Schematic Metonic Cycle

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Schematic Metonic Cycle

Palmen, KEV (Karl)
Dear Calendar People

In an earlier note, I suggested that for a Schematic Lunisolar Calendar (on in which the number of days in a month is determined by rules rather than observation), that certain cycles may be preferred because the total duration their lunations is close to a whole number of days. In particular the Coligny cycles of 5, 25 and 30 year would be preferred. Also the double Octaeteris of 16 years would be preferred.

The Metonic cycle would not be preferred, because its lunations are not close to a whole number of days. Using lunations of 29.5306 days, we get 6939.691 days in the 235 lunations of a Metonic cycle, which is about 7 hours 25 minutes short of a whole number of days, which is over 3 times the 2 hour moon-sun error of the cycle.

However some multiples of the Metonic cycle may be close to a whole number of days. Two such multiple were known to be used or proposed in ancient times: The Callypic cycle of 76 years and the Hipparchic Cycle of 304 years.

Multiple Years  Lunations   Days (of lunations)
   1       19     235         6939.691
   4       76     940        27758.764
  16      304    3760       111035.056

The Callypic cycle is not a very big improvement on the Metonic cycle, but the Hipparchic cycle is.
I expect the Callypic cycle was chose because when rounded to have whole number (27759) days, it has a mean year of exactly 365.25 days (as would Lance's A-S Octaeteris). The multiple of three Metonic cycles is more accurate than four. I list this and other multiples (up to 304 years) whose lunations are within 20% of a day of a whole number of days.

Multiple Years  Lunations   Days (of lunations)
   3       57     705        20819.073
   6      114    1410        41638.146
   7      133    1645        48577.837
  10      190    2350        69396.910
  13      247    3055        90215.983
  16      304    3760       111035.056

Clearly the 57-year cycle is closest to a whole number of days.

The 247-year cycle is also close to a whole number (12888) weeks.

The 190-year cycle of 2350 months and 69397 days would be appropriate to a Coligny-like calendar and would be constructed out of six 30-year cycles, but this would be 1 day too long.

Years: 6*30 + 2*5 = 190
Months: 6*371 + 2*62 = 2350
Days: 6*10956 + 2*1831  = 69398.


Karl
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Re: Schematic Metonic Cycle

Irv Bromberg
On Jun 22, 2006, at 08:13, Palmen, KEV (Karl) wrote:

> However some multiples of the Metonic cycle may be close to a whole
> number of days. Two such multiple were known to be used or proposed in
> ancient times: The Callypic cycle of 76 years and the Hipparchic Cycle
> of 304 years.
>
> Multiple Years  Lunations   Days (of lunations)
>    1       19     235         6939.691
>    4       76     940        27758.764
>   16      304    3760       111035.056
>
> The Callypic cycle is not a very big improvement on the Metonic cycle,
> but the Hipparchic cycle is.
> I expect the Callypic cycle was chose because when rounded to have
> whole number (27759) days, it has a mean year of exactly 365.25 days
> (as would Lance's A-S Octaeteris). The multiple of three Metonic
> cycles is more accurate than four. I list this and other multiples (up
> to 304 years) whose lunations are within 20% of a day of a whole
> number of days.
>
> Multiple Years  Lunations   Days (of lunations)
>    3       57     705        20819.073
>    6      114    1410        41638.146
>    7      133    1645        48577.837
>   10      190    2350        69396.910
>   13      247    3055        90215.983
>   16      304    3760       111035.056
>
> Clearly the 57-year cycle is closest to a whole number of days.

Karl:

Such calculations are futile, because the duration of the mean lunation
gets progressively shorter, in terms of mean solar days, and is
therefore a moving target over long cycle lengths.  You have also used
a mean synodic month that is a bit more than a second too long for the
present era.

Using the present-era mean synodic month of approximately
29.53058766633 days, my 353-year Rectified Hebrew Calendar leap cycle
with 4366 lunations is 128930.54575 days, which is very close to
128930.5, so a double cycle of 706 years is very close to an integer
number of days = about 257861.0915 mean solar days, which is 2h 11m 46s
in excess of a whole number of days.  However, if one takes into
account the progressive shortening of the mean synodic month over that
interval, by using the average mean synodic month over the 706 year
period, about 29.5305862865 mean solar days, then the 706-year period
starting this year has about 257861.0794 mean solar days, which is 1h
54m 20s in excess of a whole number of days.

This calculation says nothing about the goodness of fit of the
"rectified molad", because there is no a priori reason why the leap
cycle length should be associated with an integer number of days with
respect to the number of lunations in the cycle.  As far as I can tell,
the rectified molad has zero error.

What Karl is trying to do, it seems, is substitute a simple fixed
arithmetic leap cycle calculation for the combination of a molad with a
separate solar leap cycle.  This makes me wonder if something like my
linear approximation progressive leap rule may be a good approach here,
except that there would only be the straight diagonal line part, no
horizontal plateau at an earlier or later epoch.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>
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293-yrs & Short Tithi RE: Schematic Metonic Cycle

Brij Bhushan Vij
In reply to this post by Palmen, KEV (Karl)
Karl, all:
>.....Metonic cycle, which is about 7 hours 25 minutes short of a whole
>number of days, which is >over 3 times the 2 hour moon-sun error of the
>cycle.
Alignment of sun-Moon in 19-year Metonic cycle is easy to achieve by *short
accounting ONE tithi* as we discussed in earlier notes. [(15*19)+8]
=293-year cycle is 107015.962 days or 15288 Weeks in 107016 days. This cycle
is also good for using Sidereal days for the format of Civil Year Calendar,
is my view. This cycle is lesser than 55 minutes away from COMPLETED number
of Weeks.
Astronomy experts may have better vision to use/opine for other cycles.
>The Callypic cycle of 76 years and the Hipparchic Cycle of 304 years.
Except for the improved count, what were the other positive advantages of
these cycles over the Metonic cycle?
Brij Bhushan Vij
(Wednesday, Kali 5107-W10-03)/265+D-174 (Thursday, 2006 June
22H11:01(decimal) ET
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Jul:30; Aug:31; Sep:30; Oct:31; Nov:30; Dec:30
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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: Schematic Metonic Cycle
>Date: Thu, 22 Jun 2006 13:13:02 +0100
>
>Dear Calendar People
>
>In an earlier note, I suggested that for a Schematic Lunisolar Calendar (on
>in which the number of days in a month is determined by rules rather than
>observation), that certain cycles may be preferred because the total
>duration their lunations is close to a whole number of days. In particular
>the Coligny cycles of 5, 25 and 30 year would be preferred. Also the double
>Octaeteris of 16 years would be preferred.
>
>The Metonic cycle would not be preferred, because its lunations are not
>close to a whole number of days. Using lunations of 29.5306 days, we get
>6939.691 days in the 235 lunations of a Metonic cycle, which is about 7
>hours 25 minutes short of a whole number of days, which is over 3 times the
>2 hour moon-sun error of the cycle.
>
>However some multiples of the Metonic cycle may be close to a whole number
>of days. Two such multiple were known to be used or proposed in ancient
>times: The Callypic cycle of 76 years and the Hipparchic Cycle of 304
>years.
>
>Multiple Years  Lunations   Days (of lunations)
>    1       19     235         6939.691
>    4       76     940        27758.764
>   16      304    3760       111035.056
>
>The Callypic cycle is not a very big improvement on the Metonic cycle, but
>the Hipparchic cycle is.
>I expect the Callypic cycle was chose because when rounded to have whole
>number (27759) days, it has a mean year of exactly 365.25 days (as would
>Lance's A-S Octaeteris). The multiple of three Metonic cycles is more
>accurate than four. I list this and other multiples (up to 304 years) whose
>lunations are within 20% of a day of a whole number of days.
>
>Multiple Years  Lunations   Days (of lunations)
>    3       57     705        20819.073
>    6      114    1410        41638.146
>    7      133    1645        48577.837
>   10      190    2350        69396.910
>   13      247    3055        90215.983
>   16      304    3760       111035.056
>
>Clearly the 57-year cycle is closest to a whole number of days.
>
>The 247-year cycle is also close to a whole number (12888) weeks.
>
>The 190-year cycle of 2350 months and 69397 days would be appropriate to a
>Coligny-like calendar and would be constructed out of six 30-year cycles,
>but this would be 1 day too long.
>
>Years: 6*30 + 2*5 = 190
>Months: 6*371 + 2*62 = 2350
>Days: 6*10956 + 2*1831  = 69398.
>
>
>Karl
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Re: Schematic Metonic Cycle

Palmen, KEV (Karl)
In reply to this post by Irv Bromberg
Dear Irv and Calendar People

I'm not interested in the long term accuracy over thousands of years here.
If I were, I'd not be considering the Metonic Cycle at all!

I'm looking at short periods of a few years that could be used in a schematic lunisolar calendar. Such short periods could be concatenated to make a calendar of arbitrarily good long term accuracy. A mean lunation of 29.5306 days is sufficiently accurate for the purpose of finding such short periods.

Karl

08(04(26

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:[hidden email]]On Behalf Of Irv Bromberg
Sent: 22 June 2006 15:03
To: [hidden email]
Subject: Re: Schematic Metonic Cycle


On Jun 22, 2006, at 08:13, Palmen, KEV (Karl) wrote:

> However some multiples of the Metonic cycle may be close to a whole
> number of days. Two such multiple were known to be used or proposed in
> ancient times: The Callypic cycle of 76 years and the Hipparchic Cycle
> of 304 years.
>
> Multiple Years  Lunations   Days (of lunations)
>    1       19     235         6939.691
>    4       76     940        27758.764
>   16      304    3760       111035.056
>
> The Callypic cycle is not a very big improvement on the Metonic cycle,
> but the Hipparchic cycle is.
> I expect the Callypic cycle was chose because when rounded to have
> whole number (27759) days, it has a mean year of exactly 365.25 days
> (as would Lance's A-S Octaeteris). The multiple of three Metonic
> cycles is more accurate than four. I list this and other multiples (up
> to 304 years) whose lunations are within 20% of a day of a whole
> number of days.
>
> Multiple Years  Lunations   Days (of lunations)
>    3       57     705        20819.073
>    6      114    1410        41638.146
>    7      133    1645        48577.837
>   10      190    2350        69396.910
>   13      247    3055        90215.983
>   16      304    3760       111035.056
>
> Clearly the 57-year cycle is closest to a whole number of days.

Karl:

Such calculations are futile, because the duration of the mean lunation
gets progressively shorter, in terms of mean solar days, and is
therefore a moving target over long cycle lengths.  You have also used
a mean synodic month that is a bit more than a second too long for the
present era.

Using the present-era mean synodic month of approximately
29.53058766633 days, my 353-year Rectified Hebrew Calendar leap cycle
with 4366 lunations is 128930.54575 days, which is very close to
128930.5, so a double cycle of 706 years is very close to an integer
number of days = about 257861.0915 mean solar days, which is 2h 11m 46s
in excess of a whole number of days.  However, if one takes into
account the progressive shortening of the mean synodic month over that
interval, by using the average mean synodic month over the 706 year
period, about 29.5305862865 mean solar days, then the 706-year period
starting this year has about 257861.0794 mean solar days, which is 1h
54m 20s in excess of a whole number of days.

This calculation says nothing about the goodness of fit of the
"rectified molad", because there is no a priori reason why the leap
cycle length should be associated with an integer number of days with
respect to the number of lunations in the cycle.  As far as I can tell,
the rectified molad has zero error.

What Karl is trying to do, it seems, is substitute a simple fixed
arithmetic leap cycle calculation for the combination of a molad with a
separate solar leap cycle.  This makes me wonder if something like my
linear approximation progressive leap rule may be a good approach here,
except that there would only be the straight diagonal line part, no
horizontal plateau at an earlier or later epoch.

-- Irv Bromberg, Toronto, Canada

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