Structural complexity of Nodetide-year cycles RE: Nodetides

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Structural complexity of Nodetide-year cycles RE: Nodetides

Karl Palmen
Dear Helios, Irv and Calendar People

Here I look into the structural complexity of nodetide cycles such as 75/698 mentioned by Helios here and the shorter 13/121, which Helios has mentioned earlier on as a cycle of 121 years of 255 eclipse seasons. 255 = 2*121 + 13.

Here I list the positive half of Helios's 75/698 cycle with 37 instead of 34 from -349 to +349 symmetrical about 000. Within each row the intervals are 9 years and between rows the interval is 10 years. I extend the first row into the negative half to ensure this.

-009,  000, +009,
+019, +028, +037,
+047, +056, +065, +074,
+084, +093, +102,
+112, +121, +130,
+140, +149, +158,
+168, +177, +186, +195,
+205, +214, +223,
+233, +242, +251,
+261, +270, +279,
+289, +298, +307, +316,
+326, +335, +344,

Here I see every 4th row is long except between the last long row and its negative mirror image, which is 3 rows later.
So the cycle of the rows each taken to be a 'year' has complexity 2.
The cycle of each row also has complexity 2, because all intervals are equal, except the one interval between rows.
These two complexities add up to 4. So I expect the total cycle has a complexity of 4 (equal to the lunisolar 353-year cycle).

A sequence of cycles whose Ford circles link the Ford circle of 0/1 to 75/698 is
(1) 1/9: interval within row
(2) 3/28: short row
(3) 13/121: 4-row cycle
(4) 75/698: complete cycle.
Therefore the structural complexity of 75/698 is equal to 4 (unless you find a shorter such sequence).

The 4-row cycle is the same as the 121-year cycle Helios has mentioned before and so it has a complexity of 3.
The 65-year cycle I mentioned as occurring in the magic square is made up of one short row and one long row and also has a complexity of 3.

Karl

16(03(10
 

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 08 November 2016 16:12
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and c.p.

"34" was just a guess. I recall that year 74 was a hurdle I was trying to
get by. I had about 2 centuries mapped out, about the limits of that magic
square. "37" is correct, with the reasoning behind it.

Yesterday I turned 57 years old and I confirmed my "birth moon phase"
beginning my 4th Metonic cycle.



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Re: Structural complexity of Nodetide-year cycles RE: Nodetides

Helios
Dear Karl and Calendar People,

I now see a working method that will generate a list of lunisolar eclipse cycles, or ec-lunisolar cycles.

----------------------------------------------------------
by the intersection of the two accumulators,
( 75*Y + 37 )MOD( 698 ) < 75
( 253*Y + 126 )MOD( 687 )< 253
----------------------------------------------------------

this series begins
0, 19, 65, 84, 130, 149, 177, 223, 242, 261, 307, 326, 372, 391, . . .

It's laborious to find these years and a computer program could make a longer list.
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Re: Structural complexity of Nodetide-year cycles RE: Nodetides

Helios
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] 028 037
047 056 [065] 074 = I + 2*S Unidos
[084] 093 102
112 121 [130] = 2*I + 4*S
140 [149] 158
168 [177] 186 195 = 3*I + 5*S
205 214 [223]
233 [242] 251 = 4*I + 7*S
[261] 270 279
289 298 [307] 316 = 5*I + 9*S
[326] 335 344
354 363 [372]
382 [391] 400 409 = 16*I - 4*S G. Guinness
419 428 [437]
447 [456] 465 = 17*I - 2*S
[475] 484 493
503 512 [521] 530 = 18*I Basic Period
540 [549] 558
[568] 577 586 = 19*I + 2*S Tetradia
596 605 [614]
624 [633] 642 651 = 20*I + 3*S
661 670 [679]
689 [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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Re: Structural complexity of Nodetide-year cycles RE: Nodetides

Karl Palmen
Dear Helios and Calendar People

With the nodetides, Helios is finding eclipse cycles that are near a whole number of years. He does not evaluate how near to a whole number of years his examples are.

[019], [372] and [391] are the only ones that are near enough to a whole number of years to be considered to be considered a lunisolar cycle.

[084] was used for an Easter computus long ago.

Karl

16(04(06

-----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] 028 037
047 056 [065] 074 = I + 2*S Unidos
[084] 093 102
112 121 [130] = 2*I + 4*S
140 [149] 158
168 [177] 186 195 = 3*I + 5*S
205 214 [223]
233 [242] 251 = 4*I + 7*S
[261] 270 279
289 298 [307] 316 = 5*I + 9*S
[326] 335 344
354 363 [372]
382 [391] 400 409 = 16*I - 4*S G. Guinness
419 428 [437]
447 [456] 465 = 17*I - 2*S
[475] 484 493
503 512 [521] 530 = 18*I Basic Period
540 [549] 558
[568] 577 586 = 19*I + 2*S Tetradia
596 605 [614]
624 [633] 642 651 = 20*I + 3*S
661 670 [679]
689 [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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