Chronology of the Flood/2/en

Chronology of the Flood

Exegetical Approaches

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One Set of 150 Days Which Includes the First 40 Days

The two mentions of 150 days refer to the same time period and encompass the forty days of rain. The ark landed at the conclusion of these 150 days.  This approach subdivides regarding how the commentators understand the chapter's dating:

Months from creation

All the months mentioned in the verses are calculated according to the yearly calendar, from creation.

Sources:Qumran Scroll 4Q252Fragment 1 Columns 1-2About the Qumran Scrolls, the KaraitesYefet b. Eli Bereshit 8:4Aharon b. Yosef Bereshit 8:3-8Aharon b. Eliyahu Bereshit 8:3-6, rejected approach in Lekach TovBereshit 7:24About R. Toviah b. Eliezer, Ibn EzraBereshit 8:3-5Additional Commentary Bereshit 7:11Additional Commentary Bereshit 8:14Iggeret HaShabbatAbout R. Avraham ibn Ezra, R. Yosef Bekhor ShorBereshit 7:24Bereshit 8:3-10About R. Yosef Bekhor Shor, RidBereshit 7:12Bereshit 8:4About R. Yeshayah of Trani, approach in ChizkuniBereshit 7:17Bereshit 7:24Bereshit 8:3-13About R. Chizkiyah b. Manoach, RambanBereshit 8:4-5About R. Moshe b. Nachman #2,¹ Akeidat YitzchakBereshit 13About R. Yitzchak Arama, AbarbanelBereshit 7:24About R. Yitzchak Abarbanel, SefornoBereshit 7:24About R. Ovadyah Seforno, ShadalBereshit 7:24Bereshit 8:2Bereshit 8:4About R. Shemuel David Luzzatto, Hoil MosheBereshit 7:11About R. Moshe Yitzchak Ashkenazi, R. D"Z HoffmannBereshit 7:21-24About R. David Zvi Hoffmann, U. CassutoBereshit Introduction to the FloodAbout Prof. Umberto Cassuto

Months – These commentators assume that all the months mentioned in the account of the flood are numbered from the beginning of the calendar year.  They disagree, though, regarding whether the first month is Tishrei or Nissan.²

150 and 40 Overlap – This position assumes that the forty days of rain are subsumed in the 150 day period, reading the final verse of Chapter 7 "וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם", as a summary statement of the previously described events rather than a successive event.³

One set of 150 – They similarly assume that the statement "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם" does not mean that the water receded after a second set of 150 days, but rather that it decreased after, or the reduction was first noticeable at the end of, the 150 days previously mentioned.

"וַיִּגְבְּרוּ הַמַּיִם" and "וַיַּחְסְרוּ הַמַּיִם" – Though all these sources agree that there was but one set of 150 days, they disagree in how they understand the relationship between the periods of increasing and receding waters:⁴

• Water first decreased after 150 days ­ - The author of the Qumran Scroll, Yefet the Karaite, Ibn Ezra,⁵ Ramban, and Seforno all understand the phrase "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם" to mean that the water only began to decrease at the end of the 150 days. Until that point the water level was either still rising, or at least, maintaining its height. Seforno suggests that even though the rain stopped after 40 days, the underground sources of water were still open,⁶ causing continued water pressure during this entire period.⁷ Ramban, in contrast, seems to maintain that though the waters did not continue to increase after the fortieth day, due to the great humidity⁸ they retained their height throughout this period.
• Water decreased after the forty days of rain – Most of the other commentators in this approach⁹ assume that the water started to recede right after the rain stopped. The phrase " וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם" simply means that the waters were is a state of strength (relative to normal water heights), not that they were increasing, or even maintaining their level.  R"Y Bekhor Shor, R. Hoffmann and U. Cassuto explain the verse "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם"   to mean not that the water first began to recede at this point but that the decrease was first noticeable to Noach after this period, when the ark landed.¹⁰

Chronology

• Chronological –Those who maintain that the waters maintained their strength throughout the 150 days, read the verses chronologically, with Chapter 7 describing the rising  flood and Chapter 8 detailing the decreasing waters at the end of the 150 days.¹¹
• Achronological – The other commentators, though, assume that verses 8:2-4 (the sending of the wind and closing of the heavenly windows) occurred in the midst of the 150 days that are mentioned in the preceding chapter and that there is an element of achronology in the verses.  Akeidat Yitzchak and Shadal explain that the Torah purposely separated the images of destruction and salvation, detailing the world's collapse in Chapter 7 before describing its rebuilding in Chapter 8.

How do 150 days fit in five months? According to the fixed lunar calendar, months generally alternate between having 29 and 30 days,¹² which means that there can be at most 147-149 days in any 5 month period.¹³  As such, this position must explain how the 150 days fit into the five months between 2/17 when the rain began and 7/17 when the ark rested.   The commentators resolve this issue in different ways:

• Different calendar – Several sources question the assumption that the dates in the story are based on the Hebrew lunar calendar:
• Solar Calendar –  Some commentators¹⁴ assert that the months mentioned relate to the solar calendar. According to this reckoning there are 152 days in five months and the 150 day period ended a couple of days before the ark rested.¹⁵
• Persian/Egyptian – Ibn Ezra suggests, in contrast, that Noach was using a calendar like that of the Egyptians or Persians who would intercalate their year by adding 5 days to a certain month. The total days in the five month period can thus equal or surpass 150.¹⁶
• Lunar calendar based on eyewitnesses – Yefet and Anan the Karaites maintain that Noach did use a lunar calendar, but one in which the months were determined by eyewitnesses rather than calculations.¹⁷ They assume that if no one can see the moon (as was the case when Noach was in the Ark)¹⁸ the default length of a month is thirty days.¹⁹  This allows for exactly 150 days between 2/17 and 7/17.²⁰

• Fixed Hebrew Lunar Calendar – A last group of sources attempt to uphold the idea that the Torah is employing the Rabbinic calendar:
• Round Numbers - The Rid and U. Cassuto solve the problem by simply suggesting that 150 days is a round number. Though, in reality there were fewer days in the five month period, the Torah rounded up to the the next ten.²¹
• Full Leap Year - R. Saadia suggests that one can reach 150 days using a regular fixed lunar calendar if the year was a "full" leap year. In such a year, 4 of the 5 months( Marcheshvan, Kislev, Shevat and Adar I) would all have 30 days. As such 7/17 is the 150th day.²²

How did the ark land so soon? These commentators differ in how they understand how the ark managed to land right after the 150 days of strong waters, if the mountain tops were still covered  by fifteen cubits of water:²³

• Receded over 110 days – According to those who say that the water started to decrease already after the 40 days of rain, this is not an issue,²⁴ and is, in fact, one of the factors that motivate them to explain the verses as they do. Shadal even maintains that by this point the higher mountain tops were totally uncovered, and that 8:5 describes only the revealing of the shorter mountains.

• Receded over 2 days – According to those who maintain that the waters retained their strength throughout, but that a solar calendar was used and thus the 150 period ended by the 15th of the seventh month, there was a period of two days in which the water lessened, allowing the ark to land.
• Miraculous intervention – Ramban, in contrast, is forced to assert that Hashem sent a miraculous wind which decreased the waters substantially on the 150th day itself.²⁵ 

Shifting water levels – A simple reading of the verses suggests that the waters decreased at an extremely varied pace:

• If one posits that the water first decreased after the 150 days, then it went down only 15 cubits in the 2 1/2 months from 7/17 until 10/1 compared to thousands of cubits in the 3 months from then until the land dried on 1/1.²⁶  Ramban explains that there is no reason to think that the water needed to decrease at a steady pace. In addition, according to him, Hashem's intervention on the first day led to a vast decrease in the water, ²⁷ minimizing the difference in the rate of reduction thereafter slightly.²⁸
• Those who posit that the water began to recede after the rain stopped have an even bigger problem, since according to them, initially the water receded only 15 cubits in over 180 days!²⁹   Shadal minimizes the problem by asserting that by the time the ark landed on 7/17, more than 15 cubits of water had already receded, since Mt. Ararat was not the tallest, but rather one of the shorter mountains.³⁰  There would, nonetheless, still seem to be a significant increase in the rate of water reduction in the last 3 months, assuming that even the shorter mountains were a few thousand cubits high.

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם... וַיְשַׁלַּח אֶת הָעֹרֵב"

• Most of these commentators count the 40 days (when the raven was sent)  from the preceding date mentioned, the first of the tenth month, when the mountain tops were visible.  They explain, that despite the fact that land was already visible, the dove could not find a resting place either because it preferred the valleys or the air was too moist.³¹
• The Rid, in contrast, counts the forty days from the landing of the ark on the seventeenth of the seventh month. He assumes that the first dove was sent ten days after the raven, and the final dove twenty-one days later, on the first of the tenth month. It was only then, when the mountaintops were finally revealed that the dove could rest.

Varied Dating

Some months are counted to creation and some are dated in reference to key points during the flood.

Sources:Ramban Bereshit 8:4-5About R. Moshe b. Nachman# 1,³² R. Eliezer AshkenaziMa'asei Hashem Bereshit 26About R. Eliezer Ashkenazi (Ma'asei Hashem) #2

Months – According to Ramban and R. Ashkenazi, the beginning and end of the flood (2/17 and 2/27) are dated to the beginning of the calendar year, but the months mentioned in the middle of the story, the landing of the ark on 2/17 and the revealing of the mountain tops on 10/1, are both dated to the start of the flood.³³ The last date, though, when Noach opens the ark's cover on 1/1, is once again dated to the beginning of the year.  This approach is motivated by a desire to make sense of the various time-frames in the chapter³⁴ but its obvious disadvantage is its inconsistency.³⁵

40 and 150 overlap – Like the above approach, these commentators also view the phrase "וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם", as a summary statement of the previously described events, and assume that the 40 days are included therein.

One set of 150 – Both Ramban and R. Ashkenazi maintain that the verse "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם" does not refer to a new period of 150 days, but to the end of the previously mentioned one.

"וַיִּגְבְּרוּ הַמַּיִם" and "וַיַּחְסְרוּ הַמַּיִם" – These commentators assert that the waters maintained their strength throughout the 150 days, and only began to abate afterwards.³⁶

Chronology – This position does not need to assume any achronology in the verses deescribing the increasing and decreasing waters.

How do 150 days fit in five months? According to this approach there are really six months from the start of the flood until the resting of the ark.³⁷  This easily fits the period of 150 days of increasing waters, even leaving a month over afterwards.

How did the ark land so soon? According to this position, there was a month between the 150 days and the resting of the ark, enough time for the water to recede sufficiently to allow the ark to find a  resting place.

Shifting water levels – As this position reduces the time between the revealing of the mountain tops and the drying of the land to two months,³⁸ it exacerbates the problem of the changing rate of reduction.  It has the waters reducing 15 cubits in about 2.5 months, while they are lowered thousands of cubits in 2 months!

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם... וַיְשַׁלַּח אֶת הָעֹרֵב" – This is dated from the previously mentioned event, the revealing of the mountain tops, which this position maintains occurred on the first of the eleventh month (the tenth month being counted from start of the flood).  As such, the sending of the final dove 61 days later,³⁹ fell on the first of the first month, when the land dried.⁴⁰ This explains why it was only now, that it did not return.

One Set of 150 Days Which Does Not Include the First 40 Days

Both sets of 150 days are combined and in order to fit the separate forty days, the seven months are not counted from the beginning of the year.

Sources:Seder Olam Rabbah4About Seder Olam Rabbah, Bereshit Rabbah33:7About Bereshit Rabbah, Sefer HaArukhs.v. קלAbout R. Natan b. Yechiel, RashiBereshit 7:12Bereshit 8:3-14Ms. Leipzig 1 Bereshit 8:5-6About R. Shelomo Yitzchaki, Lekach TovBereshit 7:24Bereshit 8:3-6Bereshit 8:10-14About R. Toviah b. Eliezer, Rashbamfrom Ms. Oxford 970About R. Shemuel b. Meir, ChizkuniBereshit 7:24Bereshit 8:3-13About R. Chizkiyah b. Manoach, R. Chaim PaltielBereshit 8:6About R. Chaim Paltiel, Tosafot HaShalemBereshit 8:4:8Bereshit 8:4:8 (in footnote)Bereshit 8:5:5About Ba'alei HaTosafot, Minchat YehudaBereshit 8:6About R. Yehuda b. Elazar, Daat ZekeinimBereshit 8:3About Daat Zekeinim, first approach of R. Eliezer AshkenaziMa'asei Hashem Bereshit 26About R. Eliezer Ashkenazi (Ma'asei Hashem)

Distinct 40 and 150 days – This approach assumes that there is no overlap between the description of the flood and the 150 days of strong waters.  The verses follow sequentially, with the forty days of rain preceding the 150 days.

One Set of 150 – This position, nonetheless, agrees that there is only one set of 150 days and that the waters receded at the end of the previously mentioned 150 days (and not after a new set)..

"וַיִּגְבְּרוּ הַמַּיִם" and "וַיַּחְסְרוּ הַמַּיִם" – Though all these commentators maintain that the water started to abate only after the  150 days, they differ regarding what happened during the initial period itself:

• Waters retained strength – According to Seder Olam Rabbah and the Arukh the water stopped increasing after the forty days of rain, but maintained its level throughout.⁴¹ According to this, the phrase "וַיִּגְבְּרוּ הַמַּיִם" does not mean that the waters increased but that they stayed in their strength.
• Waters increased – According to Ba'alei HaTosafot, in contrast, even after the rain stopped, the water continued to increase throughout the 150 days. He might maintain that the waters continued to emerge from underground.⁴²

How do 190 days fit into 5 months? The verses state that the ark rested already on the 17th of the seventh month.   According to this position, though, at this point the waters were still at their height (and would be through the 29th of the eighth month)!⁴³  This forces these commentators to reinterpret this date and to assume that it is dated to an event in the flood narrative rather than the beginning of the calendar year:

• End of rain – Most of these commentators⁴⁴ assert that 7/17 is being counted from the month in which the rain stopped.  As such, it is really equivalent to 9/17,⁴⁵ which gives ample time for the waters to decrease before the ark landed.
• Beginning of Rain – The Arukh and R. Yitzchak (in a gloss found in Ms. Leipzig 1 of Rashi's commentary) get to the same date but through a different calculation. They assert that 7/17 means to count seven full months from the date the rain began (Thus, 2/17 plus 7 months brings one to 9/17).⁴⁶

The seventh and tenth months – According to these commentators the seventh and tenth months are not being counted from the beginning of the year. Three opinions are brought by the exegetes regarding these months:

• Most of these commentators explain that the seventeenth of the seventh month is really the seventeenth of the ninth month to the beginning of the year and then the 150 and 40 days fit before the ark landed.  They hold that the tenth month is the eleventh month to the beginning of the year.⁴⁷ This approach subdivides regarding how one gets to this calculation:
• Seder Olam Rabbah, Bereshit Rabbah, Lekach Tov, and Rashi hold that the seventh month is being counted from the time in which the rain stopped while the tenth month is being counted from the time in which the rain started. This inconsistency causes other exegetes to raise alternatives.
• The Arukh and R. Yitzchak in a gloss on Ms. Leipzig 1 of Rashi explain that there are seven full months from the start of the rain until the ark landed and the tenth month is also counted from the start of the rain.⁴⁸
• According to one of the Ba'alei HaTosafot the year of the flood was a leap year and both months were counted from the end of the rain.  This leaves enough time for the 150 and first 40 days to end before the ark landed but leaves two months before the end of the year from when the tops of the mountains appeared.⁴⁹
• Ba'alei HaTosafot quote an approach that the seventh month is really only the eighth month to the start of the year. This approach would say that the tenth month is really the eleventh month and both months are being counted from the start of the rain. Ba'alei HaTosafot reject this explanation since the ark would have rested before the water started decreasing. However, according to the approach it is possible that the ark got stuck and landed as Rasag quoted in Yefet suggests. This would be simpler to say if during the 150 days the water wasn't increasing anymore but just stayed in their place or were already decreasing.
• According to R. Eliezer Ashkenazi's approach both months are being counted from the end of the rain and the seventh month is the ninth month and the tenth month is the twelfth month.

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" – Three options are brought explaining from when the forty days are counted which affect when the tops of the mountains appeared:

• According to Seder Olam Rabbah, Rashbam and perhaps R. Tam the days are counted from the time the water started to abate on the first of Sivan. According to this approach the dove was sent for the last time on the day that the tops of the mountains appeared and that is were it landed. This interpretation forces the commentators to explain that the tenth month is really the tenth month from the start of the rain and not like the seventh month which is from the end of the rain in order that the dove could land after the mountains appear. The takers of this approach have to assume that before the tops of the mountains appeared there were trees that appeared and the dove could bring back an olive tree leaf.
• The Arukh, Rashi and Lekach Tov hold that the forty days are counted from the first of the tenth month which is mentioned immediately before in the verses.⁵⁰ According to them the dove landed on the first of the first month when "חָרְבוּ פְּנֵי הָאֲדָמָה" and therefore the tenth month can be no later since the dove would have landed as soon as the water disappeared. R. Tam questions this approach why the dove could not land on the tops of the mountains which appeared already when the dove was sent. R. Elyakim suggests that the dove could only land in valleys and not on mountains. Another answer suggested by Ba'alei HaTosafot is that when it says "וְלֹא מָצְאָה הַיּוֹנָה מָנוֹחַ" it means that the dove did not find any food.
• R. Tam according to R. Chayim Paltiel's version and R. Eliezer Ashkenazi hold that the forty days started when the ark landed which was a date that Noach knew when it was as oppose to the day in which the water started decreasing.  According to R. Tam the tenth month is really the eleventh month and he has to explain why the dove did not land when the mountains first appeared. R. Eliezer Ashkenazi holds that the tenth month is counted from the end of the rain like the seventh month. This would cause the dove to land before the mountains appeared. It is possible to explain like the Rid above that there was more time for the sending of the raven.

Shifting water levels – Seder Olam Rabbah calculates that if from the first of the ninth month when the water started abating until the first of the eleventh month when the mountains appeared the water decreased 15 ammot then the water went down an ammah every four days.  Therefore, on the 17th of the ninth month when the ark landed it was covered with 11 ammot of water.

Two sets of 150 Days Which Include the Second 40 Days

Sources:RadakBereshit 7:24Bereshit 8:3-14About R. David Kimchi

Relationship between numbers of days – Radak writes that for forty days there was rain and then there were two sets of 150 days one in which the water was increasing and one in which the water was decreasing. The second 150 days includes the second forty days but does not include the 21 days of sending the birds.  These numbers come out to 361 days that Noach was in the ark.

The seventh and tenth months – According to Radak the water stopped increasing on the first of the ninth month with the conclusion of the first 40 and 150 days. The seventh month in which the ark landed is being counted from the end of the rain and really is the ninth month to the beginning of the year. The ark landed on the seventeenth of the ninth month after the water started decreasing.  Radak also writes that the tenth month is the tenth month to the beginning of the flood and the tops of the mountains appeared on the first of the eleventh month.

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" – Radak writes that the second forty days started when the ark landed and not when the water started decreasing as Noach only knew when the ark landed because he felt it.  According to this explanation the 21 days were during the second 150 days in which the water was decreasing and not like his previous explanation that they were at the end of the flood after the second set of 150 days ended. This approach has to explain why the dove did not land already on the first time it was sent which was after the tops of the mountains appeared.  Ba'alei HaTosafot explain this based on Radak's explanation that Noach sent the dove which only rests once it finishes the mission it was given and therefore it only landed after it gave Noach a sign that the water was lowering by giving him an olive leaf.

¹ Ramban raises two possible understandings of the chapter.  This approach represents his second option; see below for his first possibility.

² The Rid, Ramban, Akeidat Yitzchak, Abarbanel and R. David Zvi Hoffmann all date them to Tishrei while Ibn Ezra, R"Y Bekhor Shor, and Shadal date them to Nissan.  The difference of opinion relates to the controversy in Bavli Rosh HaShanah 10b-11a regarding whether the world was created in Tishrei or Nissan.

³ They do regard the later mention of forty days as its own independent unit.  There the text writes, "וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" but it is hard to suggests that this refers to the previously mentioned 40 days of rain, since there are several other time markers in between.  See point below.

⁴ The various opinions relate to a number of other issues raised by the text, including the landing of the ark, chronology of the verses, and the shifting pace of water reduction, which will be discussed in the bullets below.

⁵ Ibn Ezra does not refer to this phrase explicitly but also agrees that the waters first abated at the end of 150 days.

⁶ Chapter 7:17 speaks of the flood lasting for 40 days, but it is only in 8:2 that the text mentions the closing of the heavenly windows and subterranean sources of water.  This leads Seforno (and Ramban) to suggest that there were two stages.  However, the continuation of 8:2 reads, "וַיִּכָּלֵא הַגֶּשֶׁם מִן הַשָּׁמָיִם" which leads others to suggest that the two events happened simultaneously.

⁷ His words are slightly ambiguous, and can be interpreted to mean either that the subterranean waters caused an increase in the water throughout this period or that they simply caused a continuous pressure, preventing the water from standing still or decreasing.

⁸ This was caused by the fact that the heavenly windows and subterranean water sources were still open.

⁹ See R"Y Bekhor Shor, Akeidat Yitzchak, Abarbanel,, Shadal, R. D"Z Hoffmann, and U. Cassuto.

¹⁰ Until then, when the ark was still moving and Noach could only see water and sky he had no way of knowing if the water level was increasing, staying the same or decreasing.

¹¹ Ramban might have been motivated to read the verses in this manner due to his general tendency to posit that the Torah is written in chronological order.

¹² There are two potential exceptions to this rule. In a "regular" year, Marcheshvan has 29 days and Kislev has 30 days, but there are some "full" years in which both have 30 days, and some "missing" years in which both have only 29 days. In addition, in a leap year, Adar I precedes the regular Adar and has 30 days.

¹³ This depends on whether the beginning and ending dates (in our case, 2/17 and 7/17) are both included in the count, whether the year is counted from Tishrei or Nissan, and whether it is a "full" or "regular" year:

• If one assumes that the flood dates are counted from Nissan, and one includes both the beginning and end dates in the count, there are 148 days.  If, though, only one end date is included there are 147 days.
• If one posits that the year began in Tishrei, included the final date of 7/17 and that both Marcheshvan and Kislev had 30 days, there would be 149 days in this five month period.  On the other hand, if one does not include the end date of 7/17 and assumes that this is a regular year in which all the months alternate between 29-30 days, there would be only 147 days.Finally, if only one or the other assumption is true, one would get to 148 days.

¹⁴ See one of the possibilities raised by R. Saadia Gaon in his arguments against the Karaitic position regarding the establishing of new months, the opinion rejected by Ibn Ezra, and the opinion of R. Yehuda HaParsi (quoted in Ibn Ezra's Iggeret HaShabbat).  The Qumran sect also use a solar calendar  which is divided into four quarters, in each of which two out of every three months has thirty days and the third month has thirty one days. They thus calculate that the 150 days lasted through the 14th of the seventh month, several days before the ark rested on 7/17. 

¹⁵ According to this approach, the 150 day period mentioned is not equivalent to the five month period and ends a couple of days before it. The Karaites, Yefet and Anan, reject this possibility for this very reason. [See Ibn Ezra as well.] They assume that there have to be exactly 150 days between 2/17 and 7/17, asserting that only the receding waters, which began after the 150 days, could possibly explain why the ark was able to finally rest. [They assume that had the water decreased earlier, the ark would have rested earlier.] See below, though, for other explanations of how and when the water level fell and how this impacted the landing of the ark.
See also R. Ashkenazi below, who makes an opposite argument, claiming that if the 150 days were equivalent to the five month period, it would be superfluous for the Torah to give both time markers.

¹⁶ Based on the math, one would think that such a calendar would allow for more than 150 days in the five month period.  Yet, Ibn Ezra himself seems to assume that such a calendar would equal exactly 150 days, since right beforehand and in his comments elsewhere, he rejects the possibility that Noach used a solar calendar because that would add two days to the 150 given in the text.

¹⁷ In their arguments with the Rabbis, they even point to this story as proof of their position that moths are determined by eyewitnesses, claiming that if one uses calculations the dating in the story does not work.

¹⁸ R. Saadia questions this assumption, suggesting that after the rain ceased on the fortieth day, the moon should have been visible.  Moreover, there is no reason to assume that the months mentioned were calculated by Noach, rather than Hashem, who does not need the moon to calculate time.

¹⁹ R. Yosef Bekhor Shor, Seforno, and Hoil Moshe also maintain that Noach was using a thirty day per month calendar, but do not elaborate regarding how they reach this conclusion.  Hoil Moshe suggests that in the time of Noach people might not have been very adept at mathematical and astronomical calculations, leading to an inexact calendar.

²⁰ This count does not include the end date, 7/17. Alternatively, according to Aharon b. Yosef and Aharon b. Eliyahu the Karaites, though 7/17 is included and there are really 151 days, the Torah was not bothered by the extra day and rounded it to 150.

²¹ U. Cassuto points out that this is common usage.  Often, a person might speak of a three month period as 90 days, even though it is not an exact calculation.  The Rid points to other examples in Torah, such as the punishment of lashes which is referred to as 40 lashes, even though there are only 39. 

²² This end date in included in the count. Cf. R D"Z Hoffmann who counts the days similarly.

²³

Since the ark was only 30 cubits high, this would mean that half of it was submerged in water!

²⁴ There was a full 110 days for the water to recede enough for the ark to land. According to Lekach Tov and R. Yosef Bekhor Shor, the water decreased at an even pace (of 1 cubit every 12 days) over the 180 days until the mountain tops were seen on 10/1. As such, in the 110 days before the ark landed, the water decreased nine cubits, allowing the ark to rest while partially submerged in the remaining 6 cubits of water.

²⁵ Ramban compares this to Yam Suf where it says: "וַיּוֹלֶךְ י"י אֶת הַיָּם בְּרוּחַ קָדִים עַזָּה כׇּל הַלַּיְלָה וַיָּשֶׂם אֶת הַיָּם לֶחָרָבָה וַיִּבָּקְעוּ הַמָּיִם".

²⁶ In these last three month the water wouldhave to have receded the height of an entire mountain!

²⁷ In his comments to 8:4, he suggests that the water fell by about 9 cubits on the first day, and assumes that the ark was submerged in 6 cubits when it rested on the mountain.  In his comments to 8:5, however, he offers a variation of the approach and asserts that since Mt. Ararat was one of the shorter mountain, the reduction in water level need not be limited to fifteen cubits and was much more than that.  He suggests that many of the taller mountain tops would have already been revealed, and had Noach been in their vicinity, he might have landed earlier.  It was only the shorter mountains in the region of Ararat that were first uncovered on the first of the tenth month.

²⁸ As the water needed to decrease the height of an entire mountain in the period from 10/1 to 1/1, while it needed to decrease just the difference in height between the taller and shorter mountains in the period between 7/17 and 10/1, there would still seem to have been a vast increase in pace at the end.

²⁹ There was 110 days until the ark rested on the 17th of the seventh month and then another 73 days until the mountain tops were revealed on the first of the tenth month.  See note above that R"Y Bekhor Shor assumes that for these 180 days there was a steady rate of reduction, 1 cubit every 12 days.  The next 90 days (until the land dried on the first of the first month), though would have to see a reduction of at least 100 cubits per day, if a mountain's height of water still needed to be lowered!

³⁰ In other areas, the tops of the tallest mountains would have already been revealed.  It was only the shorter mountain tops that were first revealed on 10/1. See Ramban in the note above who explains similarly.

³¹ According to this opinion, there is still about a month between the landing of the dove and the drying of the land and opening of the ark's cover on the first of the first month. Perhaps the dove's departure indicated the drying of treetops, but not the ground.

³² Ramban offers two different ways of understanding the chapters.  This position represents his first possibility; see above for his second option.

³³ R. Saadia Gaon brings support for the idea that one event can be dated to another, rather than to the beginning of the year, from Yirmeyahu 28:17. There Yirmeyahu prophesies that Chananyah will die by the end of the year, but he dies in the "seventh month", which would already be the new year. R. Saadia, therefore concludes that the seventh month is dated from Yirmeyahu's prophecy. This prooftext is very questionable, though, as one could easily posit Yirmeyahu counted the new year from whatever he called the first month, not the seventh.

³⁴ See below how such dating easily fits 150 days into five months and explains how the ark could land on the mountain tops though they would seem to have still been covered by 15 cubits of water.

³⁵ R. Asheknazi attempts to defend the approach, pointing out

³⁶ R. Ashkenazi even posits that the rain continued (albeit sporadically) throughout this 150 day period.

³⁷ See first point above, that both dates, 7/17 and 10/1, are dated to the beginning o fteh flood and not the calendar year.

³⁸ See above that they maintain that  "the first of the tenth month" is dated to the beginning of the flood, making it really the first of the eleventh calendar month. At the same time, the day of the drying of the land, 1/1 is dated to the calendar year, leaving but two months between the two dates.

³⁹ Apparently, R. Ashkenazi assumes that there was no waiting time between the sending of the crow and the dove.

⁴⁰ This date, unlike the previous one, is dated to the calendar year.

⁴¹ The Arukh explains that the verse, "וַיָּשֻׁבוּ הַמַּיִם מֵעַל הָאָרֶץ הָלוֹךְ וָשׁוֹב" mans that after the rain stopped, the waters went back and forth, neither increasing nor decreasing during this period. Seder Olam Rabbah explains that during these 150 days the wicked were each being sentenced for their actions.

⁴² Cf. Seforno above.

⁴³ In a regular year in which months alternate between 29 and 30 days, the 150 days of strong waters would first conclude on the 29th of the eighth month, more than a month after the ark landed!
Seder Olam Rabbah writes, "עד מתי חמשים ומאת יום עד א' בסיון" leading many to assumes that he calculated the end of the 150 days to the first of the ninth month rather than the 29th of the eighth.  It is possible, though, that he simply means that the 150 days ended before the first of the month (עד א' בסיו is not inclusive).
Rashi apparently tries to solve the problem in two different ways in his commentary (leading to an internal contradiction in Rashi).  In his comments to 7:12, he posits that the last day of the 150  really was the first of the ninth month, because the first day of rain was not included.  Elsewhere (on 8:3) though, he asserts that the first of the month is the day after the 150.

⁴⁴ See Seder Olam Rabbah, Bereshit Rabbah, Lekach Tov, Rashi, Ba'alei HaTosafot and R. Ashkenazi.

⁴⁵ Since the rain stopped at the end of the third month, one would count seven months starting from the third month (Kislev, Tevet, Shevat, Adar, Nissan, Iyyar, Sivan).

⁴⁶ A rejected opinion in Baalei haTosafot also counts from the beginning of the rain, but uses the method above, counting seven months form the second month, thereby reaching a date of 8/17.  As this is before the waters began to decrease, one must posit (as R. Saadia suggests) that the ark did not really land but rather somehow got stuck, despite the high water level.  If so, though, it was still submerged, making this improbable.  Moreover, it is not clear why the Torah would share such information if the event did not mark a difference in the flood levels. 
This opinion would work just as well if one did not recalculate the months (as during both 7/17 and 8/17 the waters were still strong), and is only motivaed by a desire to have both of the middle dates of the flood (7/17 and 101) be dated to the same point.  See the point below regarding the date of 10/1.

⁴⁷ See the point about "וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" regarding the reasoning for this explanation.

⁴⁸ They explain the language "בַּחֹדֶשׁ הַשְּׁבִיעִי בְּשִׁבְעָה עָשָׂר יוֹם לַחֹדֶשׁ" to mean that the ark landed seven months after the start of the rain on the seventeenth of the month.

⁴⁹ See later the approaches about "וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם".

⁵⁰ In the Leipzig manuscript of Rashi this explanation of the forty and 21 days and the tenth month is crossed out and instead on the margins an explanation matching Seder Olam Rabbah is given. The scribe writes that this is how the text was in the manuscript of R. Shemayah the student of Rashi. It is unclear if R. Shemayah changed this under Rashi's direction or not. After giving Rashi's approach and the questions on it Daat Zekeinim writes "לכך פירש"י" and brings this other version.  However, later on he brings Rashi explaining like the first version.  Perhaps, this is a scribal error and it is supposed to read "לכך פיר"ש" referring to R. Shemayah.