Quincunx Lunar Eclipses

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Quincunx Lunar Eclipses

Helios
Dear Calendar People,

One can define the "Quincunx" as a set of five lunar eclipses, multiple of the following

Unnamed Cycle = 1837 months = 4*( Tzolkinex ) + 11*(  Tritos ) = 7*Inex - 3*Saros

A Quincunx is presumed to be internal to some 744-year Double Gregoriana interval. The purpose of the Quincunx lunar eclipses is that they nearly mark the chasms between the correction eras needed for octaeteris luni-solar calendars, which is about 148.8 years. This gives an interesting visual ingredient to the calendar correction.

There is the special case where the Quincunx is alligned centrally.

1) 17*S - 8*I ( 927 months )
2 ) 14*S - I ( 2764 months )
3) 6*I + 11*S ( 4601 months ) = 17*( Tzolkinex ) + 23*(  Tritos )
4) 13*I + 8*S ( 6438 months )
5) 20*I + 5*S ( 8275 months )

I suppose that after the eclipse, the latter half of the next month begins.
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Re: Quincunx Lunar Eclipses

Helios
Dear Calendar People,

I've taken more of a look at the following eclipse cycle

Unnamed Cycle = 1837 months = 4*Tzolkinex + 11*Tritos = 7*Inex - 3*Saros

We recall that it is an eclipse cycle which matches the correction interval for the octaeteris luni-solar calendar. Now I find proof that it can stand alone as a correction metric.

We only have to posit that
148 & 52 / 99 years = 1837 months
= 14704 years
= 1838 ( 8 year periods )
= 181863 months
= 1838*99 - 99 months
Though the eclipse itself is expected to survive about 20 times.
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Re: Quincunx Lunar Eclipses

Karl Palmen - UKRI STFC
Dear Helios and Calendar People

Replies below.

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 04 May 2015 23:46
To: [hidden email]
Subject: Re: Quincunx Lunar Eclipses

Dear Calendar People,

I've taken more of a look at the following eclipse cycle

Unnamed Cycle = 1837 months = 4*Tzolkinex + 11*Tritos = 7*Inex - 3*Saros

KARL REPLIES:
4*Tzolkinex + 11*Tritos =  4*88 + 11*135 = 1837 months = 4*15 + 11*23 = 313 eclipse seasons
7*Inex - 3*Saros = 7*358 - 3*223 = 1837 months = 7*61 - 3*38 = 313 eclipse seasons

In terms of the hepton (H) and octon (O) it can be symmetrically expressed as:

[OHOHO|OHO|OHOHO|OHO|OHOHO|OHOHO|OHO|OHOHO|OHO|OHOHO|OHO|OHOHO|OHOHO|OHO|OHOHO|OHO|OHOHO]

With OHO at positions (2 4 7 9 11 14 16) out of 17.


HELIOS CONTINUED:
We recall that it is an eclipse cycle which matches the correction interval
for the octaeteris luni-solar calendar. Now I find proof that it can stand
alone as a correction metric.

We only have to posit that
148 & 52 / 99 years = 1837 months

KARL RELIES:
This has 12.368267138... months/years which is very accurate.
This is one month less than the (99/8)*(148 & 52/99) = 1838 months that would arise in the Octaeteris cycle
and so the octaeteris cycle is corrected by 1 month.

HELIOS CONTINUED
= 14704 years
= 1838 ( 8 year periods )
= 181863 months
= 1838*99 - 99 months
Though the eclipse itself is expected to survive about 20 times.

KARL REPLIES:
Helios gives numbers of years etc. in 99 of these 1837 month cycles. This useful for calculations rather than a realistic complete eclipse or lunisolar cycle.

The 11 Tritos in the "4*Tzolkinex + 11*Tritos" form the 120-year period of 11 Tritos and 15 Octaeterides already mentioned by Helios in his Pakal calendar.
So the remaining 4 Tzolkinex form a correction period for the Tritos and Octaeteris. It has 352 months equated as 28 & 52/99 years. Unfortunately this is not a whole number of years or even a whole number of elvears (1/11 year), instead one must divide the elvear by nine.

Helios has admitted that his cycle is only good for 20 cycles (as eclipse cycle). A second correction period is needed to correct this and eclipses can be followed perpetually with the correct mix and frequency of the two correction periods, which would slowly change.


The correction period can be divided by 4 to give a single Tzolkinex of 88 months equated to 7 & 13/99 years for smoother more frequent correction of the Tritos and Octaeteris. Other eclipse cycles could be constructed this way, this includes a Saros of 18 & 4/99 years and an Inex of 28 & 94/99 years. A cycle of S saros and I inex would have

(1786*S + 2866*I )/99 years.

The resulting mean year may only be accurate if the mix of saros & inex is similar to the Helios's 7 inex - 3 saros.
A second correction period would not follow this formula, but would correct it.

Alternative formulae for number of years may be found. For example,

Years =(3606*S + 5789*I)/200 = 18.03*S + 28.945*I,

which gives exactly 372 years to the Gregoriana (S=11 & I=6).


Karl

14(16(17


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