Alternatives to RE: Hexaseasonal yerms

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Alternatives to RE: Hexaseasonal yerms

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

I see 33 35 33 33 35 33 33 as one of several possibilities

34 33 34 33 34 33 34 (1/12 year 1 regular month)

33 35 33 33 35 33 33 (1/6 year = 2 regular months)

34 34 34 31 34 34 34 (1/4 year = 3 regular months)

33 33 33 37 33 33 33 (1/3 year = 4 regular months)

31 37 31 37 31 37 31 (1/2 year = 6 regular months)

Of which the following can be produced with hexaseason yerms

33 35 33 33 35 33 33 (1/6 year = 2 regular months)

33 33 33 37 33 33 33 (1/3 year = 4 regular months)

31 37 31 37 31 37 31 (1/2 year = 6 regular months)

The yerms can be corrected by adding 1 day about once every 31 hexaseasons.

In the other two, the 34s could be split into two yerms of 17 months so reducing the number of days that need adding.

The Metonic cycles can be corrected by  
removing a 33 (1/3 octaeteris),
adding a 34 (1/4 11-year cycle) or
adding 37 31 (1/2 11-year cycle).


Also the Hexaseason yerms reminded me of the stretch-yerms, which are formed by making every leap month have 30 days and not counted in the yerms. The resulting stretch-yerms are normally 31 regular months long, but a small minority (around 1 in 20) have 29 regular months. The number of stretch-yerms needed by a lunisolar cycle is exactly twice the number of abundant years otherwise needed.

Helios discovered that if one also does not count two months adjacent to the leap month and all three of these months have a total of 89 days, then one can have stretch-yerms of 29 counted months only. Both the 5515-year cycle and 1689-year cycle can be realised with such stretch-yerms. The number of these stretch-yerms remains twice the number of abundant years otherwise needed.


Karl

15(01(09


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 22 June 2015 16:50
To: [hidden email]
Subject: Re: Hexaseasonal yerms

yes, I correct this to

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 35 33 33
33 35 33 33 35 33 33
33 35 33 33 35 33

this totals to 1377 months = 235 + 235 + 235 + 235 + 235 + 202



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Re: Alternatives to RE: Hexaseasonal yerms

Helios
Dear Karl and Calendar People,

I looked back at the hexa-yerms and I now think this is the correct arrangement.

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 35 33
33 35 33 33 33 35 33

-----------------------------------------------------------------
With some inspection, hexaseasons seem helpful as an approximate measure of the eclipse season and some hexa-yerms can be entirely free of a Pentalunex.

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 )
better is the Double Tritos
35 33 33 35 33 33 33 35 = 270 months = 2*I - 2*S
and the Triple Inex
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

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Re: Alternatives to RE: Hexaseasonal yerms

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

Firstly a reminder of the Hexaseason yerms; I said earlier

"What Helios is effectively suggesting is a lunisolar calendar where the leap month may occur after any even-numbered month of the year. Each yerm of 33 or 35 months can be thought of as 32 or 34 regular months followed by 1 leap month. Each hexaseason consists of two months and any leap month that follows them. So a yerm of 33 months has 16 hexaseaons (2 & 2/3 years) and a yerm of 35 months has 17 hexaseaons (2 & 5/6 years).

The yerms of 33 and 35 months have too many months to produce an accurate mean month. They can be corrected by occasionally adding a day to a month. This correction is equivalent to the abundance correction shown in the third column of my lunisolar spreadsheet.
http://the-light.com/cal/kp_Lunisolar_xls.html 

The great thing about the hexaseasons is that this correction can be normally done once every 31 hexaseasons."

Helios then suggested:
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 35 33 33
33 35 33 33 35 33 33
33 35 33 33 35 33
of 1377 months of which 41 are leap months, so making 1/3 of a 334-year cycle of 3*1377=4131 months and 3*41=123 leap months.
Each row of 7 is a 19-year cycle of 235 months.

This is not a Helios cycle. It is neither symmetrical nor are the two types of yerm spread as smoothly as possible. So neither of the two defining properties of the Helios cycle is present. The latter can be seen to be absent by listing the intervals between the long yerms of 35 starting with the 1st to 2nd, which are: 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 3, 3. They are obviously not spread as smoothly as possible, hence the two yerm types are not spread as smoothly as possible.

Helios has found the Helios cycle and it is
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 35 33
33 35 33 33 33 35 33
Each row is a Helios cycle and they are arranged into a Helios cycle. This alone only proves the symmetry. The ratio of long yerms in the two row types are 2/7 & 1/3 respectively and the determinant 1*7 - 2*3 = 1, hence they are mixer cycles. So this along with the statement about the rows, makes the cycle into a Helios cycle.


For eclipse seasons, Helios takes two eclipse cycles the Hexon of 35 months = 6 eclipse seasons and the Metonic cycle of 235 months = 40 eclipse seasons.
Both are listed in http://www.staff.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm . He adds these two to get a double tritos.

So Helios allocates 6 eclipse seasons to a long yerm of 35 months, to finds out how many he allocates to a short yerm of 33 months, one takes the two long yerms and their eclipse seasons from the Metonic cycle and are left with 28 eclipse seasons for the five short yerms hence:
35-month yerm has 6 eclipses seasons
33-month yerm has 5.6 eclipse seasons.

So the row of 33 35 33 has 17.2 eclipse seasons.
Helios's 41-yerm cycle of 1/3 of a 334-year cycle has 5*40 + 2*17.2 = 234.4 eclipse seasons.
A complete 334-year cycle hence has 703.2 eclipse seasons. Adding one or two Metonic cycles and we get:
353-year cycle with 743.2 eclipse seasons and
372-year cycle (Gregoriana) with 783.2 eclipse seasons.
In both cases, this is 0.8 of an eclipse season less than the amount shown in the catalogue of eclipse cycles I have linked.

In note that the double-tritos has 270 months of which 8 are leap months (one leap month per yerm) and so has 262 regular months and hence 21 & 5/6 years. So a single Tritos has 10 & 11/12 = 10.916666... years compared to 10.915 years listed in the catalogue and reckoned by my year approximation formula for eclipse cycles.

Karl

15(01(21


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 07 July 2015 06:58
To: [hidden email]
Subject: Re: Alternatives to RE: Hexaseasonal yerms

Dear Karl and Calendar People,

I looked back at the hexa-yerms and I now think this is the correct
arrangement.

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 35 33
33 35 33 33 33 35 33

-----------------------------------------------------------------
With some inspection, hexaseasons seem helpful as an approximate measure of
the eclipse season and some hexa-yerms can be entirely free of a Pentalunex.

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 )
better is the Double Tritos
35 33 33 35 33 33 33 35 = 270 months = 2*I - 2*S
and the Triple Inex
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





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