Metonic year correcting

[Helios]

We recall how golden numbered days are distributed evenly spaced and symmetrically

N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N N @ N

and that the Metonic year can be aptly approximated by 365 & 19 / 77 days.

We have mentioned the idea of dropping a golden numbered day every 334 or 353 years. This will however necessitate a change in the leap day rule and this will diminish the accuracy of the golden numbers.

I think 2 days ( with 1 golden numbered day ) can be dropped after a 353 year period and 1 day can be dropped after a 334 year period. This scheme would apply within a 3101 year period.

3101 years = 353 + 334 + 353 + 334 + 353 + 334 + 353 + 334 + 353

This amounts to 14 days dropped every 3101 years. I think the Metonic leap day rule can then be maintained.
The mean year is then ( 365 & 19 / 77 days ) - 14 / 3101 = 365.24223857 days

[Tom Peters-6]

Op 3-okt-2009, om 12:31 heeft Helios het volgende geschreven:

> We recall how golden numbered days are distributed evenly spaced and
> symmetrically
>
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N N @ N
>
> and that the Metonic year can be aptly approximated by 365 & 19 /  
> 77 days.

No, why?  The original Metonic cycle was 6940 days, so the Metonic  
year was 365+5/19 = 365.263... days - much too long.
Since then, the Metonic relation 19 years = 235 lunations has been  
maintained with whatever year length our solar calendar had; so 19  
Julian years = 6939.75 days on average.  In 76 years (Calyppic  
cycle), that's 365+19/76 days.  So where does your 365+19/77 days  
come from?

> We have mentioned the idea of dropping a golden numbered day every  
> 334 or
> 353 years. This will however necessitate a change in the leap day  
> rule and
> this will diminish the accuracy of the golden numbers.
>
> I think 2 days ( with 1 golden numbered day ) can be dropped after  
> a 353
> year period and 1 day can be dropped after a 334 year period. This  
> scheme
> would apply within a 3101 year period.

This implies that after 353 years some 2-day error has built up.  
Then it should be corrected after 9 or 10 cycles (171 or 190 years)  
by 1 day.
But what error is that?


> 3101 years = 353 + 334 + 353 + 334 + 353 + 334 + 353 + 334 + 353
>
> This amounts to 14 days dropped every 3101 years. I think the  
> Metonic leap
> day rule can then be maintained.
> The mean year is then ( 365 & 19 / 77 days ) - 14 / 3101 =  
> 365.24223857 days

This means you make a correction for some (very) long-term accuracy,  
while allowing a 2-day error in the medium-term accuracy.

I think you are somehow confusing 2 errors here:
1) The fact that 19 years <> 235 lunations: this is corrected by  
shifting the Golden Numbers after 334 or 353 days.
2) The error in the length of the mean solar year.  That is  
independent from the ratio between the solar and lunar period (error 1).

--
Tom Peters

[Helios]

Tom Peters-6 wrote:

No, why? The original Metonic cycle was 6940 days, so the Metonic
year was 365+5/19 = 365.263... days - much too long.
Since then, the Metonic relation 19 years = 235 lunations has been
maintained with whatever year length our solar calendar had; so 19
Julian years = 6939.75 days on average. In 76 years (Calyppic
cycle), that's 365+19/76 days. So where does your 365+19/77 days
come from?

Whatever an "original Metonic cycle" is, or what other historical examples use, there is

Metonic year = ( month )*( 235 / 19 )

A Metonic year of 365 & 19 / 77 days just comes by inspection. I don't see why it matters whether it is confirmed by history.

Tom Peters-6 wrote:

This implies that after 353 years some 2-day error has built up.
Then it should be corrected after 9 or 10 cycles (171 or 190 years)
by 1 day.
But what error is that?

There is the discrepency between the Metonic year and a solar year of lesser duration. One can interupt a Metonic calendar occasionally, at 334 or 353 year instances, reset the calendar, to effectuate a net reduction in the mean year overall. If you think I have erred because I am confused, would you demonstrate this with arithmetic?

[Tom Peters-6]

Op 3-okt-2009, om 21:32 heeft Helios het volgende geschreven:

> Tom Peters-6 wrote:
>>
>> No, why? The original Metonic cycle was 6940 days, so the Metonic
>> year was 365+5/19 = 365.263... days - much too long.
>> Since then, the Metonic relation 19 years = 235 lunations has been
>> maintained with whatever year length our solar calendar had; so 19
>> Julian years = 6939.75 days on average. In 76 years (Calyppic
>> cycle), that's 365+19/76 days. So where does your 365+19/77 days
>> come from?
>>
>
> Whatever an "original Metonic cycle" is, or what other historical  
> examples
> use, there is
>
> Metonic year = ( month )*( 235 / 19 )
>
> A Metonic year of 365 & 19 / 77 days just comes by inspection. I  
> don't see
> why it matters whether it is confirmed by history.

Since we have a solar calendar, it usually works the other way round:  
month = year*19/235 .

> Tom Peters-6 wrote:
>>
>> This implies that after 353 years some 2-day error has built up.
>> Then it should be corrected after 9 or 10 cycles (171 or 190 years)
>> by 1 day.
>> But what error is that?
>>
>
> There is the discrepency between the Metonic year and a solar year  
> of lesser
> duration. One can interupt a Metonic calendar occasionally, at 334  
> or 353
> year instances, reset the calendar, to effectuate a net reduction  
> in the
> mean year overall. If you think I have erred because I am confused,  
> would
> you demonstrate this with arithmetic?

There is nothing wrong with the mean year.  The correction after 334  
or 353 years is necessary because 19 years (of whatever precise  
lenght) are not, in fact, 235 lunations.  After these 3+ centuries  
the Moon is 1,5 day out of sync with the Sun, irrespective of exactly  
how many days, months, and years have passed.  By shifting the Golden  
Numbers by 8 (or 11), a net correction of 1.5 days in the dates of  
the syzygies in the solar calendar is achieved.

Lets use a mean vernal aequinox year of 365,2424 days, and a synodic  
month of 29,53058886 days.  Ratio 12.36827... .  235/19=12,36842...  
Difference 0.00014... lunations, amounts to 1 day after 1/
(0.00014...*29.53...) = 230 years.  230 years = 84005.75 days =  
2844.703 lunations ; 235/19*230 = 2844.737 lunations; difference  
0.034 lunation; *29.53... = 1.00 days.

334 years = 121990.97 days = 4131.004 lunations ; 235/19*334 =  
4131.053 lunations; difference 0.049 lunations = 1.4 days.  As  
already stated, this can be corrected by truncating the running  
Metonic cycle after these  (334 MOD 19) = 11 years, and starting  
anew: effectively shifting the Golden Numbers by 11 (or 19-11=8)  
years.  11 years = 4017.666 days, 136 lunations  = 4016.160 days,  
difference 1.5 days.  This error of 1.5 days halfway the Metonic  
cycle compensates the error built up over 17+ cycles of the  
approximation 19 years = 235 lunations.

Similarly 353 years = 128930.58 days = 4366.0008 lunations ;  
235*19*353 = 4366.0526 lunations; difference 0.0518 lunations = 1.53  
days.  So the shift after 11 years into the 19th Metonic cycle works  
even better.

Mind that all this is about the ratio of the solar year and the  
synodic month, and the discrepancy between the two.  The integer  
number of years (230 or 334 or 353) is not an integer number of days  
nor an integer number of synodic months.  You cannot use this period  
to correct the length of the mean solar year, although of course 334  
years is close to 121991 days exactly.  But 353 years is not an  
integer number of days and is the wrong period to correct the mean  
solar year length with.

--
Tom Peters

[Karl Palmen]

Dear Helios, Tom and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 03 October 2009 11:31
To: CALNDR-L@...
Subject: Metonic year correcting

We recall how golden numbered days are distributed evenly spaced and
symmetrically

N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N N @ N

and that the Metonic year can be aptly approximated by 365 & 19 / 77
days.

KARL SAYS: This cycle does not look symmetrical to me.

Rearranging thus shows the symmetry

N @ N N @ N N @ N @ N N @ N N @ N
      N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N N @ N
      N @ N N @ N @ N N @ N
N @ N N @ N N @ N @ N N @ N N @ N

Note the N between two @ occurs only in the middle of each row.

Each N corresponds to the 1/19 of a lunar month and this pattern makes
of 73 days with 47 'N's.
Five of these 73-day cycles make up a common year with 235 'N's.
Helios did not state that he forms a leap year by adding a '@' to the
end of five 73-day cycles, so ensuring that each year has 235 'N's in
it.


HELIOS CONTINUES:
We have mentioned the idea of dropping a golden numbered day every 334
or
353 years. This will however necessitate a change in the leap day rule
and
this will diminish the accuracy of the golden numbers.

KARL SAYS: Although the leap year rule may be changed, there need be no
loss of accuracy, because it can be tailored to ANY accurate lunisolar
cycle.

Correction is done by truncating the year to 364 days. This removes the
'N' at day 365 and the '@' at day 366 if it exists.

It can be adapted to any accurate lunisolar cycle. For cycles in my
lunisolar spreadsheets at
http://www.the-light.com/cal/kp_Lunisolar_xls.html the number of 364-day
years in the cycle is simply the number of truncations (Trunc column L).
The number of leap years is equal to the number of leap years listed in
column E leap plus the number of 364-day years.

For example, the 1040-year cycle with mean month 29.5305916 days and
mean year of 365.24231 days in
http://www.the-light.com/cal/Lunisolar4.html has 1040 years of which 3
have 364 days, 252+3=255 have 366 days and the remaining 782 years have
365 days.

HELIOS CONTINUES:
I think 2 days ( with 1 golden numbered day ) can be dropped after a 353
year period and 1 day can be dropped after a 334 year period. This
scheme would apply within a 3101 year period.

3101 years = 353 + 334 + 353 + 334 + 353 + 334 + 353 + 334 + 353

This amounts to 14 days dropped every 3101 years. I think the Metonic
leap day rule can then be maintained.
The mean year is then ( 365 & 19 / 77 days ) - 14 / 3101 = 365.24223857
days

KARL SAYS:
The mean month created by the Metonic leap day rule would be changed.
This is because the 14 days removed over the course of one 3101-year
cycle is not exactly equal to the 9 periods of 1/19 month also removed.

The number of days in a 3101-year cycle varies according to its position
in the 77-year cycle of the leap year rule. The complete cycle is
11*3101=34111 years (noting the common divisor of 7 between 77 and
3101). I may work this 34111-year cycle out in by lunisolar spreadsheets
later on.

Karl

10(13(17

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[Karl Palmen]

Dear Tom, Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Tom Peters
Sent: 03 October 2009 19:29
To: CALNDR-L@...
Subject: Re: Metonic year correcting

Op 3-okt-2009, om 12:31 heeft Helios het volgende geschreven:

> We recall how golden numbered days are distributed evenly spaced and
> symmetrically
>
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N
> N @ N N @ N N @ N @ N N @ N N @ N
>
> and that the Metonic year can be aptly approximated by 365 & 19 /  
> 77 days.

No, why?  The original Metonic cycle was 6940 days, so the Metonic  
year was 365+5/19 = 365.263... days - much too long.
Since then, the Metonic relation 19 years = 235 lunations has been  
maintained with whatever year length our solar calendar had; so 19  
Julian years = 6939.75 days on average.  In 76 years (Calyppic  
cycle), that's 365+19/76 days.  So where does your 365+19/77 days  
come from?

KARL SAYS: The reason why Helios is keen on 365+19/77 days is because
1/235 of this is a very accurate approximation to the current mean
synodic month equal to 29.53058856... days.

> We have mentioned the idea of dropping a golden numbered day every  
> 334 or
> 353 years. This will however necessitate a change in the leap day  
> rule and
> this will diminish the accuracy of the golden numbers.
>
> I think 2 days ( with 1 golden numbered day ) can be dropped after  
> a 353
> year period and 1 day can be dropped after a 334 year period. This  
> scheme
> would apply within a 3101 year period.

This implies that after 353 years some 2-day error has built up.  
Then it should be corrected after 9 or 10 cycles (171 or 190 years)  
by 1 day.
But what error is that?


> 3101 years = 353 + 334 + 353 + 334 + 353 + 334 + 353 + 334 + 353
>
> This amounts to 14 days dropped every 3101 years. I think the  
> Metonic leap
> day rule can then be maintained.
> The mean year is then ( 365 & 19 / 77 days ) - 14 / 3101 =  
> 365.24223857 days

This means you make a correction for some (very) long-term accuracy,  
while allowing a 2-day error in the medium-term accuracy.

I think you are somehow confusing 2 errors here:
1) The fact that 19 years <> 235 lunations: this is corrected by  
shifting the Golden Numbers after 334 or 353 days.
2) The error in the length of the mean solar year.  That is  
independent from the ratio between the solar and lunar period (error 1).

KARL SAYS:
Helios wants to preserve the 365 19/77 leap year cycle in the belief, it
with preserves its accurate mean month.
Each of his corrections of 1 or 2 days also removes 1/19 of a month (a
golden number) so does what is required for the month to year ratio.
This correction of the 19-year cycle by 14 days over 3101 years does not
preserve the mean month, but shortens it slightly.

I reckon Helios's mean year is 365 + 19/77 - 14/3101, which I make to be
365 8263/34111 days.
The mean month is then (3101/38354)*(365 8263/34111) days =
29.530588252... days which is slightly shorter.

The complete lunisolar calendar cycle is 34,111 years and can be
produced on my lunisolar spreadsheets.

Karl

10(13(17

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