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 Lunar 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.
Both the Babylonians and Hebrews employed solar-lunar calendrics of 12 months of alternating duration of 30 and 29 days:
Nisanu 1 Aiaru 2 Tishri 1/7 Heshvan 2/8 Simanu 3 Duzu 4 Kislev 3/9 Tevet 4/10 Abu 5 Ululu 6 Shevat 5/11 Adar 6/12 Tashritu 7 Arahsamnu 8 Nisan 7/1 Iyar 8/2 Kislimnu 9 Tebetu 10 Sivan 9/3 Tammuz 10/4 Shabatu 11 Addaru 12 Av 11/5 Elul 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 "c/s" following each month's name, where "c" is that month's number in the civil calender and "s" is that month's number in the sacred calender. So the Hebrew side of the table (reading left to right, then next row) shows Tishri, Heshvan, Kislev and Tevet as the first 4 months of the civil calendar, with Nisan, Iyar, Sivan and Tammuz as the first 4 months of the sacred 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 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.
new reg common deficient -1 (30 => 29) common regular common complete +1 (29 => 30) embolismic deficient -1 (30 => 29) +1 (29 => 30) +29 embolismic regular +1 (29 => 30) +29 embolismic complete +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 they 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 the 19-year cycle, in which the 7 embolismic years were, for the:
- 4th centry B.C. Babylonian's: years 1, 4, 7, 9, 12, 15, and 18
- 3rd century A.D. Hebrew's: years 3, 6, 8, 11, 14, 17, and 19
The following table shows 9 19-year cycles of embolismic years and intercalated months for the Babylonian and Hebrew calendar systems, respectively, from the end of Judah through the rise of Nebuchadnezzar and the transition from Xerxes to Artaxerxes, covering the range of dates used by Jeremiah (Fall of Judah timeline) and Daniel (Artaxerxes I timeline). Additional cycles 10 - 14 are also shown for the Babylonian calendar to demonstrate its gradual settling into a regular intercalary pattern early 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 used a very similar computation. The merits of hypothetically extrapolating these intercalations is discussed further (see Hypothetical extrapolation of Hebrew intercalated months).
The hypothetical Hebrew embolismic or intercalary years for all 9 of the Hebrew 19-year cycles (right half of the table) are those embolismic years reported by the Fourmilab Calendar Converter1, while the actual Babylonian embolismic or intercalary years (left half of the table) were excerpted from "Babylonian Chronology, 626 B.C.-A.D. 75"2 tables for Nabopolassar - Artaxerxes II:
- Fourmilab Calendar Converter
- 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
By inspection of Parker & Dubberstein's tables, as of the 4th century B.C. the Babylonians were generally using a second Addaru (Addaru II) for their embolismic month, and had settled into a regular pattern of intercalating the 1st, 4th, 7th, 9th 12th, 15th, and 18th years of every 19. Note in the table's center columns the 19-year intercalary sequence is juxtaposed for both the Babylonian and the (hypothetical) Hebrew calendars.
Earlier, however, a second Ululu (Ululu II) was often intercalated instead of Addaru II, and occasionally 'irregular' (outside the 1st, 4th, 7th, 9th 12th, 15th, and 18th years) intercalations of either Addaru II or Ululu II were done. These variations are highlighted in the table as follows:
625 606 587 568 549 530 511 492 473 454 435 416 397 378 1 3136 3155 3174 3193 3212 3231 3250 3269 3288 624 605 586 567 548 529 510 491 472 453 434 415 396 377 2 3137 3156 3175 3194 3213 3232 3251 3270 3289 623 604 585 566 547 528 509 490 471 452 433 414 395 376 3 3138 3157 3176 3195 3214 3233 3252 3271 3290 622 603 584 565 546 527 508 489 470 451 432 413 394 375 4 3139 3158 3177 3196 3215 3234 3253 3272 3291 621 602 583 564 545 526 507 488 469 450 431 412 393 374 5 3140 3159 3178 3197 3216 3235 3254 3273 3292 620 601 582 563 544 525 506 487 468 449 430 411 392 373 6 3141 3160 3179 3198 3217 3236 3255 3274 3293 619 600 581 562 543 524 505 486 467 448 429 410 391 372 7 3142 3161 3180 3199 3218 3237 3256 3275 3294 618 599 580 561 542 523 504 485 466 447 428 409 390 371 8 3143 3162 3181 3200 3219 3238 3257 3276 3295 617 598 579 560 541 522 503 484 465 446 427 408 389 370 9 3144 3163 3182 3201 3220 3239 3258 3277 3296 616 597 578 559 540 521 502 483 464 445 426 407 388 369 10 3145 3164 3183 3202 3221 3240 3259 3278 3297 615 596 577 558 539 520 501 482 463 444 425 406 387 368 11 3146 3165 3184 3203 3222 3241 3260 3279 3298 614 595 576 557 538 519 500 481 462 443 424 405 386 367 12 3147 3166 3185 3204 3223 3242 3261 3280 3299 613 594 575 556 537 518 499 480 461 442 423 404 385 366 13 3148 3167 3186 3205 3224 3243 3262 3281 3300 612 593 574 555 536 517 498 479 460 441 422 403 384 365 14 3149 3168 3187 3206 3225 3244 3263 3282 3301 611 592 573 554 535 516 497 478 459 440 421 402 383 364 15 3150 3169 3188 3207 3226 3245 3264 3283 3302 610 591 572 553 534 515 496 477 458 439 420 401 382 363 16 3151 3170 3189 3208 3227 3246 3265 3284 3303 609 590 571 552 533 514 495 476 457 438 419 400 381 362 17 3152 3171 3190 3209 3228 3247 3266 3285 3304 608 589 570 551 532 513 494 475 456 437 418 399 380 361 18 3153 3172 3191 3210 3229 3248 3267 3286 3305 607 588 569 550 531 512 493 474 455 436 417 398 379 360 19 3154 3173 3192 3211 3230 3249 3268 3287 3306 Above, Hebrew years are Anno Mundi (A.M. with an epoch 1 Tishri 1 A.M. = -3761 Oct 7 Julian) and the Babylonian years are Julian B.C.
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):
Josephus, Wars of the Jews, Book 2 Chapter 17, paragraphs 8-9:
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.
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
- 3, 6, 8, 11, 14, 17, and 19 years, yields Julian 66 A.D. August 16 which falls on Saturday (as Josephus reported).
- 2, 5, 7, 10, 13, 16, and 18 years, yields Julian 66 A.D. September 16 which falls on Tuesday (not sabbath).
- 1, 4, 6, 9, 12, 15, and 17 years, also yields Julian 66 A.D. September 16 which falls on Tuesday (not sabbath).
The death of Herod the Great establishes another synchronism (between a Jewish fast and a lunar eclipse) reported by Josephus:
"Antiquities of the Jews", Josephus - Book XVII, Chapter 6:
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.
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:
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:
- Pre-exilic Hebrews had essentially a lunisolar calendar and such calendars must be intercalated else they quickly become useless.
- Pre-exilic Hebrews were in contact with other cultures that progressively intercalated with increasing precision since 2100 B.C. upto 540 B.C. when post-exilic Hebrews adopted much of Babylonian calendrics.
- The Rabbis as late as 2nd century B.C. had achieved precision in their intercalary calculations, and methods since then (though codified and published) have changed very little.
The narrow, specific purpose here is twofold, to argue that:
- some form of intercalation was in fact used by the Hebrews prior to the exile; and
- 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.
Siegfried H. Horn, "Chronology of Ezra 7", Seventh-day Adventist Theological Seminary, 1953.
"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)
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.
J. B. Segal, "Intercalation and the Hebrew Calendar", Vetus Testamentum, Vol. 7, Fasc. 3. (Jul., 1957), pp. 250-307:
"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." (p258)
"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?" (p259)
"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". (p267)
"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." (p273)
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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