
 1 
 3744 
 
 355
 17 BC 
 2 
 3745 
 
 354
 16 BC 
 3 
 3746 
 Adar II 
 383
 15 BC 
 4 
 3747 
 
 355
 14 BC 
 5 
 3748 
 
 354
 13 BC 
 6 
 3749 
 Adar II 
 385
 12 BC 
 7 
 3750 
 
 353
 11 BC 
 8 
 3751 
 Adar II 
 385
 10 BC 
 9 
 3752 
 
 354
   9 BC 
 10 
 3753 
 
 355
   8 BC 
 11 
 3754 
 Adar II 
 383
   7 BC 
 12 
 3755 
 
 354
   6 BC 
 1 
 
354
 13 
 3756 
 
 355
   5 BC 
 2 
 
355
 14 
 3757 
 Adar II 
 385
   4 BC 
 3 
 Addaru II 
384
 15 
 3758 
 
 353
   3 BC 
 4 
 
354
 16 
 3759 
 
 354
   2 BC 
 5 
 
354
 17 
 3760 
 Adar II 
 385
   1 BC 
 6 
 Addaru II 
384
 18 
 3761 
 
 355
 AD  1 
 7 
 
354
 19 
 3762 
 Adar II 
 383
 AD  2 
 8 
 Addaru II 
384
 1 
 3763 
 
 354
 AD  3 
 9 
 
354
 2 
 3764 
 
 355
 AD  4 
 10 
 
354
 3 
 3765 
 Adar II 
 385
 AD  5 
 11 
 Addaru II 
384
 4 
 3766 
 
 354
 AD  6 
 12 
 
355
 5 
 3767 
 
 353
 AD  7 
 13 
 
355
 6 
 3768 
 Adar II 
 385
 AD  8 
 14 
 Addaru II 
384
 7 
 3769 
 
 354
 AD  9 
 15 
 
354
 8 
 3770 
 Adar II 
 383
 10 
 16 
  
354
 9 
 3771 
 
 355
 11 
 17 
 Ululu II 
384
 10 
 3772 
 
 354
 12 
 18 
 
354
 11 
 3773 
 Adar II 
 385
 13 
 19 
 Addaru II 
384
 12 
 3774 
 
 353
 14 
 1 
 
354
 13 
 3775 
 
 354
 15 
 2 
 
355
 14 
 3776 
 Adar II 
 385
 16 
 3 
 Addaru II 
384
 15 
 3777 
 
 355
 17 
 4 
 
354
 16 
 3778 
 
 353
 18 
 5 
 
355
 17 
 3779 
 Adar II 
 384
 19 
 6 
 Addaru II 
384
 18 
 3780 
 
 355
 20 
 7 
 
354
 19 
 3781 
 Adar II 
 383
 21 
 8 
 Addaru II 
383
 1 
 3782 
 
 355
 22 
 9 
 
355
 2 
 3783 
 
 354
 23 
 10 
 
354
 3 
 3784 
 Adar II 
 385
 24 
 11 
 Addaru II 
384
 4 
 3785 
 
 355
 25 
 12 
 
355
 5 
 3786 
 
 354
 26 
 13 
 
354
 6 
 3787 
 Adar II 
 383
 27 
 14 
 Addaru II 
384
 7 
 3788 
 
 355
 28 
 15 
 
355
 8 
 3789 
 Adar II 
 383
 29 
 16 
  
354
 9 
 3790 
 
 354
 30 
 17 
 Ululu II 
383
 10 
 3791 
 
 355
 31 
 18 
 
355
 11 
 3792 
 Adar II 
 385
 32 
 19 
 Addaru II 
383
 12 
 3793 
 
 354
 33 
 1 
 
355
 13 
 3794 
 
 353
 34 
 2 
 
355
 14 
 3795 
 Adar II 
 385
 35 
 3 
 Addaru II 
384
 15 
 3796 
 
 354
 36 
 4 
 
354
 16 
 3797 
 
 355
 37 
 5 
 
354
 17 
 3798 
 Adar II 
 383
 38 
 6 
 Addaru II 
384
 18 
 3799 
 
 354
 39 
 7 
 
354
 19 
 3800 
 Adar II 
 385
 40 
 8 
 Addaru II 
 384
       
 41 
 9 
 
 354
 42 
 10 
 
 355
 43 
 11 
 Addaru II 
 384
 44 
 12 
 
 355
 45 
 13 
 
 354
 46 
 14 
 Addaru II 
 384
 47 
 15 
 
 354
 48 
 16 
  
 354
 49 
 17 
 Ululu II 
 384
 50 
 18 
 
 354
 51 
 19 
 Addaru II 
 384
 52 
 

The essential point of the foregoing is that while Babylonian and Hebrew month names are similar in spelling, and their 19-year intercalary sequences are similar (albeit 12 years apart), their intercalations do not otherwise correspond and Babylonian dates can not be assumed as equivalent or even approximate to Hebrew dates. While any given correspondence might, best case, be 1 day off for a single date, computations of time elapsed between two dates will never correspond closely, let alone accurately. For example, while, best case, their respective 1st day of the year could coincide, the last day of that same year could be 30 days apart.

Further information about Babylonian and Hebrew intercalation can be found at:

Synchronisms establishing the Hebrew 3, 6, 8, 11, 14, 17, and 19 year intercalary sequence to 5 B.C.

Josephus, recounting certain events which began the Jewish war against the Romans in A.D. 66, establishes that Elul 7th in A.D. 66 was a sabbath (Saturday):

8. ... However, those that were within sent to Manahem, and to the other leaders of the sedition, and desired they might go out upon a capitulation: this was granted to the king's soldiers and their own countrymen only, who went out accordingly; but the Romans that were left alone were greatly dejected, for they were not able to force their way through such a multitude; and to desire them to give them their right hand for their security, they thought it would be a reproach to them; and besides, if they should give it them, they durst not depend upon it; so they deserted their camp, as easily taken, and ran away to the royal towers, - that called Hippicus, that called Phasaelus, and that called Mariamne. But Manahem and his party fell upon the place whence the soldiers were fled, and slew as many of them as they could catch, before they got up to the towers, and plundered what they left behind them, and set fire to their camp. This was executed on the sixth day of the month Gorpieus [Elul].

9. But on the next day the high priest was caught where he had concealed himself in an aqueduct; he was slain, together with Hezekiah his brother, by the robbers: hereupon the seditious besieged the towers, and kept them guarded, lest any one of the soldiers should escape. ...

10. ... This loss to the Romans was but light, there being no more than a few slain out of an immense army; but still it appeared to be a prelude to the Jews' own destruction, while men made public lamentation when they saw that such occasions were afforded for a war as were incurable; that the city was all over polluted with such abominations, from which it was but reasonable to expect some vengeance, even though they should escape revenge from the Romans; so that the city was filled with sadness, and every one of the moderate men in it were under great disturbance, as likely themselves to undergo punishment for the wickedness of the seditious; for indeed it so happened that this murder was perpetrated on the sabbath day, on which day the Jews have a respite from their works on account of Divine worship.

Josephus, Wars of the Jews, Book 2, Chapter 17, paragraphs 8-9

3826 A.M. Elul 7, the next day after Elul sixth which Josephus reported as a sabbath (Saturday), falls on different days of the week: a Hebrew intercalary sequence of

The death of Herod the Great establishes another synchronism (between a Jewish fast and a lunar eclipse) reported by Josephus:

4. But the people, on account of Herod's barbarous temper, and for fear he should be so cruel and to inflict punishment on them, said what was done was done without their approbation, and that it seemed to them that the actors might well be punished for what they had done. But as for Herod, he dealt more mildly with others [of the assembly] but he deprived Matthias of the high priesthood, as in part an occasion of this action, and made Joazar, who was Matthias's wife's brother, high priest in his stead. Now it happened, that during the time of the high priesthood of this Matthias, there was another person made high priest for a single day, that very day which the Jews observed as a fast. The occasion was this:

This Matthias the high priest, on the night before that day when the fast was to be celebrated, seemed, in a dream, to have conversation with his wife; and because he could not officiate himself on that account, Joseph, the son of Ellemus, his kinsman, assisted him in that sacred office.

But Herod deprived this Matthias of the high priesthood, and burnt the other Matthias, who had raised the sedition, with his companions, alive.

And that very night there was an eclipse of the moon.

Josephus, "Antiquities of the Jews, Book 17", Chapter 6, paragraph 4

From 7 B.C. through 2 B.C. there were only three lunar eclipses visible from Jerusalem, of which two were total lunar eclipses on 5 B.C. March 23 and September 15, and the third was a partial lunar eclipse on 4 B.C. March 13 (all Julian dates). For the two total lunar eclipses in 5 B.C., the two Jewish fasts which most closely correspond to those lunar eclipses are Ta’anit Bechorim in March and Yom Kippur in September:

Hebrew fast day
Hebrew intercalated date of that fast
    3, 6, 8, 11, 14, 17, and 19 2, 5, 7, 10, 13, 16, and 18
Ta’anit Bechorim 3756 A.M. Nisan 14 -5 B.C. March 22 -5 B.C. April 21
Yom Kippur 3757 A.M. Tishri 10 -5 B.C. September 11 -5 B.C. October 11

Josephus reports that a lunar eclipsed occured on the night 'following the day on which the Jews observed an important fast'. Fred Espenak of NASA computes two total lunar eclipses as occuring on March 23 and September 15, respectively, in 5 B.C., but the only Hebrew intercalary sequence which yields Hebrew fast dates proximate to those lunar eclipses is, again, the 3, 6, 8, 11, 14, 17, and 19 year intercalary sequence.

Consequently, the only Hebrew intercalary sequence that yields dates in agreement with the historical record as reported by Josephus is that of 3, 6, 8, 11, 14, 17, and 19 years, and the above two synchronisms would seem to establish its use much earlier than 3rd century A.D., to not later than 5 B.C.

Hypothetical extrapolation of Hebrew intercalation to pre-exilic periods

To be clear, there is no irrefutable evidence that 7th, 6th and 5th century B.C. Hebrews actually intercalated as do modern Hebrews and the Fourmilab Calendar Converter, certainly not earlier than 5 B.C. There is very little historical evidence in any detail, other than:

The narrow, specific purpose here is twofold, to argue that:

  1. some form of intercalation was in fact used by the Hebrews prior to the exile; and
  2. the present Hebrew intercalation method can be extrapolated backwards without incurring significant dating errors, and certainly accurate to within a year over the centuries under consideration.

Evidence for pre-exilic Hebrew intercalation

The Hebrews or Israelites always observed feasts that were fixed to particular lunar months (of 29 or 30 days duration) and the first lunar month (Ab or Nisan) was fixed to a particular time of the solar year (365.24 days). But the motions of the earth, moon and sun are not harmoniously repetitive (certainly not in a year or even several years) and any lunisolar calendar must be frequently (every year or two) corrected by adding extra months. If this is not done after only two decades the actual observed seasons would be reversed relative to what the calendar declared, e.g. the season would actually be winter when the calendar reported summer months.

Siegfried Horn and J. B. Segal elaborate on this problem and its relevance to the pre-exilic Hebrews of Daniel's and Jeremiah's period.

"That the Jews must have had a system of intercalation by which the lunar calendar was brought into harmony with the natural solar year is implied in the law regarding the Passover feast. This law required that the feast be kept unchangeably in the middle of the first month (Leviticus 23:5), but also connected it with the barley harvest by requiring the offering of a sheaf of the first fruits (Leviticus 23:10,11). Thus the calendar was probably corrected by the insertion of embolismic months whenever needed to let the Passover occur at the beginning of the barley harvest." [p. 17]

Siegfried H. Horn, "Chronology of Ezra 7",
Seventh-day Adventist Theological Seminary, 1953.

Segal also asserts that the pre-exilic Hebrews used some form of intercalation, and further hypothesizes their observing the risings and settings of heliacal stars (a technique well within the skills of pre-exilic Hebrews) that yields a nearly Metonic sequence, and also exhibits the same intercalary pattern as the present Hebrew calendar.

"Intercalation, then, is implicit in the customs and laws of the Hebrews before the Exile, as it is fully attested among Jews of the Mishnaic period. But have we any explicit mention of intercalation in the Bible? There is a plausible reference to intercalation in the description of Hezekiah's celebration of the Passover in the second instead of the first month in 2 Chr. xxx." [pp. 256-257]

"We must seek an explanation of Jeroboam's action [1 Kings xii 32-33] elsewhere. It may best be found in the intercalation of a month in that year. We may, furthermore, infer from the circumstances of this intercalation that the people of Israel were already familiar with the deferment of a religious date by one month - that, in fact, the practice was by no means new." [p. 258]

"There is, then, evidence, both implicit and also, I have suggested, explicit, that intercalation was carried out in Israel already in the early period of the Monarchy. What were the methods employed?" [p. 259]

"I have suggested that the principal method by which the pre-Exilic Israelites adjusted their lunar calendar to the tropic year cannot have been observation of the sun or measurement of the lengths of daylight. The alternative method was observation of the heliacal risings and settings of certain fixed stars. This was both reliable and precise, for the length of the mean stellar year corresponds almost exactly to that of the tropic year". [p. 267]

"The stellar year is measured by observation of heliacal risings and settings. One stellar year began with a heliacal rising or setting at the new moon of either the spring or the autumn month. In subsequent years, if this heliacal rising or setting occurred before the end of the period of twelve lunations or during the first nine days of the following lunation, there was no need to intercalate a month. Whenever this was not so, an extra month was inserted. In the course of eight years intercalation would take place in the 1st, 4th and 7th years. A nineteen year cycle is, however, more accurate and more regular over a continuous period of time; in the course of nineteen years, intercalation would take place in the 1st, 4th, 7th, 9th, 12th, 15th and 18th years. This, it should be noted, tallies exactly with the Metonic system, and other systems of intercalation - including that in regular use in Babylonia in the 4th century B.C." [p. 273]

J. B. Segal, "Intercalation and the Hebrew Calendar",
Vetus Testamentum, Vol. 7, Fasc. 3. (Jul., 1957), pp. 250-307

So, if Jeremiah, Daniel, Ezra, Nehemiah, etc. lived among peoples that intercalated their lunisolar calendars, what method of intercalation was used by the Hebrews in the 7th, 6th and 5th century B.C.? Uncertain. But whatever technique had been used, it was likely influenced by the Egyptians and Babylonians, refined during captivity in Babylon, and seemingly has changed very little since (albeit the historical evidence is scant). Hypothetically then, experimentally extrapolating backwards using today's technique and applying it to pre-exilic and exilic periods may have some merit.

Adequacy of extrapolation from modern Hebrew intercalation

Daniel's prophecy of 69 and 70 weeks was given in multiples of seven years between specific datable events. No more, no less.

In engineering, "accuracy" means to get a correct answer, "resolution" means to get an exact answer, and "precision" means to get the same answer - these are three different concepts. Consider that the circumference of a circle divided by its diameter is "Pi", and commensurately:

Pi = 3.14 is an "accurate" answer (a correct result).
Pi = 3.14159 has greater "resolution" (more exact to within 5 decimal places).
Pi = 1.234 computed repeatedly (without variation) has "precision", though lacking "accuracy".

Daniel gives an accurate prophecy with a resolution of a year, and as its fulfillment can be forensically reexamined with the same answer, it is also precise.

Daniel's prophecy of 69 weeks was given in terms of whole years (69 "weeks" of years from a specific event to another specific event), and consequently resolution to within a year of fulfillment is required, and resolution to within a year of fulfillment is demonstrated. Conversely, the gross inaccuracy and failure of other explanations is likewise demonstrated, and while some purport a resolution to within a day they nonetheless get the entire year, month and particulars of the events wrong. By analogy again, computing "Pi" = 1.23456789 has more resolution (it is a more exact number, yes) than 3.14, but it is still an incorrect answer and no amount of exactness compensates for being wrong.

While scripture and history often provides date resolution to within a month (e.g. Artazerxes decree to Ezra, Ezra 7:8-25) and occasionally to within a day (e.g. crucifixion of Jesus on Passover) for events associated with fulfillment of Daniel's prophecies, the prophecy as given demands resolution only to within a year and Daniel is silent on any greater resolution.

Accuracy to within a year is both sufficient and necessary, whereas accuracy to within a month or day is likewise sufficient but not necessary (even though desirable and sought). Intercalation as used herein obtains the highest possible accuracy of chronologies and dates, and demonstrates that within resolution of a year, the accuracy is sufficient and unimpaired by intercalation variations of greater resolution.

In other words, any intercalation method (even though it likely has varied somewhat) that retained an accuracy of a few months over hundreds of years (as it demonstrably did) is sufficient to verify Daniel's prophecies.



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(last updated April 22 2014)