Esteemed CALNDRL members,
The Metonic Cycle, probably discovered in Babylon and either popularized or discovered independently by the Greek astronomer Meton in the 5th C. BCE, consists of a period of 235 synodic months whose duration is just two hours longer than 19 vernal equinox years, so that the Sun and the Moon return to (almost) the same position in the sky at the end of each cycle. The Metonic Cycle has formed the basis of several historical calendars, most (or all) of which consist of lunar months, where the month is deemed to begin on the day when the lunar crescent first appears (briefly) at dawn after a couple of days during which the Moon is hidden by the Sun. Such calendars are observationbased rather than rulebased. (See https://www.hermetic.ch/cal_stud/lunarcal/types.htm ) The definition of a rulebased calendar consisting of days, months and years states that a year consists of a (possibly variable) number of months, and a month consists of a (possibly variable) number of days. By associating an empirical day with a specific date (consisting of day, month and yearnumber), the date of any other day can be calculated by following a given set of rules (which are part of the definition of the calendar). A calendar may be evaluated according to various criteria, one of which is how closely the average number of days in a calendar year is to the vernal equinox year (365.2424 mean solar days). Another is how closely the average number of days in a calendar month is to the synodic month (29.35059 m.s.d.). (See https://www.hermetic.ch/cal_stud/lunarcal/luncal.htm ) Most calendars are either solar or lunar, but not both, and either their months accord with lunations or their years accord with seasonal years. Some calendars satisfy both criteria, and among these, some are observationbased and some are rulebased. I hereby propose a rulebased calendar inspired by the Metonic Cycle and which satisfies both criteria. I call this (provisionally) the RuleBased Metonic Calendar (RBMC). (After the months have been named, etc., the calendar itself will be renamed.) The average number of days in a year in the RBMC is 365.2434, and the average number of days in a month is 29.53032. The former differs from the vernal equinox year by just 0.0003% and the latter differs from the synodic month by just 0.0009%. My purpose in this post is (a) to stake a claim to the definition of this calendar and (b) to ask the members of CALNDRL to verify these values for the average number of days in the calendar month and the calendar year. The definition given below is incomplete, but I state the rules for the number of months in a year and the number of days in a month sufficient to confirm or refute the values I gave above for the average number of days in a year and the average number of days in a month. First I define a Simple RBMC, as follows: 1. Years are numbered by integers: 1, 0, 1, 2, 3, ... 2. A year consists either of 12 months or 13 months, and the months are numbered from 1 to 12 or 13. 3. A day (more exactly, a nychthemeron) begins at midnight local time. 4. Oddnumbered months have 29 days and evennumbered months have 30 days (thus 29, 30, 29, 30 and so on). 5. In any period of 19 years, seven of those years have a 13th month (a socalled 'intercalary' month), which always has 30 days. Rule #5 is what makes this calendar 'Metonic'. For the purpose of this post it is not necessary to specify which years in the 19year period have an intercalary month; this will be done later. The definition of the RBMC adds a Rule #6, given a few paragraphs below. Proposition 1: The average number of days in a calendar year in the Simple RBMC is 365.0526. Proof: Clearly the structure of the calendar repeats every 19 years, so it is sufficient to consider a period of 19 years. The twelve years without an intercalary month have 6*29 + 6*30 = 354 days. The seven years with an intercalary month have 354 + 30 = 384 days. So the total number of days in 19 years is 12*354 + 7*384 = 6936 days. (Note that this is 4 days less than the Metonic Cycle period of 6940 days.) Thus the average number of days in a year is 6936/19 = 365.0526 (to 4 decimal places). Proposition 2: The average number of days in a calendar month in the Simple RBMC is 29.51489. Proof: The number of days in 19 years is 6936. There are 12*12 + 7*13 = 235 months in 19 years, so the average length of a month is 6936/235 = 29.51489 days. Thus the Simple RBMC will get out of sync with the vernal equinox year by one day each 1/(365.2424365.0526) = 5.27 years, and it will get out of sync with lunations by one day each 1/(29.5305929.51489) = 63.69 months. Clearly the Simple RBMC is not an accurate solilunar calendar. So the RBMC has Rule #6: 6. The 7th month normally has 29 days, but it has 30 days when either (a) the year number is divisible by 5 but not by 95 or (b) the year number is divisible by 760. Proposition 3: The average number of days in a calendar year in the RBMC is 365.2434. Proof: Since 760 is divisible by 5, the structure of the calendar repeats every 760 years, so it is sufficient to consider a period of 760 years. When Rule #6 is not applied, there are 6936*(760/19) = 277,440 days = in 760 years. The application of Rule #6 adds 760/5  760/95 + 1 = 145 days, so the total number of days in 760 years = 277,440 + 145 = 277,585 days. Thus the average number of days in a year is 277,585/760 = 365.2434. Proposition 4: The average number of days in a calendar month in the RBMC is 29.53032. Proof: The number of days in 760 years is 277,585. There are (12*12 + 7*13)*(760/19) = 9400 months in 760 years, so the average length of a month is 277,585/9400 = 29.53032 days. Thus the RBMC will get out of sync with the vernal equinox year by one day every 1/(365.2434  365.2424) = 1000 years, and it will get out of sync with lunations by one day every 1/(29.5305929.53052) = 14,286 months. Since the average number of months per year is 9400/760 = 12.3684, the RBMC will get out of sync with lunations by one day every 14,285/12.3864 = 1153 years. This degree of accuracy in according with the lunar and solar cycles is quite good, or at least, not too bad. Finally we may note that 760 years in the RBMC consists of exactly 39,655 7day weeks. Comments are welcome. Regards, Peter Meyer 
Dear List,
Please see in this concern this articles: One Day Every 216 Years, Three Days Each Decan. Rebirth Cycle of Pythagoras, Phoenix, Hazon Gabriel, and Christian Dogma of Resurrection Can Be Explained by the Metonic Cycle. https://www.academia.edu/21890244/One_Day_Every_216_Years_Three_Days_Each_Decan._Rebirth_Cycle_of_Pythagoras_Phoenix_Hazon_Gabriel_and_Christian_Dogma_of_Resurrection_Can_Be_Explained_by_the_Metonic_Cycle https://www.academia.edu/2035201/Astronomischstatistische_Analyse_von_Kreissymbolen_bronzezeitlicher_Goldh%C3%BCte Best regards Sepp Rothwangl, CEP 239.610 SEAC Fellow [hidden email] www.calendersign.com Am 14.03.2019 um 07:15 schrieb Peter Meyer <[hidden email]>: > Esteemed CALNDRL members, > > The Metonic Cycle, probably discovered in Babylon and either popularized or discovered independently by the Greek astronomer Meton in the 5th C. BCE, consists of a period of 235 synodic months whose duration is just two hours longer than 19 vernal equinox years, so that the Sun and the Moon return to (almost) the same position in the sky at the end of each cycle. > > The Metonic Cycle has formed the basis of several historical calendars, most (or all) of which consist of lunar months, where the month is deemed to begin on the day when the lunar crescent first appears (briefly) at dawn after a couple of days during which the Moon is hidden by the Sun. Such calendars are observationbased rather than rulebased. (See https://www.hermetic.ch/cal_stud/lunarcal/types.htm ) > > The definition of a rulebased calendar consisting of days, months and years states that a year consists of a (possibly variable) number of months, and a month consists of a (possibly variable) number of days. By associating an empirical day with a specific date (consisting of day, month and yearnumber), the date of any other day can be calculated by following a given set of rules (which are part of the definition of the calendar). > > A calendar may be evaluated according to various criteria, one of which is how closely the average number of days in a calendar year is to the vernal equinox year (365.2424 mean solar days). Another is how closely the average number of days in a calendar month is to the synodic month (29.35059 m.s.d.). (See https://www.hermetic.ch/cal_stud/lunarcal/luncal.htm ) Most calendars are either solar or lunar, but not both, and either their months accord with lunations or their years accord with seasonal years. Some calendars satisfy both criteria, and among these, some are observationbased and some are rulebased. > > I hereby propose a rulebased calendar inspired by the Metonic Cycle and which satisfies both criteria. I call this (provisionally) the RuleBased Metonic Calendar (RBMC). (After the months have been named, etc., the calendar itself will be renamed.) The average number of days in a year in the RBMC is 365.2434, and the average number of days in a month is 29.53032. The former differs from the vernal equinox year by just 0.0003% and the latter differs from the synodic month by just 0.0009%. My purpose in this post is (a) to stake a claim to the definition of this calendar and (b) to ask the members of CALNDRL to verify these values for the average number of days in the calendar month and the calendar year. > > The definition given below is incomplete, but I state the rules for the number of months in a year and the number of days in a month sufficient to confirm or refute the values I gave above for the average number of days in a year and the average number of days in a month. > > First I define a Simple RBMC, as follows: > > 1. Years are numbered by integers: 1, 0, 1, 2, 3, ... > 2. A year consists either of 12 months or 13 months, and the months are numbered from 1 to 12 or 13. > 3. A day (more exactly, a nychthemeron) begins at midnight local time. > 4. Oddnumbered months have 29 days and evennumbered months have 30 days (thus 29, 30, 29, 30 and so on). > 5. In any period of 19 years, seven of those years have a 13th month (a socalled 'intercalary' month), which always has 30 days. > > Rule #5 is what makes this calendar 'Metonic'. For the purpose of this post it is not necessary to specify which years in the 19year period have an intercalary month; this will be done later. > > The definition of the RBMC adds a Rule #6, given a few paragraphs below. > > Proposition 1: The average number of days in a calendar year in the Simple RBMC is 365.0526. > Proof: Clearly the structure of the calendar repeats every 19 years, so it is sufficient to consider a period of 19 years. The twelve years without an intercalary month have 6*29 + 6*30 = 354 days. The seven years with an intercalary month have 354 + 30 = 384 days. So the total number of days in 19 years is 12*354 + 7*384 = 6936 days. (Note that this is 4 days less than the Metonic Cycle period of 6940 days.) Thus the average number of days in a year is 6936/19 = 365.0526 (to 4 decimal places). > > Proposition 2: The average number of days in a calendar month in the Simple RBMC is 29.51489. > Proof: The number of days in 19 years is 6936. There are 12*12 + 7*13 = 235 months in 19 years, so the average length of a month is 6936/235 = 29.51489 days. > > Thus the Simple RBMC will get out of sync with the vernal equinox year by one day each 1/(365.2424365.0526) = 5.27 years, and it will get out of sync with lunations by one day each 1/(29.5305929.51489) = 63.69 months. > > Clearly the Simple RBMC is not an accurate solilunar calendar. So the RBMC has Rule #6: > > 6. The 7th month normally has 29 days, but it has 30 days when either (a) the year number is divisible by 5 but not by 95 or (b) the year number is divisible by 760. > > Proposition 3: The average number of days in a calendar year in the RBMC is 365.2434. > Proof: Since 760 is divisible by 5, the structure of the calendar repeats every 760 years, so it is sufficient to consider a period of 760 years. When Rule #6 is not applied, there are 6936*(760/19) = 277,440 days = in 760 years. The application of Rule #6 adds 760/5  760/95 + 1 = 145 days, so the total number of days in 760 years = 277,440 + 145 = 277,585 days. Thus the average number of days in a year is 277,585/760 = 365.2434. > > Proposition 4: The average number of days in a calendar month in the RBMC is 29.53032. > Proof: The number of days in 760 years is 277,585. There are (12*12 + 7*13)*(760/19) = 9400 months in 760 years, so the average length of a month is 277,585/9400 = 29.53032 days. > > Thus the RBMC will get out of sync with the vernal equinox year by one day every 1/(365.2434  365.2424) = 1000 years, and it will get out of sync with lunations by one day every 1/(29.5305929.53052) = 14,286 months. Since the average number of months per year is 9400/760 = 12.3684, the RBMC will get out of sync with lunations by one day every 14,285/12.3864 = 1153 years. > > This degree of accuracy in according with the lunar and solar cycles is quite good, or at least, not too bad. > > Finally we may note that 760 years in the RBMC consists of exactly 39,655 7day weeks. > > Comments are welcome. > > Regards, > Peter Meyer 
In reply to this post by Peter Meyer
Actually, an observational lunar month begins when the moon is first observed at dusk, immediately after sunset. On Thu, Mar 14, 2019 at 8:15 AM Peter Meyer <[hidden email]> wrote: Esteemed CALNDRL members,  Amos Shapir

Dear, Peter, Amos and Calendar People
I see Peter has made a lunisolar calendar that follows the 19year cycle permanently by making the mean year a little too long at 365.2434 days and the mean month a too short at 29.53032 days. One gets 29.53032*(235/19) = 365.24343..... days. Peter then said "Thus the RBMC will get out of sync with the vernal equinox year by one day every 1/(365.2434  365.2424) = 1000 years, and it will get out of sync with lunations by one day every 1/(29.5305929.53052) = 14,286 months. Since the average number of months per year is 9400/760 = 12.3684, the RBMC will get out of sync with lunations by one day every 14,285/12.3864 = 1153 years. " Peter has made an error here, which is to write the mean month of his calendar as 29.53052 days instead of 29.52032 days and so gets his accuracy calculation wrong. With correct value used the months are 1 day out in just 3703.7 months, which is just 299.4 years and so the calendar is a not really accurate. No matter how you fix it, if YE is the number of years the year is a day out and YM is the number of years the month is a day out, then 1/YE + !/YM = 1/230 or thereabouts. Karl Thursday Beta March 2019
Original message 
Karl, cc sirs:
>....With correct value used the months are 1 >day out in just 3703.7 months, which is just >299.4 years and so the calendar is a not >really accurate.
I think Peter has tried to link my 19year Harappa Tithi Calendar with my 896year Modified BrijGregorian Calendar, since (299.4x3)  2.2=896years thus could easily be adjusted/reconciled.
It is my feel that Sepp. May also be interested in this link?
Regards,
Flt Lt Brij Bhushan VIJ (Retd.)
Thursday, 2019 March 14H03:87 (decimal)
Sent from my iPhone

In reply to this post by Peter Meyer
In a message to me, which Karl seems to have forgotten to post to
CALNDRL, Karl said: > Peter has made an error here, which is to write the mean month of his > calendar as 29.53052 days instead of 29.52032 days and so gets his > accuracy calculation wrong. With correct value used the months are 1 > day out in just 3703.7 months, which is just 299.4 years and so the > calendar is a not really accurate. Thanks to Karl for catching this. So the offending paragraph in my post should read: Thus the RBMC will get out of sync with the vernal equinox year by one day every 1/(365.2434  365.2424) = 1000 years, and it will get out of sync with lunations by one day every 1/(29.5305929.53032) = 3704 months. Since the average number of months per year is 9400/760 = 12.3684, the RBMC will get out of sync with lunations by one day every 3704/12.3864 = 299 years. I agree that this renders the RBMC "not really accurate". So I'll have to try to change Rule #6 so that it takes about 1000 years for the RBMC to get out of sync with lunations by one day. Regards, Peter 
Peter, Karl sirs:
Indeed happy knowing 19year cycle works, to the advantage of Harappa Lunar Tithi Calendar; and can be imbedded in 896year cycle. Thanks. Brij Bhushan VIJ (Retd.), IAF Thursday, 2019 March 14H06:37 (decimal) Sent from my iPhone > On Mar 14, 2019, at 05:26, Peter Meyer <[hidden email]> wrote: > > In a message to me, which Karl seems to have forgotten to post to CALNDRL, Karl said: > >> Peter has made an error here, which is to write the mean month of his calendar as 29.53052 days instead of 29.52032 days and so gets his accuracy calculation wrong. With correct value used the months are 1 day out in just 3703.7 months, which is just 299.4 years and so the calendar is a not really accurate. > > Thanks to Karl for catching this. So the offending paragraph in my post should read: > > Thus the RBMC will get out of sync with the vernal equinox year by one > day every 1/(365.2434  365.2424) = 1000 years, and it will get out of > sync with lunations by one day every 1/(29.5305929.53032) = 3704 > months. Since the average number of months per year is 9400/760 = > 12.3684, the RBMC will get out of sync with lunations by one day every > 3704/12.3864 = 299 years. > > I agree that this renders the RBMC "not really accurate". So I'll have to try to change Rule #6 so that it takes about 1000 years for the RBMC to get out of sync with lunations by one day. > > Regards, > Peter 
In reply to this post by Peter Meyer
In his message to me Karl also said:
> No matter how you fix it, if YE is the number of years the year is a > day out and YM is the number of years the month is a day out, then > 1/YE + 1/YM = 1/230 or thereabouts. Could Karl kindly prove this? Or is this something he simply "sees" a la Ramanujan? If true then this is bad news for making the RuleBased Metonic Calendara reasonably accurate solilunar calendar, because 1/1000 + 1/1000 = 0.002 whereas 1/230 = 0.004348 (to 6 decimal places). If 1/YE + 1/YM = 1/230 or thereabouts then YE and YM are inversely related, meaning that greater accuracy for RBMC as a lunar calendar can come only at the cost of lesser accuracy for RBMC as a solar calendar. In other words, as Karl said, there's no solution. So the problem is the problem itself. This needs further thought. Regards, Peter 
Dear Peter, Karl, and Calendar People, I think Karl is hinting at the inherent inaccuracy of the Metonic cycle. If you find an accurate yearly cycle, the drift of the monthly cycle is off by exactly the amount the Metonic cycle is off compared to the ratio of year to month lengths. If you find an accurate monthly cycle, the drift of the yearly cycle is off by exactly the amount the Metonic cycle is off compared to this ratio. Adjust each cycle a little bit and one may get better at the expense of the other. If you care about the accuracy of the year and the month and you want to use the Metonic cycle, you have to introduce periodic corrections. Victor On Thu, Mar 14, 2019 at 9:47 AM Peter Meyer <[hidden email]> wrote: In his message to me Karl also said: 
... and the correction is about 1 day in 230 years. On Thu, Mar 14, 2019 at 10:38 AM Victor Engel <[hidden email]> wrote:

In reply to this post by Peter Meyer
Dear Victor, Karl and Calendar People,
Victor said: > I think Karl is hinting at the inherent inaccuracy of the Metonic cycle. If > you find an accurate yearly cycle, the drift of the monthly cycle is off by > exactly the amount the Metonic cycle is off compared to the ratio of year > to month lengths. If you find an accurate monthly cycle, the drift of the > yearly cycle is off by exactly the amount the Metonic cycle is off compared > to this ratio. Adjust each cycle a little bit and one may get better at the > expense of the other. The basis of the Metonic Cycle is that 235 synodic months is just two hours different from 19 vernal equinox (or mean tropical) years. If we call 235 synodic months the 'lunar period' and 19 VE years the 'solar period' then it's true that they'll diverge by about 2 hours every 19 years, which is about 1 day every 12*19 = 228 years (as you said in your next message, about 230 years). But in formulating the Rulebased Metonic Calendar I am not relying on these lunar and solar periods as such. What I'm doing is using the Metonic Cycle as a basic model, namely, a period of 19 years with alternating 29day and 30day months, with an intercalary month (of 30 days) in 7 of those 19 years. (If this model is not used then it can't be called a Metonic Calendar.) Then it's a question of finding a rule for adding (and maybe subtracting) extra days. I proposed an extra day in the 7th month every 5 years to make it 30 days, but not in the 95th year, but also in the 760th year, but that does not produce a calendar which is both an accurate solar calendar and an accurate lunar calendar. So the problem is: Tweak Rule #6 for adding or subtracting days to get the desired result (a calendar which accords with both the lunar and solar cycles over 1000 years). Or replace Rule #6 with some other. At present I don't know if this is possible. Regards, Peter 
In reply to this post by Peter Meyer
Dear Peter et al Simon Cassidy came up with a system of epact shifts that puts the lunar date ahead of the solar date by one day in 228 years, for a total of one excess lunar month every 6840 years So one lunar month must be dropped every 6840 years I have thought that this might be accomplished by shifting the embolistic months smoothly from the first year of a Metonic cycle until an embolistic month occurs in the last year of the 6840 year cycle, and then shifts into the first year of the next 6840 cycle Walter Ziobro On Thursday, March 14, 2019 Peter Meyer <[hidden email]> wrote: Dear Victor, Karl and Calendar People, Victor said: > I think Karl is hinting at the inherent inaccuracy of the Metonic cycle. If > you find an accurate yearly cycle, the drift of the monthly cycle is off by > exactly the amount the Metonic cycle is off compared to the ratio of year > to month lengths. If you find an accurate monthly cycle, the drift of the > yearly cycle is off by exactly the amount the Metonic cycle is off compared > to this ratio. Adjust each cycle a little bit and one may get better at the > expense of the other. The basis of the Metonic Cycle is that 235 synodic months is just two hours different from 19 vernal equinox (or mean tropical) years. If we call 235 synodic months the 'lunar period' and 19 VE years the 'solar period' then it's true that they'll diverge by about 2 hours every 19 years, which is about 1 day every 12*19 = 228 years (as you said in your next message, about 230 years). But in formulating the Rulebased Metonic Calendar I am not relying on these lunar and solar periods as such. What I'm doing is using the Metonic Cycle as a basic model, namely, a period of 19 years with alternating 29day and 30day months, with an intercalary month (of 30 days) in 7 of those 19 years. (If this model is not used then it can't be called a Metonic Calendar.) Then it's a question of finding a rule for adding (and maybe subtracting) extra days. I proposed an extra day in the 7th month every 5 years to make it 30 days, but not in the 95th year, but also in the 760th year, but that does not produce a calendar which is both an accurate solar calendar and an accurate lunar calendar. So the problem is: Tweak Rule #6 for adding or subtracting days to get the desired result (a calendar which accords with both the lunar and solar cycles over 1000 years). Or replace Rule #6 with some other. At present I don't know if this is possible. Regards, Peter

Walter, Victor calendar folks, sirs:
My 448years/5541 Moons is a good compromise giving Year vs Lunation ratio=
12.36830357142857.
Metonic cycle has this ratio=235/19= 12.36842105263158
I have tried to correct this to TRUE Lunation period by adding slight solar duration as:
448years=163,628.5009720619 days; & 5541 Lunation=163,628.99287326 surplus by 11 h 48m & 20s.26, making my 2x448yr/ 5541 moons cycle a better lunisolar Calendar/compromise. Regards,
ExFlt Lt Brij Bhushan VIJ (Retd.),IAF
Thursday, 2019 March 14H12:05 (decimal)
Sent from my iPhone

Brij,
The number of significant digits in your calculations appears to exceed anything reasonable in terms of variables such as Delta T. But, is there a good scientific application for this mathematical calendar?  Ed ********************************

In reply to this post by Peter Meyer
Dear Peter and Calendar People
A change in rule 6, which is the rule which determines which year abundant (has one day more than in the simple calendar) is not sufficient to ensure that the calendar is accurate within one day in 1000 years. I explained that if such a calendar has its year accurate within 1 day for YS years and its month is accurate within 1 day form YM years, then 1/YM + 1/YS = about 1/230. So the 'best' one can get is 1 day in 460 years. Also 'best' assumes that the month being one day out is as bad as the year being one day out. One can argue the former is worse than the latter, because a 1 day is 3.38% of a month, but just 0.27% of a year. Then 'best' has a mean year and month a little less than in the Hipparchic 304year cycle. AN ACCURATE LUNISOLAR CALENDAR MUST CORRECT THE METONIC CYCLE. Calendars such as the MeyerPalmen Solilunar Calendar and the Archectypes Calendar follow the Metonic cycle for several centuries, then automatically correct it about once every 360 years on average. This correction takes the form of removing an octaeteris from a Metonic cycle. Karl Friday Beta March 2019 Original message From : [hidden email] Date : 14/03/2019  12:26 (GMT) To : [hidden email] Subject : Re: A rulebased Metonic calendar In a message to me, which Karl seems to have forgotten to post to CALNDRL, Karl said: > Peter has made an error here, which is to write the mean month of his > calendar as 29.53052 days instead of 29.52032 days and so gets his > accuracy calculation wrong. With correct value used the months are 1 > day out in just 3703.7 months, which is just 299.4 years and so the > calendar is a not really accurate. Thanks to Karl for catching this. So the offending paragraph in my post should read: Thus the RBMC will get out of sync with the vernal equinox year by one day every 1/(365.2434  365.2424) = 1000 years, and it will get out of sync with lunations by one day every 1/(29.5305929.53032) = 3704 months. Since the average number of months per year is 9400/760 = 12.3684, the RBMC will get out of sync with lunations by one day every 3704/12.3864 = 299 years. I agree that this renders the RBMC "not really accurate". So I'll have to try to change Rule #6 so that it takes about 1000 years for the RBMC to get out of sync with lunations by one day. Regards, Peter 
In reply to this post by Peter Meyer
Dear Peter and Calendar People
If a calendar follows the Metonic Cycle, then the mean year is equal to 235/19 mean months. If the mean month of the calendar were perfectly accurate its mean year would be 235/19 mean synodic months (about 365.24677 days). I call this the Metonic year. If after YM years the calendar mean month is running one day early compared to the mean synodic month, the calendar mean year is also running one day early compared to the Metonic year. I hope Peter sees this to be true, else I'll prove it mathematically in another note. Hence the calendar mean year runs 1/YM days earlier than the Metonic year per year. Also it runs 1/YE days later than the mean March equinox tropical year. per year. 1/YE + 1/YM is merely the difference between the mean March equinox tropical year and the Metonic year and this is a constant of about 1/230 indicating that the two years take about 230 to diverge by one day. I remind Peter that I worked out how the MeyerPalmen Solilunar Calendar follows and corrects Metonic Cycle and he kindly put this work at https://www.hermetic.ch/cal_stud/nlsc/mpslci.htm The Archetypes calendar behaves in a similar manner, but corrects the Metonic cycle once every 360.6 years instead of 360 years. This correction is the removal of an octaeteris and is equivalent to correcting by 1/19 month. Karl Friday Beta March 2019 Original message From : [hidden email] Date : 14/03/2019  14:47 (GMT) To : [hidden email] Subject : Re: A rulebased Metonic calendar In his message to me Karl also said: > No matter how you fix it, if YE is the number of years the year is a > day out and YM is the number of years the month is a day out, then > 1/YE + 1/YM = 1/230 or thereabouts. Could Karl kindly prove this? Or is this something he simply "sees" a la Ramanujan? If true then this is bad news for making the RuleBased Metonic Calendara reasonably accurate solilunar calendar, because 1/1000 + 1/1000 = 0.002 whereas 1/230 = 0.004348 (to 6 decimal places). If 1/YE + 1/YM = 1/230 or thereabouts then YE and YM are inversely related, meaning that greater accuracy for RBMC as a lunar calendar can come only at the cost of lesser accuracy for RBMC as a solar calendar. In other words, as Karl said, there's no solution. So the problem is the problem itself. This needs further thought. Regards, Peter 
In reply to this post by Walter J Ziobro
Dear Walter and Calendar People
The MeyerPalmen Solilunar Calendar does this see https://www.hermetic.ch/cal_stud/nlsc/mpslci.htm#YL The shifts occur once every 360 years and effectively removes an octaeteris from one 19year cycle. The link shows a complete table of these shifts of the long years (years with embolismic months). Karl Friday Beta March 2019 Original message 
In reply to this post by Peter Meyer
Dear Karl and Calendar People,
Thanks for the comments on the Metonic Cycle, the MeyerPalmen Solilunar Calendar ( https://www.hermetic.ch/cal_stud/nlsc/nlsc.htm ) and the Archetypes Calendar ( https://www.hermetic.ch/cal_stud/arch_cal/arch_cal.htm ). It seems my attempt to construct a RuleBased Metonic Calendar which accords with the lunar and solar cycles over more than a few hundred years is futile, so I hereby abandon the attempt. The only good thing about it was a clear example (for students of calendar science) of how to calculate the average length of the calendar month and the calendar year in a proposed rulebased calendar, and to compare these with astronomical values, thereby evaluating the accuracy of the proposed calendar, but that is not a particularly difficult thing to do, unless the proposed calendar is especially complex. Regards, Peter 
It is highly recommended for anyone attempting to create any new calendar scheme, to read Irv Bromberg's Kelendis site first. (Maybe this list should have a Welcome message containing all such recommended links, to be sent to new subscribers  and possibly to old subscribers once a year) On Sat, Mar 16, 2019 at 8:14 AM Peter Meyer <[hidden email]> wrote: Dear Karl and Calendar People,  Amos Shapir

In reply to this post by Peter Meyer
Dear Amos et al: Yet, Irv himself had proposed a revision of the Jewish calendar, which is rule based. I recall that I once suggested that the Jewish calendar could be easily reformed by dropping an embolistic month in the year 6840 AM and he got annoyed at this because it was more jittery than his proposal Walter Ziobro On Saturday, March 16, 2019 Amos Shapir <[hidden email]> wrote: It is highly recommended for anyone attempting to create any new calendar scheme, to read Irv Bromberg's Kelendis site first. (Maybe this list should have a Welcome message containing all such recommended links, to be sent to new subscribers  and possibly to old subscribers once a year) On Sat, Mar 16, 2019 at 8:14 AM Peter Meyer <[hidden email]> wrote: Dear Karl and Calendar People,  Amos Shapir

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