Alternating Periodic Sequences

[Helios]

I can make a batch of these "mean yerm" formulas. I think they work because 730 = 2*365. There's a sort of range between 727 and 733 which are prime numbers. So, there are certain subdivisions of the year and a "mean yerm" to each...

mean yerm of 8ths = 1 / [  2 - ( 728 / Y ) ]

mean yerm of 3rds = 1 / [  2 - ( 729 / Y ) ]

mean yerm of 10ths = 1 / [  2 - ( 730 / Y ) ]

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]

mean yerm of 12ths = 1 / [ ( 732 / Y ) - 2 ]

Some years lead to integer mean yerms. These 5 "elegant" years are

365 & 71 / 293
365 & 119 / 491
365 & 365 / 1507
365 & 8 / 33
365 & 39 / 161

and all cycles are odd numbered. These cycles are then described by their alternating periodic sequence,

8ths = 46, 45, ...
3rds = 122, 121, ...
10ths = 37, 36, ...
17ths = 21, 22, ...
12ths = 30, 31, ...

[Karl Palmen]

Dear Helios and Calendar People

Interesting.

We have a mean yerm of ABS( 1/ [ N/Y - 2]) days, where Y is the mean
year and N is an integer close to 2*Y. The nearer N is to 2*Y the longer
the yerm. N is multiple of the number of days in two consecutive
subdivisions in a yerm and the multiplier is the number of subdivisions
in a year

728 = 8*(46+45)
729 = 3*(122+121)
730 = 10*(37+36)
731 = 17*(21+22)
732 = 12*(30+31)

There is no example for N=733, because 733 is prime. 734 is twice a
prime number so can be used only for a half year subdivision.

The subdivision must be considerably smaller than the mean yerm, else
the yerms will have too few subdivisions to be useful. T For N=729, The
1/3 year (of 122 or 121 days) whose mean yerm is about 246 days is
rather too long. However N=729 will work with smaller divisions as shown
below.

N=728 can also be done for 1/56 and 1/104 of a year with subdivisions
alternating between (7,6) and (4,3) days respectively.

N=729 can also be done for subdivisions of a 1/9 and 1/27 of a year with
subdivisions alternating between (41,40) and (14,13) days respectively.
Also 1/81 for (5,4) and 1/243 for (2,1).

N=730 can be done for 1/146 of a year with subdivisions alternating
between (3,2) days.

N=731 can be done for 1/43 of a year with subdivisions alternating
between (9,8) days.

N=732 can also be done for 1/4 year with quarters alternating between
(91,92) days. It can also be done with 1/244 part of a year with
subdivisions alternating between (1,2) days and Victor has done this for
the 161-year cycle that has a mean year of exactly 241 days.

The mean yerm is often close to a whole number of days and furthermore
is equal to a whole number of days for one of the 5 "elegant" solar
calendar cycles listed by Helios

N=728 has a 147 day mean yerm for mean year 365 71/293 days
N=729 has a 246 day mean yerm for mean year 365 119/491 days
N=730 has a 754 day mean yerm for mean year 365 365/1507 days
N=731 has a 709 day mean yerm for mean year 365 8/33 days
N=732 has a 241 day mean yerm for mean year 365 39/161 days

Furthermore, N=731 can have every yerm with 709 days, if a subdivision
of 1/17 year (a term) is used.
Also, N=732 can have every yerm with 241 days, if a subdivision of 1/244
year is used (discovered by Victor).

Karl

10(06(17

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 12 March 2009 10:34
To: CALNDR-L@...
Subject: Alternating Periodic Sequences

I can make a batch of these "mean yerm" formulas. I think they work
because
730 = 2*365. There's a sort of range between 727 and 733 which are prime
numbers. So, there are certain subdivisions of the year and a "mean
yerm" to
each...

mean yerm of 8ths = 1 / [  2 - ( 728 / Y ) ]

mean yerm of 3rds = 1 / [  2 - ( 729 / Y ) ]

mean yerm of 10ths = 1 / [  2 - ( 730 / Y ) ]

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]

mean yerm of 12ths = 1 / [ ( 732 / Y ) - 2 ]

Some years lead to integer mean yerms. These 5 "elegant" years are

365 & 71 / 293
365 & 119 / 491
365 & 365 / 1507
365 & 8 / 33
365 & 39 / 161

and all cycles are odd numbered. These cycles are then described by
their
alternating periodic sequence,

8ths = 46, 45, ...
3rds = 122, 121, ...
10ths = 37, 36, ...
17ths = 21, 22, ...
12ths = 30, 31, ...

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[Karl Palmen]

Dear Helios, Victor and Calendar People

Similar to this, one could have a calendar where each year contains a
fixed number of parts and each part with a small number of exceptions
have the same number of days and the exceptions either all have one day
more or all have one day less than the normal part.

One can then work out the mean interval between the exception parts for
a mean year of Y

1 / [ 1 - (365/Y) ] applies to
1/5 years normally of 73 days exceptions of 74 days and
1/73 years normally of 5 days exceptions of 6 days.

1/ [ 1 - (364/Y) ] applies to
1/4 years normally of 91 days exceptions of 92 days,
1/7 years normally of 52 days exceptions of 53 days,
1/13 years normally of 28 days exceptions of 29 days (e.g. Victor's
293/28 calendar)
1/14 years normally of 26 days exceptions of 27 days and numerous
others.

1/ [ (366/Y) - 1 ] applies to
1/3 years normally of 122 days exceptions of 121 days,
1/6 years normally of 61 days exceptions of 60 days,
1/61 years normally of 6 days exceptions of 5 days,
1/122 years normally of 3 days exceptions of 2 days.

These three intervals are approximately, 1508, 294 and 482 days
respectively.

Karl

10(06(22

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 12 March 2009 10:34
To: CALNDR-L@...
Subject: Alternating Periodic Sequences

I can make a batch of these "mean yerm" formulas. I think they work
because
730 = 2*365. There's a sort of range between 727 and 733 which are prime
numbers. So, there are certain subdivisions of the year and a "mean
yerm" to
each...

mean yerm of 8ths = 1 / [  2 - ( 728 / Y ) ]

mean yerm of 3rds = 1 / [  2 - ( 729 / Y ) ]

mean yerm of 10ths = 1 / [  2 - ( 730 / Y ) ]

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]

mean yerm of 12ths = 1 / [ ( 732 / Y ) - 2 ]

Some years lead to integer mean yerms. These 5 "elegant" years are

365 & 71 / 293
365 & 119 / 491
365 & 365 / 1507
365 & 8 / 33
365 & 39 / 161

and all cycles are odd numbered. These cycles are then described by
their
alternating periodic sequence,

8ths = 46, 45, ...
3rds = 122, 121, ...
10ths = 37, 36, ...
17ths = 21, 22, ...
12ths = 30, 31, ...

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[Helios]

I see two cases of alternating periodic sequences that seem compelling to calculate

Case 1) 9th of year = 11 / 8 months
mean yerm of 9ths = 1 / [ 2 - ( 729 / Y ) ]
octaeteris year = Y = 365 & 187 / 424 days
mean yerm of 9ths = 194 & 45 / 266 days = 212 / 399 Y
--------------------------------------------
Case 2) 8th of year = 17 / 11 months
mean yerm of 8ths = 1 / [ 2 - ( 728 / Y ) ]
11-L'S year = Y = 365 & 29 / 275 days
mean yerm of 8ths = 165 & 21 / 152 days
--------------------------------------------
and these cases, used for a calendar, reckon for the 11 and 17 month interval

[Karl Palmen]

Dear Helios

Neither of these suggestions is as accurate as

Case 3) 5th of a year 47 / 19 months
Mean interval between leap 5ths 1 / [ 1 - ( 365/ Y ) ] in days
Metonic year = Y = 365 & 149/600 days
Mean interval between leap 5ths 1470 & 119/149 days

But even this does not provide a very accurate year.


The 334-year cycle has 243*17 lunar months hence 243*(11/8) = 334 + 1/8
11-L's years. Hence we can get an accurate mean year by making every
334th year have nine of these '8ths' instead of the usual 8. Using the
same mean '8th' as Helios, we get a mean year of (2673/2672)*( 365 & 29
/ 275 days ), which is approximately 365.2420958 days.

But the mean yerm of 8ths is only about 165 days or 3.6 '8ths' and so is
too short for the yerm concept to be of much use. Also the correction of
one '8th' every 334 years would produce a jitter in excess of one month.


Instead, we could abandon the alternating sequence and divide each 8th
into 17 parts and have 136 parts in 317 years and 137 parts in 17 years
each 334-year cycle. Each part corresponds to 1/11 lunar month. We still
need a rule for the number of days in each part. One such rule may have
been suggested to the list.

For case three once can divide the 47th into 5 parts and have 235 parts
in most years and correct by removing a part about once every 300 to 400
years. Each part corresponds to 1/19 of a lunar month. We still need a
rule for the number of days in each part. I believe at least one such
rule has been suggested to the list.

Karl

10(08(11

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 02 May 2009 05:40
To: CALNDR-L@...
Subject: Re: Alternating Periodic Sequences

I see two cases of alternating periodic sequences that seem compelling
to
calculate

Case 1) 9th of year = 11 / 8 months
mean yerm of 9ths = 1 / [ 2 - ( 729 / Y ) ]
octaeteris year = Y = 365 & 187 / 424 days
mean yerm of 9ths = 194 & 45 / 266 days = 212 / 399 Y
--------------------------------------------
Case 2) 8th of year = 17 / 11 months
mean yerm of 8ths = 1 / [ 2 - ( 728 / Y ) ]
11-L'S year = Y = 365 & 29 / 275 days
mean yerm of 8ths = 165 & 21 / 152 days
--------------------------------------------
and these cases, used for a calendar, reckon for the 11 and 17 month
interval

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[Helios]

Dear Karl P and Calendar People

This might be an improvement. There are notable properties of the cell that is set to 1 / 243rd year when applied to the mean year of the 95-year. By first using the year

Y = 365 & 23 / 95 days

There are 23065 cells in the 95-year cycle of 34698 days. With
1 cell = 1 & 3871 / 7695 days = 1 / 243rd year

We are equipt to evaluate

mean yerm of 243rds = 1 / [ 2 - ( 729 / Y ) ] = X

X = cell-yerm = 246 & 4 / 47 days

So there are 141 cell-yerms in the 95-year cycle of 34698 days.
-------------------------------------------------------------------------

Now it may be noted that

334 cells = 17 months

4131 months = 334 years

[Karl Palmen]

Dear Helios and Calendar People

The resulting cycle has a mean year of about 365.242105 days and a mean
lunar month of 29.5305889997 days.

It would take 95 334-year cycles lasting 31,730 years to repeat exactly.

Each cell yerm has either 163 cells totalling 245 days or 165 cells
totalling 248 days. They occur is the proportion needed to average 246
4/47 days.

Karl

10(08(11

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 05 May 2009 16:36
To: CALNDR-L@...
Subject: Re: Alternating Periodic Sequences

Dear Karl P and Calendar People

This might be an improvement. There are notable properties of the cell
that
is set to 1 / 243rd year when applied to the mean year of the 95-year.
By
first using the year

Y = 365 & 23 / 95 days

There are 23065 cells in the 95-year cycle of 34698 days. With
1 cell = 1 & 3871 / 7695 days = 1 / 243rd year

We are equipt to evaluate

mean yerm of 243rds = 1 / [ 2 - ( 729 / Y ) ] = X

X = cell-yerm = 246 & 4 / 47 days

So there are 141 cell-yerms in the 95-year cycle of 34698 days.
------------------------------------------------------------------------
-

Now it may be noted that

334 cells = 17 months

4131 months = 334 years

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[Karl Palmen]

Dear Helios and Calendar People

If for this we had a mean cell yerm of exactly 246 days, then we'd have
a 3-yerm cycle of 491 cells lasting 738 days. 243 of these 3-yerm cycle
would make 491 years of 179,334 days with a mean year of exactly 365
119/491 days about 365.24236 days.

The mean month is 334/(243*17) this mean year (738*334/(491*17)) so is
exactly 29 4429/8347 days about 29.5306098 days.
The resulting 8347-month cycle forms a simple lunar yerm cycle of 511
lunar yerms (one lunar yerm era).

I'll leave it the Helios and other calendar people to investigate the
case were the mean cell yerm has exactly 246 3/34 days.

Karl

10(08(11 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 05 May 2009 16:36
To: CALNDR-L@...
Subject: Re: Alternating Periodic Sequences

Dear Karl P and Calendar People

This might be an improvement. There are notable properties of the cell
that
is set to 1 / 243rd year when applied to the mean year of the 95-year.
By
first using the year

Y = 365 & 23 / 95 days

There are 23065 cells in the 95-year cycle of 34698 days. With
1 cell = 1 & 3871 / 7695 days = 1 / 243rd year

We are equipt to evaluate

mean yerm of 243rds = 1 / [ 2 - ( 729 / Y ) ] = X

X = cell-yerm = 246 & 4 / 47 days

So there are 141 cell-yerms in the 95-year cycle of 34698 days.
------------------------------------------------------------------------
-

Now it may be noted that

334 cells = 17 months

4131 months = 334 years

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[Helios]

I now count three cases in which there is an endless repetition of the same yerm. Quite surely these are the only possibilities of this scheme.

17ths = 21, 22, 21, 22, 21, 22, 21, 22, 21, ... ( 33 cells )
146ths = 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, ... ( 301 cells )
244ths = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... ( 161 cells )

To calculate the mean yerm

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 730 / Y ) ]
mean yerm of 244ths = 1 / [ ( 732 / Y ) - 2 ]

or ( C = cell )

mean yerm of 17ths = 1 / [ ( 43 / C ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 5 / C ) ]
mean yerm of 244ths = 1 / [ ( 3 / C ) - 2 ]

[Karl Palmen]

Dear Helios and Calendar People

The 146ths and 244ths can be paired up to form 73rds and 122nds that are
normally of constant length.

I've already suggested the 122nds to Victor, where each 122nd has 3 days
except every 161st 122nd which has one day less. Giving a 161-year cycle
with mean year 365 39/161 days (365.242236 days)

For the 73rds, each has 5 days except every 301st, which has one day
more.
Giving a 301-year cycle mean year of 365 73/301 days (365.242525 days).

The 17ths cannot be so paired, because 17 is an odd-number.


Karl

10(08(27

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 21 May 2009 14:47
To: CALNDR-L@...
Subject: Re: Alternating Periodic Sequences

I now count three cases in which there is an endless repetition of the
same
yerm. Quite surely these are the only possibilities of this scheme.

17ths = 21, 22, 21, 22, 21, 22, 21, 22, 21, ... ( 33 cells )
146ths = 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, ... ( 301 cells )
244ths = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... ( 161 cells )

To calculate the mean yerm

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 730 / Y ) ]
mean yerm of 244ths = 1 / [ ( 732 / Y ) - 2 ]

or ( C = cell )

mean yerm of 17ths = 1 / [ ( 43 / C ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 5 / C ) ]
mean yerm of 244ths = 1 / [ ( 3 / C ) - 2 ]

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[Brillig]

And the 122nds can be further paired to make 61sts.

Here is a picture of one Metonic cycle crocheted out of such 244ths. The color changes after each run of 161 cells. The row joins the previous row at the start of a new year, so there are 19 rows making up the Metonic cycle.

http://the-light.com/cal/vecrochet0.jpg

This was crocheted using single stitches and double stitches alternating with each other. For more detail, consult the archives.

Victor

On Thu, May 21, 2009 at 11:14 AM, Palmen, KEV (Karl) <karl.palmen@...> wrote:

Dear Helios and Calendar People

The 146ths and 244ths can be paired up to form 73rds and 122nds that are
normally of constant length.

I've already suggested the 122nds to Victor, where each 122nd has 3 days
except every 161st 122nd which has one day less. Giving a 161-year cycle
with mean year 365 39/161 days (365.242236 days)

For the 73rds, each has 5 days except every 301st, which has one day
more.
Giving a 301-year cycle mean year of 365 73/301 days (365.242525 days).

The 17ths cannot be so paired, because 17 is an odd-number.


Karl

10(08(27

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 21 May 2009 14:47
To: CALNDR-L@...
Subject: Re: Alternating Periodic Sequences

I now count three cases in which there is an endless repetition of the
same
yerm. Quite surely these are the only possibilities of this scheme.

17ths = 21, 22, 21, 22, 21, 22, 21, 22, 21, ... ( 33 cells )
146ths = 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, ... ( 301 cells )
244ths = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... ( 161 cells )

To calculate the mean yerm

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 730 / Y ) ]
mean yerm of 244ths = 1 / [ ( 732 / Y ) - 2 ]

or ( C = cell )

mean yerm of 17ths = 1 / [ ( 43 / C ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 5 / C ) ]
mean yerm of 244ths = 1 / [ ( 3 / C ) - 2 ]

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[Karl Palmen]

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

Dear Victor and Calendar People

 

Victor said: And the 122nds can be further paired to make 61sts.

 

The 122nds are not alternating and  the effect of pairing is not the same as for pairing an alternating sequence.

 

When pairing the alternating sequences described by Helios, the period between the exceptional paired cells is two yerm periods of the alternating cells. So the number of cells in the period is unchanged by the pairing.

 

When pairing non-alternating cells the period between the exceptional pairs cells is the same as for the unpaired cells and hence the number of cells in the period is halved. This results in a simple regular period only if the number of unpaired cells in the period is an even number. In the case of the 122nds this is not the case (period 161 cells). So pairing them does not result in a simple regular period of exceptional cells (161/122 cannot become INT/61).

 

 

Victor said: Here is a picture of one Metonic cycle crocheted out of such 244ths.

 

I don’t see how the crocheted cells of 244^th year make up a Metonic cycle. You’d need cells of 235^th year to make a Metonic cycle. Perhaps, it could be done with  9*19=171 of the 235 months have 20 cells while the remaining 64 months have 19 cells, but this would not produce a good correspondence with the moon phases, owing to considerable variation of the month length.

The year can be divided into 235 cells by having 26 fortnights of 9 cells each followed by one  more cell.

See http://www.the-light.com/cal/kp_yermette.html .

 

 

Karl

 

10(08(27 till noon

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Victor Engel
Sent: 21 May 2009 17:23
To: CALNDR-L@...
Subject: Re: Pairing: Alternating Periodic Sequences

 

And the 122nds can be further paired to make 61sts.

Here is a picture of one Metonic cycle crocheted out of such 244ths. The color changes after each run of 161 cells. The row joins the previous row at the start of a new year, so there are 19 rows making up the Metonic cycle.

http://the-light.com/cal/vecrochet0.jpg

This was crocheted using single stitches and double stitches alternating with each other. For more detail, consult the archives.

Victor

On Thu, May 21, 2009 at 11:14 AM, Palmen, KEV (Karl) <karl.palmen@...> wrote:

Dear Helios and Calendar People

The 146ths and 244ths can be paired up to form 73rds and 122nds that are
normally of constant length.

I've already suggested the 122nds to Victor, where each 122nd has 3 days
except every 161st 122nd which has one day less. Giving a 161-year cycle
with mean year 365 39/161 days (365.242236 days)

For the 73rds, each has 5 days except every 301st, which has one day
more.
Giving a 301-year cycle mean year of 365 73/301 days (365.242525 days).

The 17ths cannot be so paired, because 17 is an odd-number.


Karl

10(08(27

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 21 May 2009 14:47
To: CALNDR-L@...
Subject: Re: Alternating Periodic Sequences

I now count three cases in which there is an endless repetition of the
same
yerm. Quite surely these are the only possibilities of this scheme.

17ths = 21, 22, 21, 22, 21, 22, 21, 22, 21, ... ( 33 cells )
146ths = 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, ... ( 301 cells )
244ths = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... ( 161 cells )

To calculate the mean yerm

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 730 / Y ) ]
mean yerm of 244ths = 1 / [ ( 732 / Y ) - 2 ]

or ( C = cell )

mean yerm of 17ths = 1 / [ ( 43 / C ) - 2 ]
mean yerm of 146ths = 1 / [ 2 - ( 5 / C ) ]
mean yerm of 244ths = 1 / [ ( 3 / C ) - 2 ]

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[Brillig]

Dear Karl and Calendar People,

On Fri, May 22, 2009 at 3:19 AM, Palmen, KEV (Karl) <karl.palmen@...> wrote:

Dear Victor and Calendar People

 

Victor said: And the 122nds can be further paired to make 61sts.

 

The 122nds are not alternating and  the effect of pairing is not the same as for pairing an alternating sequence.

 

True.

Victor said: Here is a picture of one Metonic cycle crocheted out of such 244ths.

 

I don’t see how the crocheted cells of 244^th year make up a Metonic cycle. You’d need cells of 235^th year to make a Metonic cycle.

Perhaps I misspoke. The crocheted pattern has nothing to do with the moon. It is simply 19 years (19 rounds), which I was calling a Metonic cycle. I do have a similar pattern of stitches with moon indicated using a tan line at http://the-light.com/cal/vecrochet1.jpg and http://the-light.com/cal/vecrochet2.jpg There the stitches are arranged in a shorter cycle (2 fortnights with adjustments, to keep yearly events columnar, if memory serves).

Victor