Ominous Years of the 365 & 292 / 1207 day Year

[Helios]

I. The Solar Year = 365 & 292 / 1207 days

There are "ominous years" which are years for which a 3-day epagomenal period is dropped from a 366-day year, usually every 4 years but seldomly after 3 years. These ominous years happen when;

( 305*Y + 603 )MOD( 1207 )< 305

An alternative is ominous years for which a 5-day epagomenal period is dropped from a 366-day year every 6 or 7 years. These ominous years happen when;

( 183*Y + 603 )MOD( 1207 )< 183

II. The Lunar Year = 366 days

There is a lunar year of 366 days that is reckoned to equal 12 & 13 / 33 months. The 366 days are labeled with 409 golden numbers ( of 33 kinds ). There are 43 days that the golden numbers must be doubled up. The lunar year is grafted directly to the solar year. The lunar year is then shifted forward by 3 or 5 days after every ominous year.

[Karl Palmen]

Dear Helios and Calendar People

I recall the 1207-year solar calendar cycle appearing as part of a
lunisolar cycle, which is a multiple of a lunar cycle.

2414 years of 73 lunar 409-month cycles with mean month of 29.530562
days and mean year of 365.24192 days 2*(1207 years with 292 leap days)

This suggests running a lunar calendar on the 409-month cycle rather
than grafting on the solar calendar. Then the solar calendar could be
grafted on this lunar calendar.

I thought of two possible ways of grafting a solar calendar on a lunar
calendar, which can be applied to any accurate lunisolar cycle,
including all those listed by Helios.

(1) designate 19 near evenly spread days* of each lunar month as solar
days. Make each year have 235 solar days and the following non-solar
days, except for seven ominous years every 2414-year cycle, which have
234 solar days.
(2) have a common solar year of 371 dates and a leap year of 372 dates.
Skip one date at the end of each 29-day month so each month has 30
dates.  58 leap years are needed every 1207-year cycle.

In general he number of ominous years of method (1) and the number of
leap years of method (2) can be worked out by my lunisolar spreadsheets
http://www.the-light.com/cal/kp_Lunisolar_xls.html . Column L Trunc
gives the number of ominous years of method 1, which simply correct the
19-year cycle. Column K Saltus gives the number of leap years of method
2, which are Saltus Lunae corrections of a 30-year cycle.

For the 2414-year lunisolar cycle A=2414, B=889 and C=468.

* In method (1) once could designate all days 1 to 29 that do not have a
remainder of 1 when divided by 3 as solar days (i.e. 2nd, 3rd, 5th, 6th,
8th, 9th, 11th, 12th, 14th, 15th, 17th, 18th, 20th, 21st, 23rd, 24th,
26th, 27th & 29th).

Karl

10(13(20

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 08 October 2009 11:43
To: CALNDR-L@...
Subject: Ominous Years of the 365 & 292 / 1207 day Year

I. The Solar Year = 365 & 292 / 1207 days

There are "ominous years" which are years for which a 3-day epagomenal
period is dropped from a 366-day year, usually every 4 years but
seldomly
after 3 years. These ominous years happen when;

( 305*Y + 603 )MOD( 1207 )< 305

An alternative is ominous years for which a 5-day epagomenal period is
dropped from a 366-day year every 6 or 7 years. These ominous years
happen
when;

( 183*Y + 603 )MOD( 1207 )< 183

II. The Lunar Year = 366 days

There is a lunar year of 366 days that is reckoned to equal 12 & 13 / 33
months. The 366 days are labeled with 409 golden numbers ( of 33 kinds
).
There are 43 days that the golden numbers must be doubled up. The lunar
year
is grafted directly to the solar year. The lunar year is then shifted
forward by 3 or 5 days after every ominous year.

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[Helios]

Dear Karl et al.,

In just this case, I think distributing 409 golden numbers into 366 days is absolutely jitterless and therefore is as perfect as it gets. Though Karl's suggestions will likely work, either there will be some microscopic amount of jitter ( because of irregular yerms ), or there will be no jitter and then there will just be an equivalence or isomorphism to the method I describe which seems simpler. I cannot then see a reason to attempt Karl's alternate formulations.

Here is the modular series to the 33-year golden numbers as they would go into the 366-day year;

01-06-11-16-21-26-31-03-08-13-18-23-28-33-05-10-15-20-25-30-02-07-12-17-22-27-32-04-09-14-19-24-29

( corrected )

[Karl Palmen]

Dear Helios and Calendar People


-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 08 October 2009 16:22
To: CALNDR-L@...
Subject: Re: Ominous Years of the 365 & 292 / 1207 day Year

Dear Karl et al.,

In just this case, I think distributing 409 golden numbers into 366 days
is
absolutely jitterless and therefore is as perfect as it gets.

KARLS SAYS:
I thought Helios was suggesting grafting a lunar calendar on a solar
calendar and thereby making it dependent on the solar calendar. This
would produce additional jitter on the lunar calendar.

However I more careful read suggests otherwise.

The 366-day cycle is uninterrupted and completely independent of any
solar calendar. Also the lunisolar 2414-year cycle is a multiple (2409)
of 366 days. If this indeed does produce minimum jitter (not proven),
then the resulting lunar months would be equivalent to those of a
409-yerm cycle.

In either case a solar calendar can be grafted on as I have suggested
(as it works for any lunar calendar with months of 29 or 30 days). This
solar calendar would have some additional jitter, but not as much as
Helios's suggested solar calendar where 3 or 5 days are removed from 366
in an ominous year.

The possible advantage of Helios's solar calendar is that the
relationship between the solar year and the 366-day cycle could be
easier to calculate.

Helios has not fully specified how his 409 golden numbers would be
distributed around the 366-day cycle so I cannot tell whether it would
produce minimum jitter. I could reverse engineer such a sequence of
golden numbers from a 409-month cycle of 25 yerms.
I note that the number 2409 of 366-day cycles in the 2414-year cycle
equal to 33*73, so perhaps this is how Helios's 33 gets in.

Karl

10(13(20



Though Karl's
suggestions will likely work, either there will be some microscopic
amount
of jitter ( because of irregular yerms ), or there will be no jitter and
then there will just be an equivalence or isomorphism to the method I
describe which seems simpler. I cannot then see a reason to attempt
Karl's
alternate formulations.

Here is the modular series to the 33-year golden numbers as they would
go
into the 366-day year;

01-29-24-19-14-09-04-32-27-22-17-12-07-02-30-25-20-15-10-05-33-28-23-18-
13-08-03-31-26-21-16-11-06

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[Karl Palmen]

Dear Helios and Calendar People

I now realise that Helios exploits the fact that a lunar 409-month cycle
is made up of 33 periods of 366 days as well as the fact that 73 of
these lunar cycles makes up a lunisolar 2414-year cycle with the said
mean year.

Consequently, it is possible to give 409 golden numbers with values from
1 to 33 to the days of the 366-day cycle (referred to as a lunar year)
so that the resulting lunar months form a 25-yerm cycle and so have
minimum jitter. It is worthy of investigation despite the not very
accurate mean lunar month.


I've also thought of a solar calendar that Helios might be interested
in.  In each cycle of 33 lunar years of 366 days, eight lunar years
(e.g. 3, 7, 11, 15, 19, 23, 27 and 31) are designated as leap years.
This designation of course does not affect the number of days in the
lunar year, but instead makes the last day of a leap lunar year into a
leap day in the solar calendar. So in the complete 2414-year lunisolar
cycle there are 8*73=584 leap days and so 292 leap days per 1207 years
as required.

The leap days drift slowly later in the solar year. They occur at
intervals of 4 years and 3 days within each 33-lunar-year cycle and
intervals of 5 years and 4 days between the 33-lunar-year cycles. The
lunar new years jumps one day later in the solar year after each common
lunar year and stays put after each leap lunar year. So the leap years
actually stop leaping from occurring. The leap days and the lunar new
years drift five times across the solar year over one lunisolar
2414-year cycle.

The solar calendar exploits the fact that the number of 33-lunar-year
cycles in the lunisolar cycle (73) is one fifth the number of days in a
common solar year. In each 33-lunar-year cycle, the lunar year drifts
exactly 25 non-leap days later in the solar calendar. Over a complete
lunisolar cycle this accumulates to 73*25=365*5 non-leap days, which is
exactly five years.


In his solar calendars, Helios exploits the fact that the number of
common years in the 1207-year cycle (915) is divisible by 3 and 5. In my
solar calendar, the 2414-year cycle has 73*25=1825 solar common years
that contain the last day of a lunar common year and 5 solar common
years without the last day of any lunar year. This makes a total
1830=2*915 common years.


Karl

10(13(21

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 08 October 2009 11:43
To: CALNDR-L@...
Subject: Ominous Years of the 365 & 292 / 1207 day Year

I. The Solar Year = 365 & 292 / 1207 days

There are "ominous years" which are years for which a 3-day epagomenal
period is dropped from a 366-day year, usually every 4 years but
seldomly
after 3 years. These ominous years happen when;

( 305*Y + 603 )MOD( 1207 )< 305

An alternative is ominous years for which a 5-day epagomenal period is
dropped from a 366-day year every 6 or 7 years. These ominous years
happen
when;

( 183*Y + 603 )MOD( 1207 )< 183

II. The Lunar Year = 366 days

There is a lunar year of 366 days that is reckoned to equal 12 & 13 / 33
months. The 366 days are labeled with 409 golden numbers ( of 33 kinds
).
There are 43 days that the golden numbers must be doubled up. The lunar
year
is grafted directly to the solar year. The lunar year is then shifted
forward by 3 or 5 days after every ominous year.

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[Helios]

Dear Karl et al.,

I have found that

mean yerm of ( 10ths, 146ths ) of year = 1 / [ 2 - ( 730 / Y ) ] = 754.875 days

and therefore

409 months = 25 lunar yerms = 16 yerms of 10th-years

Is this ratio of yerms a rule maybe? Another cycle,

1634 months = 100 lunar yerms = 64 yerms of 10th-years

W = 1 / [ 2 - ( 730 / Y ) ]
Y = 730 / [ 2 - ( 1 / W ) ]

365 1634-month cycles = 48221 years

so possibly there's a theorem in the making.

[Karl Palmen]

Dear Helios and Calendar People

I've decided to have a go at producing the Golden number myself, so that
the resulting months and yerms produce a Helios cycle, which is
symmetrical and has minimum jitter. I've so far put in the golden
numbers 1, 2 and 3 for the whole 366-day cycle. I did this by running
the yerm calendar over the first three 366-day cycles.
Here I show the part for the first 33 days to compare with what Helios
has shown below.

001: 01 14(01
007: 03 15(09
019: 02 14(14
031: 01 14(02

I've placed the corresponding month of the 25-yerm cycle after each
golden number. I started at yerm 14 to ensure symmetry.
I see Helios's sequence has only one occurrence of 01 when I'd expect a
second one 30 days later. Then I realise no day has two Golden numbers
and this is because Helios has just shown the Golden numbers in order
and not which days they occur.
Helios's scheme is not workable for a calendar (which labels days),
unless the day of each golden number is specified.

I see that 02 occurs before 03, which is unexpected! It implies that one
can get a whole number (13) lunar months by adding less than half a
lunar month to 366 days, when in fact you need to add about 18 days (to
get 384 days). Given that each golden number corresponds to 1/33 lunar
month it implies 13 lunar months have 366 days plus 13/33 lunar months,
which is too short.
In his first note Helios reckoned 366 days to be 12 13/33 lunar months.
Hence 13 lunar months have 366 days and 20/33 lunar month. This suggests
that Helios's Golden numbers just need reversing (before allocating to
days).

I see the Golden numbers increasing in steps of 5 mod 33 (when
reversed). This corresponds to 62 lunar months, which have an average of
1830 366/409 days in the 409-month cycle. The 62 lunar months normally
have 1831 days, which is five lunar years and a day, but they sometimes
have just 1830 days, which is exactly five lunar years. These 62-month
intervals of just 1830 days mark the 43 days of the 366-day lunar year
that have two golden numbers. Such intervals begin with the 2nd month of
a yerm or the 4th month a 17-month yerm or the last 15-month yerm (yerm
24).

This helps me list the golden numbers for the first few days of the
366-day cycle and their corresponding yerm months:

001: 01 14(01
002: 06 17(14
003: 11 21(10
004: 16 25(08
005: 21 04(04  005: 26 07(17
006: 31 11(13
007: 03 15(09
008: 08 19(07
009: 13 23(03
010: 18 01(16
011: 23 05(12
012: 28 09(10
013: 33 13(06
014: 05 17(02  014: 10 20(15
015: 15 24(11
016: 20 03(07
017: 25 07(05
018: 30 11(01
019: 02 14(14

Karl

10(13(21

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 08 October 2009 16:22
To: CALNDR-L@...
Subject: Re: Ominous Years of the 365 & 292 / 1207 day Year

Dear Karl et al.,

In just this case, I think distributing 409 golden numbers into 366 days
is
absolutely jitterless and therefore is as perfect as it gets. Though
Karl's
suggestions will likely work, either there will be some microscopic
amount
of jitter ( because of irregular yerms ), or there will be no jitter and
then there will just be an equivalence or isomorphism to the method I
describe which seems simpler. I cannot then see a reason to attempt
Karl's
alternate formulations.

Here is the modular series to the 33-year golden numbers as they would
go
into the 366-day year;

01-29-24-19-14-09-04-32-27-22-17-12-07-02-30-25-20-15-10-05-33-28-23-18-
13-08-03-31-26-21-16-11-06

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[Karl Palmen]

Dear Helios and Calendar People

I've checked out these solar yerms.
I find that the yerms of 1/10 year repeat once every 584 yerms, which is
a whole 1207-year cycle. So each 409-month cycle has either more or less
than 16 of these yerms.
The yerms of 1/146 days repeat one every eight yerms, which is half a
409-month cycle. So each 409-month cycle has EXACTLY 16 of these yerms.

Why do we get exactly 16 yerms for the 1/146 years, but not the 1/10
years? The answer is simple. The 409-month cycle has exactly 2414/73 =
33 5/73 years so has a whole number (4828) 1/146 years, but does not
have a whole number of 1/10 years.

This suggests to me having a solar calendar based on a 1/73 of a year.
This can be done by pairing together the 146ths in their eight-yerm
cycle. Then we get 73rds of five days each plus leap days at three
intervals 4 10/73 years to one interval of 4 9/73 years. Eight of these
intervals always add up to 33 5/73 years.

The year could be divided into twelve months of six 73rds plus an
epagomenal 73rd. Then a 409-month cycle would last 33 years one month
less a 73rd.
Alternatively, the year could have fourteen months of five 73rds and one
short month of three 73rds. Then a 409-month cycle would last 33 years
and one full month.


This has made me become aware of a general idea about a solar calendar
for a lunisolar cycle that is a multiple of a short lunar cycle. It is a
good idea to divide the year into near equal parts whose number is equal
to the number of lunar cycles in the lunisolar cycle. For example, for
the 5515-year cycle of 39 lunar 1749-year cycles, the year would be
divided into 39 parts. These parts could be grouped into threes to form
a 13-month calendar as I have already suggested.

One thing notable about the 2414-year cycle is that the number of leap
days is a multiple (8) of the number (73) of lunar cycles. I don't think
any of the other examples quoted by Helios has such a property. If I am
wrong with this an counter-example would be appreciated.


Karl

10(13(24

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 09 October 2009 16:59
To: CALNDR-L@...
Subject: Re: Lunar 409-month cycle and solar calendar with 365 & 292 /
1207 day Year

Dear Karl et al.,

I have found that

mean yerm of ( 10ths, 146ths ) of year = 1 / [ 2 - ( 730 / Y ) ] =
754.875
days

and therefore

409 months = 25 lunar yerms = 16 yerms of 10th-years

Is this ratio of yerms a rule maybe? Another cycle,

1634 months = 100 lunar yerms = 64 yerms of 10th-years

W = 1 / [ 2 - ( 730 / Y ) ]
Y = 730 / [ 2 - ( 1 / W ) ]

365 1634-month cycles = 48221 years

so possibly there's a theorem in the making.

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[Karl Palmen]

Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 09 October 2009 16:59
To: CALNDR-L@...
Subject: Re: Lunar 409-month cycle and solar calendar with 365 & 292 /
1207 day Year

Dear Karl et al.,

I have found that

mean yerm of ( 10ths, 146ths ) of year = 1 / [ 2 - ( 730 / Y ) ] =
754.875
days

and therefore

409 months = 25 lunar yerms = 16 yerms of 10th-years

Is this ratio of yerms a rule maybe? Another cycle,

1634 months = 100 lunar yerms = 64 yerms of 10th-years

W = 1 / [ 2 - ( 730 / Y ) ]
Y = 730 / [ 2 - ( 1 / W ) ]

365 1634-month cycles = 48221 years

so possibly there's a theorem in the making.

KARL SAYS: No.
The number of these solar yerms in simply twice the number of leap days.
Helios's supposed theorem would imply that every lunisolar cycle has
exactly 8 leap days to every 25 yerms. Checking my lunisolar
spreadsheets at
http://www.the-light.com/cal/kp_Lunisolar_xls.html or just those cycles
that Helios has listed shows that the ratio of leap days to lunar yerms
varies.

How can one change one lunisolar cycle to another without changing its
ratio of leap days to yerms?
Take the 59-year cycle of 730 lunar months where all the years have 365
days and the lunar months alternate between 29 and 30 days. Add or
subtract a number of these to/from a number of your lunisolar cycle and
then divide by any common divisor of years, months and days.

Karl

10(13(24
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[Helios]

Dear Karl et al.,

Here is the distribution of 409 months by 33 golden numbers.
The resulting months and yerms produce a Helios cycle, which is
symmetrical and has minimum jitter.

http://www.helios.netne.net/year366.htm

Instead of the 365 & 292 / 1207 day year, a 483-year cycle that is divisable by 366 days might be used. The 483 years is the drift cycle of the 366-day year.
 

[Karl Palmen]

Dear Helios, Victor and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 17 October 2009 06:59
To: CALNDR-L@...
Subject: Re: Ominous Years of the 365 & 292 / 1207 day Year

Dear Karl et al.,

Here is the distribution of 409 months by 33 golden numbers.
The resulting months and yerms produce a Helios cycle, which is
symmetrical and has minimum jitter.

http://www.helios.netne.net/year366.htm 

KARL SAYS: They agree with the golden numbers I worked out for the first
19 days in the note of 9 October.

HRLIOS CONTINUED:
Instead of the 365 & 292 / 1207 day year, a 483-year cycle that is
divisable
by 366 days might be used. The 483 years is the drift cycle of the
366-day
year.

KARL SAYS: Actually the solar 483-year solar calendar cycle breaks into
three 161-year cycles each with 39 leap days, which was discovered by
Victor. The mean year is (482/483)*366 = 365.242236 days.

Victor suggested alternating periods of 1 and 2 days grouped into solar
yerms of 241 days = 161 periods.. Helios should note that 241 days is
not just the mean length of Victors' solar yerms, but is the actual
length of every one of his solar yerms. There are 244 of them each
161-year cycle.

I have suggested pairing these 1 or 2 day periods to create a period
normally of 3 days that is shortened to 2 days once every 482 days = 161
periods. There are 122 of these short periods each 161-year cycle.

One could have a solar calendar with 366 dates each year from which one
date is skipped once every 482 days (= 483 dates).
The solar year could have 12 months of 30 dates or 33 dates. Then it
could be arranged that only the dates whose day of month is divisible by
three is ever skipped.

The 409-month cycle is made up of 33 years of 366 days. 33 has no common
divisor with 482 and so 482 409-month cycles are needed to make a
lunisolar cycle that is also a multiple of the 161-year cycle. This
lunisolar cycle has 33*483 = 15,939 years
(A=15,939, B=5870, C=3090 for
http://www.the-light.com/cal/kp_Lunisolar_xls.html ).


Karl

10(14(01


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[Karl Palmen]

Dear Helios and Calendar People

Helios provided his golden numbers in the 366-day cycle for the
409-month cycle at
http://www.helios.netne.net/year366.htm .

I check the days that have two golden numbers against the 25-yerm cycle.
I put the corresponding month of the 25-yerm cycle next to each of these
golden numbers. I start the first 366-day year one the first day of
month 14(01 in the 25-yerm cycle to keep symmetry. I find a perfect
match and so all golden numbers are as derived from the 25-yerm cycle.

005: 21 04(04  -  26 07(17
014: 05 17(02  -  10 20(15
022: 17 01(04  -  22 04(17
031: 01 14(02  -  06 17(15
039: 13 23(04  -  18 01(17
048: 30 11(02  -  02 14(15
056: 09 20(04  -  14 23(17
065: 26 08(02  -  31 11(15
073: 05 17(04  -  10 20(17
082: 22 05(02  -  27 08(15
090: 01 14(04  -  06 17(17
099: 18 02(02  -  23 05(15
107: 30 11(04  -  02 14(17
116: 14 24(02  -  19 02(15
124: 26 08(04  -  31 11(17
133: 10 21(02  -  15 24(15
141: 22 05(04  -  27 08(17
150: 06 18(02  -  11 21(15
158: 18 02(04  -  23 05(17
167: 02 15(02  -  07 18(15
175: 14 24(04  -  19 02(17
184: 31 12(02  -  03 15(15
193: 15 25(02  -  20 03(13
201: 27 09(02  -  32 12(15
210: 11 22(02  -  16 25(15
218: 23 06(02  -  28 09(15
227: 07 19(02  -  12 22(15
235: 19 03(02  -  24 06(15
244: 03 16(02  -  08 19(15
252: 15 25(04  -  20 03(15
261: 32 13(02  -  04 16(15
269: 11 22(04  -  16 25(17
278: 28 10(02  -  33 13(15
286: 07 19(04  -  12 22(17
295: 24 07(02  -  29 10(15
303: 03 16(04  -  08 19(17
312: 20 04(02  -  25 07(15
320: 32 13(04  -  04 16(17
329: 16 01(02  -  21 04(15
337: 28 10(04  -  33 13(17
346: 12 23(02  -  17 01(15
354: 24 07(04  -  29 10(17
363: 08 20(02  -  13 23(15

I worked this out by noting that the 3-yerm cycle of 49 months (1447
days) is 17 days short of a whole number (4) of 366-day years and so
worked the yerm months over intervals of 17 days (equivalent to 3-yerm
cycle backwards). Note that the 17-day intervals are broken only when
the 3-yerm cycle is broken (from day 175 to day 193). Day 193 has the
only occurrence of the 13th month of a yerm for a second golden number
of a day.

The 62-month interval for each such day has only three new yerms in it
rather than the usual four (for 62 months) and the first month is
even-numbered while the second month is odd-numbered (within its yerm).
The statement " Such intervals begin with the 2nd month of a yerm or the
4th month a 17-month yerm or the last 15-month yerm (yerm 24)." is
confirmed.

I also found that the golden numbers for day 337 and 346 were not
correct. They don't follow the drop of 4 mod 33 over 17 days. I
corrected this above. I later saw on the web site that Helios had missed
out an 18 for day 335 and day 347 is incorrectly 24, but subsequent
golden numbers are correct.

Karl

10(14(02

-----Original Message-----
From: Palmen, Karl (STFC,RAL,CICT)
Sent: 09 October 2009 17:10
To: 'East Carolina University Calendar discussion List'
Subject: RE: Ominous Years of the 365 & 292 / 1207 day Year

Dear Helios and Calendar People

I've decided to have a go at producing the Golden number myself, so that
the resulting months and yerms produce a Helios cycle, which is
symmetrical and has minimum jitter. I've so far put in the golden
numbers 1, 2 and 3 for the whole 366-day cycle. I did this by running
the yerm calendar over the first three 366-day cycles.
Here I show the part for the first 33 days to compare with what Helios
has shown below.

001: 01 14(01
007: 03 15(09
019: 02 14(14
031: 01 14(02

I've placed the corresponding month of the 25-yerm cycle after each
golden number. I started at yerm 14 to ensure symmetry.
I see Helios's sequence has only one occurrence of 01 when I'd expect a
second one 30 days later. Then I realise no day has two Golden numbers
and this is because Helios has just shown the Golden numbers in order
and not which days they occur.
Helios's scheme is not workable for a calendar (which labels days),
unless the day of each golden number is specified.

I see that 02 occurs before 03, which is unexpected! It implies that one
can get a whole number (13) lunar months by adding less than half a
lunar month to 366 days, when in fact you need to add about 18 days (to
get 384 days). Given that each golden number corresponds to 1/33 lunar
month it implies 13 lunar months have 366 days plus 13/33 lunar months,
which is too short.
In his first note Helios reckoned 366 days to be 12 13/33 lunar months.
Hence 13 lunar months have 366 days and 20/33 lunar month. This suggests
that Helios's Golden numbers just need reversing (before allocating to
days).

I see the Golden numbers increasing in steps of 5 mod 33 (when
reversed). This corresponds to 62 lunar months, which have an average of
1830 366/409 days in the 409-month cycle. The 62 lunar months normally
have 1831 days, which is five lunar years and a day, but they sometimes
have just 1830 days, which is exactly five lunar years. These 62-month
intervals of just 1830 days mark the 43 days of the 366-day lunar year
that have two golden numbers. Such intervals begin with the 2nd month of
a yerm or the 4th month a 17-month yerm or the last 15-month yerm (yerm
24).

This helps me list the golden numbers for the first few days of the
366-day cycle and their corresponding yerm months:

001: 01 14(01
002: 06 17(14
003: 11 21(10
004: 16 25(08
005: 21 04(04  005: 26 07(17
006: 31 11(13
007: 03 15(09
008: 08 19(07
009: 13 23(03
010: 18 01(16
011: 23 05(12
012: 28 09(10
013: 33 13(06
014: 05 17(02  014: 10 20(15
015: 15 24(11
016: 20 03(07
017: 25 07(05
018: 30 11(01
019: 02 14(14

Karl

10(13(21

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 08 October 2009 16:22
To: CALNDR-L@...
Subject: Re: Ominous Years of the 365 & 292 / 1207 day Year

Dear Karl et al.,

In just this case, I think distributing 409 golden numbers into 366 days
is
absolutely jitterless and therefore is as perfect as it gets. Though
Karl's
suggestions will likely work, either there will be some microscopic
amount
of jitter ( because of irregular yerms ), or there will be no jitter and
then there will just be an equivalence or isomorphism to the method I
describe which seems simpler. I cannot then see a reason to attempt
Karl's
alternate formulations.

Here is the modular series to the 33-year golden numbers as they would
go
into the 366-day year;

01-29-24-19-14-09-04-32-27-22-17-12-07-02-30-25-20-15-10-05-33-28-23-18-
13-08-03-31-26-21-16-11-06

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[Karl Palmen]

Dear Helios and Calendar People

This gives me the idea of not only having 409 Golden numbers (1 to 33)
in the 366-day cycle to indicate new lunar months, but also 483 Silver
numbers (1 to 482) to indicate new solar years. These silver numbers
could go in the sequence 1, 482, 481, 480, ... 3, 2, 1.  with one silver
number per day, except for 39 days each third of 122 days, which have
two silver numbers.

It is possible to assign Golden and Silver numbers to periods other the
366 days, but each day may have many gold and silver numbers. Such a
system would be of interest if it is fairly accurate and each day has
very few Golden and Silver numbers.

Karl

10(14(05

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 17 October 2009 06:59
To: CALNDR-L@...
Subject: Re: Ominous Years of the 365 & 292 / 1207 day Year

Dear Karl et al.,

Here is the distribution of 409 months by 33 golden numbers.
The resulting months and yerms produce a Helios cycle, which is
symmetrical and has minimum jitter.

http://www.helios.netne.net/year366.htm

Instead of the 365 & 292 / 1207 day year, a 483-year cycle that is
divisable
by 366 days might be used. The 483 years is the drift cycle of the
366-day
year.
 

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[Helios]

Also the Julian calendar cycle can be indicated because

488 years = 366*487

Maybe the Gregorian calendar could be mapped out in centuries as it is just a concatenated Julian calendar.

The other scheme 483 years = 366*482 could be made into a gear mechanism with

1) double gear on same axial with small gear = 409 teeth and large gear = 483 teeth
2) annual gear = 482 teeth engaged to the large gear
3) lunar gear = 33 teeth engaged to the small gear

[Karl Palmen]

Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 October 2009 11:50
To: CALNDR-L@...
Subject: Re: Golden and Silver Numbers RE: Ominous Years of the ...

Also the Julian calendar cycle can be indicated because

488 years = 366*487

Maybe the Gregorian calendar could be mapped out in centuries as it is
just
a concatenated Julian calendar.

KARL SAYS: No.
It is a Julian calendar with some dates skipped (such a 29 Feb 1900).

The number of days in a Gregorian 400 year cycle is 146097 = (3^3) * 7 *
773.
So like the Julian, its cycle is divisible by three and so gives 48,800
years = 366*48,699.
These figures are possibly too large to be useful.

One could use instead of 366 days some other period with a large common
divisor in days with the Gregorian 400-year cycle.
For example,
a 364-day cycle (common divisor 7),  20,800 years = 364*20,871
a 360-day cycle (common divisor 9),  16,000 years = 360*16,233,
a 378-day cycle (common divisor 189),   800 years = 378*773.

The 364-day cycle can have 293 years = 364*294, while
The 365-day cycle can have 1506 years = 365*1507 or 1507 years =
365*1508. The latter is also a lunisolar cycle with a rather short mean
month (about 29.53054 days), which could be of interest to Helios.

Karl

10(14(08

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[Karl Palmen]

Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 October 2009 11:50
To: CALNDR-L@...
Subject: Re: Golden and Silver Numbers RE: Ominous Years of the ...

<snip>

The other scheme 483 years = 366*482 could be made into a gear mechanism
with

1) double gear on same axial with small gear = 409 teeth and large gear
=
483 teeth
2) annual gear = 482 teeth engaged to the large gear
3) lunar gear = 33 teeth engaged to the small gear

KARL SAYS: Helios neglected to say that the double gear wheel (1) would
be turned at exactly one revolution every 366 days.

Karl

10(14(08

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[Karl Palmen]

Dear Helios, Victor and Calendar People

Helios's gears can be generalised as follows:

1) double gear on same axle turning once every D days
with lunar gear with L teeth and solar gear with S teeth
2) annual gear with T teeth engaged to the solar gear
3) monthly gear with M teeth engaged to the lunar gear

Find integers D, S, T, L and M a small as possible yet giving an
accurate year for D*S/T and an accurate lunation for D*L/M.

Helios has found D=366, S=483, T=482, L=409, M=33.

Karl

10(14(08 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 October 2009 11:50
To: CALNDR-L@...
Subject: Re: Golden and Silver Numbers RE: Ominous Years of the ...

Also the Julian calendar cycle can be indicated because

488 years = 366*487

Maybe the Gregorian calendar could be mapped out in centuries as it is
just
a concatenated Julian calendar.

The other scheme 483 years = 366*482 could be made into a gear mechanism
with

1) double gear on same axial with small gear = 409 teeth and large gear
=
483 teeth
2) annual gear = 482 teeth engaged to the large gear
3) lunar gear = 33 teeth engaged to the small gear
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5801727p26058316.html
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[Brillig]

Dear Karl, Helios, and Calendar People,

I've created a spreadsheet to find values for these variable in general. I've sent a copy of the spreadsheet to Karl to review. Meanwhile, the most efficient combination I've been able to find is:

D=122, S=482, T=161, L=99, M=409

Actually, my spreadsheet thinks the following is more efficient, but only because it's so accurate. My definition of efficiency takes accuracy and teeth count into account.

D=346, S=1006, T=953, L=60, M=703

Victor

On Tue, Oct 27, 2009 at 4:04 AM, Karl Palmen <karl.palmen@...> wrote:

Dear Helios, Victor and Calendar People

Helios's gears can be generalised as follows:

1) double gear on same axle turning once every D days
with lunar gear with L teeth and solar gear with S teeth
2) annual gear with T teeth engaged to the solar gear
3) monthly gear with M teeth engaged to the lunar gear

Find integers D, S, T, L and M a small as possible yet giving an
accurate year for D*S/T and an accurate lunation for D*L/M.

Helios has found D=366, S=483, T=482, L=409, M=33.

Karl

10(14(08 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 October 2009 11:50
To: CALNDR-L@...
Subject: Re: Golden and Silver Numbers RE: Ominous Years of the ...

Also the Julian calendar cycle can be indicated because

488 years = 366*487

Maybe the Gregorian calendar could be mapped out in centuries as it is
just
a concatenated Julian calendar.

The other scheme 483 years = 366*482 could be made into a gear mechanism
with

1) double gear on same axial with small gear = 409 teeth and large gear
=
483 teeth
2) annual gear = 482 teeth engaged to the large gear
3) lunar gear = 33 teeth engaged to the small gear
--
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5801727p26058316.html
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[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear Victor Helios and Calendar People

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Victor Engel
Sent: 27 October 2009 16:32
To: CALNDR-L@...
Subject: Re: Helios type Gears RE: Golden and Silver Numbers RE: Ominous Years of the ...

 

Dear Karl, Helios, and Calendar People,

I've created a spreadsheet to find values for these variable in general. I've sent a copy of the spreadsheet to Karl to review. Meanwhile, the most efficient combination I've been able to find is:

D=122, S=482, T=161, L=99, M=409

This one is equivalent to Helios’s D=366, S=482, T=483, L=33, M=409, but uses fewer teeth.

Actually, my spreadsheet thinks the following is more efficient, but only because it's so accurate. My definition of efficiency takes accuracy and teeth count into account.

D=346, S=1006, T=953, L=60, M=703

This one follows the 703-month lunar cycle of 43 yerms whose number of days is divisible by 30, which appears in the L rather than the D.  Victor’s calculations stop at D=455, so miss out D=692, S=503, T=953, L=30, M=703, which is equivalent, but has fewer teeth.

Karl

10(14(09

 

Victor

On Tue, Oct 27, 2009 at 4:04 AM, Karl Palmen <karl.palmen@...> wrote:

Dear Helios, Victor and Calendar People

Helios's gears can be generalised as follows:

1) double gear on same axle turning once every D days
with lunar gear with L teeth and solar gear with S teeth
2) annual gear with T teeth engaged to the solar gear
3) monthly gear with M teeth engaged to the lunar gear

Find integers D, S, T, L and M a small as possible yet giving an
accurate year for D*S/T and an accurate lunation for D*L/M.

Helios has found D=366, S=483, T=482, L=409, M=33.

Karl

10(14(08 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 October 2009 11:50
To: CALNDR-L@...
Subject: Re: Golden and Silver Numbers RE: Ominous Years of the ...

Also the Julian calendar cycle can be indicated because

488 years = 366*487

Maybe the Gregorian calendar could be mapped out in centuries as it is
just
a concatenated Julian calendar.

The other scheme 483 years = 366*482 could be made into a gear mechanism
with

1) double gear on same axial with small gear = 409 teeth and large gear
=
483 teeth
2) annual gear = 482 teeth engaged to the large gear
3) lunar gear = 33 teeth engaged to the small gear
--
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http://www.nabble.com/Ominous-Years-of-the-365---292---1207-day-Year-tp2
5801727p26058316.html
Sent from the Calndr-L mailing list archive at Nabble.com.

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[Karl Palmen]

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Dear Victor, Helios and Calendar People

 

I checked a spread sent to me by Victor for all values that give rise to a mean year between 365.2416 and 365.2428 days and a mean month between those of the Tabular Islamic Calendar (29.530555..) and the 3-yerm 49-month cycle (29.530612…).  I then eliminated all equivalent cycles (with same mean year and month), except the one with the fewest teeth (not including any teeth that may be used for D).

 

D=120, S=627, T=206, L=173, M=703

D=122, S=482, T=161, L=99, M=409  - equivalent to Helios’s gears

D=161, S=946, T=417, L=177, M=965

D=177, S=1397, T=677, L=161, M=965

D=198, S=190, T=103, L=61, M=409

D=413, S=543, T=614, L=69, M=965

D=421, S=583, T=672, L=39, M=556

D=509, S=531, T=740, L=55 M=948  - uses 79-lunar-year cycle

D=519, S=677, T=703, L=40, M=703

D=692, S=503, T=953, L=30, M=703

 

You may note that T is the number of years in the solar calendar cycle and M is the number months in the lunar calendar cycle (or a multiple thereof).

 

The following use Helios’s 409-month cycle:

D=122, S=482, T=161, L=99, M=409  - equivalent to Helios’s gears

D=198, S=190, T=103, L=61, M=409

 

The following use the 703-month cycle of 692 30-day periods:

D=120, S=627, T=206, L=173, M=703

D=519, S=677, T=703, L=40, M=703

D=692, S=503, T=953, L=30, M=703

 

The following use the 103-year cycle with mean year of 365.24272 days:

D=120, S=627, T=206, L=173, M=703

D=198, S=190, T=103, L=61, M=409

 

The following use the Vij Tithi of 2/59 month equated to 966/965 days

D=161, S=946, T=417, L=177, M=965

D=177, S=1397, T=677, L=161, M=965

D=413, S=543, T=614, L=69, M=965

 

 

Karl

 

10(14(10

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Victor Engel
Sent: 27 October 2009 16:32
To: CALNDR-L@...
Subject: Re: Helios type Gears RE: Golden and Silver Numbers RE: Ominous Years of the ...

 

Dear Karl, Helios, and Calendar People,

I've created a spreadsheet to find values for these variable in general. I've sent a copy of the spreadsheet to Karl to review. Meanwhile, the most efficient combination I've been able to find is:

D=122, S=482, T=161, L=99, M=409

Actually, my spreadsheet thinks the following is more efficient, but only because it's so accurate. My definition of efficiency takes accuracy and teeth count into account.

D=346, S=1006, T=953, L=60, M=703

Victor

On Tue, Oct 27, 2009 at 4:04 AM, Karl Palmen <karl.palmen@...> wrote:

Dear Helios, Victor and Calendar People

Helios's gears can be generalised as follows:

1) double gear on same axle turning once every D days
with lunar gear with L teeth and solar gear with S teeth
2) annual gear with T teeth engaged to the solar gear
3) monthly gear with M teeth engaged to the lunar gear

Find integers D, S, T, L and M a small as possible yet giving an
accurate year for D*S/T and an accurate lunation for D*L/M.

Helios has found D=366, S=483, T=482, L=409, M=33.

Karl

10(14(08 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Helios
Sent: 26 October 2009 11:50
To: CALNDR-L@...
Subject: Re: Golden and Silver Numbers RE: Ominous Years of the ...

Also the Julian calendar cycle can be indicated because

488 years = 366*487

Maybe the Gregorian calendar could be mapped out in centuries as it is
just
a concatenated Julian calendar.

The other scheme 483 years = 366*482 could be made into a gear mechanism
with

1) double gear on same axial with small gear = 409 teeth and large gear
=
483 teeth
2) annual gear = 482 teeth engaged to the large gear
3) lunar gear = 33 teeth engaged to the small gear
--
View this message in context:
http://www.nabble.com/Ominous-Years-of-the-365---292---1207-day-Year-tp2
5801727p26058316.html
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