Calculating the new moon and creating a calendar of months

Calculating the new moon and creating a calendar of months:

As explained in the previous Halachos, calculating the Molad and the expected time of the appearance of the new moon, was part of the process of the sanctification even in the times of the Sanhedrin. However, in the times of the Sanhedrin, the calculation at times had to be verified by witnesses, while in today’s times we rely only on the calculation of the new moon. Likewise, the form of calculation in the times of the Sanhedrin differed from the calculation of today, in previous times the purpose of the calculation was to establish when the moon can be visible [See A and B], while in today’s times the purpose of the calculation is to establish the time of the Molad. [See A and C] Even today, the new month is not based on calculation alone but takes into account certain rules of dates that we cannot allow Rosh Chodesh to fall on, and hence not by all months does Rosh Chodesh coincide with the Moled. [See D-E] This Halacha will explain the calculations that are done and the dates on which Rosh Chodesh is avoided. Understanding the details of this section is the foundation for having a grasp on understanding the makeup of the Jewish calendar.

A. Cheshbon Haamiti and Cheshbon Haemtzai:[1]
There are two types of calculations that exist regarding the new moon. One is called Cheshbon Haamiti while the second is called Cheshbon Haemtzai or Ibbur. In Cheshbon Haamiti one calculates the precise time of when the moon will become visible to earth. In Cheshbon Haemtzai one calculates the Molad, which is exactly 29 days 12 hours and 793 Chalakim from the previous Molad. So while Cheshbon Haemtzai calculates the start of the new orbit, Cheshbon Haamiti calculates the time that the moon will be visible. The Cheshbon Haemtzai is a simple calculation, and setting Rosh Chodesh based on this can be understood even by young children in a few days. However Cheshbon Haamiti is a very deep and difficult calculation.[2] In the times of the Sanhedrin the calculation was done in the form of Cheshbon Haamiti, while in today’s times we calculate using Cheshbon Haemtzai. It is not possible to establish the Cheshbon Haamiti without first knowing the Cheshbon Haemtzai, which is the Molad, and hence in the times of the Sanhedrin, both calculations had to be done. Nevertheless, Rosh Chodesh was only set based on the calculation of Cheshbon Haamiti.[3]      

B. Cheshbon Haamiti-How to calculate when the moon will be visible:
Knowing whether the moon can be visible on the 30th day is dependent on when the Molad of the new moon will occur, as well as the distance of the moon from the sun. At times, the Molad ends up being between the 30th and 31st day, which effects that the moon cannot be visible anytime throughout the 30th day, and at times it falls on the 29th-30th day, in which case the moon can be visible on the 30th.[4] However, in addition to having knowledge of the Molad, knowing whether the moon would be visible required knowledge of the positioning of the moon under the sun. This matter is elaborated in length in the Rambam’s Kiddush Hachodesh chapters 11-17.

C. Cheshbon Haemtzai-How to calculate the Molad:[5]
The calculation of Cheshbon Haemtzai, which is the precise time that the moon’s new orbit will begin, otherwise known as the Molad, is explained in length in Rambam Kiddush Hachodesh chapter 6. Once one has established the Molad of a single month he can then establish the Molad of every month post or prior to that month by adding or subtracting exactly 29 days 12 hours and 793 Chalakim from the known Molad, as there is always precisely this amount of time between orbits. Thus, having down even one Molad is the key to calculating the Molads for all the other months for an infinite amount of years.[6] 

D. Malei and Chaseir-Why in some months are there 29 days while in others 30 days?[7]

Definition of Malei Chaseir:[8] A month in which Rosh Chodesh is on the 30th day is called a Chaseir/lacking month, being that the month only had 29 days. A month in which Rosh Chodesh was on the 31st day is called a Malei/full month, being that the month had 30 days.

In times of testimony:[9] In times of testimony, being that the moon had to be seen on the night of the 30th in order for the new month to be sanctified, therefore, whether the month would have 29 days or 30 days was determined by whether the new moon was seen on the 30th day. If witnesses came and their testimony was accepted then the month would have only 29 days. If however witnesses did not come, or their testimony was declined, then the month would have thirty days. The length of days in each month was hence held in question until the end of the month.[10] The only time in which it was automatically predetermined that the month would have thirty days is if the Sages calculation showed that the moon could not be seen on the 30th day.[11] Likewise, in the event that the moon was not seen for one month after the other, such as due to clouds or positioning, and hence no witnesses came forward, the future months are set not in accordance to testimony of the moon but rather in accordance to calculation, having one month Malei and one month Chaseir, or two months Malei or two months Chaseir, in accordance to the calculation.[12]  

In times of calendar:[13] The orbit of the moon takes 29.5+ days as explained above in Halacha 3B, and in today’s time when we don’t set the months based on testimony, the month is set based on the Molad, which is the start of the new orbit, as will be explained in Halacha F. Now, since it is not possible to begin Rosh Chodesh in middle of the day of the Molad, and every month must begin in proximity to the Molad, therefore the calendar was set up in a way that when a given month is Malei the next month is Chaseir, and when a given month is Chaseir the next month is Malei. Thus, throughout the months of the year we follow a cycle of Malei, Chaseir, Malei, Chaseir, Male, Chaseir etc. This rotation of adding and subtracting one day to the month, every other month, allows Rosh Chodesh to remain in proximity to the Molad. If however we had all Malei months or all Chaseir Months then after a few months the extra half day of the Molad [29.5 days] would cause the Molad to fall many days before the 30th [if we did many months Malei] or many days after the 30th [if we did every month Chaseir]. See the chart below! The above cycle has two months of exception in which these two months can be either both Malei or both Chaseir, as will be explained next!

Molad calendar with one month Malei and one month Chaseir

Date of month Molad
Nissan 30th=1st Iyar Midday of 30th
Iyar 31st=1st Sivan Beginning of 31st 
Sivan 30th=1st Tamuz Midday of 30th
Tamuz 31st=1st Elul Beginning of 31st 

Molad calendar with all months Malei

Date of month Molad
Nissan 31st=1st Iyar Beginning of 31st 
Iyar 31st=1st Sivan Midday of 30th
Sivan 31st=1st Tamuz Beginning of 30th
Tamuz 31st=1st Av Middle of 29th   
Av 31st=1st Elul Beginning of 29th
Elul 31st=1st Tishrei Midday of 28th
Tishrei 31st=1st Cheshvan Beginning of 28th  
Cheshvan 31st=1st Kisleiv Middle of 27th   

 Molad calendar with all months Chaseir

Date of month Molad
Nissan 31st=1st Iyar Midday of 30th
Iyar 31st=1st Sivan Beginning of 31st  
Sivan 31st=1st Tamuz Middle of 31st    
Tamuz 31st=1st Av Beginning of 32nd
Av 31st=1st Elul Midday of 32nd
Elul 31st=1st Tishrei Beginning of 33rd
Tishrei 31st=1st Cheshvan Middle of 33rd  

 E. The order of months that are Malei/Chaseir and the months of Cheshvan/Kisleiv:[14]

The month of Teves is always Chaseir and from it begins the cycle of one month Malei and one month Chaseir until the month of Cheshvan. Starting from the month of Cheshvan this cycle ends, depending on the year. In some years both Cheshvan and Kisleiv are Malei, thus having three consecutive Malei months [Tishrei, Cheshvan, and Kisleiv]. This type of year is called a Shleima/complete year. In this year the lunar year contains 355 days. In other years both Cheshvan and Kisleiv are Chaseir, thus having three consecutive Chaseir months [Cheshvan, Kisleiv, and Teves]. This type of year is called a Chaseira/incomplete year. In this year the lunar year contains 353 days. In other years Cheshvan is Chaseir and Kisleiv is Malei, thus following regular pattern of Chaseir/Malei throughout all 12 months of the year. This type of year is called a Kesidran [proper order] year. In this year the lunar year contains 354 days. In no year is Cheshvan Malei and Kisleiv Chaseir. In total there are ten predetermined fixed months of Chaseir/Malei, five months being always Malei and five months being always Chaseir, with two months remaining interchangeable. [Look at Table]

A leap year:[15] In a leap year the first Adar is always Malei while the second Adar is always Chaseir. This breaks the cycle of Chaseir/Malei that was started in Teves, as we now have two consecutive months of Malei [Shevat and Adar 1]

The reason Cheshvan and Kisleiv are interchangeable: The reason that the two months of Cheshvan and Kisleiv do not follow the set pattern is because of the Molad. The Molad is not exactly 29.5 days, but is 29 days 12 hours and 44 minutes. Now, if we would make six months of the year Malei and 6 months Chaseir it would cause the Molad to fall 44 minutes later each month. This would eventually cause the Molad to fall many days after Rosh Chodesh and the entire proximity of the Molad to Rosh Chodesh would be lost. Because of this surplus of minutes to the Molad we need to have some years contain 8 Malei months, and some years contain 8 Chaseir months and some years contain an equal amount of Malei and Chaseir months.[16] Another reason for the interchangeable months of Cheshvon and Kisleiv is to allow the calendar to position Rosh Hashanah on its proper day of the week. In Halacha F-G we will learn that Rosh Hashanah can never fall on Sunday, Wednesday and Friday, and cannot fall on the Molad Zakein. If we set the two months of Cheshvan/Kisleiv to always follow the pattern of Kesidron or to always be either both Malei or both Chaseir, then it would cause Rosh Hashanah to fall on one of the undesirable days.[17]

Determining whether Cheshvan/Kisleiv is Shaleim/Chaseir/Kesidran:[18] There are various methods that can be used to determine the patterns of the months of Cheshvan and Kisleiv. One method is by determining the number of weekdays between the falling of the previous Rosh Hashanah and the falling of next year’s Rosh Hashanah. If it is a non-leap year and there are four weekday’s in-between the two Rosh Hashanahs then the two months are Malei. If there are three weekdays in-between then the two months are Kesidran, and if there are two weekdays in-between the two months are Chaseir. If it is a leap year and there are four weekdays in-between then the two months are Chaseir. If there are five weekdays in-between then the two months are Kesidran, and if there are six weekdays in-between then the two months are Malei.[19] Another method of calculation [which is the same form of calculation to determine the next years Rosh Hashanah] is to take the weekday of last Rosh Hashanah and by a non-leap year to add two, three, or four weekdays to that day to determine next year’s Rosh Hashanah. If any of the added days falls on either Sunday, Wednesday or Friday then it eliminates the possibility of the months following that order of days. This will usually leave us with two possibilities remaining for the months of Cheshvon and Kisleiv. One then sees if any of the remaining two possibilities cause Rosh Hashanah to fall on the Molad Zakein, or prior to the Molad, and if it does then that possibility is also eliminated, thus leaving us with only one possibility for the order of the months of that year.[20]

The list of the Malei and Chaseir months:[21]

Month 29/30
Tishrei 30
Cheshvan Variable
Kisleiv Variable
Teves 29
Shevat 30
Adar 29
Adar 1 30
Adar 2 29
Nissan 30
Iyar 29
Sivan 30
Tamuz 29
Av 30
Elul 29

 

Always Malei [30 days] Always Chaseir [29 days] Interchangeable
Tishrei Teves Marcheshvan
Shevat Adar Kisleiv
Nissan Iyar  
Sivan Tamuz  
Av Elul  
Adar 1 Adar 2  

 F. Days on which Rosh Chodesh will never fall:[22]

The Sages established that Rosh Hashanah, which is Rosh Chodesh Tishrei, cannot fall out on certain days of the week even if the Molad is on that day. It cannot fall on Sunday, Wednesday or Friday. In the event that the Molad falls on one of these days, Rosh Chodesh is pushed off a further day.[23] Some Rishonim[24] explain the reason for the avoidance of the above weekdays is because doing so would cause Yom Kippur to fall a day before or a day after Shabbos and they did not want there to be two consecutive days of prohibited food preparation. Likewise, they did not want Hoshana Rabbah to fall on Shabbos which would nullify the Mitzvah of Arava. The Rambam[25] however explains the reason this is done is in order to have Rosh Chodesh fall on the Cheshbon Haamiti of when the moon will be seen and not on the Molad, which is the Cheshbon Haemtzai. This decree effected that the Rosh Chodesh of every month cannot fall on certain days, as once we force Rosh Chodesh Tishrei to only fall on certain days, and the remaining months follow an automatic pattern of Malei and Chaseir, it effects that every Rosh Chodesh has only certain days on which it can fall. Practically, the following are the dates in which Rosh Chodesh can and cannot fall on:

  • All days of Rosh Chodesh refer to the first day of Rosh Chodesh, in a month in which Rosh Chodesh is two days. This is with exception to Marcheshvan.[26]
  • Rosh Chodesh Nissan: Can only fall on Sunday, Tuesday, Thursday, Shabbos. Cannot fall on Monday, Wednesday, and Friday.
  • Rosh Chodesh Iyar: Can only fall on Monday, Tuesday, Thursday, Shabbos. Cannot fall on Sunday, Wednesday, and Friday.
  • Rosh Chodesh Sivan: Can only fall on Sunday, Tuesday, Thursday, and Friday. Cannot fall on Monday, Wednesday, Shabbos.
  • Rosh Chodesh Tamuz: Can only fall on Sunday, Tuesday, Wednesday, and Friday. Cannot fall on Monday, Thursday, Shabbos.
  • Rosh Chodesh Av: Can only fall on Monday, Wednesday, Friday, Shabbos. Cannot fall on Sunday, Tuesday, and Thursday.
  • Rosh Chodesh Elul: Can only fall on Sunday, Monday, Wednesday, and Friday. Cannot fall on, Tuesday, Thursday, Shabbos.
  • Rosh Chodesh Tishrei: Can only be set on Monday, Tuesday, Thursday, Shabbos. Cannot be set on Sunday, Wednesday, or Friday.
  • Rosh Chodesh MarCheshvan: The first[27] day can only fall on Sunday, Tuesday, Wednesday, and Friday. It cannot fall on Monday, Thursday, Shabbos. Consequently, the second day can only fall on Monday, Wednesday, Thursday, Shabbos.[28]
  • Rosh Chodesh Kisleiv: Can only fall on Sunday, Monday, Tuesday, Wednesday, Thursday, and Friday. Cannot fall on Shabbos.
  • Rosh Chodesh Teves: Can only fall on Sunday, Monday, Tuesday, Wednesday, and Friday. Cannot fall on Thursday or Shabbos.
  • Rosh Chodesh Shevat: Can only fall on Monday, Tuesday, Wednesday, Thursday, Shabbos. Cannot fall on Sunday, or Friday.
  • Rosh Chodesh Adar [Non-leap year]: Can only fall on Monday, Wednesday, Friday, Shabbos. Cannot fall on Sunday, Tuesday, and Thursday.
  • Rosh Chodesh Adar 1 [Leap year]: Can only fall on Monday, Wednesday, Thursday, Shabbos. Cannot be set on Sunday, Tuesday, and Friday.
  • Rosh Chodesh Adar 2 [Leap year]: Can only fall on Monday, Wednesday, Friday, Shabbos. Cannot fall on Sunday, Tuesday, and Thursday.

Chart of days that Rosh Chodesh can/cannot fall on

  Sunday Monday Tuesday Wednesday Thursday Friday Shabbos
Tishrei xxxxxxxxx Ö         Ö         xxxxxxxxx Ö         xxxxxxx Ö        
Marcheshvan xxxxxxxxx Ö         xxxxxxxxxxx Ö         Ö         xxxxxxx Ö        
Kisleiv Ö         Ö         Ö         Ö         Ö         Ö         xxxxxxx
Teves Ö         Ö         Ö         Ö         xxxxxxx Ö         xxxxxxx
Shevat xxxxxxx Ö         Ö         Ö         Ö         xxxxxxx Ö        
Adar xxxxxxx Ö         xxxxxxx Ö         xxxxxxx Ö         Ö        
Adar 1 xxxxxxx Ö         xxxxxxx Ö         Ö         xxxxxxx Ö        
Adar 2 xxxxxxx Ö         xxxxxxx Ö         xxxxxxx Ö         Ö        
Nissan Ö         xxxxxxx Ö         xxxxxxx Ö         xxxxxxx Ö        
Iyar xxxxxxx Ö         Ö         xxxxxxx Ö         xxxxxxx Ö        
Sivan Ö         xxxxxxx Ö         xxxxxxx Ö         Ö         xxxxxxx
Tamuz Ö         xxxxxxx Ö         Ö         xxxxxxx Ö         xxxxxxx
Av xxxxxxx Ö         xxxxxxx Ö         xxxxxxx Ö         Ö        
Elul Ö         Ö         xxxxxxx Ö         xxxxxxx Ö         xxxxxxx

 G. Molad Zakein:[29]

Molad Zakein literally means the old Molad. This refers to when the Molad falls between midday and night of a given day, which is approximately the last six hours of the day. It is called an old Molad because it is falling at the latest possible opportunity of that date. The Molad Zakein has special significance in the calendar of today, as although in general we try to set the day of Rosh Chodesh on the day of the Molad [if it falls on a permissible day], nevertheless this only applies if the Molad falls prior to midday. If however the Molad falls after midday, then we differ Rosh Chodesh until the next day. In other words, we do not set Rosh Chodesh on the day of a Molad Zakein even if it is a day that is permitted to have Rosh Chodesh fall on.[30]

To which months does the above rule apply? The above rule of Molad Zakein only applies to the month of Tishrei, as once we have set the date of Rosh Chodesh Tishrei, we follow from there a set order of Malei and Chaseir, as explained in the previous Halacha.

Exact definition of Molad Zakein: See Halacha 7 for the exact definition of a Molad Zakein, regarding how much time after midday it must fall.

H. The calendar of today:

After having summarized all of the above aspects, one now has all the information taken into account when making a calendar today [i.e. the Molad, Malei/Chaseir, Molad Zakein, and days of avoidance]. The following is the makeup of our calendar based on all the above rules: In general, the calendar of today is based on the Molad, the time when we calculate that the new orbit will begin, and not upon when the moon will be visible. At times the new month is established on the day that the moon will be seen, and at times a day earlier and at times a day later.[31] Likewise, at times Rosh Chodesh falls on the day of the Molad, although most often it falls on the day after the Molad, and in some months even two days after the Molad. The calendar is organized as follows: Once one establishes when the Molad of Tishrei will fall he then has to calculate whether this would cause Rosh Chodesh Tishrei to fall on one of the days in which we do not set Rosh Chodesh Tishrei on, as explained above in F-G. If it does, then we delay Rosh Chodesh Tishrei until the next day. This causes Rosh Chodesh Tishrei to be a day or two later than the actual Molad. Despite this delay however, the calendar is set up in such a way that there is never a month with less than 29 days and never a month with more than 30 days. Thus, in conclusion, every month in our calendar contains 29-30 days and Rosh Chodesh always falls in the closest proximity to the Molad after taking into account the above days of avoidance. For example, if the Molad of the next month falls in the end of the 29th day of the month, we do not establish Rosh Chodesh on the 29th day, but rather on the 30th. Likewise, if the Molad falls on the 30th day of the month but that day is a day that we do not establish Rosh Chodesh on, then we establish Rosh Chodesh on the 31st day of the month. Practically, starting from the month of Tishrei, the calendar is established in an organized fashion of one month of Malei and one month of Chaseir, besides for the months of Cheshvan and Kisleiv, as explained in Halacha E. This system proximates the Molad to Rosh Chodesh as much as possible, while abiding by the rules mentioned in F-G regarding days of avoidance. In our calendar, the Molad always falls anywhere between the 28th-31st day of the month.

Examples of Molad during months in years 5775-5779:

5775: The Molad of the months in year 5775 in some months fell on the 29th day [Molad of Tishrei; Kisleiv; Shevat; Nissan; Sivan; Av which were all after 29 day months] and in some months on the 30th day [Molad of MarCheshvan; Teves; Adar; Iyar; Tamuz; Elul which were all after 30 day months].

5776: The Molad of the months in year 5776 in some months fell on the 28th day [Nissan, Sivan; Av] and in some months on the 29th day [Molad of Tishrei; Teves; Shevat; Adar 1-2; Iyar; Tamuz; Elul; in which some were after 29 day months and some 30 day months] and in some months on the 30th day [Molad of MarCheshvan; Kisleiv].

5777: The Molad of the months in year 5777 in some months falls on the 28th day [Tishrei, Kisleiv] and in some months on the 29th day [Molad of Cheshvan; Teves; Shevat;] and in some months on the 30th day [Molad of Adar; Nissan; Iyar; Sivan; Tamuz; Av; Elul].

5778: The Molad of the months in year 5778 in some months falls on the 30th day [Molad of Tishrei; Cheshvan; Kisleiv; Teves; Shevat; Adar; Nissan; Iyar; Sivan; Av] and in some months on the 31st day [Molad of Tamuz; Elul]

5779: The Molad of the months in year 5778 in some months falls on the 29th day [Molad of Shevat; Adar 2; Nissan; Iyar; Sivan; Tamuz; Av; Elul] and in some months on the 30th day [Molad of Tishrei; Kisleiv; Teves; Adar1; Iyar].

Day of Molad

  Molad on 28th day Molad on 29th day Molad on 30th day Molad on 31st day
5775 0 months 6 months 6 months 0 months
5776 3 months 8 months 2 months 0 months
5777 2 months 3 months 7 months 0 months
5778 0 months 0 months 10 months 2 months
5779 0 months 8 months 5 months 0months

In summary:
In the above five years 30 months had the Molad fall prior to the 30th, while 32 months had the Molad fall on the 30th or later. However only a small minority of months had the first of the month begin on the day of the Molad.

 
Sparks of Chassidus
 

How can Rosh Chodesh be set on a day that is not the Molad?[32]
The entire meaning behind Rosh Chodesh is that on this day the moon began its new cycle [the Molad] or that on this day the moon was seen. How can Rosh Chodesh be set for a day that in truth is not the head of the month, but a day earlier or a day later? How can we have Rosh Chodesh on the 30th if the Molad was on the 28th? The Rebbe explains that while it is true that the physical Molad below in this world is on a fixed date, the spiritual Molad in the upper worlds which corresponds to the physical Molad here below may be on a different date. It is for this reason that Hashem gave over the establishing of the new month to the Sages, as they are able to set Rosh Chodesh in accordance to the spiritual Molad above.

I. Determining the weekday of Rosh Chodesh:

Once you know the day of the week that Rosh Chodesh of the current month fell on you can easily calculate on which day of the week the next Rosh Chodesh will fall, as the 30th day is always Rosh Chodesh, and will hence always fall four weeks later on the next weekday. For example, if Rosh Chodesh Nissan fell on Shabbos then Rosh Chodesh Iyar will fall four weeks later on Sunday. If the next month is Malei then the next weekday is the first day of Rosh Chodesh. If the next month is Chaseir then the next weekday is the only day of Rosh Chodesh.

_________________________________________

[1] See Rambam ibid 6/1; 11/4; Sefer Yeshua Beyisrael on Rambam 5/2

[2] See Rambam ibid 11/4; It is this calculation that the Rambam discusses in length in chapters 11-17

[3] Naaveh Kodesh on Rambam ibid 6/1

[4] See Rambam Kiddush Hachodesh 1/3

[5] Rambam ibid 6

[6] Rambam ibid 6/7

[7] Rambam Kiddush Hachodesh 8/1-6; Tur 427-428; Understanding the calendar p. 65

[8] Rambam ibid 1/4

[9] Rambam Kiddush Hachodesh 1/3-4

[10] Rambam ibid

[11] Rambam ibid 1/7

[12] Rambam ibid 18/8; Tur 427

[13] Rambam 8/4-6

[14] Rambam ibid 8/5-6; Tur 428; Understanding the Jewish calendar chapter 12-13

[15] Rambam ibid; Tur 428

[16] Tur 427

[17] Tur 428

[18] Rambam ibid 8/7

[19] Rambam ibid 8/8; Tur 428

[20] See Tur ibid and Understanding the Jewish calendar chapter 13

[21] Rambam Kiddush Hachodesh 8/5-6

[22] Michaber 428/1-3; Rambam Kiddush Hachodesh 7

[23] Rambam ibid 7/1

[24] Raavad 7/7

[25] Rambam ibid 7/7

[26] Taz 428/2

[27] Taz 428/1; Machatzis Hashekel on M”A 428/2

[28] M”A 428/2; See Machatzos Hashekel ibid

[29] Rambam ibid 7/2; See Torah Sheleima ibid 8/2 [p. 114-115]; Understanding the Jewish Calendar chapter 10

[30] The reason: See Torah Sheleima ibid for various reasons given behind this deferral.

[31] Rambam ibid 5/2; See Rambam Kiddush Hachodesh 1/3 that the period of time that the moon remains in the dark is approximately two days, give or take some. It is in the dark for approximately one day before the orbit is complete and one day after the orbit is complete.

[32] Sichas Acharon Shel Pesach 1953

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