293-year Leap Week Calendar Cycle

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293-year Leap Week Calendar Cycle

Palmen, KEV (Karl)
Dear Irv and Calendar People

Irv has suggested using a 293-year cycle of 52 leap weeks for his Symmetry454 leap week calendar. The 52 leap week years are as evenly spread as possible and so there exists an integer K such that year Y has a leap week if and only if

(52*Y + K) mod 293) < 52

Irv has chosen K=166,  which I think was to align contemporary leap week years with the ISO-week leap week years.

However the sequence of leap week years in this or any other accurate rule that places them as evenly as possible is not simple. The intervals are a complicated sequence of 6 and 5 years.

I've found that constraining the leap weeks to occur on even-numbered years results in a simpler sequence at the price of about 1.25 days extra variation of the calendar year over the seasons and the loss of the 293-year cycle option (586-year cycle instead). Most intervals are 6 years and the other intervals are 4 years and usually alternate between 28 and 34 years after the previous short interval.


One interesting variation of the 293-year cycle, which I have already mentioned, allows the leap week to occur at the end of any quarter year. This is compatible with the structure of the Symmetry454 calendar. One has the leap week occurring alternately once every 23 and 22 quarters starting with a 23-quarter interval until 13 leap weeks have occurred, then repeat for each 13 leap weeks.

The leap week years are as evenly spread as possible, hence, there exists an integer constant K such that a year Y has a leap week if and only if
(52*Y + K) mod 293 < 52
The accumulator (52*Y + K) mod 293 tells you which quarter the leap week occurs.
If the accumulator is less than 13, it's the last quarter,
if the accumulator is at least 13, but less than 26, it's the 3rd quarter,
if the accumulator is at least 26, but less than 39, it's the 2nd quarter and
if the accumulator is at least 39, but less than 52, it's the 1st quarter.

To make year 0 end with a leap week that is 23 quarters from both its neighbours, one sets K=6. Whatever the value of K, the year that ends with a leap week that is 23 quarters from both is neighbours is the year whose accumulator is 6.

I show this with 16 leap weeks for K=6:

Acc Year Quarter  #quarters to next
06    0     4     23
25    6     3     22
44   12     1     23
11   17     4     22
30   23     2     23
49   29     1     22
16   34     3     23
35   40     2     22
02   45     4     23
21   51     3     22
40   57     1     23
07   62     4     22
26   68     2     23
--------------------
45   74     1     23
12   79     4     22
31   85     2    
 
Such a calendar is structurally much simpler but computationally less simple than the same with the leap weeks at the end of the year.

Karl

08(04(27
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Re: 293-year Leap Week Calendar Cycle

Irv Bromberg
On Jun 23, 2006, at 08:23, Palmen, KEV (Karl) wrote:
> Irv has suggested using a 293-year cycle of 52 leap weeks for his
> Symmetry454 leap week calendar. The 52 leap week years are as evenly
> spread as possible and so there exists an integer K such that year Y
> has a leap week if and only if
>
> (52*Y + K) mod 293) < 52
>
> Irv has chosen K=166,  which I think was to align contemporary leap
> week years with the ISO-week leap week years.

Bromberg replies:

No, as stated in the documentation, the K values were chosen to align
the average northward equinox at the middle of the 79th day of the
calendar at the Prime Meridian.  In the present era this happens to
usually coincide with the ISO calendar when Sym454 is set to start the
year on Monday, but there is an occasional year when one inserts the
leap week
in the year before or after the other.  In the future they diverge
because ISO, based on the Gregorian mean year, is about 12 seconds too
long per year, whereas the 52/293 cycle seems to be about 3.5 seconds
too short for the present-era northward equinoctial mean year.

Also, the K=166 is valid only for a calendar year that starts on the
recommended Monday.  For calendars starting on Sunday it would be 208
to produce the same equinox alignment.  The determination of the K
value is documented in the currently posted Sym454 Arithmetic.  The new
version of Kalendis, to be released soon, supports any choice of
starting day for Sym454 or Classical Symmetry (or leap week at end
variants of The World Calendar and the 13-Month Calendar), and gives
the K value for that leap rule and that starting weekday.

> However the sequence of leap week years in this or any other accurate
> rule that places them as evenly as possible is not simple. The
> intervals are a complicated sequence of 6 and 5 years.

Bromberg replies:

The "complexity of the intervals is irrelevant.  The simple expression
(52*Y + K) mod 293) < 52 takes care of it.

> I've found that constraining the leap weeks to occur on even-numbered
> years results in a simpler sequence at the price of about 1.25 days
> extra variation of the calendar year over the seasons and the loss of
> the 293-year cycle option (586-year cycle instead). Most intervals are
> 6 years and the other intervals are 4 years and usually alternate
> between 28 and 34 years after the previous short interval.

Bromberg replies:

The SEQUENCE is "simpler" in terms of the perceived pattern of leap
year intervals.  But the CALENDAR ARITHMETIC is actually MORE COMPLEX
doing it the way Karl proposes.

> One interesting variation of the 293-year cycle, which I have already
> mentioned, allows the leap week to occur at the end of any quarter
> year. This is compatible with the structure of the Symmetry454
> calendar. One has the leap week occurring alternately once every 23
> and 22 quarters starting with a 23-quarter interval until 13 leap
> weeks have occurred, then repeat for each 13 leap weeks.

Bromberg replies:

That variation is an interesting idea, but certainly NOT compatible
with the Sym454 structure, which explicitly by design strives to
permanently fix the ordinal day number and ordinal week number of every
regular date in the year, to keep the calendar arithmetic as simple as
possible and for consistency in business date statistics and
scheduling.  Therefore the only valid place to insert the leap week
within this design is at the END of the year.

Furthermore, there is no compelling BENEFIT from a roving leap week.  
It doesn't reduce the wobble of the target celestial event.  Having the
leap week inserted at an inconsistent position would likely confuse the
public, and big mistakes could be made if someone plans an event or
production schedule or trip without taking into account the leap week
position.  Well, actually, there is a minor benefit:  some people
protest against having the leap week at the end of year.  Can't please
everybody.  But a roving leap week will give everybody a chance to
occasionally (only a few times in a lifetime) enjoy a leap week in any
of 4 possible positions in the calendar year.  I don't consider this a
compelling benefit.

> The leap week years are as evenly spread as possible, hence, there
> exists an integer constant K such that a year Y has a leap week if and
> only if
> (52*Y + K) mod 293 < 52
> The accumulator (52*Y + K) mod 293 tells you which quarter the leap
> week occurs.
> If the accumulator is less than 13, it's the last quarter,
> if the accumulator is at least 13, but less than 26, it's the 3rd
> quarter,
> if the accumulator is at least 26, but less than 39, it's the 2nd
> quarter and
> if the accumulator is at least 39, but less than 52, it's the 1st
> quarter.

Bromberg replies:

I imagine that a simple MOD expression would yield the quarter number,
that would be more convenient in calendar arithmetic, instead of
executing the list of 4 double-comparison IF statements.

Simply determining a year's leap status and where to insert the quarter
is simple enough.  Try on the other hand to work out the entire
calendar arithmetic package and you will see that the roving leap week
makes the arithmetic much more complex.  Required functions:

- isSymLeapYear
- FixedToSymYear
- FixedToSym
- SymToFixed
- SymNewYear (unaffected by the roving leap week position)

By "Fixed", I am referring to the Reingold & Dershowitz rata die, but
any desired fixed day count could be substituted, such as Julian Day
Number, Modified JDN, J2000-relative day number, or a day count
relative to any arbitrary epoch.

Both FixedToSym and SymToFixed would have to be aware of the roving
leap week position.  That could be encapsulated in the
SymDaysBeforeMonth function.  In the existing Sym454 arithmetic the
SymDaysBeforeMonth function is trivial, not needing to know leap status
for the year.  The roving leap week version, however, would need to
know leap status, and if it is a leap year would have to determine the
quarter after which the leap week will be inserted, then, if necessary,
adjust it return value to account for the possible insertion of that
leap week prior to the month asked for.

> To make year 0 end with a leap week that is 23 quarters from both its
> neighbours, one sets K=6. Whatever the value of K, the year that ends
> with a leap week that is 23 quarters from both is neighbours is the
> year whose accumulator is 6.

Bromberg replies:

Setting K to a value other than my recommended 166 will change the
equinox alignment.  As Simon Cassidy has pointed out, the choice of
alignment meridian is arbitrary, and not particularly important.  My
purpose was to define how a meridian can be selected, and how to change
the arithmetic to shift the average equinox or solstice to any desired
meridian, in the case where for whatever reason some future authority
that is adopting such a leap rule prefers to select a specific
meridian.  For the Bahai Calendar variants I have used the meridian of
Haifa, location of their World Temple.

> Such a calendar is structurally much simpler but computationally less
> simple than the same with the leap weeks at the end of the year.

Bromberg replies:

Oh, yes, your bottom line agrees!


-- Irv Bromberg, University of Toronto, Canada

<http://www.sym454.org/>
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Re: 293-year Leap Week Calendar Cycle

Palmen, KEV (Karl)
Dear Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:[hidden email]]On Behalf Of Irv Bromberg
Sent: 23 June 2006 15:07
To: [hidden email]
Subject: Re: 293-year Leap Week Calendar Cycle


On Jun 23, 2006, at 08:23, Palmen, KEV (Karl) wrote:
> Irv has suggested using a 293-year cycle of 52 leap weeks for his
> Symmetry454 leap week calendar. The 52 leap week years are as evenly
> spread as possible and so there exists an integer K such that year Y
> has a leap week if and only if
>
> (52*Y + K) mod 293) < 52
>
> Irv has chosen K=166,  which I think was to align contemporary leap
> week years with the ISO-week leap week years.

Bromberg replies:

No, as stated in the documentation, the K values were chosen to align
the average northward equinox at the middle of the 79th day of the
calendar at the Prime Meridian.  In the present era this happens to
usually coincide with the ISO calendar when Sym454 is set to start the
year on Monday, but there is an occasional year when one inserts the
leap week
in the year before or after the other.  In the future they diverge
because ISO, based on the Gregorian mean year, is about 12 seconds too
long per year, whereas the 52/293 cycle seems to be about 3.5 seconds
too short for the present-era northward equinoctial mean year.

Also, the K=166 is valid only for a calendar year that starts on the
recommended Monday.  For calendars starting on Sunday it would be 208
to produce the same equinox alignment.  The determination of the K
value is documented in the currently posted Sym454 Arithmetic.  The new
version of Kalendis, to be released soon, supports any choice of
starting day for Sym454 or Classical Symmetry (or leap week at end
variants of The World Calendar and the 13-Month Calendar), and gives
the K value for that leap rule and that starting weekday.

KARL SAYS: Thank you Irv, for correcting me over the matter of the choice of K.


IRV CONTINUES (first quoting me):
> However the sequence of leap week years in this or any other accurate
> rule that places them as evenly as possible is not simple. The
> intervals are a complicated sequence of 6 and 5 years.

Bromberg replies:

The "complexity of the intervals is irrelevant.  The simple expression
(52*Y + K) mod 293) < 52 takes care of it.

KARL SAYS:
I disagree strongly with this and Irv's implied assertion that a calendar if simple only if it is simple to compute. There are other types of simplicity such a structural simplicity.

Karl

08(04(27