Five-Year vs. Eight-Year Lunisolar Cycle

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Five-Year vs. Eight-Year Lunisolar Cycle

Palmen, KEV (Karl)
Dear Calendar People

Thinking about ancient lunisolar calendars such as Greek, Germanic or Celtic (Coligny) Calendars, it occurred to me that the best cycles for a 'schematic' calendar where number of days in each month is determined by rules as well as the number of months in each year could be different from the best cycles for a 'non-schematic' calendar where the months are determined by observation.

For a 'schematic' calendar it would be desirable for the years to be grouped into periods or cycles, where the mean length of the number of lunations in the period is close to a whole number of days. This would enable a fairly accurate days in month rule to apply over the period.

Examining the length of the number of lunations (assumed to be 29.5306 days each) over several possible periods, I find that the
lunations of five year cycle used by the Coligny calendar are fairly close to a whole number of days and so are its two possible corrections, the 30-year cycle and 25-year cycle:

Years  Months  Length of Lunations   Days
 5      62       1830.8972           1831
30     371      10955.8526          10956
25     309       9124.9554           9125

and closer than either the Octaeteris or Metonic cycle:

Years  Months   Length of Lunations
 8      99      2923.5294
19     235      6939.691

but the double Octaeteris is also fairly close:

Years  Months  Length of Lunations   Days
16     198      5847.0588            5847

The double Octaeteris could be constructed out of three five-year cycles and one twelve month year of 354 days.

Karl

08(04(17
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Re: Five-Year vs. Eight-Year Lunisolar Cycle

Lance Latham
RE:
> Thinking about ancient lunisolar calendars such as
> Greek, Germanic or Celtic (Coligny) Calendars, it
> occurred to me that the best cycles for a
> 'schematic' calendar where number of days in each
> month is determined by rules as well as the number
> of months in each year could be different from the
> best cycles for a 'non-schematic' calendar where the
> months are determined by observation.

Lance replies:
The Coligny calendar is an interesting challenge. I
have worked with the numbers and Olmsted's text in the
past, and have not had time to resume that work.

I generally agree with Karl, however, that larger
cycles of 25 or 30 years would be useful in analyzing
it.

I have not looked at the Coligny calendar from the
perspective of the octaeteris, so I can't state that
no association exists, but it seems unlikely that the
calendar is octaeteris-based.

-Lance


Lance Latham
[hidden email]
Phone:    (518) 274-0570
Address: 78 Hudson Avenue/1st Floor, Green Island, NY 12183
 




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Re: Five-Year vs. Eight-Year Lunisolar Cycle

Palmen, KEV (Karl)
Dear Lance and Calendar People

I think a five-year cycle was chosen for the Coligny calendar because, its lunations have a better approximation to a whole number of days than the lunations of alternatives such as the Octaeteris and furthermore, this also applies to its 30-year correction.

The 25-year correction whose 309 lunations are even more closer to a whole number of days is equal to 9125 days, which is equivalent to 25 years of 365 days has been used by the Egyptians.

Karl

08(04(18

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:[hidden email]]On Behalf Of Lance Latham
Sent: 13 June 2006 22:08
To: [hidden email]
Subject: Re: Five-Year vs. Eight-Year Lunisolar Cycle


RE:
> Thinking about ancient lunisolar calendars such as
> Greek, Germanic or Celtic (Coligny) Calendars, it
> occurred to me that the best cycles for a
> 'schematic' calendar where number of days in each
> month is determined by rules as well as the number
> of months in each year could be different from the
> best cycles for a 'non-schematic' calendar where the
> months are determined by observation.

Lance replies:
The Coligny calendar is an interesting challenge. I
have worked with the numbers and Olmsted's text in the
past, and have not had time to resume that work.

I generally agree with Karl, however, that larger
cycles of 25 or 30 years would be useful in analyzing
it.

I have not looked at the Coligny calendar from the
perspective of the octaeteris, so I can't state that
no association exists, but it seems unlikely that the
calendar is octaeteris-based.

-Lance


Lance Latham
[hidden email]
Phone:    (518) 274-0570
Address: 78 Hudson Avenue/1st Floor, Green Island, NY 12183
 




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Re: Five-Year vs. Eight-Year Lunisolar Cycle

Palmen, KEV (Karl)
In reply to this post by Lance Latham
Dear Calendar People

I'm aware that for accuracy in the long term, one requires three types of period to build up an accurate lunisolar calendar cycle. Thinking about this and these periods whose lunations come close to a whole number of days, I came up with the following three building blocks

Years Months Days
  4     49   1447
  5     62   1831
  6     74   2185

Each of these periods can be constructed with one year of 355 days with and all other years of 354 or 384 days.

The 5-year period would be the most common period and it would be occasionally corrected by a 6-year period or a 4-year period.

From the point of view of lunar short-term accuracy, the 4-year period is so accurate that it does not matter when they occur, but the 6-year periods should be as evenly distributed amongst the 5-year periods as possible ignoring any 4-year periods.

Solar short-term accuracy requires that the 4-year periods need to be kept away from the 6-year periods. Simplicity likes to have the 4-year periods next to the 6-year periods to generate a 10-year period, which is a multiple of the 5-year period, but has one less month.

Some cycles can be constructed thus:

Years   4-yrs  5-yrs 6-yrs
 30       1      4     1
 25       1      3     1
 16       0      2     1
 19       1      3     0
 57       1      7     3
 76       2     10     3
304       6     38    15
334       7     42    16

Note that the 8-year cycle (Octaeteris) cannot be constructed out of these periods, because its 99 lunations are too far from a whole number of days. The Metonic cycle of 6940 days can be constructed with three 5-year periods and one 4-year period.

If a cycle has A years B leap months and C days in excess of 354 or 384 in each year, then the number N4, N5, and N6 of these periods (4,5,6 years respectively) are as follows:

N4 =  2C-B
N5 =  2B+2C-A
N6 =  A-B-3C

Let's see what we get for the Callypic cycle where A=76, B=28 and C=15. We get N4=2, N5=10, N6=3. For the 8-year cycle (A=8, B=3) either N4 or N6 must be negative (for any integer value of C).

However I doubt whether these building blocks were ever actually used. It would be nice to be proven wrong here.

Karl

08(04(18

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:[hidden email]]On Behalf Of Lance Latham
Sent: 13 June 2006 22:08
To: [hidden email]
Subject: Re: Five-Year vs. Eight-Year Lunisolar Cycle


RE:
> Thinking about ancient lunisolar calendars such as
> Greek, Germanic or Celtic (Coligny) Calendars, it
> occurred to me that the best cycles for a
> 'schematic' calendar where number of days in each
> month is determined by rules as well as the number
> of months in each year could be different from the
> best cycles for a 'non-schematic' calendar where the
> months are determined by observation.

Lance replies:
The Coligny calendar is an interesting challenge. I
have worked with the numbers and Olmsted's text in the
past, and have not had time to resume that work.

I generally agree with Karl, however, that larger
cycles of 25 or 30 years would be useful in analyzing
it.

I have not looked at the Coligny calendar from the
perspective of the octaeteris, so I can't state that
no association exists, but it seems unlikely that the
calendar is octaeteris-based.

-Lance
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Brij Bonavian vs Sidereal days Re: Five-Year vs. Eight-Year Lunisolar Cycle

Brij Bhushan Vij
Karl, Lance & CC:
>However I doubt whether these building blocks were ever actually used.
>The Metonic cycle of 6940 days can be constructed with three 5-year periods
>and one 4-year >period.
Whatever the objective to develope 'software' that suit 5-yr & 8-yr blocks,
I feel there is a need to review using *Sidereal Days* for construction of
the Civil Calendar that promises to keep Seasonal precession in check. I
place my calculations:
1-Year = 365.242189669781 days =366.24218981 Sidreal days =12.368266411
lunation (52.17746W);
5-Years =1826.211 days =1831.21094904 Sidereal days =61.8413321 lunation
(260.88728 Weeks);
8-Years = 2921.938 days =2929.9375184579 Sidereal days =98.9461313 lunation
(417.41965 weeks);
16-Years =5843.875 days =5859.875037 Sidereal days=197.892263 lunation
(834.83929 weeks);
19-Years =6939.602 days =6958.60161 Sidereal days = 234.9971 lunation
(991.371682 weeks);
25-Years =9131.055 days =9156.055 Sidereal days =309.20666027 lunation
(1304.436392 weeks);
28-years =10226.781311 days =10254.781314603 Sidereal days =346.3114595
lunation =1460.96876W);
30-Years =10957.26569 days =10987.26569 Sidereal days =371.04799 lunation
(1565.32367 weeks).
33-Years =12052.99226 days =12085.9923 Sidereal days =408.15279 lunation
(1721.85604 Weeks).
128-Years =46751.00023 days =46879.00023 Sidereal days =1583.1381006
lunation (6678.7143254 W).
896-Years =327257.001944 days =328153.002067 Sid.Days =11081.9667041
lunation (46751.00028 W).
During discussion on REFORM of the Metonic Cycle, during 2004-2005 - I
projected [19-years=5*47 lunation=235 lunation] that can form the basis for
accurate 'EPACT count'; and 128-year cycle can become the basis for Reform
of Gregorian Era calendar [128-years=19*5+33-years; and
896-years=(11*19+16)+(11*19)+28+(11*19)+(11*19+16)].
Brij Bhushan Vij
(Tuesday, Kali 5107-W09-02)/265+D-166 (Wednesday, 2006 June
14H18:37(decimal) ET
Aa Nau Bhadra Kritvo Yantu Vishwatah -Rg Veda
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Jul:30; Aug:31; Sep:30; Oct:31; Nov:30; Dec:30
(365th day of Year is World Day)
******As per Kali V-GRhymeCalendaar*****
"Koi bhi cheshtha vayarth nahin hoti, purshaarth karne mein hai"
Contact # 001(201)675-8548


>From: "Palmen, KEV (Karl)" <[hidden email]>
>Reply-To: East Carolina University Calendar discussion List              
><[hidden email]>
>To: [hidden email]
>Subject: Re: Five-Year vs. Eight-Year Lunisolar Cycle
>Date: Wed, 14 Jun 2006 16:23:55 +0100
>
>Dear Calendar People
>
>I'm aware that for accuracy in the long term, one requires three types of
>period to build up an accurate lunisolar calendar cycle. Thinking about
>this and these periods whose lunations come close to a whole number of
>days, I came up with the following three building blocks
>
>Years Months Days
>   4     49   1447
>   5     62   1831
>   6     74   2185
>
>Each of these periods can be constructed with one year of 355 days with and
>all other years of 354 or 384 days.
>
>The 5-year period would be the most common period and it would be
>occasionally corrected by a 6-year period or a 4-year period.
>
>From the point of view of lunar short-term accuracy, the 4-year period is
>so accurate that it does not matter when they occur, but the 6-year periods
>should be as evenly distributed amongst the 5-year periods as possible
>ignoring any 4-year periods.
>
>Solar short-term accuracy requires that the 4-year periods need to be kept
>away from the 6-year periods. Simplicity likes to have the 4-year periods
>next to the 6-year periods to generate a 10-year period, which is a
>multiple of the 5-year period, but has one less month.
>
>Some cycles can be constructed thus:
>
>Years   4-yrs  5-yrs 6-yrs
>  30       1      4     1
>  25       1      3     1
>  16       0      2     1
>  19       1      3     0
>  57       1      7     3
>  76       2     10     3
>304       6     38    15
>334       7     42    16
>
>Note that the 8-year cycle (Octaeteris) cannot be constructed out of these
>periods, because its 99 lunations are too far from a whole number of days.
>The Metonic cycle of 6940 days can be constructed with three 5-year periods
>and one 4-year period.
>
>If a cycle has A years B leap months and C days in excess of 354 or 384 in
>each year, then the number N4, N5, and N6 of these periods (4,5,6 years
>respectively) are as follows:
>
>N4 =  2C-B
>N5 =  2B+2C-A
>N6 =  A-B-3C
>
>Let's see what we get for the Callypic cycle where A=76, B=28 and C=15. We
>get N4=2, N5=10, N6=3. For the 8-year cycle (A=8, B=3) either N4 or N6 must
>be negative (for any integer value of C).
>
>However I doubt whether these building blocks were ever actually used. It
>would be nice to be proven wrong here.
>
>Karl
>
>08(04(18
>
>-----Original Message-----
>From: East Carolina University Calendar discussion List
>[mailto:[hidden email]]On Behalf Of Lance Latham
>Sent: 13 June 2006 22:08
>To: [hidden email]
>Subject: Re: Five-Year vs. Eight-Year Lunisolar Cycle
>
>
>RE:
> > Thinking about ancient lunisolar calendars such as
> > Greek, Germanic or Celtic (Coligny) Calendars, it
> > occurred to me that the best cycles for a
> > 'schematic' calendar where number of days in each
> > month is determined by rules as well as the number
> > of months in each year could be different from the
> > best cycles for a 'non-schematic' calendar where the
> > months are determined by observation.
>
>Lance replies:
>The Coligny calendar is an interesting challenge. I
>have worked with the numbers and Olmsted's text in the
>past, and have not had time to resume that work.
>
>I generally agree with Karl, however, that larger
>cycles of 25 or 30 years would be useful in analyzing
>it.
>
>I have not looked at the Coligny calendar from the
>perspective of the octaeteris, so I can't state that
>no association exists, but it seems unlikely that the
>calendar is octaeteris-based.
>
>-Lance