Re: Alternating Periodic Sequences

Dear Helios and Calendar People

Interesting.

We have a mean yerm of ABS( 1/ [ N/Y - 2]) days, where Y is the mean
year and N is an integer close to 2*Y. The nearer N is to 2*Y the longer
the yerm. N is multiple of the number of days in two consecutive
subdivisions in a yerm and the multiplier is the number of subdivisions
in a year

728 = 8*(46+45)
729 = 3*(122+121)
730 = 10*(37+36)
731 = 17*(21+22)
732 = 12*(30+31)

There is no example for N=733, because 733 is prime. 734 is twice a
prime number so can be used only for a half year subdivision.

The subdivision must be considerably smaller than the mean yerm, else
the yerms will have too few subdivisions to be useful. T For N=729, The
1/3 year (of 122 or 121 days) whose mean yerm is about 246 days is
rather too long. However N=729 will work with smaller divisions as shown
below.

N=728 can also be done for 1/56 and 1/104 of a year with subdivisions
alternating between (7,6) and (4,3) days respectively.

N=729 can also be done for subdivisions of a 1/9 and 1/27 of a year with
subdivisions alternating between (41,40) and (14,13) days respectively.
Also 1/81 for (5,4) and 1/243 for (2,1).

N=730 can be done for 1/146 of a year with subdivisions alternating
between (3,2) days.

N=731 can be done for 1/43 of a year with subdivisions alternating
between (9,8) days.

N=732 can also be done for 1/4 year with quarters alternating between
(91,92) days. It can also be done with 1/244 part of a year with
subdivisions alternating between (1,2) days and Victor has done this for
the 161-year cycle that has a mean year of exactly 241 days.

The mean yerm is often close to a whole number of days and furthermore
is equal to a whole number of days for one of the 5 "elegant" solar
calendar cycles listed by Helios

N=728 has a 147 day mean yerm for mean year 365 71/293 days
N=729 has a 246 day mean yerm for mean year 365 119/491 days
N=730 has a 754 day mean yerm for mean year 365 365/1507 days
N=731 has a 709 day mean yerm for mean year 365 8/33 days
N=732 has a 241 day mean yerm for mean year 365 39/161 days

Furthermore, N=731 can have every yerm with 709 days, if a subdivision
of 1/17 year (a term) is used.
Also, N=732 can have every yerm with 241 days, if a subdivision of 1/244
year is used (discovered by Victor).

Karl

10(06(17

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 12 March 2009 10:34
To: CALNDR-L@...
Subject: Alternating Periodic Sequences

I can make a batch of these "mean yerm" formulas. I think they work
because
730 = 2*365. There's a sort of range between 727 and 733 which are prime
numbers. So, there are certain subdivisions of the year and a "mean
yerm" to
each...

mean yerm of 8ths = 1 / [  2 - ( 728 / Y ) ]

mean yerm of 3rds = 1 / [  2 - ( 729 / Y ) ]

mean yerm of 10ths = 1 / [  2 - ( 730 / Y ) ]

mean yerm of 17ths = 1 / [ ( 731 / Y ) - 2 ]

mean yerm of 12ths = 1 / [ ( 732 / Y ) - 2 ]

Some years lead to integer mean yerms. These 5 "elegant" years are

365 & 71 / 293
365 & 119 / 491
365 & 365 / 1507
365 & 8 / 33
365 & 39 / 161

and all cycles are odd numbered. These cycles are then described by
their
alternating periodic sequence,

8ths = 46, 45, ...
3rds = 122, 121, ...
10ths = 37, 36, ...
17ths = 21, 22, ...
12ths = 30, 31, ...

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