Old Earth Calendar

[Palmen, KEV (Karl)]

Dear Irv, Victor and Calendar People

Here is the result for a 133-day cycle, which is equivalent to both a 19-day cycle and a week. It is applicable to all three intervals used in
http://www.the-light.com/cal/kp_NdaySolarCyc.html
http://www.the-light.com/cal/kp_NdaySolarCyc1.html
http://www.the-light.com/cal/kp_NdaySolarCyc2.html


 N   Days  Years LongYrs  YrLength
-----------------------------------
133  119434  327   244    365.24159    
133  191387  524   391    365.24237
133   71953  197   147    365.24365
-----------------------------------

The number of long years is in this case the number of years of three 133-day cycles rather than two 133-day cycles.

I produce this by guessing and then checking that
(1) the mean years are respectively less than, within and more than the range
(2) the middle cycle is the sum of the other two cycles
(3) Years1*LongYrs3 - Years3*LongYrs1 = 1

In this case, I guessed the 327-year cycle by subtracting the 524-year cycle from the 851-year cycle. I then got the 197-year cycle by subtracting the 327-year cycle from the 524-year cycle.

LongYrs would not of course be directly used in any Bahai leap month variant cycle that is a whole number of weeks, but can be used as follows:
The number of leap months is 7*LongYrs - 5*Years,
The number of leap weeks is 19*LongYrs - 14*Years and
The number of leap days is 133*LongYrs - 99*Years.

The coefficient of LongYrs is the number of months/weeks/days in 133 days and the coefficient of Years is the number of months/weeks/days in a common year in excess of two periods of 133 days.

So we have

Years LeapMonths LeapWeeks LeapDays  YrLength
--------------------------------------------
327      73        58         79     365.24159
524     117        93        127     365.24237
197      44        35         48     365.24365
-----------------------------------------------

All more accurate Bahai leap month cycles that are also a whole number of weeks can be constructed by adding whole numbers these cycles together.

The 327-year, 524-year and 197-year cycles are composed out of periods of alternating 4&5-year intervals as follows:
(327,73)=(112,25)+(112,25)+(103,23)
(524,117)=(112,25)+(103,23)+(103,23)+(103,23)+(103,23)
(197,44)=(103,23)+(94,21)

I've also noticed that in each of these periods of alternating 4&5-year intervals the number of equivalent leap days is always two more than the number of leap months.

Years LeapMonths LeapDays  YrLength
-----------------------------------
112      25        27      365.2410
103      23        25      365.2427
 94      21        23      365.2447
-----------------------------------

Also if a cycle has a whole number of 19-day months and weeks, then the number of leap months less the number of leap weeks is always divisible by three.

Karl

08(02(14

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 11 April 2006 14:39
To: CALNDR-L@...
Subject: Re: Bahai leap month variant


On Apr 11, 2006, at 07:53, Palmen, KEV (Karl) wrote:

> IRV CONTINUES:
> How about seeing even more cycles by adding those cycles that are the
> nearest LONGER than Victor's with a lesser and greater mean year?
>
> KARL SAYS: There is no such thing as the nearest longer cycle!
> If there were such thing, one could find a nearer cycle that is also
> longer, which contradicts the 'nearest' in the if clause.
>
> The two shorter cycles can generate ALL the cycles whose mean years
> lie in between there's by adding them together. This includes ALL the
> cycles whose mean year is within the range of 365.2416 days and
> 365.2428 days. I have the two bounding cycles outside the range so
> that they generate all cycles within the range.

<snip>

Irv replies:  Thanks for the explanation!

[Palmen, KEV (Karl)]

Dear Irv, Victor and Calendar People

Concerning cycles that are also a whole number of weeks:

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 09 April 2006 06:49
To: CALNDR-L@...
Subject: Bahai leap month variant


Dear Calendar People:

Re:  Leap Month Variant of the Bahai Calendar


A Bahai Calendar leap month variant could be made by replacing the 4 or
5 extra days with a 19-day leap month.
I searched for leap cycles shorter than 1000 years.

The leap month could be inserted at the end of the year for simplest
calendar arithmetic.  I could not, however, find a cycle suitable for
use with a rotating positional scheme such as Karl suggested, because
no useful cycles have an integer ratio of months to leap years.

An excellent fixed arithmetic leap month cycle would have 524 years,
with 9956 regular months and 117 leap months for a total of 10073
months.  This cycle has 191387 days or exactly 27341 weeks.  The
average leap year interval would be 524/117 or almost 4+1/2 years, that
is almost equally alternating between 4- and 5-year intervals.

This proposed cycle has exactly the same mean year as a leap day
calendar with 127 leap days in 524 years or a leap week calendar with
93 leap weeks in 524 years, that is 365 days 5 hours 49 minutes 1/2
second, which is an almost perfect fit to the present era mean
northward equinoctial year length.

73 leaps in 327 years = 365d 5h 47m 53s (would be excellent for the
next 4 millennia), each cycle would start on the same weekday.

I would have to say the 327-year cycle is the north solstice "winner",
being conveniently short, an excellent fit at present, and a whole
number of weeks.
The equivalent cycles would be 79 leap days in 327 years, or 58 leap
weeks in 327 years -- another for the Symmetry "soup mix"!

KARL SAYS:
In my previous note I discovered that all accurate cycles that also have a whole number of weeks can be made up from the following:

Years LeapMonths LeapWeeks LeapDays  YrLength
----------------------------------------------
327      73        58         79     365.24159
524     117        93        127     365.24237
197      44        35         48     365.24365
-----------------------------------------------

I see that Irv has already discovered the 327-year cycle, but only because its mean year is close to the Northern Solstice year.

We can make more cycles out of these

Mix   Years LeapMonths LeapWeeks LeapDays  YrLength
---------------------------------------------------
1:0   327      73        58         79     365.24159
2:1   851     190       151        206     365.24207
3:2  1375     307       244        333     365.24218
4:3  1899     424       337        460     365.24223
1:1   524     117        93        127     365.24237
3:4  1769     395       314        429     365.24251
2:3  1245     278       221        302     365.24257
1:2   721     161       128        175     365.24272
0:1   197      44        35         48     365.24365
----------------------------------------------------

Karl

08(02(14

[Irv Bromberg]

On Apr 12, 2006, at 07:55, Palmen, KEV (Karl) wrote:
>  N   Days  Years LongYrs  YrLength
> 133   71953  197   147    365.24365

Bromberg says:

OK, I added the 197-year cycle to the Bahai leap month cycles
spreadsheet, with also fast and slow leap year interval groupings.

> The number of long years is in this case the number of years of three
> 133-day cycles rather than two 133-day cycles.
>
> I produce this by guessing and then checking that
> (1) the mean years are respectively less than, within and more than
> the range
> (2) the middle cycle is the sum of the other two cycles
> (3) Years1*LongYrs3 - Years3*LongYrs1 = 1

> In this case, I guessed the 327-year cycle by subtracting the 524-year
> cycle from the 851-year cycle. I then got the 197-year cycle by
> subtracting the 327-year cycle from the 524-year cycle.
>
> LongYrs would not of course be directly used in any Bahai leap month
> variant cycle that is a whole number of weeks, but can be used as
> follows:
> The number of leap months is 7*LongYrs - 5*Years,
> The number of leap weeks is 19*LongYrs - 14*Years and
> The number of leap days is 133*LongYrs - 99*Years.

> The coefficient of LongYrs is the number of months/weeks/days in 133
> days and the coefficient of Years is the number of months/weeks/days
> in a common year in excess of two periods of 133 days.

> So we have
>
> Years LeapMonths LeapWeeks LeapDays  YrLength
> --------------------------------------------
> 327      73        58         79     365.24159
> 524     117        93        127     365.24237
> 197      44        35         48     365.24365

Bromberg says:

Too bad I didn't read the above few lines before struggling for 30
minutes trying to figure out why you gave LongYrs=147 for the 197-year
cycle above, ultimately figuring out that the number should 44, and
then later learning what you meant.  Memo to all departments:  Always
finish reading Karl's messages first!

Interesting observations (above and below) ...

> All more accurate Bahai leap month cycles that are also a whole number
> of weeks can be constructed by adding whole numbers these cycles
> together.
>
> The 327-year, 524-year and 197-year cycles are composed out of periods
> of alternating 4&5-year intervals as follows:
> (327,73)=(112,25)+(112,25)+(103,23)
> (524,117)=(112,25)+(103,23)+(103,23)+(103,23)+(103,23)
> (197,44)=(103,23)+(94,21)
>
> I've also noticed that in each of these periods of alternating
> 4&5-year intervals the number of equivalent leap days is always two
> more than the number of leap months.
>
> Years LeapMonths LeapDays  YrLength
> -----------------------------------
> 112      25        27      365.2410
> 103      23        25      365.2427
>  94      21        23      365.2447
> -----------------------------------
>
> Also if a cycle has a whole number of 19-day months and weeks, then
> the number of leap months less the number of leap weeks is always
> divisible by three.

Bromberg says:

OK, I've to highlight that observation, I've rearranged the columns in
the spreadsheet, added the number of leap weeks, and added a column for
Leap Months - Leap Weeks.  The difference Leap Days - Leap Months is
always divisible by 2 for all cycles, and it is only 2 more than the
number of leap months for the short mixing cycles, so I didn't add a
column for that.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Irv Bromberg]

On Apr 12, 2006, at 11:13, Palmen, KEV (Karl) wrote:

> Concerning cycles that are also a whole number of weeks:
> In my previous note I discovered that all accurate cycles that also
> have a whole number of weeks can be made up from the following:
>
> Years LeapMonths LeapWeeks LeapDays  YrLength
> 327      73        58         79     365.24159
> 524     117        93        127     365.24237
> 197      44        35         48     365.24365
>
> I see that Irv has already discovered the 327-year cycle, but only
> because its mean year is close to the Northern Solstice year.
>
> We can make more cycles out of these
>
> Mix   Years LeapMonths LeapWeeks LeapDays  YrLength
> 1:0   327      73        58         79     365.24159
> 2:1   851     190       151        206     365.24207
> 3:2  1375     307       244        333     365.24218
> 4:3  1899     424       337        460     365.24223
> 1:1   524     117        93        127     365.24237
> 3:4  1769     395       314        429     365.24251
> 2:3  1245     278       221        302     365.24257
> 1:2   721     161       128        175     365.24272
> 0:1   197      44        35         48     365.24365

It's nice to see the easier way of deriving those cycles as Karl has
shown.  All of the above were already in my spreadsheet except for the
721-year cycle, which equals 7 x 103 years, having the same mean year,
but converting the 103-year cycle into a whole number of weeks.  The
721-year cycle made my slow leap year interval grouping algorithm go
into an endless loop!  (Fixed by having it also signal it is finished
when there is no minor type.)  I updated the spreadsheet, now 169 KB,
which as before is at:

<http://individual.utoronto.ca/kalendis/Bahai_Leap_Month_cycles.xls>


-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 12 April 2006 16:25
To: CALNDR-L@...
Subject: Re: Bahai leap month variant

IRV SAID (first quoting me):

> All more accurate Bahai leap month cycles that are also a whole number
> of weeks can be constructed by adding whole numbers these cycles
> together.
>
> The 327-year, 524-year and 197-year cycles are composed out of periods
> of alternating 4&5-year intervals as follows:
> (327,73)=(112,25)+(112,25)+(103,23)
> (524,117)=(112,25)+(103,23)+(103,23)+(103,23)+(103,23)
> (197,44)=(103,23)+(94,21)
>
> I've also noticed that in each of these periods of alternating
> 4&5-year intervals the number of equivalent leap days is always two
> more than the number of leap months.
>
> Years LeapMonths LeapDays  YrLength
> -----------------------------------
> 112      25        27      365.2410
> 103      23        25      365.2427
>  94      21        23      365.2447
> -----------------------------------
>
> Also if a cycle has a whole number of 19-day months and weeks, then
> the number of leap months less the number of leap weeks is always
> divisible by three.

Bromberg says:

OK, I've to highlight that observation, I've rearranged the columns in
the spreadsheet, added the number of leap weeks, and added a column for
Leap Months - Leap Weeks.  The difference Leap Days - Leap Months is
always divisible by 2 for all cycles, and it is only 2 more than the
number of leap months for the short mixing cycles, so I didn't add a
column for that.

KARL SAYS:
I now reckon that LeapMonths - LeapWeeks = 3/7*(FiveYearIntervals).

This realisation arose soon after I realised that a 4-year interval not only has a whole number of weeks ( 19*(4*19+1)/7 = 209 = 4*52+1 ), but has the same number (1) of leap weeks as leap months. Therefore 4-year intervals have no effect when reckoning the difference between the number of leap months and leap weeks. Only the intervals of five years count.

Now consider seven 5-year intervals, which make up 35 years with 7 leap months. This cycle has  19*(5*19+1) = 1824 = 35*52+4 weeks, so has 4 leap weeks. Therefore there are 3 more leap months than leap weeks.

This leads to the conclusion that there are 3 more leap months than leap weeks for every seven intervals of five years in the cycle.

This I verify for the three cycles:

Years LeapMonths LeapWeeks Diffr   Fives
--------------------------------------------------------
 327      73        58     15=3*5  12+12+11=35=7*5
 524     117        93     24=3*8  12+11+11+11+11=56=7*8
 197      44        35     9=3*3   11+10=21=7*3
--------------------------------------------------------

Karl

08(02(14 till noon

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

I've realised that my method of creating Bahai Leap Month cycles that also have a whole number of weeks sometimes generates an overlong cycle.
Irv has modified his grouping algorithms to cope with such a overlong cycle.

All such overlong cycles are 7 times overlong, in order to have a whole number of weeks. Such a cycle is overlong w.r.t.. which years have a leap month, but not necessarily overlong with respect to individual leap months, if the leap months occur at differing times of the year.


Suppose you make all the leap months to begin on the same day of week. This can be done by having each leap month either 4 years 7 months (84 months) or 4 years 14 months (91 months) after the previous leap month.
Let's call these leap months that are 4 years 14 months after the previous leap month (rather than 4 years 7 months) Late Leap Months.
How many Late Leap Months are there in a cycle?

I make it (19*Years - 83*LeapMonths)/7.

For the 721-year cycle of 161 leap months, we get 19*103-83*23=48. So 48 of the 161 leap months are late leap months. Note that this number (48) is not divisible by 7, so this particular cycle is not overlong if individual leap months (not just leap years) are taken into account.


For the 327-year, 524-year and 197-year cycles we get

Years LeapMonths  LateLeapMonths
---------------------------------
327      73         22
524     117         35
197      44         13

From these one can work them out for the other cycles:

 Mix  Years LeapMonths  LateLeapMonths
---------------------------------------
 1:0   327      73       22
 2:1   851     190       57    
 3:2  1375     307       92    
 4:3  1899     424      127
 1:1   524     117       35
 3:4  1769     395      118    
 2:3  1245     278       83
 1:2   721     161       48
 0:1   197      44       13


Happy Easter Full Moon!

Karl

08(02(15

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 12 April 2006 18:38
To: CALNDR-L@...
Subject: Re: Bahai leap month variant


On Apr 12, 2006, at 11:13, Palmen, KEV (Karl) wrote:

> Concerning cycles that are also a whole number of weeks:
> In my previous note I discovered that all accurate cycles that also
> have a whole number of weeks can be made up from the following:
>
> Years LeapMonths LeapWeeks LeapDays  YrLength
> 327      73        58         79     365.24159
> 524     117        93        127     365.24237
> 197      44        35         48     365.24365
>
> I see that Irv has already discovered the 327-year cycle, but only
> because its mean year is close to the Northern Solstice year.
>
> We can make more cycles out of these
>
> Mix   Years LeapMonths LeapWeeks LeapDays  YrLength
> 1:0   327      73        58         79     365.24159
> 2:1   851     190       151        206     365.24207
> 3:2  1375     307       244        333     365.24218
> 4:3  1899     424       337        460     365.24223
> 1:1   524     117        93        127     365.24237
> 3:4  1769     395       314        429     365.24251
> 2:3  1245     278       221        302     365.24257
> 1:2   721     161       128        175     365.24272
> 0:1   197      44        35         48     365.24365

It's nice to see the easier way of deriving those cycles as Karl has
shown.  All of the above were already in my spreadsheet except for the
721-year cycle, which equals 7 x 103 years, having the same mean year,
but converting the 103-year cycle into a whole number of weeks.  The
721-year cycle made my slow leap year interval grouping algorithm go
into an endless loop!  (Fixed by having it also signal it is finished
when there is no minor type.)  I updated the spreadsheet, now 169 KB,
which as before is at:

<http://individual.utoronto.ca/kalendis/Bahai_Leap_Month_cycles.xls>


-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Palmen, KEV (Karl)]

Dear Irv, Victor and Calendar People

Thinking about calendars where every day belongs to one month of 19 days (based on Bahai Calendar) has lead me
to think about calendars where every day belongs to one month of 30 days (based on Egyptian Calendar).

In
http://www.the-light.com/cal/kp_NdaySolarCyc.html
I showed the 30-day cycle as an example on how to all accurate cycles less than a 1000 years from those shown in the table:

 N  Days  Years LongYrs  YrLength    Mix
-----------------------------------------
30 135870   372      65  365.24194  (1,0)
30 173490   475      83  365.24211  (1,1)
30  37620   103      18  365.24272  (0,1)
-----------------------------------------
30 309360   847     148  365.24203  (2,1)
30 173490   475      83  365.24211  (1,1)
30 211110   578     101  365.24221  (1,2)
30 248730   681     119  365.24229  (1,3)
30 286350   784     137  365.24235  (1,4)
30 323970   887     155  365.24239  (1,5)
30 361590   990     173  365.24242  (1,6)
-----------------------------------------

Of particular interest are those cycles with a whole number of weeks, because these cycles also have a whole number of Balinese 210-day cycles.

Victor's
http://www.the-light.com/cal/VECyc.txt
shows that there is no such cycle less than 1000 years whose mean year is within the range 365.2420 and 365.2427 days. It seems to be exceedingly bad luck that none of the seven cycles listed above and also within that range has a whole number of weeks!

I show you the remainders of the number of days divided by seven for all these cycles:

 N  Days  Years LongYrs  YrLength    Sum  WeekRemainder
-------------------------------------------------------
30 135870   372      65  365.24194  (1,0)    0
30 173490   475      83  365.24211  (1,1)    2
30  37620   103      18  365.24272  (0,1)    2
-------------------------------------------------------
30 309360   847     148  365.24203  (2,1)    2
30 173490   475      83  365.24211  (1,1)    2
30 211110   578     101  365.24221  (1,2)    4
30 248730   681     119  365.24229  (1,3)    6
30 286350   784     137  365.24235  (1,4)    1
30 323970   887     155  365.24239  (1,5)    3
30 361590   990     173  365.24242  (1,6)    5
-------------------------------------------------------

The bad luck arises the fact that ONE of the two cycles to mix (the 372-year cycle) does have a whole number of weeks. This prevents any short mixes within range from having a whole number of weeks.

So it's either the 372-year Gregoriana or a cycle with more than 1000 years.
From the above table it can be seen that adding another 103-year cycle to the 990-year cycle will generate such a cycle. So we have:

 N  Days  Years LongYrs  YrLength    Sum  
-----------------------------------------
30 135870   372      65  365.24194  (1,0)
30 399210  1093     191  365.24245  (1,7)
-----------------------------------------

Furthermore the number of days in the resulting 1093-year cycle has 210 which it is divisible by as its last three digits.
 
Also just out of the range is the 721-year cycle with 126 leap years formed from seven 103-year cycles. This gives us

  Days  Years  YrLength  
------------------------
 135870   372  365.24194  
 399210  1093  365.24245  
 263340   721  365.24272
------------------------

for the 210-day cycle.

This 721-year cycle is also a whole number of 19-day cycles. It's also good for cycles of 7,9,10,11,12 and 28 days. Twice the 721-year cycle (1442 years) is good form all cycles up to 12 days and many more.


Finally, let's compare the number of leap months with the number of leap weeks:

 Days  Years LMths LWks Diffr YrLength  
---------------------------------------
135870   372   65   66    1   365.24194  
399210  1093  191  194    3   365.24245
263340   721  126  128    2   365.24272
---------------------------------------
See how small the difference is!


Karl

08(02(15

[Lance Latham]

RE:

> > Interesting, although I am having trouble getting
> my brane to come to
> > grips with a calendar that has a 1-day week!

Lance replies:
The simplest way to imagine it is to conceive it as a
sort of complement to the Harmonic Week, which is just
the sum of the terms (1/N), for day lengths N, N = 1
to infinity...

-Lance



Lance Latham
scdtl@...
Phone:    [phone]
Address: 78 Hudson Avenue/1st Floor, Green Island, NY 12183
 




__________________________________________________
Do You Yahoo!?
Tired of spam?  Yahoo! Mail has the best spam protection around
http://mail.yahoo.com 

[Irv Bromberg]

On Apr 13, 2006, at 10:46, Palmen, KEV (Karl) wrote:
> Thinking about calendars where every day belongs to one month of 19
> days (based on Bahai Calendar) has lead me
> to think about calendars where every day belongs to one month of 30
> days (based on Egyptian Calendar).

Irv says:

Karl's work on the uniformly 30-day leap month calendar is also an
intriguing idea!
Regular year = 360 days, Leap year = 390 days.  How elegant!

His other comments about the relationships between leap weeks and Bahai
leap months are also interesting.
One could propose a Bahai leap week calendar that has 3 "Days of God"
between the 18th and 19th month in a regular year (=361+3=364 days=52
weeks), or 10 such days in a leap year (=361+10=371 days=53 weeks).  
That would be a perpetual calendar, starting each year on the same
weekday -- presumably the Bahai would prefer Saturday because that is
their start-of-week weekday.  I personally prefer the more uniform
always 19 days per month in the Leap Month Variant, but I'll bet that
the Bahai would rather have such a leap week variant, because the "Days
of God" disappear in non-leap years of the Leap Month Variant!  (or
disappear altogether if month zero is simply called "Leap Month"
instead, as I have provisionally implemented in Kalendis)

Meanwhile, I've been working on implementing the Bahai Leap Month
Variant in Kalendis, it is progressing nicely and looking good,
currently all of the basic functions are working, I'm just testing it
as much as possible and polishing off features and reports.  En passant
I'm also implementing some new features such as letting the user switch
the weekday/month/year names between Persian and English (previously
Persian only).  Until now I hadn't needed a leap year list export for
the Bahai Calendar (although it would have been nice for the
astronomical "future" Bahai calendar), but now that is mandatory.  
Fortunately I can reuse code that is already written for my Hebrew leap
year list.  When time permits, I will post a draft web page documenting
it, then others may have comments to refine it.  However, I was offline
for several recent days, and will be for several more this week.

This work, which started out as an academic exercise, has had a
spin-off benefit in leading to some arithmetic simplifications of the
Rectified Hebrew Calendar, and possibly also the Symmetry454 Calendar.  
This may eliminate the dreaded "New Year Offset" that Karl objected to
in the Sym454 arithmetic, and its equivalent "Delta" factor in the
Rectified Hebrew arithmetic, replaced in both cases by a calendar epoch
adjustment similar to that used for the Dee and Revised Julian
calendars, and which I have already used in for the Bahai Leap Month
Variant.  The public version of Kalendis currently doesn't display
Rectified or Astronomical Hebrew dates, but the work on the Bahai Leap
Month Variant has pointed the way to implementing those options.

The problem with my "New Year Offset" or "Delta" factor approach is
that when the equinox or solstice alignment is fine tuned the list of
leap years change.  Using an epoch adjustment instead avoids that
problem, and the leap rule simplifies to:  it is a leap year only if
modulus(LeapYearsPerCycle*(Year+1), YearsPerCycle)<LeapYearsPerCycle.  
On the other hand, the "New Year Offset" or "Delta" factor each allow
the equinox or solstice alignment to be incremented in finer steps that
are equal to the length of the leap unit (leap day or leap week or leap
month of whatever length) divided by the number of years in the cycle
(accomplished by shifting the leap year intervals sequence earlier or
later, without changing the calendar epoch), whereas the epoch
adjustment can only shift the alignment in units of a whole day.  Such
precise fine tuning, however, does not seem worthwhile.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Irv Bromberg]

Dear Calendar People:

I wasn't planning to do this yet, but then I sort of started and then
didn't stop until it was done:

I implemented the Bahai leap WEEK variant in my home version of
"Kalendis", along the lines of my previous message in this thread.  It
was fairly straightforward, and simply synchronizes with whichever
Symmetry454 leap week cycle that the user has selected (this was the
same way that I previously implemented "leap week at end of year"
variants of The World Calendar and the 13-Month Calendar, which are in
the public version of Kalendis).  The only slightly tricky part was
that the Bahai leap week, inserted between the 18th and 19th months,
comes AFTER the Sym454 leap week, in fact it comes at a time that
corresponds to the NEXT Sym454 year, so the leap status of the Bahai
leap week variant year is given by the leap status of the PRIOR Sym454
year.

So far I did nothing special to make the Bahai leap week variant
calendar start on Saturday.  The Bahai Naw Ruz (New Year Day) is taken
as the Symmetry New Year Day + 79 days.  Therefore the Bahai leap week
variant will always start modulus(79,7)=2 weekdays after the starting
weekday of Sym454 months.  Thus if the user chooses "Start On:" =
Thursday in the Sym454 window then the Bahai leap week variant will
always start its calendar year on Saturday (called Independence, or
Sovereignty by the Bahai, in English translation from the original
Persian "Jalál").

After further testing and report enhancements as previously outlined, I
will include this variant in the web page documentation and a new
public release of Kalendis, when time permits.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 17 April 2006 19:47
To: CALNDR-L@...
Subject: Re: Egyptian leap month variant

IRV SAID:
The problem with my "New Year Offset" or "Delta" factor approach is
that when the equinox or solstice alignment is fine tuned the list of
leap years change.  Using an epoch adjustment instead avoids that
problem, and the leap rule simplifies to:  it is a leap year only if
modulus(LeapYearsPerCycle*(Year+1), YearsPerCycle)<LeapYearsPerCycle.  
On the other hand, the "New Year Offset" or "Delta" factor each allow
the equinox or solstice alignment to be incremented in finer steps that
are equal to the length of the leap unit (leap day or leap week or leap
month of whatever length) divided by the number of years in the cycle
(accomplished by shifting the leap year intervals sequence earlier or
later, without changing the calendar epoch), whereas the epoch
adjustment can only shift the alignment in units of a whole day.  Such
precise fine tuning, however, does not seem worthwhile.

KARL SAYS:
So if we have a leap year rule that states that a year is a leap year if and only if

modulus(LeapYearsPerCycle*Year + Koffset, YearsPerCycle) < LeapYearsPerCycle

one can eliminate Koffset by renumbering the years so that year 0 is a year whose accumulator (LHS of above expression) is zero.

Irv is wrong when he states that an epoch adjustment can only shift the alignment in units of a whole day. Except for overlong cycles, there will be a year for which the accumulator (with Koffset=0) is 1 and a year for which the accumulator is YearsPerCycle-1. Renumbering the years to change the epoch (year 0) to either of these years would make the minimum adjustment, which is equivalent to changing Koffset by 1.

Perhaps Irv may be referring to the situation where after many years of running the calendar, one needs to adjust the leap year rule to keep it accurate. In such a situation, one may not be at liberty to renumber the years. Then one may need to introduce Koffset to make the adjustment. If the number of leap units in a year is large (e.g. 365 for a leap day calendar), one could still get away with not using Koffset in such an adjustment.

Karl

08(02(20

[Irv Bromberg]

On Apr 18, 2006, at 07:40, Palmen, KEV (Karl) wrote:

> So if we have a leap year rule that states that a year is a leap year
> if and only if
>
> modulus(LeapYearsPerCycle*Year + Koffset, YearsPerCycle) <
> LeapYearsPerCycle
>
> one can eliminate Koffset by renumbering the years so that year 0 is a
> year whose accumulator (LHS of above expression) is zero.
>
> Irv is wrong when he states that an epoch adjustment can only shift
> the alignment in units of a whole day. Except for overlong cycles,
> there will be a year for which the accumulator (with Koffset=0) is 1
> and a year for which the accumulator is YearsPerCycle-1. Renumbering
> the years to change the epoch (year 0) to either of these years would
> make the minimum adjustment, which is equivalent to changing Koffset
> by 1.

Irv replies:

I tried a fractional epoch offset and found that it generated an odd
equinox distribution pattern when I evaluated the long-term alignment
of the 117/524 Bahai leap month variant at the meridian of Haifa.  I
did not investigate the matter, but that was why I said the increment
has to be whole days.  It could be that I'm flooring a value too early
in the calendar arithmetic, or maybe the pattern only "looks" odd, but
really isn't odd!

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 18 April 2006 14:25
To: CALNDR-L@...
Subject: Re: Epoch & Offset RE: Egyptian leap month variant


On Apr 18, 2006, at 07:40, Palmen, KEV (Karl) wrote:

> So if we have a leap year rule that states that a year is a leap year
> if and only if
>
> modulus(LeapYearsPerCycle*Year + Koffset, YearsPerCycle) <
> LeapYearsPerCycle
>
> one can eliminate Koffset by renumbering the years so that year 0 is a
> year whose accumulator (LHS of above expression) is zero.
>
> Irv is wrong when he states that an epoch adjustment can only shift
> the alignment in units of a whole day. Except for overlong cycles,
> there will be a year for which the accumulator (with Koffset=0) is 1
> and a year for which the accumulator is YearsPerCycle-1. Renumbering
> the years to change the epoch (year 0) to either of these years would
> make the minimum adjustment, which is equivalent to changing Koffset
> by 1.

Irv replies:

I tried a fractional epoch offset and found that it generated an odd
equinox distribution pattern when I evaluated the long-term alignment
of the 117/524 Bahai leap month variant at the meridian of Haifa.  I
did not investigate the matter, but that was why I said the increment
has to be whole days.  It could be that I'm flooring a value too early
in the calendar arithmetic, or maybe the pattern only "looks" odd, but
really isn't odd!

KARL RESPONDS:
What I suggested is not a fractional epoch offset.
Try shifting the epoch in the 117/524 Bahai leap month variant by 103 years (which is the majority type of the last row of the groupings) and exactly one leap year in the cycle should change (by one year). If there is either no change or a big change in the leap years, you may indeed be flooring in the wrong place.

Karl

08(02(20

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 17 April 2006 19:47
To: CALNDR-L@...
Subject: Re: Egyptian leap month variant


On Apr 13, 2006, at 10:46, Palmen, KEV (Karl) wrote:
> Thinking about calendars where every day belongs to one month of 19
> days (based on Bahai Calendar) has lead me
> to think about calendars where every day belongs to one month of 30
> days (based on Egyptian Calendar).

Irv says:

Karl's work on the uniformly 30-day leap month calendar is also an
intriguing idea!
Regular year = 360 days, Leap year = 390 days.  How elegant!

KARL SAYS:
I also mentioned that the number of leap months required is close to the number of leap weeks required in a leap week calendar. It takes over 360 years for the count of leap weeks to exceed the count of leap months by an average of one.

 Days  Years LMths LWks Diffr YrLength  Years/Diffr
---------------------------------------------------
135870   372   65   66    1   365.24194   372
399210  1093  191  194    3   365.24245   364.3333
263340   721  126  128    2   365.24272   360.5
---------------------------------------------------

Nevertheless, if you were foolish enough to apply a good leap week rule as a leap month rule, the calendar would be 30 days out in 360 to 372 years and so 1 day out in about 12 years.

If you add the 372-year cycle to the 1093-year cycle, you get the 1465-year cycle, which is equivalent to five 293-year cycles.

 Days  Years LMths LWks Diffr YrLength  Years/Diffr
---------------------------------------------------
135870   372   65   66    1   365.24194   372
535080  1465  256  260    4   365.24232   366.25
399210  1093  191  194    3   365.24245   364.3333
---------------------------------------------------

Also of curiosity, is the 4000-year cycle equivalent to ten Gregorian 400-year cycles. It has 48699 months hence 699 leap months. It can be made from
three 1093-year cycles and one 721-year cycle

  Days  Years LMths LWks Diffr YrLength  Years/Diffr
----------------------------------------------------
 399210  1093  191  194    3   365.24245   364.3333
1460970  4000  699  710   11   365.2425    363.6363
 263340   721  126  128    2   365.24272   360.5
----------------------------------------------------

The Gregorian calendar drops 30 leap days every 4000 years. Therefore adding one month (to give 700 leap months) would make the cycle equivalent to the Julian calendar cycle, which can be made from a 40-year cycle with 7 leap months.


Looking at all accurate cycles with 30-day months and not just those that are a whole number of weeks, I examine how the intervals between leap years would occur if spaced as evenly as possible with Koffset = 0, showing how these intervals group.

For the 103-year cycle of 18 leap months, the groupings go thus:
6 6 6 5  6 6 6 5  6 6 5  6 6 6 5  6 6 5
23 23 17 23 17
63 40

For the others that are

 N  Days  Years LongYrs  YrLength    Mix
-----------------------------------------
30 309360   847     148  365.24203  (2,1)
30 173490   475      83  365.24211  (1,1)
30 211110   578     101  365.24221  (1,2)
30 248730   681     119  365.24229  (1,3)
30 286350   784     137  365.24235  (1,4)
30 323970   887     155  365.24239  (1,5)
30 361590   990     173  365.24242  (1,6)
30 399210  1093     191  365.24245  (1,7)
-----------------------------------------

the later rows go
372: 166 103 103
475: 166 103 103 103
578: 166 103 103 103 103
681: 166 103 103 103 103 103
887: 166 103 103 103 103 103 103
990: 166 103 103 103 103 103 103 103
1093: 166 103 103 103 103 103 103 103 103

847:
166 103 103 166 103 103 103
372 475

where 166 is grouped from 63 63 40, which in turn are grouped as in the 103-year cycle (63 40) as already shown.

The group of 40 has 7 leap years and the same mean year as the Julian calendar
(40*5+10=210=7*30)

6 6 6 5  6 6 5
23 17
40

The group of 23 (6 6 6 5) has 8400 days and so is a whole number of weeks. If you were 3 years old by your first leap day, you'd be 8400 days old on your 23rd birthday, which would also be on the same day of the week as your birth.

Karl

08(02(20

[VictorEngel]

Dear Brij,

You corrected the wrong item. Instead of "After 1505 normal years" I should
have typed "After 1506 normal years". With this change my calculations are
correct.

Victor

> Please see:
> http://www.brijvij.com/bb_sidrl-to-civil.doc
> based on my work done during 1992-93; and comments related to
> Victor's
> insertion.
> Regards,
> Brij Bhushan Vij

[VictorEngel]

Dear Brij,

> ...  to make the world a place that ancients worshipped.

Did the ancients worship their future?

Victor

[Irv Bromberg]

On Apr 18, 2006, at 10:07, Palmen, KEV (Karl) wrote:

> On Apr 18, 2006, at 07:40, Palmen, KEV (Karl) wrote:
>> So if we have a leap year rule that states that a year is a leap year
>> if and only if
>>
>> modulus(LeapYearsPerCycle*Year + Koffset, YearsPerCycle) <
>> LeapYearsPerCycle
>>
>> one can eliminate Koffset by renumbering the years so that year 0 is a
>> year whose accumulator (LHS of above expression) is zero.
>>
>> Irv is wrong when he states that an epoch adjustment can only shift
>> the alignment in units of a whole day. Except for overlong cycles,
>> there will be a year for which the accumulator (with Koffset=0) is 1
>> and a year for which the accumulator is YearsPerCycle-1. Renumbering
>> the years to change the epoch (year 0) to either of these years would
>> make the minimum adjustment, which is equivalent to changing Koffset
>> by 1.
>
> Irv replies:
>
> I tried a fractional epoch offset and found that it generated an odd
> equinox distribution pattern when I evaluated the long-term alignment
> of the 117/524 Bahai leap month variant at the meridian of Haifa.  I
> did not investigate the matter, but that was why I said the increment
> has to be whole days.  It could be that I'm flooring a value too early
> in the calendar arithmetic, or maybe the pattern only "looks" odd, but
> really isn't odd!
>
> KARL RESPONDS:
> What I suggested is not a fractional epoch offset.
> Try shifting the epoch in the 117/524 Bahai leap month variant by 103
> years (which is the majority type of the last row of the groupings)
> and exactly one leap year in the cycle should change (by one year). If
> there is either no change or a big change in the leap years, you may
> indeed be flooring in the wrong place.

IRV REPLIES:

Indeed, I had not considered a year count offset.  That accomplishes
the same thing, but most cultures would object to renumbering the
years.  The calendar arithmetic could employ a renumbered year
internally, but apply a shift for otuput/display purposes.

The day-wise epoch adjustment for equinox alignment (as was done for
the Dee Calendar and the Revised Julian Calendar) won't work with the
Rectified Hebrew Calendar because years are constrained to start in a
fixed relationship to the molad or rectified molad.

The day-wise epoch adjustment also doesn't work for the Symmetry454
Calendar or essentially ANY leap week calendar because at the epoch the
first calendar year was constrained to start on the calendar's starting
weekday.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

More about Epoch and Offset adjustments

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 21 April 2006 03:01
To: CALNDR-L@...
Subject: Re: Epoch & Offset RE: Egyptian leap month variant


On Apr 18, 2006, at 10:07, Palmen, KEV (Karl) wrote:

> On Apr 18, 2006, at 07:40, Palmen, KEV (Karl) wrote:
>> So if we have a leap year rule that states that a year is a leap year
>> if and only if
>>
>> modulus(LeapYearsPerCycle*Year + Koffset, YearsPerCycle) <
>> LeapYearsPerCycle
>>
>> one can eliminate Koffset by renumbering the years so that year 0 is a
>> year whose accumulator (LHS of above expression) is zero.
>>
>> Irv is wrong when he states that an epoch adjustment can only shift
>> the alignment in units of a whole day. Except for overlong cycles,
>> there will be a year for which the accumulator (with Koffset=0) is 1
>> and a year for which the accumulator is YearsPerCycle-1. Renumbering
>> the years to change the epoch (year 0) to either of these years would
>> make the minimum adjustment, which is equivalent to changing Koffset
>> by 1.
>
> Irv replies:
>
> I tried a fractional epoch offset and found that it generated an odd
> equinox distribution pattern when I evaluated the long-term alignment
> of the 117/524 Bahai leap month variant at the meridian of Haifa.  I
> did not investigate the matter, but that was why I said the increment
> has to be whole days.  It could be that I'm flooring a value too early
> in the calendar arithmetic, or maybe the pattern only "looks" odd, but
> really isn't odd!
>
> KARL RESPONDS:
> What I suggested is not a fractional epoch offset.
> Try shifting the epoch in the 117/524 Bahai leap month variant by 103
> years (which is the majority type of the last row of the groupings)
> and exactly one leap year in the cycle should change (by one year). If
> there is either no change or a big change in the leap years, you may
> indeed be flooring in the wrong place.

IRV REPLIES:

Indeed, I had not considered a year count offset.  That accomplishes
the same thing, but most cultures would object to renumbering the
years.  The calendar arithmetic could employ a renumbered year
internally, but apply a shift for otuput/display purposes.

KARL SAYS:
The renumbering of the years (whether internal or not) is equivalent to changing the Koffset value.

If you renumber the years by adding an integer called Shift to them and leaving Koffset absent or unchanged. The effect is equivalent to adding LeapYearsPerCycle*Shift to Koffset.

If you add an integer Kshift to Koffset the effect is equivalent to leaving Koffset unchanged or absent and adding MagicNumber*Kshift to the year number, where MagicNumber is the first number in the last row of the slow grouping (which must have two numbers in it).


IRV CONTINUED:
The day-wise epoch adjustment for equinox alignment (as was done for
the Dee Calendar and the Revised Julian Calendar) won't work with the
Rectified Hebrew Calendar because years are constrained to start in a
fixed relationship to the molad or rectified molad.

The day-wise epoch adjustment also doesn't work for the Symmetry454
Calendar or essentially ANY leap week calendar because at the epoch the
first calendar year was constrained to start on the calendar's starting
weekday.

KARL SAYS:
I don't really understand what Irv means here about day-wise adjustment in this context.

Karl

08(02(23

[Irv Bromberg]

On Apr 21, 2006, at 07:41, Palmen, KEV (Karl) wrote:

> IRV CONTINUED:
> The day-wise epoch adjustment for equinox alignment (as was done for
> the Dee Calendar and the Revised Julian Calendar) won't work with the
> Rectified Hebrew Calendar because years are constrained to start in a
> fixed relationship to the molad or rectified molad.
>
> The day-wise epoch adjustment also doesn't work for the Symmetry454
> Calendar or essentially ANY leap week calendar because at the epoch the
> first calendar year was constrained to start on the calendar's starting
> weekday.
>
> KARL SAYS:
> I don't really understand what Irv means here about day-wise
> adjustment in this context.

IRV EXPLAINS:

The Dee Calendar has an epoch that is one day earlier than the
Gregorian Calendar.
The Revised Julian Calendar has an epoch that is two days later than
the Julian Calendar, and thus equals the Gregorian Calendar.
These calendar reforms adjusted equinox alignment by applying an epoch
adjustment of an integer number of days.

By contrast, a leap week calendar can't make epoch adjustments of
single day units, because then the calendar will not start at the
beginning of its week at the epoch.  Adjustments of 7-day intervals are
possible, but far too coarse to be of use for "fine tuning" the equinox
alignment.  Thus I concluded that the fine tuning can only be done by
shifting the epoch year or adding what I call a "New Year Offset" to
the KOffset, which has the disadvantage that the leap year list changes
every time that the fine tuning is fiddled with.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Palmen, KEV (Karl)]

Dear Irv and Calendar People

Thank you Irv for your explanation.

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...]On Behalf Of Irv Bromberg
Sent: 21 April 2006 14:21
To: CALNDR-L@...
Subject: Re: Epoch & Offset RE: Egyptian leap month variant


On Apr 21, 2006, at 07:41, Palmen, KEV (Karl) wrote:

> IRV CONTINUED:
> The day-wise epoch adjustment for equinox alignment (as was done for
> the Dee Calendar and the Revised Julian Calendar) won't work with the
> Rectified Hebrew Calendar because years are constrained to start in a
> fixed relationship to the molad or rectified molad.
>
> The day-wise epoch adjustment also doesn't work for the Symmetry454
> Calendar or essentially ANY leap week calendar because at the epoch the
> first calendar year was constrained to start on the calendar's starting
> weekday.
>
> KARL SAYS:
> I don't really understand what Irv means here about day-wise
> adjustment in this context.

IRV EXPLAINS:

The Dee Calendar has an epoch that is one day earlier than the
Gregorian Calendar.
The Revised Julian Calendar has an epoch that is two days later than
the Julian Calendar, and thus equals the Gregorian Calendar.
These calendar reforms adjusted equinox alignment by applying an epoch
adjustment of an integer number of days.

By contrast, a leap week calendar can't make epoch adjustments of
single day units, because then the calendar will not start at the
beginning of its week at the epoch.  Adjustments of 7-day intervals are
possible, but far too coarse to be of use for "fine tuning" the equinox
alignment.  

KARL SAYS: Thank you, I see.

Thus I concluded that the fine tuning can only be done by
shifting the epoch year or adding what I call a "New Year Offset" to
the KOffset, which has the disadvantage that the leap year list changes
every time that the fine tuning is fiddled with.

KARL SAYS:
This is what I've been suggesting and it does of course alter which years are leap years in a fixed year numbering or naming system.

I still do not understand what Irv means by "New Year Offset" which he may add to KOffset.

I am aware that the modification of Koffset can be combined with a unit-wise adjustment, where unit is whatever (day, week or month) is added in the intercalation. Indeed for fine tuning, this sometimes needs to be done to ensure that the majority of years still start on the same day after the fine tuning.

This would be the case if the fine tuning were to result in a large jump of the value of Koffset modulo YearsPerCycle. In such a case, the leap unit would jump over the start of year 1, so causing the new year's day of year 1 to change by a unit relative to most of the other new years. This change would need to put in as an accompanying unit-wise adjustment to keep the majority of new years on the same day.

Karl

08(02(23