Lunisolar Cycle Multiples of Short Lunar Cycle

[Karl Palmen]

Dear Calendar People

Helios and I have found several lunisolar calendar cycles that are each a multiple of a lunar calendar cycle both short and accurate enough to appear in http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm and are themselves less than 10,000 years long.

These are

3511 years of 45 lunar 965-month cycles with mean month of 29.530570 days and mean year of 365.24210 days (is based on Vij tithi).

5515 years of 39 lunar 1749-month cycles with mean month of 29.530589 days and mean year of 365.24225 days.

6251 years of 86 lunar 899-month cycles with mean month of 29.530590 days and mean year of 365.242204 days.

6707 years of 118 lunar 703-month cycles with mean month of 29.530583 days and mean year of 365.24228 days (19 353-year cycles).

Also worth mentioning is

16,700 years of 243 lunar 850-month cycles with a mean month of 29.530588 days and mean year of 365.24210 days (50 334-year cycles).

Karl

10(08(21

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Scanned by iCritical.

[Irv Bromberg]

On 2009 May 15, at 14:22 , ELITE 3000 wrote:

But which cycles are the most accurate in terms of lunation period and mean year compared to the mean solar year (365.242 189 67 days (365 d 5 h 48 min 45 s))

Irv replies:

That is 365+31/128 days, which doesn't approximate any equinox or solstice -- why do you care about it?

That average also keeps getting shorter due to tidal slowing of the Earth rotation rate, whereas in the present era the mean northward equinoctial year and the mean north solstitial year are nearly constant, and will remain that way for the next 4-5 millennia and 11-12 millennia, respectively, enabling good approximation with simple fixed arithmetic leap cycles.

Check out the various solar year lengths that are evaluated for drift relative to each of the equinoxes and solstices in my "Solar_Calendar_Drift" spreadsheet at:

http://individual.utoronto.ca/kalendis/leap/Solar_Calendar_Drift.xls

You can replace any of the entries with your own to see how they look, just follow the format of the other entries.

ELITE continued:  and the mean lunation period (29.530589 days (29 d 12 h 44 min 2.9 s)) (correct me if I have the mean solar year/syndonic month slightly off)?

Irv replies:  That mean lunation period is a tad long for today.  The mean synodic month is getting progressively shorter.  If one insists on using a fixed lunation period for a lunar calendar or a lunisolar calendar, then it is better to choose a period that is intentionally too short by about 1/2 second, so that its era of best accuracy will endure longer.  Check out the various lunation periods that are evaluated for drift in my "Lunar_Calendar_Drift" spreadsheet at:

http://individual.utoronto.ca/kalendis/lunar/Lunar_Calendar_Drift.xls

In this case you will find that there is a very broad and comprehensive range of lunation periods represented by the many cycles evaluated in the spreadsheet.  Those that are not represented, however, are the very long lunisolar cycles that are the topic of this thread, so again, feel free to replace any of the entries with your own to see how they look, just follow the format of the other entries.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/leap/>

<http://www.sym454.org/lunar/>

<http://www.sym454.org/seasons/>

[Irv Bromberg]

On 2009 May 15, at 16:55 , Brij Bhushan Vij wrote:

Elite 3000, Irv, CC sirs:
>On 2009 May 15, at 14:22 , ELITE 3000 wrote:

But which cycles are the most accurate in terms of lunation period and mean year compared to the mean solar year (365.242 189 67 days (365 d 5 h 48 min 45 s))

>Irv replies:

>That is 365+31/128 days, which doesn't approximate any equinox or solstice -- why do you care about it?

Brij says:  My arguments since 1980's have been directed towards a 128-year (2*64-year Metric calendar) cycle

http://www.brijvij.com/bb-bonavian_brijorChris.pdf

leading to LWks & Keplers' LWks, NEVER tried by man on *div.six(6) with added KLWks* has remained under examination ever since, I joined Calndr-L list in mid-2002. The Mean Year =(365+31/128)=365.2421875 days =7*(52+159/896) days. The same is also obtained as: 7*[(52+1/6+29/2688)] days.

 The 834-year/148 LWks give Mean Year =7*[(52+1/6+9/834)] =365.242206235012 days.

It may be worth noting that Mean Year=365.2421875 days is CLOSEST to the Mean Average Atomic Year Value, obtained as the average of Year values at FOUR cardinal points.

Irv replies:

Averaging the four cardinal points, which I assume Brij is referring to the mean lengths of the years measured at each equinox and solstice, will not equal the average value that you seek, because those points are not equally spaced.  One can however average the longest and shortest year lengths and that will yield the average that you seek.  The shortest is instantaneous mean solar year length at the ecliptic longitude of aphelion, and the longest is the instantaneous mean solar year length at the ecliptic longitude of perihelion.  [These are not the same as the year length from aphelion to aphelion or from perihelion to perihelion, which is the appreciably longer solar anomalistic year.]

Anyhow, although it is true that 365+31/128 days is essentially equal to the long-term mean solar year length in terms of atomic days, that is of no relevance to any calendar.

Calendars need mean solar time, not atomic time.  Don't be deceived by the small difference between atomic time and mean solar time in the present era, that is just because the atomic second currently happens to be nearly equal to the mean solar second.  As the Earth rotation rate is progressively slowed by the tidal transfer of angular momentum to Moon the mean solar second will become progressively longer than the atomic second.  Atomic time is essential for celestial mechanical calculations, but it is useless for calendrical calculations, except very near to the present era.

In terms of atomic time, each of the equinoctial and solstitial years take turns enjoying eras of stability in mean length, but those eras are much shorter than the correspondingly stable eras in terms of mean solar time:

Atomic time:  http://individual.utoronto.ca/kalendis/solar/Solar_Year_Lengths_30K.pdf

Mean solar time:  http://individual.utoronto.ca/kalendis/solar/Mean_Solar_Years_50K.pdf

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/leap/>

<http://www.sym454.org/lunar/>

<http://www.sym454.org/seasons/>

[Irv Bromberg]

On 2009 May 16, at 22:03 , Brij Bhushan Vij wrote:

 " REVISING Average Mean Atomic Year Value (Y_2007): 

According to <http://en.wikipedia.org/wiki/Tropical_year> : The current values and their annual change of the time of return to the cardinal ecliptic points are: (1) vernal equinox: 365.24237404 + 0.00000010338×a days; (2) northern solstice: 365.24162603 + 0.00000000650×a days; (3) autumn equinox: 365.24201767 −0.00000023150×a days; and (4) southern solstice: 365.24274049 − 0.00000012446×a days, where a is number of years after Y2000. I consider the YEAR length can be fixed at the Average Mean Atomic Year i.e. [365.24237404 + 365.24162603 + 365.24201767 + 365.24274049] / 4 =365.2421895575 days. Astronomers, however, have fixed the Year value, Y2000 = 365.242189669781 days. This during the past century, Y1900 was 365d 5h 48m 45s.9747 i.e. 365.24219878125 days".

Irv replies:  Like I wrote before, you can't calculate the average of those 4 "cardinal" points because they are not equally spaced, so when averaged the result is not the true average length of the solar cycle.

Also, if you look again at the atomic time chart that I cited previously:

http://individual.utoronto.ca/kalendis/solar/Solar_Year_Lengths_30K.pdf

... please note that the "Tropical Year" curve (in black) cyclically varies -- it is not a fixed constant length of 365+31/128 days, although that is its value today (near year 2000 on the x-axis).

The period of the cycle is 41000 years = the Earth axial tilt cycle.  During that cycle Earth's axial tilt (=obliquity of the ecliptic) varies from a minimum of about 22.6° to a maximum of about 24.2°.  This axial tilt cycle is both caused by and stabilized within these limits by Moon.  The cycle is not perfectly predictable, because ice mass depression of Earth's poles and the resultant equatorial bulge interact with Moon to produce long-term variations.

The problem with using an average solar year length for any calendar is that there is no observable astronomical event available to evaluate or verify calendar drift.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/leap/>

<http://www.sym454.org/lunar/>

<http://www.sym454.org/seasons/>

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear Calendar People

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of ELITE 3000
Sent: 15 May 2009 19:22
To: CALNDR-L@...
Subject: Re: Lunisolar Cycle Multiples of Short Lunar Cycle

 

But which cycles are the most accurate in terms of lunation period and mean year compared to the mean solar year (365.242 189 67 days (365 d 5 h 48 min 45 s)) and the mean lunation period (29.530589 days (29 d 12 h 44 min 2.9 s)) (correct me if I have the mean solar year/syndonic month slightly off)?  This is a question that is meaningful only if you make the incorrect assumption that these values are constant. Why does he ask this question?

 

Karl

 

10(08(23 till noon

 

 

From: karl.palmen@...

Sent: Friday, May 15, 2009 11:52 AM

To: CALNDR-L@...

Subject: Lunisolar Cycle Multiples of Short Lunar Cycle

 

Dear Calendar People

Helios and I have found several lunisolar calendar cycles that are each a multiple of a lunar calendar cycle both short and accurate enough to appear in http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm and are themselves less than 10,000 years long.

These are

3511 years of 45 lunar 965-month cycles with mean month of 29.530570 days and mean year of 365.24210 days (is based on Vij tithi).

5515 years of 39 lunar 1749-month cycles with mean month of 29.530589 days and mean year of 365.24225 days.

6251 years of 86 lunar 899-month cycles with mean month of 29.530590 days and mean year of 365.242204 days.

6707 years of 118 lunar 703-month cycles with mean month of 29.530583 days and mean year of 365.24228 days (19 353-year cycles).

Also worth mentioning is

16,700 years of 243 lunar 850-month cycles with a mean month of 29.530588 days and mean year of 365.24210 days (50 334-year cycles).

Karl

10(08(21

 

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Scanned by iCritical.

 

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Scanned by iCritical.

[Irv Bromberg]

On 2009 May 18, at 04:05 , Palmen, KEV (Karl) wrote:

ELITE wrote:  But which cycles are the most accurate in terms of lunation period and mean year compared to the mean solar year (365.242 189 67 days (365 d 5 h 48 min 45 s)) and the mean lunation period (29.530589 days (29 d 12 h 44 min 2.9 s)) (correct me if I have the mean solar year/syndonic month slightly off)?

Karl says:  This is a question that is meaningful only if you make the incorrect assumption that these values are constant. Why does he ask this question?

Irv replies:

My previous message in this thread discussed the accuracy of his proposed solar year and lunation period.

ELITE perhaps assumed that the values are constant, but even though the values vary it is easy to answer his question, just by evaluating the appropriate lengths relative to the changing astronomical lengths.  The easiest way to carry out a solar and lunar astronomical drift analysis, if I do say so myself, is using my "Solar_Calendar_Drift" spreadsheet at:

http://individual.utoronto.ca/kalendis/leap/Solar_Calendar_Drift.xls

and my "Lunar_Calendar_Drift" spreadsheet at:

http://individual.utoronto.ca/kalendis/lunar/Lunar_Calendar_Drift.xls

as I cited previously in this thread.

The solar spreadsheet uses actual SOLEX-integrated equinox and solstice moments (listed at one-century intervals).

The lunar spreadsheet uses my mean lunation polynomials from <http://www.sym454.org/lunar/> which were fitted to SOLEX-integrated lunar conjunction moments.

The SOLEX integrations were carried out in terms of Terrestrial Time (atomic time).

For conversion to mean solar time both spreadsheets use Meeus-Espenak Delta T expressions from the NASA Eclipses web site.

Both are open source.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear ELITE, Irv and Calendar People

 

Why would one ask, which one of these cycles is the most accurate.

There is the implication that I should have listed just one cycle (the most accurate).

 

I listed several cycles for the following reasons:

(1)    There is no consensus about which would be the most accurate cycle (as shown in other notes)

(2)    There are criteria other than accuracy, such as shortness of the lunar cycle or odd-number of years for Helios cycle.

 

This is also the approach I take in my lunisolar spreadsheets: http://www.the-light.com/cal/kp_Lunisolar_xls.html .

 

Karl

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 18 May 2009 15:05
To: CALNDR-L@...
Subject: Re: Lunisolar Cycle Multiples of Short Lunar Cycle

 

On 2009 May 18, at 04:05 , Palmen, KEV (Karl) wrote:

ELITE wrote:  But which cycles are the most accurate in terms of lunation period and mean year compared to the mean solar year (365.242 189 67 days (365 d 5 h 48 min 45 s)) and the mean lunation period (29.530589 days (29 d 12 h 44 min 2.9 s)) (correct me if I have the mean solar year/syndonic month slightly off)?

 

Karl says:  This is a question that is meaningful only if you make the incorrect assumption that these values are constant. Why does he ask this question?

 

Irv replies:

 

My previous message in this thread discussed the accuracy of his proposed solar year and lunation period.

 

ELITE perhaps assumed that the values are constant, but even though the values vary it is easy to answer his question, just by evaluating the appropriate lengths relative to the changing astronomical lengths.  The easiest way to carry out a solar and lunar astronomical drift analysis, if I do say so myself, is using my "Solar_Calendar_Drift" spreadsheet at:

 

http://individual.utoronto.ca/kalendis/leap/Solar_Calendar_Drift.xls

 

and my "Lunar_Calendar_Drift" spreadsheet at:

 

http://individual.utoronto.ca/kalendis/lunar/Lunar_Calendar_Drift.xls

 

as I cited previously in this thread.

 

The solar spreadsheet uses actual SOLEX-integrated equinox and solstice moments (listed at one-century intervals).

The lunar spreadsheet uses my mean lunation polynomials from <http://www.sym454.org/lunar/> which were fitted to SOLEX-integrated lunar conjunction moments.

The SOLEX integrations were carried out in terms of Terrestrial Time (atomic time).

For conversion to mean solar time both spreadsheets use Meeus-Espenak Delta T expressions from the NASA Eclipses web site.

 

Both are open source.

 

-- Irv Bromberg, Toronto, Canada

 

<http://www.sym454.org/>

 

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Scanned by iCritical.

[Irv Bromberg]

On 2009 May 18, at 17:53 , Brij Bhushan Vij wrote:

I cite and draw your attention to THREE of my worked cycles.

Irv asks:  Huh?  But there were FOUR listed!  Perhaps Brij miscounted because of the way that he spaced the cycles, which I fixed in the text quoted below, although he did originally color-code them.

Brij continued:

1. 896-yrs/11082 Ln;  327257d.002 (46751W); MY=365.2421875(365^d5^h48^m45^s.0); MLn=29.53049991 (29^d12^h43^m55^s.19)
2. 1749-yrs/21632 Ln; 638808d.59 (91258W+2d); MY=365.241852487(365^d5^h48^m16^s.05); MLn=29.530695266 (29^d12^h44^m12^s.071)

3. 8019-yrs/99181 Ln; 2928877d.12 (418411W); MY=365.242174835 (365^d5^h48^m43^s.906); MLn=29.53062582551 (29^d12^h44^m06^s.1)
4. 373632-yrs/4621180L; 136466169d.8(19495167W); MY=365.2421875(365^d5^h48^m45^s.0); MLn=29.5305893733 (29^d12^m44^m2^s.92)

Irv replies:  All of the above have a mean year equal to or close to 365+31/128 days, which doesn't approximate any equinox or solstice.  I don't understand why Brij consistently prefers to be in this "never-never-land".

For today, the lunation period of #1 is far too short, of #2 is way too long, of #3 is still too long, and #4 is a bit too long.  For a fixed mean month cycle, a mean month in the range of 2 to 2.5 seconds in excess of 29d 12h 44m would be slightly too short and therefore essentially optimal for a millennium or two.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/seasons>

<http://www.sym454.org/lunar>

[Helios]

There is a 5181 year lunisolar cycle with 80 repeats of a 49 yerm cycle

5181 years = [ 2*2*2*2*5 | 2*11827 ] = 64080 months

mean year = 365 & 1255 / 5181 = 365.2422312 days
mean month = 29 & 425 / 801 = 29.53058677 days


5181 years is three of this cycle:
1727 years = 21360 months

5181 years is 334 years less than the 5515 year cycle.

There is also a 8464 year cycle using mean month = 29 & 529 / 997 days but I think anything over a pictun is out of bounds.

[Irv Bromberg]

On 2009 May 19, at 14:10 , Brij Bhushan Vij wrote:

But, my goal has been for correction of the Gregorian calendar, which is a 'solar day calendar':

400-years =146089.68758679124 days BUT works with Mean Year =365+97/400 =365.2425 days (146097 days). Thus, the controversy of corrections needed now.

the Mean Year=365.2421875 days (365^d5^h48^m45^s.0)

=(365+31/128) days is a practical unhindered value.

Irv replies:  OK, well the primary goal of the Gregorian calendar reform was to keep the vernal equinox on March 21st, so the relevant length of year to evaluate it against is not 365+31/128 days but rather the vernal equinox year, which is presently very close to 365+127/524 days, or 365d 5h 49m 0.5s (and getting slightly longer over the coming centuries), about 15.5 seconds per year longer than the mean year that you are using.  The Gregorian mean year is 365d 5h 49m 12s, so it's mean year is about 11.5 seconds too long, and so the Gregorian mean year is actually more accurate with respect to the vernal equinox year than your proposed reform!  Nevertheless, because of tidal slowing of the Earth rotation rate, in the long term it is better to have a mean year that is too short than a mean year that is too long.

For my calculations I use the value 29.53058881 days (29d 12h 44m 2s.873184). Mean values as shown by me: MeanLunation =29.5305893733 (29^d12^m 44^m2^s.92) is close enough to be practical, UNLESS there is a better value achievable, sir.

Irv replies:  I looked into this further and found that improving on the accuracy is very method dependent, so I have to say that that lunation period is about as accurate as one can use.  I was just suggesting that it is not ideal to use an accurate lunation period for a lunar calendar intended to be used today and into the future, because the astronomical mean lunation period is getting progressively shorter.  Instead, I suggest intentionally using a slightly shorter lunation period, which will allow the lunar approximation to serve with better accuracy for a longer duration until correction is necessary.

I had suggested a shorter mean month somewhere in the range of 2.0 to 2.5 seconds in excess of 29d 12h 44m.  Compare with the Gregorian Easter computus which is about 2.7 seconds in excess, or the even shorter mean month used for the Dee Easter computus which is less than 2.4 seconds in excess, or the shorter still mean month used for the modern Hindu lunisolar calendar (Arya) of less than 2.3 seconds in excess.  These shorter cycles will endure further into the future, provided that they are set up to be a bit late in the present era, then in the future they will drift earlier until they are accurate, and later they will drift late again (when the astronomical mean synodic month gets shorter than the cycle mean month).  These cycles are shown in my lunar calendar drift spreadsheet at:

http://individual.utoronto.ca/kalendis/lunar/Lunar_Calendar_Drift.xls

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/seasons>

<http://www.sym454.org/lunar>

[Irv Bromberg]

On 2009 May 19, at 14:10 , Brij Bhushan Vij wrote:

For my calculations I use the value 29.53058881 days (29d 12h 44m 2s.873184). Mean values as shown by me: MeanLunation =29.5305893733 (29^d12^m 44^m2^s.92) is close enough to be practical, UNLESS there is a better value achievable, sir.

On 2009 May 19, at 15:31 , Irv Bromberg wrote:

I looked into this further and found that improving on the accuracy is very method dependent, so I have to say that that lunation period is about as accurate as one can use.  I was just suggesting that it is not ideal to use an accurate lunation period for a lunar calendar intended to be used today and into the future, because the astronomical mean lunation period is getting progressively shorter.  Instead, I suggest intentionally using a slightly shorter lunation period, which will allow the lunar approximation to serve with better accuracy for a longer duration until correction is necessary.

I had suggested a shorter mean month somewhere in the range of 2.0 to 2.5 seconds in excess of 29d 12h 44m.  Compare with the Gregorian Easter computus which is about 2.7 seconds in excess, or the even shorter mean month used for the Dee Easter computus which is less than 2.4 seconds in excess, or the shorter still mean month used for the modern Hindu lunisolar calendar (Arya) of less than 2.3 seconds in excess.  These shorter cycles will endure further into the future, provided that they are set up to be a bit late in the present era, then in the future they will drift earlier until they are accurate, and later they will drift late again (when the astronomical mean synodic month gets shorter than the cycle mean month).  These cycles are shown in my lunar calendar drift spreadsheet at:

http://individual.utoronto.ca/kalendis/lunar/Lunar_Calendar_Drift.xls

Irv adds, using Hindu mean month fractions from "Calendrical Calculations: Third Edition" by Dershowitz and Reingold:

(I hope I got the the following right, because I don't know much about the Hindu calendars.  Any corrections would be appreciated!)

The Hindu synodic month used for the old Hindu calendar before 1100 AD, as documented around the 4th century in the Surya Siddhanta as 29 + 7087771 / 13358334 days = 29 days 12h 44m 2.8s, was about 1/2 second too short for the era when Hindus started to use that calendar, but is actually almost perfect for the present era.

Using the Lunar Calendar Drift spreadsheet cited above, with an epoch of 400 AD, the Surya mean month affords a mean lunar drift of <8 hours from 2400 BC to 6200 AD.

By contrast, the Arya lunar month, used as the mean synodic month for the modern Hindu calendar since 1000 AD, is 1577917500 / 53433336 days = 29 days 12h 44m 2 + 597062 / 2226389 seconds, or about 4/5 second too short for the era when Hindus started to use that calendar, and still about 1/2 second too short for the present era.

Using the Lunar Calendar Drift spreadsheet cited above, with an epoch of 1000 AD, the Arya mean month affords a mean lunar drift of <8 hours from 1000 BC to 8300 AD.

Anyone who is designing a fixed cycle lunar calendar for use today and for as long as possible into the future would be well advised to follow the Hindu examples, by choosing a mean lunation period that is intentionally slightly too short, to maximize the duration that the calendar will serve with reasonable accuracy, because the astronomical mean synodic month is getting progressively shorter in terms of the mean solar days that calendars need.

I arbitrarily picked the mean lunar drift limit of <8 hours.  Anybody can view the charts in the spreadsheet after updating them for any desired epoch and choose whatever cutoff limit is desired.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/seasons>

<http://www.sym454.org/lunar>

[Irv Bromberg]

On 2009 May 20, at 13:44 , Irv Bromberg wrote:

The Hindu synodic month used for the old Hindu calendar before 1100 AD, as documented around the 4th century in the Surya Siddhanta as 29 + 7087771 / 13358334 days = 29 days 12h 44m 2.8s, was about 1/2 second too short for the era when Hindus started to use that calendar, but is actually almost perfect for the present era.

By contrast, the Arya lunar month, used as the mean synodic month for the modern Hindu calendar since 1000 AD, is 1577917500 / 53433336 days = 29 days 12h 44m 2 + 597062 / 2226389 seconds, or about 4/5 second too short for the era when Hindus started to use that calendar, and still about 1/2 second too short for the present era.

Correction:  Please disregard my message quoted above.  I got the Arya (old) and Surya (modern) Hindu calendars mixed up, and that kills my point.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Irv Bromberg]

On 2009 May 20, at 19:02 , Brij Bhushan Vij wrote:

 I was not aware of these fractional values for length of the lunation.

Irv replies:  Sorry for causing confusion with my previous message.

The corrected fractions, obtained by actually looking them up in "Calendrical Calculations:  Third Edition" are:

The Hindu synodic month used for the old Hindu calendar before 1100 AD, as documented around 499 AD in the Arya Siddhanta, was 29+2362563/4452778 days = 29 days 12h 44m 2+597062/2226389s (the fraction is just over 1/4 second), or just over a second shorter than the astronomical mean synodic month when Hindus started to use that calendar.

By contrast, the mean synodic month used for the modern Hindu calendar, as documented in the Surya Siddhanta around 1000 AD, is 29+7087771/13358334 days = 29 days 12h 44m 2+1777862/2226389s (the fraction is almost 4/5 second), which was almost 1/3 second too short for the era when Hindus started to use that calendar, but is actually essentially perfect for the present era.

(I posted a revised Lunar_Calendar_Drift.xls spreadsheet which now correctly indicates Arya as the old and Surya as the modern Hindu calendar.)

I find it interesting that the denominators, as fraction of a day or fraction of a second are the same in both cases.  (The Arya fraction is reduced 3-fold, that is 4452778 x 3 = 13358334.)

I don't suppose that there is any evidence that the Hindus intentionally selected slightly short mean synodic month lengths?

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Helios]

3511 years of 45 lunar 965-month cycles with mean month of 29.530570 days and mean year of 365.24210 days
5181  years of 80 lunar 801-month cycles with mean month of 29.530587 days and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589 days and mean year of 365.24225 days.
6251 years of 86 lunar 899-month cycles with mean month of 29.530590 days and mean year of 365.242204 days.
6707 years of 118 lunar 703-month cycles with mean month of 29.530583 days and mean year of 365.24228 days
6745  years of 88 lunar 948-month cycles with mean month of 29.530591 days and mean year of 365.24240 days
8464  years of 105 lunar 997-month cycles with mean month of 29.530592 days and mean year of 365.24220 days
-------------------------------------------
For the 6745 year luni-solar cycle
= 5104 yerms
= 6745 years
= 83424 months
= 2463560 days = 88*27995
-------------------------------------------
mean year = 365 & 327 / 1349 days
mean month = 29 & 503 / 948 days
note that 79 months = 55990 hours
-------------------------------------------
This year could be divided into 88ths of 4 & 203 / 1349 days
then 6745 cells = 948 months
and 1 cell = 948 / 6745 months
96 cells = 398.446256 days ( Jovain synodic period = 398.88 days )

[Karl Palmen]

Dear Helios

A good piece of work!
I've yet to check the year figures.

Here I list the cycles in the order of the mean month an give the yerm
calendar structure that would generate the lunar calendar cycle:

Please see http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm ,
 http://www.hermetic.ch/cal_stud/palmen/yerm1.htm and
http://www.hermetic.ch/cal_stud/palmen/yerm1.htm#tool for more details
about yerms.

3511 years of 45 lunar 965-month cycles has each lunar cycle
constructible from alternating cycles of 28 and 31 yerms (or whole week
cycles of 19 and 40 yerms). Mean month of 29.530570 days and mean year
of 365.24210 days. The lunar cycle fits in with Brij's tithi of 2/59
month equated to 966/965 days and has a whole number of weeks.
 
6707 years of 118 lunar 703-month cycles has each lunar cycle
constructible from a single cycle of 43 yerms. Mean month of 29.530583
days and mean year of 365.24228 days. The number of days in each lunar
cycle is a multiple of 30.

5181  years of 80 lunar 801-month cycles has each lunar cycle
constructible from a single cycle of 49 yerms. Mean month of 29.530587
days and mean year of 365.24223 days

5515 years of 39 lunar 1749-month cycles has each lunar cycle
constructible from alternating cycles of 52 and 55 yerms. Mean month of
29.530589 days and mean year of 365.24225 days.

6251 years of 86 lunar 899-month cycles has each lunar cycle
constructible from a single cycle of 55 yerms. Mean month of 29.530590
days and mean year of 365.242204 days

6745 years of 88 lunar 948-month cycles has each lunar cycle
constructible from a single cycle of 58 yerms equal to 79 twelve-month
lunar years. Mean month of 29.530591 days and mean year of 365.24240
days. The whole cycle is equal to 88*79=6959 twelve-month years and so
has 207 seasonal drift cycles of the twelve-month year.

8464 years of 105 lunar 997-month cycles has each lunar cycle
constructible from a single cycle of 61 yerms. Mean month of 29.530592
days and mean year of 365.24220 days. Each lunar cycle has a whole
number of weeks.


The suggested division of the year into 88 cells for the 6745-year cycle
won't work unless a simple rule can found to construct lunar months from
1 cell = 948 / 6745 months. I think it would be simpler to use a yerm
lunar calendar with a 58-yerm cycle or a modification of the Tabular
Islamic calendar that uses a 79-year cycle.



Karl

10(11(20

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 07 August 2009 15:48
To: CALNDR-L@...
Subject: 6745 Year Lunar Cycle

3511 years of 45 lunar 965-month cycles with mean month of 29.530570
days and mean year of 365.24210 days
5181  years of 80 lunar 801-month cycles with mean month of 29.530587
days and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589
days and mean year of 365.24225 days.
6251 years of 86 lunar 899-month cycles with mean month of 29.530590
days and mean year of 365.242204 days.
6707 years of 118 lunar 703-month cycles with mean month of 29.530583
days and mean year of 365.24228 days
6745  years of 88 lunar 948-month cycles with mean month of 29.530591
days and mean year of 365.24240 days
8464  years of 105 lunar 997-month cycles with mean month of 29.530592
days and mean year of 365.24220 days
-------------------------------------------
For the 6745 year luni-solar cycle
= 5104 yerms
= 6745 years
= 83424 months
= 2463560 days = 88*27995
-------------------------------------------
mean year = 365 & 327 / 1349 days
mean month = 29 & 503 / 948 days
note that 79 months = 55990 hours
-------------------------------------------
This year could be divided into 88ths of 4 & 203 / 1349 days
then 6745 cells = 948 months
and 1 cell = 948 / 6745 months
96 cells = 398.446256 days ( Jovain synodic period = 398.88 days )

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[Karl Palmen]

Dear Helios and Calendar People

I've realised that three of these cycles are a multiple of the Metonic
19-year cycle of 235 months with one month removed.

6251 = 19*329 and 86*899 = 329*235-1
6707 = 19*353 and 118*703 = 353*235-1 = 19*4366
6745 = 19*355 and 88*948 = 355*235-1 = 12*79*88

Also the 6707-year cycle is a multiple of the 353-year cycle one of
several multiples of the 353-year cycle that have been mentioned on this
list.

I have already listed the 6707-year cycle and a 6745-year cycle in
http://www.the-light.com/cal/Lunisolar19.html of
http://www.the-light.com/cal/kp_Lunisolar_xls.html ,
but the 6745-year cycle that I listed is a day short of Helios's.

Only the 6745-year, 8464-year and 5181-year cycles are new to the list
all others have been mentioned as multiples of a lunar calendar cycle as
shown beforehand.

Here are the values of Number of years (A), number of leap months (B)
and abundant days (C) for each cycle:

A=3511, B=1293, C=681.  45 lunar cycles of 965 months
A=5181, B=1908, C=1006. 80 lunar cycles of 801 months
A=5515, B=2031, C=1071. 39 lunar cycles of 1749 months
A=6251, B=2302, C=1214. 86 lunar cycles of 899 months and 325 Metonic
cycles less a month
A=6707, B=2470, C=1302. 118 lunar cycles of 703 months, 353 Metonic
cycles less a month and 19 353-year cycles
A=6745, B=2484, C=1310. 88 lunar cycles of 948 months equal to 79
twelve-month years, 355 Metonic cycles less a month
A=8464, B=3117, C=1644. 105 lunar cycles of 997 months.

My lunisolar spreadsheet gives the mean years as calculated by Helios. I
have E-mailed Helios a copy.
Mean months are as for the lunar cycles, which are all listed in
http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm .

Karl

10(11(21

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 07 August 2009 15:48
To: CALNDR-L@...
Subject: 6745 Year Lunar Cycle

3511 years of 45 lunar 965-month cycles with mean month of 29.530570
days and
mean year of 365.24210 days
5181  years of 80 lunar 801-month cycles with mean month of 29.530587
days
and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589
days
and mean year of 365.24225 days.
6251 years of 86 lunar 899-month cycles with mean month of 29.530590
days
and mean year of 365.242204 days.
6707 years of 118 lunar 703-month cycles with mean month of 29.530583
days
and mean year of 365.24228 days
6745  years of 88 lunar 948-month cycles with mean month of 29.530591
days
and mean year of 365.24240 days
8464  years of 105 lunar 997-month cycles with mean month of 29.530592
days
and mean year of 365.24220 days
-------------------------------------------
For the 6745 year luni-solar cycle
= 5104 yerms = 88*58
= 6745 years = 19*335
= 83424 months = 88*79*12
= 2463560 days = 88*27995
-------------------------------------------
mean year = 365 & 327 / 1349 days
mean month = 29 & 503 / 948 days
note that 79 months = 55990 hours
-------------------------------------------
This year could be divided into 88ths of 4 & 203 / 1349 days
then 6745 cells = 948 months
and 1 cell = 948 / 6745 months
96 cells = 398.446256 days ( Jovain synodic period = 398.88 days )

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[Helios]

3014 years of 57 lunar 654-month cycles with mean month of 29.530581 days and mean year of 365.24253 days
3473 years of 71 lunar 605-month cycles with mean month of 29.530579 days and mean year of 365.24215 days
3511 years of 45 lunar 965-month cycles with mean month of 29.530570 days and mean year of 365.24210 days
4437 years of 46 lunar 1193-month cycles with mean month of 29.530595 days and mean year of 365.24228 days
5181 years of 80 lunar 801-month cycles with mean month of 29.530587 days and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589 days and mean year of 365.24225 days
6251 years of 86 lunar 899-month cycles with mean month of 29.530590 days and mean year of 365.242204 days
6707 years of 118 lunar 703-month cycles with mean month of 29.530583 days and mean year of 365.24228 days
6745 years of 88 lunar 948-month cycles with mean month of 29.530591 days and mean year of 365.24240 days
8119 years of 74 lunar 1357-month cycles with mean month of 29.530582 days and mean year of 365.24227 days
8464 years of 105 lunar 997-month cycles with mean month of 29.530592 days and mean year of 365.24220 days

------------------------------------------------------------------------
I received e-mail but haven't opened file, reluctant of .xls format

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

I attach it in HTML and Text (tab separated) formats.

Karl

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 11 August 2009 18:45
To: CALNDR-L@...
Subject: Re: 6745 Year Lunar Cycle

3014 years of 57 lunar 654-month cycles with mean month of 29.530581
days and
mean year of 365.24253 days
3473 years of 71 lunar 605-month cycles with mean month of 29.530579
days
and mean year of 365.24215 days
3511 years of 45 lunar 965-month cycles with mean month of 29.530570
days
and mean year of 365.24210 days
4437 years of 46 lunar 1193-month cycles with mean month of 29.530595
days
and mean year of 365.24228 days
5181 years of 80 lunar 801-month cycles with mean month of 29.530587
days
and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589
days
and mean year of 365.24225 days
6251 years of 86 lunar 899-month cycles with mean month of 29.530590
days
and mean year of 365.242204 days
6707 years of 118 lunar 703-month cycles with mean month of 29.530583
days
and mean year of 365.24228 days
6745 years of 88 lunar 948-month cycles with mean month of 29.530591
days
and mean year of 365.24240 days
8119 years of 74 lunar 1357-month cycles with mean month of 29.530582
days
and mean year of 365.24227 days
8464 years of 105 lunar 997-month cycles with mean month of 29.530592
days
and mean year of 365.24220 days

------------------------------------------------------------------------
I received e-mail but haven't opened file, reluctant of .xls format

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3562589p24922412.html
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Lunisolar Cycles of Multiple Lunar Cycles

Years	Long	Abundant		Leap	Yerms	Mean Yr	Mean Mth	Days	Months	Saltus	Trunc
725	267	141		176	549	365.24276	29.5306122	264801	8967	35	2		183 49-month cycles
3511	1293	681		850	2655	365.2421	29.5305699	1282365	43425	169	10		45 Vij Tithi 965-month cycles
5181	1908	1006		1255	3920	365.24223	29.5305868	1892320	64080	249	15		80 801-month cycles
5515	2031	1071		1336	4173	365.24225	29.5305889	2014311	68211	265	16		39 1749-month cycles (Helios)
6251	2302	1214		1513	4730	365.24204	29.5305895	2283128	77314	299	19		86 899-month cycles
6707	2470	1302		1625	5074	365.24228	29.5305832	2449680	82954	323	19		118 703-month cycles and 19 535-year cycles
6745	2484	1310		1635	5104	365.2424	29.5305907	2463560	83424	325	19		88 948-month cycles
8464	3117	1644		2050	6405	365.2422	29.5305918	3091410	104685	406	25		105 997-month cycles

Years Long Abundant Leap Yerms Mean Yr Mean Mth Days Months Saltus Trunc
725 267 141 176 549 365.2427586 29.53061224 264801 8967 35 2 183 49-month cycles
3511 1293 681 850 2655 365.2420963 29.53056995 1282365 43425 169 10 45 Vij Tithi 965-month cycles
5181 1908 1006 1255 3920 365.2422312 29.53058677 1892320 64080 249 15 80 801-month cycles
5515 2031 1071 1336 4173 365.2422484 29.53058891 2014311 68211 265 16 39 1749-month cycles (Helios)
6251 2302 1214 1513 4730 365.2420413 29.53058954 2283128 77314 299 19 86 899-month cycles
6707 2470 1302 1625 5074 365.2422842 29.53058321 2449680 82954 323 19 118 703-month cycles and 19 535-year cycles
6745 2484 1310 1635 5104 365.2424018 29.53059072 2463560 83424 325 19 88 948-month cycles
8464 3117 1644 2050 6405 365.2422023 29.53059178 3091410 104685 406 25 105 997-month cycles

[Helios]

a few more to the Lunisolar Cycle Multiples of Short Lunar Cycle list


725 years of 183 lunar 49-month cycles with mean month of 29.530612 days and mean year of 365.24276 days
2414 years of 73 lunar 409-month cycles with mean month of 29.530562 days and mean year of 365.24192 days
3014 years of 57 lunar 654-month cycles with mean month of 29.530581 days and mean year of 365.24253 days
3473 years of 71 lunar 605-month cycles with mean month of 29.530579 days and mean year of 365.24215 days
3511 years of 45 lunar 965-month cycles with mean month of 29.530570 days and mean year of 365.24210 days
4437 years of 46 lunar 1193-month cycles with mean month of 29.530595 days and mean year of 365.24228 days
5181 years of 80 lunar 801-month cycles with mean month of 29.530587 days and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589 days and mean year of 365.24225 days
6251 years of 86 lunar 899-month cycles with mean month of 29.530590 days and mean year of 365.242204 days
6707 years of 118 lunar 703-month cycles with mean month of 29.530583 days and mean year of 365.24228 days
6745 years of 88 lunar 948-month cycles with mean month of 29.530591 days and mean year of 365.24240 days
7364 years of 253 lunar 360-month cycles with mean month of 29.530556 days and mean year of 365.24212 days
8119 years of 74 lunar 1357-month cycles with mean month of 29.530582 days and mean year of 365.24227 days
8464 years of 105 lunar 997-month cycles with mean month of 29.530592 days and mean year of 365.24220 days
8518 years of 137 lunar 769-month cycles with mean month of 29.530559 days and mean year of 365.24219 days

[Karl Palmen]

Dear Helios and Calendar People

Again I order them by mean month, so that the lunar cycle can be more
easily found in
http://www.hermetic.ch/cal_stud/lunarcal/lunacycl.htm and choice can be
made according the mean month.
I also give yerm figures for each cycle.

7364 years of 253 lunar 360-month cycles with mean month of 29.530556
days
and mean year of 365.24212 days (253x(22 yerms)) 7590 lunar years
8518 years of 137 lunar 769-month cycles with mean month of 29.530559
days
and mean year of 365.24219 days (137x(22+25 yerms))
2414 years of 73 lunar 409-month cycles with mean month of 29.530562
days
and mean year of 365.24192 days (73x(25 yerms))
3511 years of 45 lunar 965-month cycles with mean month of 29.530570
days
and mean year of 365.24210 days (45x(28+31 yerms)) a Vij Tithi cycle
3473 years of 71 lunar 605-month cycles with mean month of 29.530579
days
and mean year of 365.24215 days (71x(37 yerms))
3014 years of 57 lunar 654-month cycles with mean month of 29.530581
days
and mean year of 365.24253 days (57x(40 yerms)) 3106.5 lunar years
8119 years of 74 lunar 1357-month cycles with mean month of 29.530582
days
and mean year of 365.24227 days (74x(40+43 yerms))
6707 years of 118 lunar 703-month cycles with mean month of 29.530583
days
and mean year of 365.24228 days (118x(43 yerms)) nineteen 353-year
cycles
5181 years of 80 lunar 801-month cycles with mean month of 29.530587
days
and mean year of 365.24223 days (80x(49 yerms))
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589
days
and mean year of 365.24225 days (39x(52+55 yerms))
6251 years of 86 lunar 899-month cycles with mean month of 29.530590
days
and mean year of 365.242204 days (86x(55 yerms))
6745 years of 88 lunar 948-month cycles with mean month of 29.530591
days
and mean year of 365.24240 days (88x(58 yerms)) 6952 lunar years
8464 years of 105 lunar 997-month cycles with mean month of 29.530592
days
and mean year of 365.24220 days (105x(61 yerms))
4437 years of 46 lunar 1193-month cycles with mean month of 29.530595
days
and mean year of 365.24228 days (46x(70 yerms)) close to Hebrew mean
month
725 years of 183 lunar 49-month cycles with mean month of 29.530612 days
and
mean year of 365.24276 days (183x(3 yerms))

Most of the pure-yerm (single-era) cycles in my list are included.

Notably absent is the 850-month cycle of 52 yerms used in my yerm
calendar.
It has a lunisolar cycle of
16,700 years of 243 lunar cycles with a mean month of 29.530588 days and
mean year of 365.24210 days (243x(52 yerms)) fifty 334-year cycles.

Karl

10(13(10

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 September 2009 09:47
To: CALNDR-L@...
Subject: Lunisolar Cycle Multiples of Short Lunar Cycle

a few more to the Lunisolar Cycle Multiples of Short Lunar Cycle list


725 years of 183 lunar 49-month cycles with mean month of 29.530612 days
and
mean year of 365.24276 days
2414 years of 73 lunar 409-month cycles with mean month of 29.530562
days
and mean year of 365.24192 days
3014 years of 57 lunar 654-month cycles with mean month of 29.530581
days
and mean year of 365.24253 days
3473 years of 71 lunar 605-month cycles with mean month of 29.530579
days
and mean year of 365.24215 days
3511 years of 45 lunar 965-month cycles with mean month of 29.530570
days
and mean year of 365.24210 days
4437 years of 46 lunar 1193-month cycles with mean month of 29.530595
days
and mean year of 365.24228 days
5181 years of 80 lunar 801-month cycles with mean month of 29.530587
days
and mean year of 365.24223 days
5515 years of 39 lunar 1749-month cycles with mean month of 29.530589
days
and mean year of 365.24225 days
6251 years of 86 lunar 899-month cycles with mean month of 29.530590
days
and mean year of 365.242204 days
6707 years of 118 lunar 703-month cycles with mean month of 29.530583
days
and mean year of 365.24228 days
6745 years of 88 lunar 948-month cycles with mean month of 29.530591
days
and mean year of 365.24240 days
7364 years of 253 lunar 360-month cycles with mean month of 29.530556
days
and mean year of 365.24212 days
8119 years of 74 lunar 1357-month cycles with mean month of 29.530582
days
and mean year of 365.24227 days
8464 years of 105 lunar 997-month cycles with mean month of 29.530592
days
and mean year of 365.24220 days
8518 years of 137 lunar 769-month cycles with mean month of 29.530559
days
and mean year of 365.24219 days
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