27 Month Lunar Unit

classic Classic list List threaded Threaded
10 messages Options
Reply | Threaded
Open this post in threaded view
|

27 Month Lunar Unit

Helios
Dear Calendar People,

I am finding that a 27-month period is useful enough to need its own name so I have called it a "Pakal".
Its value is approximated by continued fraction, here arranged by tiers of accuracy,

1) 797 days ( 27-month yerm )
2) 797 & 1 / 3 days ( 15 Pakals = 46 Tzolkins = Mayan eclipse cycle )
3) 797 & 15 / 46 days
4) 797 & 29 / 89 days
5) 797 & 44 / 135 days ( converges to 5010 years = 2295 Pakals )

See how months within the Pakal can be grouped

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons

[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

This is why the Pakal is interesting, that 27 months carry two kinds of information. The math leads to 15 Octaeterides = 120 Years = 55 Pakals = 11 Tritos.
The grouping scheme results in 3 pentalunex every 23 eclipse seasons and 3 leap months every 8 years. I consider these basic cycles likely to have been known since neolithic times. Just with 27 stones ( "moon stones" ) and the knowledge of how to arrange them, it's a simple matter of how to reckon things by months. This is an easy chronological system. By the eleven partion year, every 8th partion gets two months just as the actual numeral "8" resembles two moons.

The 23 season grouping cycle has common seasons of 6 months, but with a pentalunex every 7 or 8 seasons. These are seasons 4, 12, and 20 -- odd multiples of 4. These can be named tetrahedron, dodecahedron, and icosahedron. The arrangement of months in this fashion can be called the Tritos method or Tritos style. There are 135 months in the Tritos cycle.

The only possible octaeteris magic square prescribes certain leap months per 8 years. These are years 1, 4, and 7. This makes Olympic years symmetical in the eleven-fold scheme.

The 11-fold Tritos eclipse ( 15 Octaeterides ) would be expected to repeat 6 times. However, the Octaeteris is off by a month between 18 and 19 repeats as a solar measure.
Reply | Threaded
Open this post in threaded view
|

Re: 27 Month Lunar Unit

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

To get a mean Pakal of 797 & 44/135 days, one needs not only a 27-month yerm of 797 days, but also a leap pakal of 798 days which can be made from three 9-month yerms of 266 days.

Each 9-month yerm in a leap pakal is equal to 38 weeks.

The 5010-year cycle of 2295 Pakals occurs in
http://the-light.com/cal/LunisolarML.htm 
It has 17 lunar calendar cycles of 3645 months = 27*135 months. It is also equal to fifteen 334-year cycles. Helios has mentioned this cycle before.

Karl

14(15(03 till noon

________________________________________
From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Helios [[hidden email]]
Sent: 23 March 2015 01:04
To: [hidden email]
Subject: 27 Month Lunar Unit

Dear Calendar People,

I am finding that a 27-month period is useful enough to need its own name so
I have called it a "Pakal".
Its value is approximated by continued fraction, here arranged by tiers of
accuracy,

1) 797 days ( 27-month yerm )
2) 797 & 1 / 3 days ( 15 Pakals = 46 Tzolkins = Mayan eclipse cycle )
3) 797 & 15 / 46 days
4) 797 & 29 / 89 days
5) 797 & 44 / 135 days ( converges to 5010 years = 2295 Pakals )

See how months within the Pakal can be grouped

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons

[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

This is why the Pakal is interesting, that 27 months carry two kinds of
information. The math leads to 15 Octaeterides = 120 Years = 55 Pakals = 11
Tritos.
The grouping scheme results in 3 pentalunex every 23 eclipse seasons and 3
leap months every 8 years. I consider these basic cycles likely to have been
known since neolithic times. Just with 27 stones ( "moon stones" ) and the
knowledge of how to arrange them, it's a simple matter of how to reckon
things by months. This is an easy chronological system. By the eleven
partion year, every 8th partion gets two months just as the actual numeral
"8" resembles two moons.

The 23 season grouping cycle has common seasons of 6 months, but with a
pentalunex every 7 or 8 seasons. These are seasons 4, 12, and 20 -- odd
multiples of 4. These can be named tetrahedron, dodecahedron, and
icosahedron. The arrangement of months in this fashion can be called the
Tritos method or Tritos style. There are 135 months in the Tritos cycle.

The only possible octaeteris magic square prescribes certain leap months per
8 years. These are years 1, 4, and 7. This makes Olympic years symmetical in
the eleven-fold scheme.

The 11-fold Tritos eclipse ( 15 Octaeterides ) would be expected to repeat 6
times. However, the Octaeteris is off by a month between 18 and 19 repeats
as a solar measure.




--
View this message in context: http://calndr-l.10958.n7.nabble.com/27-Month-Lunar-Unit-tp15706.html
Sent from the Calndr-L mailing list archive at Nabble.com.
Reply | Threaded
Open this post in threaded view
|

Re: 27 Month Lunar Unit

Helios
Dear Karl and Calendar People,

From 1973, an octaeteris calendar can be implemented which begins in January. After 120 years it could be retired. We can test if it's an empirical fit. Helpful is the fact that;
24 years = 50 & 3 / 5 seasons

Eleven Part Years
2015) 1 1 1 1 2 1 1 1 1 1 1
2016) 1 2 1 1 1 1 1 1 1 2 1 Leap
2017) 1 1 1 1 1 1 2 1 1 1 1
2018) 1 1 1 2 1 1 1 1 1 1 1
2019) 2 1 1 1 1 1 1 1 2 1 1 Leap
2020) 1 1 1 1 1 2 1 1 1 1 1

after rearrangement into eclipse seasons...
2015) 1 1 ][ 2 1 1 1 1 ][ 1 2 1 ( Mar 20 Total, Sep 13 Partial )
2016) 1 1 ][ 1 1 2 1 1 ][ 1 1 2 1 ( Mar 9 Total, Sep 1 Annular )
2017 & 2018 ) 1 ][ 1 1 1 2 1 ][ 1 1 1 1 2 ][ 1 1 1 1 1 ][ 2 1 1 1 1 ]
( Feb 26 Annular, Aug 21 Total, Feb 15 Partial, Jul 13 Partial and Aug 11 Partial )
2019) [ 2 1 1 1 1 ][ 1 2 1 1 1 ][ 1  ( Jan 6 Partial, Jul 2 Total, Dec 26 Annular )
2020) 1 2 1 1 ][ 1 1 1 2 1 ][ 1  ( Jun 21 Annular, Dec 14 Total )

Note that there are double partial eclipses in the last season of 2018.
Reply | Threaded
Open this post in threaded view
|

Re: 27 Month Lunar Unit

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 23 March 2015 19:56
To: [hidden email]
Subject: Re: 27 Month Lunar Unit

Dear Karl and Calendar People,

From 1973, an octaeteris calendar can be implemented which begins in
January. After 120 years it could be retired. We can test if it's an
empirical fit. Helpful is the fact that;
24 years = 50 & 3 / 5 seasons

KARL REPLIES: 24 years = 297 lunar months in the Octaeteris and so 1485 lunar months = 253 eclipse seasons. This divides by 11 to give
135 lunar months = 23 eclipse seasons, which is a tritos cycle.
See http://www.staff.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm 
The tritos has (8/99)*135 = 120/11 octaeteris years.

HELIOS CONTINUED:
Eleven Part Years
2015) 1 1 1 1 2 1 1 1 1 1 1
2016) 1 2 1 1 1 1 1 1 1 2 1 Leap
2017) 1 1 1 1 1 1 2 1 1 1 1
2018) 1 1 1 2 1 1 1 1 1 1 1
2019) 2 1 1 1 1 1 1 1 2 1 1 Leap
2020) 1 1 1 1 1 2 1 1 1 1 1

KARL SAYS: Note that every 8th 1/11 year has 2 months, while the others have 1 month. This creates an Octaeteris (8 years = 99 months).

after rearrangement into eclipse seasons...
2015) 1 1 ][ 2 1 1 1 1 ][ 1 2 1 ( Mar 20 Total, Sep 13 Partial )
2016) 1 1 ][ 1 1 2 1 1 ][ 1 1 2 1 ( Mar 9 Total, Sep 1 Annular )
2017 & 2018 ) 1 ][ 1 1 1 2 1 ][ 1 1 1 1 2 ][ 1 1 1 1 1 ][ 2 1 1 1 1 ]
( Feb 26 Annular, Aug 21 Total, Feb 15 Partial, Jul 13 Partial and Aug 11
Partial )
2019) [ 2 1 1 1 1 ][ 1 2 1 1 1 ][ 1  ( Jan 6 Partial, Jul 2 Total, Dec 26
Annular )
2020) 1 2 1 1 ][ 1 1 1 2 1 ][ 1  ( Jun 21 Annular, Dec 14 Total )

Note that there are double partial eclipses in the last season of 2018.

KARL REPLIES:
Helios does not state how he creates his eclipse seasons out of this 1/11 years. The tritos has 23 eclipse seasons lasting 120/11 years. This leads to 18 eclipse seasons of 5/11 years and 5 eclipse seasons of 6/11 years. This is different from the division of 27-month unit into 23 1/5-eclipse-seasons shown in previous note.

Helios mentioned the 5010-year cycle. It is equal to 17 334-year cycles and each 334-year cycle has 153 27-month units. The octaeteris places 24/11 years in each 27-month unit. I think this is why Helios chose the octaeteris. If this were continued 153 27-month units, which make one 334-year cycle we get 153*24/11 years = 3672/11 years = 333 9/11 years. So we'd need to add 2/11 years. We'd get the 334-year cycle if we were to add 1/11 year to every 76th & 77th 27-month unit alternating (replace one 2 with 1 1). These intervals exceed the 120 year lifetime of the octaeteris calendar.

The tritos itself is not very accurate and will not last much more than 120 years. It may not even last 120 years.

Karl

14(15(05




--
View this message in context: http://calndr-l.10958.n7.nabble.com/27-Month-Lunar-Unit-tp15706p15709.html
Sent from the Calndr-L mailing list archive at Nabble.com.
Reply | Threaded
Open this post in threaded view
|

55 Pakal cycle calendar

Helios
Dear Karl and Calendar People,

Since a Pakal doesnt have any use other use than in the 120 year cycle, we can call it the 55 Pakal cycle calendar and assign it a simple value of

1 Pakal = 797 & 18 / 55 days

Any Pakal can be picked out and assume either of two crystallizations,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons
[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

or one form can be dissolved and made into the other.

Tritos = 23 seasons = 10 & 10 / 11 years = 5 Pakals
Stitor = 24 years = 50 & 3 / 5 seasons = 11 Pakals

5 Stitors = 55 Pakals = 11 Tritos ( Stitor is an invented word )

so at 3 Stitors from 1973 for example,

using [ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] eclipse style

2045) 1 1 2 1 1 1 1 1 2 1 1
furthermore we can group seasons by fives, knowing that
3 Stitors = 151 & 4 / 5 seasons

2045) 1 ][ 1 2 1 1 1 ][ 1 1 2 1 1

I find on tables that indeed ( feb 16 annular, aug 12 total ). I do not think this method can fail inside the bounds since the endpoints of the 5 Stitor cycle are also Tritos eclipses. The 55 Pakal cycle calendar should be retired after one use.
Reply | Threaded
Open this post in threaded view
|

Re: 55 Pakal cycle calendar

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 24 March 2015 23:27
To: [hidden email]
Subject: 55 Pakal cycle calendar

Dear Karl and Calendar People,

Since a Pakal doesnt have any use other use than in the 120 year cycle, we
can call it the 55 Pakal cycle calendar and assign it a simple value of

1 Pakal = 797 & 18 / 55 days

KARL REPLIES: The Pakal can be used for a 334-year cycle which has 153 Pakals.


HELIOS CONTINUED:
Any Pakal can be picked out and assume either of two crystallizations,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons
[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

KARL REPLIES:
I think these periods of 1 or 2 months need names and think of pentipse for 1/5 eclipse season and elvear for 1/11 of octaeteris year. So a Pakal has 23 pentipses = 24 elvears made of months as shown by Helios. Now I can describe the pattern of 1s and 2s Helios has shown.

The 2-month pentipses occur once every six pentipses (= 7 months) within the Pakal and after five pentipses (= 6 months) between Pakals.
The 2-month elvears occur once every eight elvears (= 9 months) without exception.



HELIOS CONTINUED:
or one form can be dissolved and made into the other.

Tritos = 23 seasons = 10 & 10 / 11 years = 5 Pakals
Stitor = 24 years = 50 & 3 / 5 seasons = 11 Pakals

KARL REPLIES
Tritos = 23 seasons = 10 years & 10 elvears = 120 elvears = 5 Pakals
Stitor = 24 years = 50 seasons & 3 pentipses = 253 pentipses = 11 Pakals = 3 Octaeterides


HELIOS CONTINUED:
5 Stitors = 55 Pakals = 11 Tritos ( Stitor is an invented word )

so at 3 Stitors from 1973 for example,

using [ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] eclipse style

2045) 1 1 2 1 1 1 1 1 2 1 1
furthermore we can group seasons by fives, knowing that
3 Stitors = 151 & 4 / 5 seasons

2045) 1 ][ 1 2 1 1 1 ][ 1 1 2 1 1

KARL REPLIES: Helios groups the pentipses into fives to show the eclipse seasons.


HELIOS CONTINUED:
I find on tables that indeed ( feb 16 annular, aug 12 total ). I do not
think this method can fail inside the bounds since the endpoints of the 5
Stitor cycle are also Tritos eclipses. The 55 Pakal cycle calendar should be
retired after one use.

KARL SAYS:
After retirement, a new 55 Pakal calendar would be needed. This calendar can be started before retirement to provide and overlap period when both calendars run. The new calendar would be the same as the old calendar, except for its start date. I have found out that the 55 Pakal calendar can follow a 3-Pakal cycle of 2392 days and I will explain why later on. Therefore, the 3-Pakal cycle of the new calendar could begin a certain number of days before (or after) the 3-Pakal cycle of the old calendar. This number of days would of course be near a whole number of lunar months, no more than 80 lunar months.

The important question is how many days does the 3-Pakal cycle of new calendar begin before the 3-Pakal cycle of the old calendar?


The old & new calendars would not always agree during the overlap period, but if the overlap period is long enough I expect both calendars would sometime begin a month, elvear and pentipse on the same day. This day could be used for a smooth transition from the old to the new calendar.

The Octaeteris/Stitor gives too many 2s in the elvears and so there are only 24*153=3672 elvears in a 334-year cycle rather than the 3674 required. So a smooth transition would create a long run of 1-month elvears.

The Tritos gives too many 2s in the pentipses. Both 23 Saros cycles and 38 Tritos cycles have 23*38 = 874 eclipse seasons, but the 23 Saros cycles have 23*223=5129 months, while the 38 Tritos cycles have 38*135=5130 months. So a smooth transition would create a long run of 1-month pentipses.


Helios has not stated how many days each month would have, but did state that the mean Pakal is 797 & 18 / 55 days. This cycle has an error of less than 2 hours over the 55-Pakal lifetime of the calendar. Each 55-Pakal cycle has 18 short long Pakals of 798 days and 37 short Pakals of 797 days. These can be arranged so that every 3rd Pakal is a long Pakal. Because the calendar is retired after 55 Pakals, we can regard the 55th Pakal as the first Pakal of an unfinished 3-Pakal cycle and therefore a 3-Pakal cycle is effectively used throughout the lifetime. This means that the calendar effectively uses an 81-month cycle of 3 Pakals and of 2392 days (multiple of 23 days and also 13 days).

A smooth transition would create a long run of short Pakals, normally as the 55-Pakal cycle would.

I expect Helios would be delighted to find out that five of these 3-Pakal cycles is equal to the Maya 405-month cycle of 11960 days = 46 Tzolkin cycles of 260 days. An alternative way of defining the number of days of the months of the 3-Pakal cycle is by five yerms of 17, 15, 17, 15, 17 months rather than 27, 27, 9, 9, 9 months.


Karl

14(15(06




--
View this message in context: http://calndr-l.10958.n7.nabble.com/27-Month-Lunar-Unit-tp15706p15719.html
Sent from the Calndr-L mailing list archive at Nabble.com.
Reply | Threaded
Open this post in threaded view
|

Re: 55 Pakal cycle calendar

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

I wrote:
"After retirement, a new 55 Pakal calendar would be needed. This calendar can be started before retirement to provide and overlap period when both calendars run. The new calendar would be the same as the old calendar, except for its start date. I have found out that the 55 Pakal calendar can follow a 3-Pakal cycle of 2392 days and I will explain why later on. Therefore, the 3-Pakal cycle of the new calendar could begin a certain number of days before (or after) the 3-Pakal cycle of the old calendar. This number of days would of course be near a whole number of lunar months, no more than 80 lunar months."

I was wrong about the 80 lunar months. It has to be up to one month short of the full 27*55=1485 lunar months of the 55-Pakal cycle. I forgot that the grouping of the pentipses into eclipse seasons and the elvears into years form parts of the calendar and these parts do not follow the 3-Pakal cycle.

For example,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons

is short for
[ { 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }{ 2 1 1 ]
[ 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 ]
[ 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 ]
[ 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 ]
[ 1 1 2 }{ 1 1 1 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 } ]
for 5 Pakals = 1 tritos.

Similarly

[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

is short for
[ { 1 1 2 1 1 1 1 1 1 1 2 1 }{ 1 1 1 1 1 1 2 1 1 1 }{ 1 1  ]
[   1 1 2 1 1 1 1 1 1 1 }{ 2 1 1 1 1 1 1 1 2 1 }{ 1 1 1 1  ]
[   1 1 2 1 1 1 1 1 }{ 1 1 2 1 1 1 1 1 1 1 }{ 2 1 1 1 1 1  ]
etc.. for 11 Pakals = 1 stitor

Karl

14(15(06


-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 25 March 2015 13:44
To: 'East Carolina University Calendar discussion List'
Subject: RE: 55 Pakal cycle calendar

Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 24 March 2015 23:27
To: [hidden email]
Subject: 55 Pakal cycle calendar

Dear Karl and Calendar People,

Since a Pakal doesnt have any use other use than in the 120 year cycle, we
can call it the 55 Pakal cycle calendar and assign it a simple value of

1 Pakal = 797 & 18 / 55 days

KARL REPLIES: The Pakal can be used for a 334-year cycle which has 153 Pakals.


HELIOS CONTINUED:
Any Pakal can be picked out and assume either of two crystallizations,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons
[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

KARL REPLIES:
I think these periods of 1 or 2 months need names and think of pentipse for 1/5 eclipse season and elvear for 1/11 of octaeteris year. So a Pakal has 23 pentipses = 24 elvears made of months as shown by Helios. Now I can describe the pattern of 1s and 2s Helios has shown.

The 2-month pentipses occur once every six pentipses (= 7 months) within the Pakal and after five pentipses (= 6 months) between Pakals.
The 2-month elvears occur once every eight elvears (= 9 months) without exception.



HELIOS CONTINUED:
or one form can be dissolved and made into the other.

Tritos = 23 seasons = 10 & 10 / 11 years = 5 Pakals
Stitor = 24 years = 50 & 3 / 5 seasons = 11 Pakals

KARL REPLIES
Tritos = 23 seasons = 10 years & 10 elvears = 120 elvears = 5 Pakals
Stitor = 24 years = 50 seasons & 3 pentipses = 253 pentipses = 11 Pakals = 3 Octaeterides


HELIOS CONTINUED:
5 Stitors = 55 Pakals = 11 Tritos ( Stitor is an invented word )

so at 3 Stitors from 1973 for example,

using [ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] eclipse style

2045) 1 1 2 1 1 1 1 1 2 1 1
furthermore we can group seasons by fives, knowing that
3 Stitors = 151 & 4 / 5 seasons

2045) 1 ][ 1 2 1 1 1 ][ 1 1 2 1 1

KARL REPLIES: Helios groups the pentipses into fives to show the eclipse seasons.


HELIOS CONTINUED:
I find on tables that indeed ( feb 16 annular, aug 12 total ). I do not
think this method can fail inside the bounds since the endpoints of the 5
Stitor cycle are also Tritos eclipses. The 55 Pakal cycle calendar should be
retired after one use.

KARL SAYS:
After retirement, a new 55 Pakal calendar would be needed. This calendar can be started before retirement to provide and overlap period when both calendars run. The new calendar would be the same as the old calendar, except for its start date. I have found out that the 55 Pakal calendar can follow a 3-Pakal cycle of 2392 days and I will explain why later on. Therefore, the 3-Pakal cycle of the new calendar could begin a certain number of days before (or after) the 3-Pakal cycle of the old calendar. This number of days would of course be near a whole number of lunar months, no more than 80 lunar months.

The important question is how many days does the 3-Pakal cycle of new calendar begin before the 3-Pakal cycle of the old calendar?


The old & new calendars would not always agree during the overlap period, but if the overlap period is long enough I expect both calendars would sometime begin a month, elvear and pentipse on the same day. This day could be used for a smooth transition from the old to the new calendar.

The Octaeteris/Stitor gives too many 2s in the elvears and so there are only 24*153=3672 elvears in a 334-year cycle rather than the 3674 required. So a smooth transition would create a long run of 1-month elvears.

The Tritos gives too many 2s in the pentipses. Both 23 Saros cycles and 38 Tritos cycles have 23*38 = 874 eclipse seasons, but the 23 Saros cycles have 23*223=5129 months, while the 38 Tritos cycles have 38*135=5130 months. So a smooth transition would create a long run of 1-month pentipses.


Helios has not stated how many days each month would have, but did state that the mean Pakal is 797 & 18 / 55 days. This cycle has an error of less than 2 hours over the 55-Pakal lifetime of the calendar. Each 55-Pakal cycle has 18 short long Pakals of 798 days and 37 short Pakals of 797 days. These can be arranged so that every 3rd Pakal is a long Pakal. Because the calendar is retired after 55 Pakals, we can regard the 55th Pakal as the first Pakal of an unfinished 3-Pakal cycle and therefore a 3-Pakal cycle is effectively used throughout the lifetime. This means that the calendar effectively uses an 81-month cycle of 3 Pakals and of 2392 days (multiple of 23 days and also 13 days).

A smooth transition would create a long run of short Pakals, normally as the 55-Pakal cycle would.

I expect Helios would be delighted to find out that five of these 3-Pakal cycles is equal to the Maya 405-month cycle of 11960 days = 46 Tzolkin cycles of 260 days. An alternative way of defining the number of days of the months of the 3-Pakal cycle is by five yerms of 17, 15, 17, 15, 17 months rather than 27, 27, 9, 9, 9 months.


Karl

14(15(06




--
View this message in context: http://calndr-l.10958.n7.nabble.com/27-Month-Lunar-Unit-tp15706p15719.html
Sent from the Calndr-L mailing list archive at Nabble.com.
Reply | Threaded
Open this post in threaded view
|

Re: 55 Pakal cycle calendar

Karl Palmen - UKRI STFC
In reply to this post by Helios
Dear Helios and Calendar People

Applying my eclipse calendar ideas, which I explained in my previous note, I find the that each unitos (hepton or octon) ends with an eclipse season with no '2' in it. Also a unitos is a hepton if and only if it contains the start of a tritos.

I show Helios's tritos again
[ { 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }{ 2 1 1 ]
[ 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 ]
[ 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 ]
[ 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 ]
[ 1 1 2 }{ 1 1 1 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 } ]

Now I put a 'U' after each eclipse season with no '2' in it.
[ { 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }U{ 2 1 1 ]
[ 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 ]
[ 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }U{ 2 1 1 1 1 }{ 1 2 1 1 ]
[ 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 ]
[ 1 1 2 }{ 1 1 1 1 1 }U{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 } ]

Then I rearrange into unitoses, putting T at the start of Helios's tritos.
{ 2 1 1 ][ 1 1 }{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 ][ 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }U=O
{ 2 1 1 1 1 }{ 1 2 1 1 ][ 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 ][ 1 1 2 }{ 1 1 1 1 1 }U=O
{ 2 1 1 1 1 }{ 1 2 1 1 1 }{ 1 1 2 1 1 } ]T[ { 1 1 2 1 1 }{ 1 1 1 2 1 }{ 1 1 1 1 2 }{ 1 1 1 1 1 }U=H
From this I can see that the Hepton is the unitos that contains the start of a Tritos. It is also the unitos that has the start of only one Pakal in it.

I look for eclipse cycles that have a whole number of Pakals in them and from
http://www.staff.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm 
I do not find any that are not a multiple of a Tritos or 27 shorter cycles. The total number of Inexes and Saroses must be a multiple of 27.
However, three consecutive cycles (Trihex, Half Babylonian Period, Unnamed(246)) do add up to such a cycle (16I+11S), which has 8181 months, 303 Pakals, 1394 eclipse seasons.
Such a cycle could span several of Helios's 55-Pakal calendars using the same Pakals, but different eclipse seasons within them.

Karl

14(15(07


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 24 March 2015 23:27
To: [hidden email]
Subject: 55 Pakal cycle calendar

Dear Karl and Calendar People,

Since a Pakal doesnt have any use other use than in the 120 year cycle, we
can call it the 55 Pakal cycle calendar and assign it a simple value of

1 Pakal = 797 & 18 / 55 days

Any Pakal can be picked out and assume either of two crystallizations,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 / 5 eclipse seasons
[  1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 ] = 24 / 11 years

or one form can be dissolved and made into the other.

Tritos = 23 seasons = 10 & 10 / 11 years = 5 Pakals
Stitor = 24 years = 50 & 3 / 5 seasons = 11 Pakals

5 Stitors = 55 Pakals = 11 Tritos ( Stitor is an invented word )

so at 3 Stitors from 1973 for example,

using [ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] eclipse style

2045) 1 1 2 1 1 1 1 1 2 1 1
furthermore we can group seasons by fives, knowing that
3 Stitors = 151 & 4 / 5 seasons

2045) 1 ][ 1 2 1 1 1 ][ 1 1 2 1 1

I find on tables that indeed ( feb 16 annular, aug 12 total ). I do not
think this method can fail inside the bounds since the endpoints of the 5
Stitor cycle are also Tritos eclipses. The 55 Pakal cycle calendar should be
retired after one use.




--
View this message in context: http://calndr-l.10958.n7.nabble.com/27-Month-Lunar-Unit-tp15706p15719.html
Sent from the Calndr-L mailing list archive at Nabble.com.
Reply | Threaded
Open this post in threaded view
|

Re: Pakals and Pakaloids

Helios
Dear Karl and Calendar People,

Apart from a very long continuum of Pakals, there will be an abberant grouping, a Pakaloid, every so often. I find this to be a 34 month period, here compared to the Pakal,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 pentipse ( Tritos I - S  )

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 29 pentipse
( 34m*5 = 170m 12*S - 7*I  )

Now add Pakals to both sides of a Pakaloid to make a larger symmetrical entity, As it turns out, these are the simple eclipses which emerge,

Inex = 358 months = 1 Pakaloid + 12 Pakals
Tritos = 135 months = 5 Pakals
Tzolkinex = 88 months = 1 Pakaloid + 2 Pakals
------------------------------------------------------
There are these estimates,
1 Tzolkinex = 7 & 2 / 17 years ( 121 years = 11 hendecaeterides = 17 Tzolkinex )
1 Tritos = 10 & 10 / 11 years ( 120 years = 15 octaeterides = 11 Tritos )
Reply | Threaded
Open this post in threaded view
|

Re: Pakals and Pakaloids

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

I considered what period could be placed between two Pakal calendars each lasting about a 120 years and came up with short Pakaloid of 15 months and of 10 fewer Pentipses and elvears than a Pakal.
I let Helios know of this in private.
Two types of Pakaloid are needed to regulate both the mean eclipse season and the mean year.

Helios has not shown the elvear pattern for his 34-month Pakaloid nor even told us how many elvears it has. One elvear is 1/11 year of which there are 24 in a Pakal. Such a period would have about 30.24 elvears for an accurate year/month ratio.

Helios's said
Inex = 358 months = 1 Pakaloid + 12 Pakals
but missed out
Saros = 223 months = 1 Pakaloid + 7 Pakals
This shows that the Pakaloid is intended to be used around once every 10 Pakals, which is much less than 120 years = 55 Pakals intended for the short Pakaloid.

I find 6 Pakaloids = 7 Pakals + one short Pakaloid = 204 months = 174 Pentipses.

So one short Pakaloid of 15 months has the same correction effect on the Pakal for eclipses as 6 Pakaloids. Hence we get

6 Inex = 1 short Pakaloid + 79 Pakals
6 Saros = 1 short Pakaloid + 49 Pakals

The Pakals and short Pakaloid then make

6 Inex = 1910 elvears = 173.636363.. years
6 Saros = 1190 elvears = 108.18181.. years

http://www.staff.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm 
gives 86.835 years to 3 Inex 54.090 years to 3 Saros.
When doubled we get
173.67 years for 6 Inex and
108.18 years for 6 Saros.

At the end, Helios did do some work on the year, but this would require mixing units of 1/11 year from the Octaeteris (8 years of 99 months) with a unit of 1/17 year from the 11-year cycle of 136 months, so requiring a unit of 1/187 year from which both can be made.

A second type of correction period (Pakaloid) is still required to enable regulation of both the mean eclipse season and the mean year.
One possibility is 1 Pakal + 1 short Pakaloid - 1 Pakaloid =
8 months = 7 pentipses to which I'll give 8 elvears. This is equivalent to five Pakaloids on the Pakal for eclipses, but has a stronger correction of the Octaeteris.

Added to 55 Pakals this very short Pakaloid makes 120.727272... years of 1493 months (ratio 12.3667...)
Added to 55 Pakals the short Pakaloid makes 121.272727... years of 1500 months (ratio 12.3688...)  

More work is needed on these correction periods (Pakaloids).

Karl

14(15(20


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 05 April 2015 22:40
To: [hidden email]
Subject: Re: Pakals and Pakaloids

Dear Karl and Calendar People,

Apart from a very long continuum of Pakals, there will be an abberant
grouping, a Pakaloid, every so often. I find this to be a 34 month period,
here compared to the Pakal,

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 23 pentipse ( Tritos I -
S  )

[ 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 ] = 29 pentipse
( 34m*5 = 170m 12*S - 7*I  )

Now add Pakals to both sides of a Pakaloid to make a larger symmetrical
entity, As it turns out, these are the simple eclipses which emerge,

Inex = 358 months = 1 Pakaloid + 12 Pakals
Tritos = 135 months = 5 Pakals
Tzolkinex = 88 months = 1 Pakaloid + 2 Pakals
------------------------------------------------------
There are these estimates,
1 Tzolkinex = 7 & 2 / 17 years ( 121 years = 11 hendecaeterides = 17
Tzolkinex )
1 Tritos = 10 & 10 / 11 years ( 120 years = 15 octaeterides = 11 Tritos )




--
View this message in context: http://calndr-l.10958.n7.nabble.com/27-Month-Lunar-Unit-tp15706p15737.html
Sent from the Calndr-L mailing list archive at Nabble.com.