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Dan 
Posted: Apr 3 2013, 06:50 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
We've had several proposals on this forum for new solar calendars. But so far, nobody has done a lunisolar calendar (although Wendy has sort of done so with a new algorithm for Easter). So, I'll take up the challenge, by finding a way to make the Hebrew Calendar a little more accurate.
The existing calendar is based on three cycles: The Week Is 7 days long, ending with Shabbat. This won't change, so there's nothing to discuss. The Month Each month begins with a new moon. Originally, this was based on observation, but today, it's based on a calculated synodic month of 29 days, 12 hours, and 793 halakim (1080ths of an hour), which is exactly 765433/25920 days. That's 29.53059413580247 in decimal (25;644X49724972.... dozenal). For comparison, the astronomical synodic month (based on USNO data) is 29.5305873949 days (25;644X3157B283). So, a Hebrew month is 582 ms (3;43 Tm) too long. That's impressive accuracy, but still, it adds up to an hour every 500 years or so. The Year The Hebrew year is based on a strict Metonic cycle of 19 years = 235 months (that is, an intercalary month [Adar I] is added 7 times every 19 years). The average length of a year is therefore 235/19 * 765433/25920 = 35975351/98496 = 365.24682220597794. That's 6.4 minutes longer than an astronomical equinoctial year. The error adds up to a day every 219 years. This is an improvement over the Julian calendar, with 1 day of error every 131 years, but large enough to be problematic. The Combined Cycle The calendar repeats itself every 689 472 years. This figure is exactly equal to 8 527 680 months, 35 975 351 weeks, or 251 827 457 days. The first step in developing the new calendar is to establish new values for these cycles. 
Dan 
Posted: Apr 3 2013, 07:05 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
How long should a month be?
The two criteria I shall use to determine the mean length of a calendar month are:
The candidates for the fractional part of the number of days in a month are thus:

Dan 
Posted: Apr 4 2013, 12:36 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Note, however, that the length of a day is gradually getting longer, and this causes the number of days in a lunar month to decrease. Dr. Irv Bromberg's Rectified Hebrew calendar deals with this phenomenon by making each successive month about 27 Âµs (3;4 _{4}Tm) shorter than the last.
I, however, will ignore it, except to give myself more flexibility in determining the length of the molad interval. As I calculated in another thread based on USNO data, the approximate time of the new moon is the Julian date 2.0051826865596922e10 * x^{2} + 29.530586740228046 * x + 2451550.0952949869 where x is the number of months that have elapsed since Y2K. Taking the derivative gives the length of a month as: 29.530586740228046  4.0103653731193845e10 * x The rate of decrease works out to 34.65 Âµs per month, slightly higher than calculated by Bromberg. Because the length of a month is decreasing, and the current approximation is slightly too long anyway, then the requirement of "a better approximation than the status quo" gives a wider interval of allowed approximations in the future. For my reference point, I will choose Rosh Hashana of the Jewish year 6000 (using the existing calendar). This corresponds to the gregorian date 22390930, and is 2965 synodic months (239.72 Metonic years) after Y2K. By the formula above, the length of a month then will be 29.53058555115471 days. The length of the approximated month, then, should be between 29.530576966506953 and 29.53059413580247 days. And the valid (with denominators no more than *1000) fractional approximations for the noninteger part of the number are:

Dan 
Posted: Apr 4 2013, 01:04 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
An alternative way to ask the question is: How many weeks should be in the mean calendar month? This might not necessarily have a onetoone correspondence with the previous list, due to the arbitrary limit on denominators.
Using the same calculations as the previous post, the approximated month should be between 4.218653852358136 and 4.218656305114639 weeks. Fractions that fall into this range are:
Converting to day fractions by adding 4, multiplying by 7, and subtracting 29, gives:
Which is a strict subset of the previous set. It doesn't give us any additional options. 
Dan 
Posted: Apr 4 2013, 04:20 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The other natural cycle that needs to be approximated is the solar year.
A year can be defined by the time between two equinoxes, or two solstices, or an average thereof. For the Hebrew calendar, the event that matters is the March equinox. This is because the intercalary month is explicitly intended to ensure that Pesach (Passover) doesn't occur too early. Based on the formula I derived in the Calendar Calculations thread from this table, the length of an vernal equinoctial year is 365.24236968356126  5.7533554809197085e08 * x, where x is the number of years since the March 20, 2000 equinox. The Gregorian calendar (which I shall use as a minimum standard of accuracy) approximates the year as 365.2425 days. Again, we have an existing approximation that's too long, for a natural cycle that's decreasing in length, so in the future, the range of valid approximations will be wider. Using the vernal equinox of the Jewish year 5999 (Gregorian 2239) as the reference point, the extrapolated astronomical length of a year will be 365.24235593304167 days. In order to meet the requirement of being more accurate than the Gregorian calendar, the approximated year will need to be between 365.2422118660833 and 365.2425 days. Let's also require the denominator to be less than the Gregorian calendar's 400. That gives us the following choices for the noninteger part of the number:
If the calculation is done in terms of weeks instead, the valid remainders are 8/33, 71/293, and 86/355. 
Dan 
Posted: Apr 4 2013, 05:08 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The posts above give 17 possible approximations for the synodic month, and 11 possible approximations for the equinoctial year. When these are treated as independent cycles, that gives 187 possible combinations.
To choose between them, I shall pick the one whose full repeating cycle (least common multiple of year, month, and week) is the shortest. Month Year Repeating Cycle (29+x) (365+x) Years Months Weeks Days 347/654 63/260 56420 697818 2943853 20606971 347/654 70/289 180047 2226870 9394395 65760765 529/997 71/293 205393 2540356 10716888 75018216 373/703 71/293 253445 3134677 13224120 92568840 876/1651 71/293 291535 3605784 15211560 106480920 555/1046 8/33 419727 5191298 21900301 153302107 668/1259 8/33 505197 6248417 26359911 184519377 347/654 8/33 637329 7882662 33254227 232779589 347/654 71/293 808387 9998352 42179592 295257144 529/997 8/33 971586 12016841 50694918 354864426 347/654 86/355 979445 12114042 51104957 357734699 Unsurprisingly, the shortest repeating cycle has the simplest synodic month approximation. Surprisingly, it does not have the simplest (8/33) year approximation, but a 260year cycle instead. 
Dan 
Posted: Apr 5 2013, 04:46 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
To make sure I didn't miss any better choices by arbitrarily limiting the denominators, I just wrote a computer program to help calculate:
Running the command with the arguments 1728 and 399 gives the same result as calculated earlier:
By allowing denominators up to *1000 for both, I get a shorter cycle:


Dan 
Posted: Apr 5 2013, 01:01 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Increasing the limit to *10000 reveals an even shorter cycle:


Dan 
Posted: Apr 6 2013, 01:54 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The 1803year cycle seems to be the best one possible with the YEAR_MIN and MONTH_MIN values listed above. Let's see if we can improve this by increasing the error tolerance just a tad. In the program above, the range of acceptable values for a month has a range of about 0.445 helek. Let's expand this to half a helek by reducing MONTH_MIN to 29.530574845679013. This is equivalent to assuming that the current Hebrew calendar approximation is correctly rounded, but rounded up. By applying the same idea to the Gregorian approximation of the year, the YEAR_MIN should be 365.24125. (Otherwise, it would be more accurate without the 400year leap year exception.) The best result I can get from my program is
The approximation of the synodic month is just 121 ms less than the value used by the existing Hebrew calendar. The approximation of the year is about 70 seconds too short, so will accumulate 1 day of error every 1230 years (until the earth slows down to match). This is 6 times as accurate as the existing calendar, and 9 times as accurate as the Julian calendar. It is, however, less accurate than the Gregorian calendar. Despite this, it's probably "good enough", so it's the cycle I'll use for the rest of the calculations. 

Dan 
Posted: Apr 7 2013, 03:57 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The intervals between molads and equinoxes having been established, the next question to answer is when they are.
Returning to the quadratic approximations of these events: MoladJD = 2.0051826865596922e10 * x^{2} + 29.530586740228046 * x + 2451550.0952949869 where x = number of synodic months since 20000106 EquinoxJD = 2.8766777404598542e08 * x^{2} + 365.24236968356126 * x + 2451623.8110382501 where x = number of years since 20000302 Now, pick arbitrary x's on which to base the calendar. Choosing two examples at random: May 15, 1948
By adding 639 approximated months (of 29.53059273422562 days) and 52 approximated years (of 365.24155405405406 days), we arrive at the Y2K values:
June 7, 1967
Adding 404 months and 33 years to get the corresponding Y2Kbased values:

Dan 
Posted: Apr 8 2013, 04:17 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
So far, all calculations have been done with Julian dates (with the day starting at 12:00) at the UTC timezone. It would be more convenient for Hebrew calendar calculations to have the day start at sunset (or 18:00) Jerusalem mean time. The longitude of the Temple Mount is 35.2354Â° E, which works out to a UTC offset of +2:20:56.5. So, 18:00 Jerusalem mean time is equivalent to 15:39:03.5 UTC.
Define the Jerusalem Julian Date (JJD) as the Julian Date minus 0.1521238425925926. This adjusts for the 3:39:03.5 difference between 12:00 UTC and 18:00 JMT. Using the 1967 reference point from the previous post, the dates of the first molad and vernal equinox after Y2K are:
The Y2K molad and equinox times expressed in terms of ticks are:

Dan 
Posted: Apr 12 2013, 04:25 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The next question to ask is "On what date does the year begin?" Assuming we preserve the existing calendar structure with ten of the months having fixed lengths, a simplistic answer is "The year begins exactly 163 days after the first night of Pesach." Of course, that raises the obvious question of when Pesach is.
This year (2013/5773), Pesach started on March 26. This happens to currently be the earliest possible date. Moreover, it was the secondtolast time that Pesach will ever start on March 26; the next time this will happen will be in 2089/5849, after which calendar drift will push it to the 27th and later. In the past, Pesach could start on earlier dates:
If the current calendar rules were indeed designed by Hillel II, then March 19 would be the earliest Gregorian date on which Pesach ever started since their adoption. This could be a day before the vernal equinox. But for the sake of simplicity, I shall assume that Pesach must start after the vernal equinox. The Molad of Tishri can then be calculated as follows:
The adjustments will be covered later. 
Dan 
Posted: Apr 14 2013, 03:58 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The Molad of Tishri may occur on any day of the week. However, for reasons of Jewish law that are outside the scope of this thread, it's inconvenient for Rosh Hashana to fall on a Sunday, Wednesday, or Friday. So, if this would happen, we delay it by a day. The calculation of the reformed Rosh Hashana data, expressed in Python code, is:
However, for technical reasons, some more adjustments may still be required... 

Dan 
Posted: Apr 14 2013, 04:58 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
In the current Hebrew calendar, there are 3 intercalation points:
The remaining ten months have a fixed length of 295 days. Therefore, there are 6 possible lengths of a year:
Using the calculation in my previous post, the frequency count of calendar year lengths over the full 592year cycle are:
As in the existing Hebrew calendar, the illegal year lengths will be "fixed" by adding more postponement rules. First, let's address the 356day years. These occur when the year (anno mundi) modulo 592 falls in the set {45, 65, 72, 92, 143, 163, 170, 241, 312, 319, 339, 390, 410, 417, 437, 488, 515, 559, 586}, and start on a Tuesday. We can shorten the year to a valid length of 355, 354, or 353 days by postponing Rosh Hashana to Wednesday, Thursday, or Friday, respectively. Since Rosh Hashana cannot occur on Wednesday or Friday, the only choice is Thursday. So, Rosh Hashana will be postponed by 2 days. After making this change, the distribution of year lengths becomes:
The 382day years occur when the anno mundi year, modulo 592, is 210 or 457. These years start Thurdays. We could increase their length to a legal 384 days by making Rosh Hashana 2 days earlier, but months aren't supposed to start before the molad. Instead, we will postpone the start of the following years, i.e., 211 and 458 of the 592year cycle. These years would otherwise start on Monday and be 355 days long. Postponing Rosh Hashana to Tuesday (which is valid) makes them 354 days long (which is also valid). The final Rosh Hashana computation with all the postponement rules in place is:
and the frequency count of year lengths is:


Kodegadulo 
Posted: Apr 14 2013, 05:53 PM

Obsessive poster Group: Moderators Posts: 4,187 Member No.: 606 Joined: 10September 11 
Wow talk about complexity! And I thought the Gregorian calendar was a nightmare...

Dan 
Posted: Apr 14 2013, 07:22 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
The Gregorian Calendar is simple because it's a purely solar calendar and doesn't have to track the phases of the moon. Also, its month/year structure is completely independent of the week; i.e., it doesn't have rules to prevent holidays from falling on inconvenient days. OTOH, if you include the computation of Easter, the Gregorian Calendar is arguably more complex than the Hebrew one. 

Dan 
Posted: Apr 14 2013, 07:45 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Most of the time, the "new" Rosh Hashana date coincides with the "old" date. The years between Gregorian 2000 and 2100 (inclusive) when it doesn't are:
Most of the differences are 1 month. These occur in years 1, 4, 9, and 12 of the Metonic cycle (Y mod 19) when Rosh Hashana (and thus, the preceding Pesach) occurs "late" under the current calendar. Note that the year 2095/5856 will be the first time that a onemonth difference between the two calendars will affect year 4 in the Metonic cycle. For now, only 3 years in the 19year cycle start "late":
There are also 1day and 2day differences. The ones adjacent to the leap year changes are due to the postponement rules. 

dgoodmaniii 
Posted: Apr 14 2013, 08:40 PM


Dozens Demigod Group: Admin Posts: 1,927 Member No.: 554 Joined: 21May 09 
I think that's right; the Gregorian calendar is a simpler way to keep the seasons at the same time of year. It's equally complex as the Hebrew calendar, at least, when we add in religious holidays, presidential birthdays, and the like. 

Dan 
Posted: Apr 15 2013, 01:21 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
It doesn't do an ideal job of that, though. Assuming that the 365.2425day average year length is correct, the standard deviation of the equinox date/time is 10 h 41 min 28 s. If, however, the 97 leap days in each 400year cycle were distributed as evenly as possible, then the standard deviation would be only 6 h 55 min 41 s (almost the same as a continuous uniform distribution with a range of 1 day). However, it's simple in the sense that it's easy to distinguish leap years from common years â€” if you use baseten. (In dozenal, the 4year rule is simplified, but the 100year and 400year rules become a more awkward 84 and 294.)
I work in the private sector, so have to work on presidential birthdays. But you've got a point. (For example, with our "floating companydeclared holiday" in December.) 

Dan 
Posted: Apr 15 2013, 02:41 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
It's possible to simplify the calculation of the Rosh Hashana date. For example, instead of explicitly calculating the equinox:


Dan 
Posted: Apr 16 2013, 06:05 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
As stated earlier, the average length of a year in the 592year cycle is 365+143/592 = 365.24155405405406 days. This is 81.729729... seconds shorter than the mean Gregorian year of 365.2425 days, and 70.4 seconds less than the true equinoctial year (365.242368931396 days) derived from the astronomical data posted earlier.
This means that the calendar drifts one day earlier every 1227 years. This is an improvement over the existing Metonic calendar (365.24682220597794 days), which drifts one day later every 225 years. However, that the drift is backwards means that eventually, Pesach will occur before the equinox. The drift can be lessened by using a longer cycle. The shortest cycle that gives a (slightly) more accurate mean length of the year is 1556 yr = 19245 mo = 81188 wk = 568316 d, for a year of 365.24164524421593 days. A calendar based on this cycle would drive one day earlier every 1382 years. This is a marginal improvement over the 592year cycle, and may not be worth bothering with. A large accuracy improvement can be obtained by using the cycle 1803 yr = 22300 mo = 94076 wk = 658532 d. As shown earlier, this is the shortest cycle which has both a more accurate approximation of the synodic month than the existing Hebrew calendar, and a more accurate approximation of the equinoctial year than the Gregorian calendar. The approximated year is 365.24237382140876 days, a mere 422 ms longer than the current equinoctial year, and exactly matched it in 1928. The approximated month is 29.530582959641254 days, only 320 ms less than the current true value of 29.53058667437678 days (derived from the USNO data), which will become exact in 2762. 
Oschkar 
Posted: Jun 3 2017, 07:45 AM

Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
How bad, comparatively, would a slightly shorter cycle of 1040 years = 12863 months = 379852 days be? Assuming it was perfectly in sync on the June solstice of 9564 BC, how far would it have drifted by now?

Dan 
Posted: Jun 3 2017, 03:16 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Well, let's see. On such a calendar,
This is very good accuracy. But a 379852cycle has the disadvantage of not being a whole number of 7day weeks. When this is considered, your proposal has a cycle of 7280 years. 

Dan 
Posted: Jun 5 2017, 06:20 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Since the 592year cycle has the flaw of making Passover slowly drift earlier, I'll redo the calculations with the 1803year cycle. Recall that the average year length is 365+437/1803 days and the average month length is 29+2958/5575 days.
To simplify the math, I will divide the day into 80 413 800 "ticks" — the lowest common multiple of 24 (hours in a day), 1803 (denominator for year length), and 5575 (denominator for month length). One hour is 3 350 575 ticks, and one second is 930.7152777777... ticks, so the tick approximates a millisecond. 
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