Re: Nodetides

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Re: Nodetides

Karl Palmen
Dear Helios and Calendar People

All the years Helios listed belong to the 698-year cycle:

( 75*Y + 37 ) MOD ( 687 ) < 75

Those in [] also satisfy

( 253*Y + 126 )MOD( 687 ) < 253

which ensures that the given number of years is not far from an eclipse cycle.
Each Inex-Saros expression applies to the number of years in the [] of the row.


To make the eclipse cycle closer lunisolar, one needs a stricter condition instead of

( 253*Y + 126 )MOD( 687 )< 253

and I thought of

( 7*Y - NINT(Y/349) ) MOD ( 19 ) = 0

which is easy to use with the 698-year cycle.
If the 698-year cycle is run seven times to 4886, then this stricter condition produces the cycles in {}, which I have added to the note below.

As expected all the shorter cycles occur with a number in []. I list them here and calculate some Inex-Saros formulae:
[019]{0019}
[372]{0372}
[391]{0391} = 16*I - 4*S G. Guinness

[065]{0763} = 22*I + 7*S Unidos + 698-years
[084]{0782}
[437]{1135}
[456]{1154} = 38*I + 3*S
[475]{1173}

[130]{1526} = 44*I + 14*S
[149]{1545}
[521]{1917} = 60*I + 10*S Basic period + 2*698 years

All these cycles are a mix of 372 & 391 (with 19 = 391-372) including the number of nodetides or eclipse seasons and have
(200*Years - 16*Months) eclipse seasons and also twice as many nodetides as Metonic cycles, including truncated.
The next longer cycle after {1917} is over a 1000 years longer.

In every example, the number in {} differs from the preceding number by a multiple of 698 and so occupies the same place as the preceding number in the 698-year cycle.

Also I find out if I cut off at 4537 instead of 4886, I'd get exactly one example in each row. If I were to extend to 19 698-year cycles (to 13262) then every member of the 698-year cycle would have one example, but I don't think the 698-year cycle is accurate enough for this.

Karl

16(04(10

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 08 December 2016 14:42
To: [hidden email]
Subject: Re: Structural complexity of Nodetide-year cycles RE: Nodetides

Dear Karl and Calendar People,

Here is the nodetide list extended up to 698 years. I used

( 253*Y + 126 )MOD( 687 )< 253

as a criterion to check for eclipses. It turns out that 698 years itself is
an eclipse. The 698 years is that good enough a lunisolar cycle as to make
the eclipses symmetrical on this list about midway. Here eclipses are
bracketed.

009
[019]{0019} 028 037
047 056 [065]{0763} 074 = I + 2*S Unidos
[084]{0782} 093 102
112 121 [130]{1526} = 2*I + 4*S
140 [149]{1545} 158
168{1564} [177] 186 195{2987} = 3*I + 5*S
205 214{3006} [223]
233{3025} [242] 251 = 4*I + 7*S
[261] 270 279{3769}
289 298{3750} [307] 316 = 5*I + 9*S
[326] 335 344{4532}

354 363{4551} [372]{0372}
382{4570} [391]{0391} 400 409 = 16*I - 4*S G. Guinness
419 428 [437]{1135}
447 [456]{1154} 465 = 17*I - 2*S
[475]{1173} 484 493
503 512 [521]{1917} 530 = 18*I Basic Period
540{2634} [549] 558
[568] 577 586{3378} = 19*I + 2*S Tetradia
596 605{3397} [614]
624 [633] 642 651{4141} = 20*I + 3*S
661 670{4160} [679]
689{4179} [698] = 21*I + 5*S


Every row (rows are separated by 10 years ) has its own eclipse, at least up
to the end this list. I have given the Inex-Saros value to only some rows.
These eclipses look like they are more accurate than their fringe neighbors
which are 19 years away. Rows without description can be evaluated by
finding a more major eclipse nearby.
By using the intersection of the two accumulators, we create a list which
does include the Basic Period and the Tetradia, which are featured on the
Eclipse Cycle webpage we reference.




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