Counting eclipse seasons in a lunisolar calendar RE: Alternatives to RE: Hexaseasonal yerms

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Counting eclipse seasons in a lunisolar calendar RE: Alternatives to RE: Hexaseasonal yerms

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

I've come up with a good way of counting eclipse seasons in a lunisolar calendar, thanks to thought provoked my Helios's attempts to do this for his hexaseasons.

Allocate 1/6 eclipse season to each month and an additional 1/8 eclipse season to each leap month.

This works exactly for the Grattan-Guinness cycle of 391 years = 4836 months = 824 eclipse seasons.
For the Gregoriana, it gives 1/24 eclipse seasons too few.


I show how I got to this next replying to Helios on the way.

Firstly I'll define a leap month period LMP as a period of 33 or 35 months one of which is a leap month. Helios constructs these with yerms that do not count the extra days added to abundant years. I'll refer to them as LMPs, because the yerm-like properties are not relevant to eclipses.

Helios said:
There are even some hexa-yerm eclipse cycles;
35 months = 13*S - 8*I ( 6 eclipse seasons )
235 months = 10*I - 15*S ( 40 eclipse seasons )

From this I worked out that Helios allocates 6 eclipse seasons to an LMP of 35 months and 5.6 eclipse seasons to an LMP of 33 months.
This is the ONLY allocation that gives BOTH 6 eclipse seasons to an LMP of 35 months and 40 eclipse seasons to a Metonic cycle of 235 months.

In my previous reply, I overlooked the triple Inex that Helios mentioned.
33 35 33 33 33 35 33
33 35 33
33 35 33 33 33 35 33
33 35 33 33 33 35 33
33 35 33 33 33 35 33
33
Helios did not check that that the number of eclipse seasons as stated earlier led to the correct number of eclipse seasons for this triple Inex.
I count eclipse seasons as follows:
33 35 33 33 33 35 33 (40)
33 35 33             (5.6 + 6 + 5.6 = 17.2)
33 35 33 33 33 35 33 (40)
33 35 33 33 33 35 33 (40)
33 35 33 33 33 35 33 (40)
33                   (5.6)
So the total number of eclipse seasons is then 188.8, which is 5.2 than the required 3*61 = 183 eclipse seasons.


Now suppose instead of allocating 6 eclipse seasons to the LMP of 35 months, we allocate 183 eclipse seasons to the triple Inex and we continue to allocate 40 eclipse seasons to the Metonic cycle of 235 months. We then get 23 eclipse seasons in
33 35 33
33
which is one month less than a tritos.

By subtracting this from the 235 months we get 17 eclipse seasons in
33 35 33

Hence we have 6 eclipse seasons in an LMP of 33 months instead of 35 months. This leads to an LMP of 35 months getting 5 eclipse seasons to ensure that the Metonic cycle of 235 months has 40 eclipse seasons. So we get
6 eclipse seasons in an LMP of 33 months and
5 eclipse seasons in an LMP of 35 months!


An alternative to this is to allocate 46 eclipse seasons to the double tritos and 183 eclipse seasons to the triple inex.

35 33 33 35 33 33 33 35 = 270 months = 46 eclipse seasons

33 35 33 33 33 35 33
33 35 33
33 35 33 33 33 35 33
33 35 33 33 33 35 33
33 35 33 33 33 35 33
33 = 1074 months = 183 eclipse seasons.

Let x be the number of eclipse seasons allocated to an LMP of 33 months and y be the number of eclipse seasons allocated to an LMP of 35 months. Then we get:
5x + 3y = 46
23x + 9y = 183
Subtracting 3 times the first equation from the second equation leads to
8x = 45 hence x = 5.625 .
Then substituting x in the first equation we get
28.125 + 3y = 46
3y = 17.875
y =  6 - 1/24
Hence this would allocate
5 & 15/24 eclipse seasons to an LMP of 33 months and
5 & 23/24 eclipse seasons to an LMP of 35 months.

If one subtracts three double tritos cycles from two triple inex cycles, one get six Saros cycles of 31 LMPs of 33 months and 9 LMPs of 35. I'll leave it to Helios to arrange these. It is equal to one triple Inex (with 183 eclipse seasons) + eight LMPs of 33 months (with 45 eclipse seasons), hence 183+45=228=6*38 eclipse seasons as required.

This can be applied to any of my alternatives to the hexaseasonal yerms as shown in my original note "Alternatives to RE: Hexaseasonal yerms".

Allocate 1/6 eclipse season to each month and 1/8 eclipse season to each LMP.

Or put another way, allocate 1/6 eclipse season to each month and an additional 1/8 eclipse season to each leap month.

Karl

15(01(23


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Re: Counting eclipse seasons in a lunisolar calendar RE: Alternatives to RE: Hexaseasonal yerms

Helios
Dear Karl and Calendar People,

I got to the Triple Inex by knowing that 8*33 months = 3 Tzolkinex. It wasn't until 3 Double Tritos that I could get them all to intersperse. Karl's math preserves the eclipse seasons for these two;

33 33 33 33 33 33 33 33 = Triple Tzolkinex = 264 months
264*( 1 / 6 ) + 8*( 1 / 8 ) = 45 eclipse seasons

35 33 33 35 33 33 33 35 = Double Tritos = 270 months
270*( 1 / 6 ) + 8*( 1 / 8 ) = 46 eclipse seasons

This is proof that it's a good scheme, and with easy math. I predict it will work better on the hexaseasonal yerm calendar than on a conventional luni-solar calendar.

Since both the Triple Tzolkinex and the Double Tritos both contain 6 Pentalunex and both contain 8 hexaseasonal yerms, it is very common that 1 in 4 hexaseasonal yerms will not contain a Pentalunex.

I've been looking at starting the hexaseasonal yerm cycle on March 21, 1985. This puts us now on the 9th hexaseasonal yerm which has 17 hexaseasons, beginning January 20, 2015.

AQU-ARI-GEM-LEO-LIB-SAG- etc.

The hexaseasons seem to start accurately in telling the time of the year, but degrade as they go until the leap month sets things straight.

We know of the recent spring eclipse ( ARIES 2015 ).
( mar20 T )( sep13 P )( mar09 T )( sep01 A )( feb26 A )( aug21 T -last hexaseason )
These are all of the eclipses of the 9th hexaseasonal yerm. It does not contain a Pentalunex.
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Re: Counting eclipse seasons in a lunisolar calendar RE: Alternatives to RE: Hexaseasonal yerms

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

Thank you Helios for your reply.

Reply below.

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 10 July 2015 06:07
To: [hidden email]
Subject: Re: Counting eclipse seasons in a lunisolar calendar RE: Alternatives to RE: Hexaseasonal yerms

Dear Karl and Calendar People,

I got to the Triple Inex by knowing that 8*33 months = 3 Tzolkinex. It
wasn't until 3 Double Tritos that I could get them all to intersperse.
Karl's math preserves the eclipse seasons for these two;

33 33 33 33 33 33 33 33 = Triple Tzolkinex = 264 months
264*( 1 / 6 ) + 8*( 1 / 8 ) = 45 eclipse seasons

35 33 33 35 33 33 33 35 = Double Tritos = 270 months
270*( 1 / 6 ) + 8*( 1 / 8 ) = 46 eclipse seasons

This is proof that it's a good scheme, and with easy math. I predict it will
work better on the hexaseasonal yerm calendar than on a conventional
luni-solar calendar.

KARL REPLIES:
It will work slightly better on hexaseasonal calendar than a conventional lunisolar calendar with leap months at the same time of year, because the leap months and hence the 1/8s are spread more smoothly. For the same reason, it will work even better on a lunisolar calendar with leap months every 33 or 34 months.

This 1/6 eclipse season per month with addition 1/8 season per leap month is accurate only if the hexaseasonal calendar follows the Grattan-Guinness cycle of 391 years of 144 leap months = 824 eclipse seasons, else the cycle will be out by a number of 1/24 eclipse season.

The Grattan-Guinness cycle can be divided into six parts of:

24 hexaseasonal yerms of 391 hexaseasons with 806 months = 137 & 1/3 eclipse seasons.

This part can be made by extending Helios's cycle of 41 hexaseasonal yerms by one more Metonic cycle, to get a cycle of 48 hexaseasonal yerms, then dividing this into two equal parts.

33 35 33 33 33 35 33
33 35 33
33 35 33 33 33 35 33
33 35 33 33 33 35 33
 
For other lunisolar cycles, a correction of 1/24 eclipse season would occasionally need to be made.


HELIOS CONTINUED:
Since both the Triple Tzolkinex and the Double Tritos both contain 6
Pentalunex and both contain 8 hexaseasonal yerms, it is very common that 1
in 4 hexaseasonal yerms will not contain a Pentalunex.

KARL REPLIES:
A triple tzolkinex has 264 months = 45 eclipse seasons and the double tritos has 270 months = 46 eclipse seasons, so they differ by one eclipse Season of six months. Therefore the number of eclipse seasons shortened to 5 months is the same in both. This is what Helios means by a pentalunex, also referred to as a short semester.

Giving 8 hexaseasonal yerms to each makes the triple tzolkinex contain (264-8)/12 = 21 & 1/3 years and the double tritos contain 21 & 5/6 years.

Half a Grattan-Guinness cycle can be constructed out of 7 Double tritos and 2 triple tzolkinex.


HELIOS CONTINUED:
I've been looking at starting the hexaseasonal yerm cycle on March 21, 1985.
This puts us now on the 9th hexaseasonal yerm which has 17 hexaseasons,
beginning January 20, 2015.

AQU-ARI-GEM-LEO-LIB-SAG- etc.

The hexaseasons seem to start accurately in telling the time of the year,
but degrade as they go until the leap month sets things straight.

KARL REPLIES: This is an inevitable feature of a lunisolar calendar.

HELIOS CONTINUED:
We know of the recent spring eclipse ( ARIES 2015 ).
( mar20 T )( sep13 P )( mar09 T )( sep01 A )( feb26 A )( aug21 T -last
hexaseason )
These are all of the eclipses of the 9th hexaseasonal yerm. It does not
contain a Pentalunex.

KARL REPLIES: I think Helios means that the six eclipses listed here all occur in intervals of 6 months. The 1/6, 1/8 scheme gives 3/4 pentalunex to each hexaseasonal yerm.
The number of pentalunex in M months of S eclipse seasons is equal to 6*S - M.
For the yerms S = M/6 + 1/8 and so we get
6*(M/6 + 1/8) - M =
M + 6/8 - M = 3/4.
Hence Helios is correct in stating that 1 in 4 hexaseasonal yerms will not contain a Pentalunex, provided he also counts any Pentalunex that runs through two consecutive hexaseasonal yerms as being contained in ONE of these two hexaseasonal yerms.

I've found a more accurate scheme that works well for lunisolar calendars with leap month every 2 or 3 years. By interpolation it can be extended to the hexaseasonal yerms, it won't give exactly the correct number of eclipse seasons to the Helios's triple tzolkinex and double tritos, because the numbers of years that Helios implicitly gives to these are not exactly correct.

Karl

15(01(27




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Re: Counting eclipse seasons in a lunisolar calendar RE: Alternatives to RE: Hexaseasonal yerms

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

KARL PREVIOUSLY SAID:
I've found a more accurate scheme that works well for lunisolar calendars with leap month every 2 or 3 years. By interpolation it can be extended to the hexaseasonal yerms, it won't give exactly the correct number of eclipse seasons to the Helios's triple tzolkinex and double tritos, because the numbers of years that Helios implicitly gives to these are not exactly correct.

KARL NOW SAYS:
So I'll now mention it.

Simply count 8 eclipse seasons for each year with a leap month that occurs 3 years after a year with a leap month. It is assumed that every year with a leap month occurs 2 or 3 years after the previous such year.

This will give the correct number of eclipse seasons in the Metonic Cycle, The Gregoriana, The Grattan-Guinness cycle and the unnamed (353).

For a lunisolar cycle of Y years and L leap months. The number E of eclipse seasons counted is then given by

E = 24*L - 8*Y

This only counts the number of eclipse seasons and says nothing about how to distribute them amongst the months. It is only accurate if 12 + L/Y is an accurate number of lunisolar months per year. It won't give the correct number of eclipse seasons for either Helios's triple tzolkinex or double tritos, because neither has 12 + L/Y sufficiently accurate.

For hexaseasons, it can be interpolated to give 5 & 1/3 eclipse seasons to a 33 and 6 & 2/3 eclipse seasons to a 35. For 33s & 34s the 34 has 6 eclipse seasons exactly. Note that L is the number of 33s, 34s or 35s and Y is the number of months minus L all divided by 12. 12 + Y/L must be accurate.

Karl

15(01(27




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