Joining list. Calendar reforms.

[Irv Bromberg]

On 2009 Jun 30, at 19:49 , MIKE OSSIPOFF wrote:

Karl wrote> It is not clear what Mike means by a symmetrical leap year rule.

Mike replied:  I mean a leapyear rule that tries to keep the calendrical drift minimized, making each year a leap year, or not, whichever will result in the least drift for the following year.
 
By "drift", in this context, I mean the amount by which the solar ecliptic longitude for some particular designated calendar date differs from the solar ecliptic longitude that you want to keep that date close to.  I thought that "symmetrical leapyear system" or "symmetrical" leapyear rule" was the term, on this mailing-list, for such a leapyear system.

Mike, there was no need for you to guess what the term means.  Several times in recent messages I gave you the URL where I have documented the attributes and arithmetic for a symmetrical leap rule, referring to a leap rule that distributes its leap years symmetrically within each cycle.  Here it is yet again:

<http://www.sym454.org/leap/> and then click on the heading "Symmetrical Leap Cycles".

My documentation also refers to smoothly spread symmetrically distributed leap years, although it is possible to implement a symmetrical leap cycle that is not smoothly spread.

One controls calendar drift by appropriate choice of calendar mean year relative to whatever you consider your gold standard.

One minimizes short-term equinox or solstice wobble by spreading the leap years as smoothly as possible.

Symmetrical distribution of leap years makes it easier to choose the epoch for the calendar, and easier to astronomically evaluate long-term drift, because the first year of each cycle will always fall at the average for the cycle, so it is only necessary to evaluate the drift for the first year of each cycle, using interpolation between cycles.  Such a drift analysis will be the same as starting at the epoch then adding the calendar mean year multiplied by the number of years per cycle, compare that moment with the target astronomical moment, and so on for the desired number of cycles.

-- Irv Bromberg, Toronto, Canada

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

[Mark J. Reed]

> On 2009 Jun 30, at 19:49 , MIKE OSSIPOFF wrote:
> By "drift", in this context, I mean the amount by which the solar ecliptic
> longitude for some particular designated calendar date differs from the
> solar ecliptic longitude that you want to keep that date close to.  I
> thought that "symmetrical leapyear system" or "symmetrical" leapyear rule"
> was the term, on this mailing-list, for such a leapyear system.

AIUI, a symmetrical leap year system is just one where the
distribution of leap years within a cycle is symmetrical about the
center of the cycle.

The Gregorian cycle has an odd number of leap years within an even
number of years, so it can't be made perfectly symmetrical, but you
could get close with 48 leap years in one half and 49 in the other.

--
Mark J. Reed <markjreed@...>

[Karl Palmen]

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Dear Calendar People

 

The article that Irv refers to about Symmetrical Calendar Rules is not concise and does not differentiate between the definition and properties.

 

The Symmetrical cycle Irv is referring to is one of an odd-number of years C where each pair of mirror image years (1,C), (2,C-1), … (Y,C-Y+1) has both years either common or leap years. The key property of such a symmetrical cycle is that the first calendar year starts at the same time as the first mean calendar year.

So if you want the average North season to begin on the ecliptic longitude of June 1, make the North season of year 1 begin on the ecliptic longitude of June 1 (and intercalate at the end of the year).

 

Irv also writes about such Symmetrical cycles in which the leap years are spread as smoothly as possible. I’ve called such a cycle a Helios cycle, because Helios has produced a large number of them in his posts. If such a cycle has C years of which L years are leap years, then year Y is a leap year if and only if

 

(L*Y + (C-1)/2) mod ( C ) < L

 

There are other types of symmetrical cycle, (such as the Gregorian Calendar cycle, which is symmetrical about the 200^th and 400^th year), but they do not have the key property that I have given above.

 

Karl

 

10(10(08 till noon

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 01 July 2009 02:16
To: CALNDR-L@...
Subject: Re: Symmetrical Leap Year Rule RE: June-September season

 

On 2009 Jun 30, at 19:49 , MIKE OSSIPOFF wrote:

Karl wrote> It is not clear what Mike means by a symmetrical leap year rule.

 

Mike replied:  I mean a leapyear rule that tries to keep the calendrical drift minimized, making each year a leap year, or not, whichever will result in the least drift for the following year.
 
By "drift", in this context, I mean the amount by which the solar ecliptic longitude for some particular designated calendar date differs from the solar ecliptic longitude that you want to keep that date close to.  I thought that "symmetrical leapyear system" or "symmetrical" leapyear rule" was the term, on this mailing-list, for such a leapyear system.

 

Mike, there was no need for you to guess what the term means.  Several times in recent messages I gave you the URL where I have documented the attributes and arithmetic for a symmetrical leap rule, referring to a leap rule that distributes its leap years symmetrically within each cycle.  Here it is yet again:

 

<http://www.sym454.org/leap/> and then click on the heading "Symmetrical Leap Cycles".

 

My documentation also refers to smoothly spread symmetrically distributed leap years, although it is possible to implement a symmetrical leap cycle that is not smoothly spread.

 

One controls calendar drift by appropriate choice of calendar mean year relative to whatever you consider your gold standard.

 

One minimizes short-term equinox or solstice wobble by spreading the leap years as smoothly as possible.

 

Symmetrical distribution of leap years makes it easier to choose the epoch for the calendar, and easier to astronomically evaluate long-term drift, because the first year of each cycle will always fall at the average for the cycle, so it is only necessary to evaluate the drift for the first year of each cycle, using interpolation between cycles.  Such a drift analysis will be the same as starting at the epoch then adding the calendar mean year multiplied by the number of years per cycle, compare that moment with the target astronomical moment, and so on for the desired number of cycles.

 

 

-- Irv Bromberg, Toronto, Canada

 

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

 

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[Karl Palmen]

Dear Mark and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Mark J. Reed
Sent: 01 July 2009 05:07
To: CALNDR-L@...
Subject: Re: Symmetrical Leap Year Rule RE: June-September season

> On 2009 Jun 30, at 19:49 , MIKE OSSIPOFF wrote:
> By "drift", in this context, I mean the amount by which the solar ecliptic
> longitude for some particular designated calendar date differs from the
> solar ecliptic longitude that you want to keep that date close to.  I
> thought that "symmetrical leapyear system" or "symmetrical" leapyear rule"
> was the term, on this mailing-list, for such a leapyear system.

AIUI, a symmetrical leap year system is just one where the
distribution of leap years within a cycle is symmetrical about the
center of the cycle.

The Gregorian cycle has an odd number of leap years within an even
number of years, so it can't be made perfectly symmetrical, but you
could get close with 48 leap years in one half and 49 in the other.

KARL SAYS: If you remove the 400th year the remaining 399-year cycle is symmetrical, about the centre (200th) year.

Karl

10(10(08 till noon

[MIKE OSSIPOFF]

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 Irv--
 
Thanks again for the URL. I'll check it out. As you can tell, I haven't before looked at the mathematics of the most efficient ways of predicting, in the far future, which years will be leapyears.
 
Minimum Drift is my effort to minimize calendar-drift with respect to a desired position of the calendar with respect to the solar ecliptic longitude that is desired for a certain calendar date.
 
Mike Ossipoff
 

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[MIKE OSSIPOFF]

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Dear Christopher and Calendar People,
 
 
"If you agree with me, then you're speaking for yourself. If you disagree with me, then you must be culturally influenced." :-)
 
That doesn't sound very convincing, does it? And yet it's what I implied in my recent reply. But let me show some justification:
 
The _TimesBooks World Weather Guide_ has temperature records for eight temperate (non-tropical) Australian cities. That book gives, for each month, the average daily temperature for that month, averaged over many years. It gives two daily temperatures--minimum and maximum.
 
I have averaged, for each month, the average daily maximum and average daily minimum figures, over all of the eight cities, in diverse representative places in temperate Australia. These cities include Sydney, Canaberra, Melbourne, Perth, etc. I'll post a list of them if desired.
 
Here are the Australian averages that I got for six of those months, by averaging the records for those eight cities:
 
(in Fahrenheit)
 
November: 66.31
December: 70.25
January: 72.50
February: 72.31
March: 68.69
April: 62.56
 
What do these records say about the typical or average middle of summer in Australia?
 
The middle of summer, judging by the time of apparent maximum, and by the method that I used for Santa Cruz (equal temperature sums for two months before and after the middle) is at or very near to the boundary between January and February. Judging by that, if summer arrives with December, then summer is four months long,
 
Or you might say that a temperature roughly equal to December's average is found about halfway between the averages for February and March. That would suggests that the end of summer is about halfway into March.
 
Either way, the records suggest that, if summer arrives with December, then, on average, in temperate Australia, summer extends well into March.
 
This has been only a rough look at the records. Average Australian temperature records would be a worthwhile thing for Irv's technique to look at, either with daily temperature records, or, somehow, with the averages listed in this posting.
 
My point is this: There unavoidably is subjectivity about when some seasons begin. We have the widespread subjective perceptions that the high north-declination season arrives with June, and that the high south-declination season arrives with December. But, given that, and given some temperature records, we can infer what season-length is consistent with those subjective starting-dates and with the temperature records.
 
A three-month summer is not consistent with the combination of the following two things: 1) Summer arrives with December; and 2) The temperature averages that I've posted here. That's why I suggest that the perception of three-month summer, or other equal-seasons assumptions, are cultural in origin.
 
In this country (the U.S.) our media tell us that officially the seasons start on the equinoxes and solstices. Of course it never occurs to any media talker to question why that is. Those guys are specialists at repeating eachother, and they do it exceptionally consistently. Even our astronomers say that the seasons begin on equinoxes and solstices. If the astronomers say it then it must be true, right? Science couldn't be wrong. Sorry, but the astronomers have no scientific justification for their claim about seasons beginning on equinoxes and solstices.
 
As June 21 approached, I spoke to an astronomer about this. He said that astronomers like the equinoxes and solstices as season-boundaries because it gives them a precise, astronomically-based definition of the seasons. Forgive me, but who appointed an astronomer to define the seasons? I replied to the astronmer, "Fine, it would be reasonable for you to define 'astronomical quarters' in that or any other way you want to, but "seasons" is a word that already has a meaning, and it doesn't sound very valid when you give it a new meaning of your own."
 
Then the astronomer then backed off, saying that he wasn't responsible for the official definition. As I said, the official definition amounts to people repeating eachother without knowing why.
 
Now, if Australia's media announce the arrival of summer on December 1 instead of December 21, then I highly commend them for that. That would be a tremendous improvement over what our media here tell us four times a year. Do the Australian media officially agree that summer arrives with December?
 
That leads me to this question: In your country, when do the media announce an official beginning of the seasons, whether that country is the U.K., Canada, Australia, China, Russia or India, or a different country?
 
Mike Ossipoff
 
 
 
 
 
 
 
 
 
 
 
 

> My point was that around here (in the temperate parts) the general
> understanding is that no season has parts in both June and September,
> or in both December and March for that matter. Every season begins and
> ends on (Gregorian) month boundaries and last for three months each.
>

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[MIKE OSSIPOFF]

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Yesterday I meant to thank Michael Deckers for the information about the efficient longterm application of Minimum Drift. So thanks to Michael Decker and Karl for the information.
 
As I was saying, I'd never looked at the problem of efficient prediction of leapyears far into the future, for a leapyear rule such as Minimum Drift, and so it's something that I knew nothing about.
 
Mike Ossipoff
 

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[Christopher Vance-8]

Mike, hi.

Weather is so variable, that I won't claim a personal perception.
(It's unseasonably warm today, but that doesn't make it summer yet.)

What perception there is for me, is reinforced by education and local
media. (World media is irrelevant, since they almost always get the
hemisphere wrong.)

I do not recall ever hearing anyone in an educational or media context
in this country using solstices or equinoxes to form boundaries of
seasons, but the month boundaries are not infrequently referred to,
particularly with regard to weather. Aggregating monthly weather
statistics into seasons is trivial if a season boundary is also a
month boundary.

Even if you do use solstices and equinoxes, you still have to decide
whether they form the boundaries or the centres of your seasons. I
just don't think any kind of general agreement can happen.

-- Christopher

On Wed, Jul 1, 2009 at 9:55 AM, MIKE OSSIPOFF<nkklrp@...> wrote:

>
> Christopher and Calendar People,
>
> You wrote:
>
>> My point was that around here (in the temperate parts) the general
>> understanding is that no season has parts in both June and September,
>> or in both December and March for that matter. Every season begins and
>> ends on (Gregorian) month boundaries and last for three months each.
>>
>
> It's interesting to hear from people faraway places about seasonal
> perceptions. My question is: Are you sure that this perception of 3-month
> seasons is from personal direct perception, or, rather, of a cultural
> nature, a traditional or media-perpeturated notion?
>
> Mike Ossipoff
>
>
>
>
> ________________________________
> Windows Live™ SkyDrive™: Get 25 GB of free online storage. Get it on your
> BlackBerry or iPhone.

--
Christopher Vance

[Karl Palmen]

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Dear Mike and Calendar People

 

Mike’s rule is really a minimum jitter rule rather than a minimum drift rule.

 

Drift refers to the long term changes owing to the calendar mean year not being exactly equal to the tropical year.

It does not refer to the backwards and forwards movement arising from the particular placement of leap years. This has been referred to as jitter on this list.

The Gregorian calendar has a maximum  jitter of 2.1975 days, while any minimum jitter rule has a maximum jitter just under 1 day for a leap day calendar or 1 week for a leap week calendar.

So Mike’s rule is a minimum jitter rule. All such rules can be expressed in the form: year Y is a leap year if and only if

 

(L*Y + K) mod ( C ) < L

 

where K is a constant that can be an integer from 0 to C-1.

It forms a cycle of C year and L leap years. Mike’s example has C=700,000 and L=124,219 and K can be worked out.

The resulting cycle is also symmetrical about the centre if C is odd and K=(C-1)/2. Then it is a Helios cycle and year 1 has an average start half way through the jitter range.

 

Because of the precession of the equinoxes and solstices relative to the perihelion, the equinoxes and solstices don’t all drift at the same speed relative to the calendar year and indeed for an accurate solar calendar, drift in different directions. In the Gregorian calendar, the South Solstice is drifting later in the calendar year, while the other solstice and the equinoxes are drifting earlier. This drift can be observed by looking at solstices and equinoxes exactly 400 years apart. Intervals not a multiple of 400 years won’t show this because they are affected by jitter.

 

Because the equinoxes and solstices drift in different directions in an accurate solar calendar, there are two seasons that don’t drift at all. These have been referred to on the list as Calendar Seasons. Irv has studied these and found that one of the two calendar seasons is stable. For the Gregorian calendar, the stable calendar season occurs at the beginning of March.

 

Karl

 

10(10(08 till noon

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of MIKE OSSIPOFF
Sent: 01 July 2009 23:36
To: CALNDR-L@...
Subject: Re: June-September season

 

Yesterday I meant to thank Michael Deckers for the information about the efficient longterm application of Minimum Drift. So thanks to Michael Decker and Karl for the information.
 
As I was saying, I'd never looked at the problem of efficient prediction of leapyears far into the future, for a leapyear rule such as Minimum Drift, and so it's something that I knew nothing about.
 
Mike Ossipoff
 

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[MIKE OSSIPOFF]

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Christopher--
 
You wrote:

> I do not recall ever hearing anyone in an educational or media context
> in this country using solstices or equinoxes to form boundaries of
> seasons, but the month boundaries are not infrequently referred to,
> particularly with regard to weather. Aggregating monthly weather
> statistics into seasons is trivial if a season boundary is also a
> month boundary.
 
I reply:
 
That's what I'd hoped. I'm glad that the rest of the world isn't officially saying that the South Declination season starts on December 21.
 
It makes a lot more sense, in terms of our own experience, to officially refer to southern hemisphere summer and northern hemisphere winter arriving with December; and to southern hemisphere winter and northern hemisphere summer arriving with June.
 
Mike Ossipoff

 
 

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[MIKE OSSIPOFF]

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(I have company from out-of-state arriving here in a few days, and so I'll be offline between now and a little after mid-July)

Dear Karl and Calendar People,
 
I'd be the first to admit that I don't know the accepted meaning of "drift", and so it could well mean something other than what I was using it to mean.
 
Here is what I mean to minimize:
 
The leapyear rule that I've been calling "Minimum Drift" is intended to minimize the amount by which the solar ecliptic longitude at the midnight that starts NorthI/1 differs from a certain desired value of that solar ecliptic longitude. (And that value is the value that I've referred to as the middle of the designated lyc).
 
Mike Ossipoff
 
 
 

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[MIKE OSSIPOFF]

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I just want to add that I realize that the leapyear rule that I've called "Minimum Drift" doesn't _precisely and perfectly_ minimize what it is intended to minimize, due to the fact that the sun's motion per day on the ecliptic isn't precisely the same on different days, even within a fairly small region of the ecliptic.
 
But, for the purposes of a calendar, it minimizes it perfectly well enough. Only a very negligibly slight improvement could be gotten at a very great cost in complication.
 
Just one definition (and I'll probablly not find out about any replies till mid-July):
 
Here's how I've been defining "drift":
 
I've been using "drift" to mean the difference between the solar ecliptic longitude at a particular time and calendar date, and the _desired_ ecliptic longitude for that time and date.
 
That's what Minimum Drift is intended to minimize.
 
Till around mid-July,
 
Mike Ossipoff
 

---

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[Brillig]

Drift vs. jitter:

Mike,

Here is how drift and jitter are used on this forum.

Jitter is the measure of how much the calendar date misses the target
date during a calendar cycle.
Drift is the measure of how much this error changes over subsequent
calendar cycles.

Take the Gregorian and Julian calendars, for example. There is jitter
each time an adjustment is made. There are more complex adjustments to
the Gregorian calendar than to the Julian calendar. The Julian
calendar consequently has very little jitter when compared to the
Gregorian calendar. The mean year Julian year, however, is too long.
Over subsequent cycles, the calendar date drifts away from the target
date.

Victor

On Thu, Jul 2, 2009 at 5:28 PM, MIKE OSSIPOFF<nkklrp@...> wrote:

>
> I just want to add that I realize that the leapyear rule that I've called
> "Minimum Drift" doesn't _precisely and perfectly_ minimize what it is
> intended to minimize, due to the fact that the sun's motion per day on the
> ecliptic isn't precisely the same on different days, even within a fairly
> small region of the ecliptic.
>
> But, for the purposes of a calendar, it minimizes it perfectly well
> enough. Only a very negligibly slight improvement could be gotten at a very
> great cost in complication.
>
> Just one definition (and I'll probablly not find out about any replies till
> mid-July):
>
> Here's how I've been defining "drift":
>
> I've been using "drift" to mean the difference between the solar ecliptic
> longitude at a particular time and calendar date, and the _desired_ ecliptic
> longitude for that time and date.
>
> That's what Minimum Drift is intended to minimize.
>
> Till around mid-July,
>
> Mike Ossipoff
>
>
> ________________________________
> Lauren found her dream laptop. Find the PC that’s right for you.

[Karl Palmen]

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Dear Mike and Calendar People

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of MIKE OSSIPOFF
Sent: 02 July 2009 23:06
To: CALNDR-L@...
Subject: Re: Drift and Jitter RE: June-September season

 

 
(I have company from out-of-state arriving here in a few days, and so I'll be offline between now and a little after mid-July)

Dear Karl and Calendar People,
 
I'd be the first to admit that I don't know the accepted meaning of "drift", and so it could well mean something other than what I was using it to mean.
 
Here is what I mean to minimize:
 
The leapyear rule that I've been calling "Minimum Drift" is intended to minimize the amount by which the solar ecliptic longitude at the midnight that starts NorthI/1 differs from a certain desired value of that solar ecliptic longitude. (And that value is the value that I've referred to as the middle of the designated lyc).

Such a “drift” is actually a mixture of jitter and drift.

However the 124,219/700,000 rule as interpreted by Michael Deckers is minimum jitter rule because it minimises the amount by which the start of the North Season differs from the corresponding point in a mean year of exactly 365.24219 days. The difference between this corresponding point of the mean year and the certain desired value of the solar ecliptic longitude is what makes up the drift.

Karl

10(10(11

 

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[Karl Palmen]

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Dear Mike and Calendar People

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of MIKE OSSIPOFF
Sent: 02 July 2009 23:28
To: CALNDR-L@...
Subject: Re: Drift and Jitter RE: June-September season

 

 
I just want to add that I realize that the leapyear rule that I've called "Minimum Drift" doesn't _precisely and perfectly_ minimize what it is intended to minimize, due to the fact that the sun's motion per day on the ecliptic isn't precisely the same on different days, even within a fairly small region of the ecliptic.

It minimises the jitter.

But, for the purposes of a calendar, it minimizes it perfectly well enough. Only a very negligibly slight improvement could be gotten at a very great cost in complication.

The drift is not linear. The drift rate changes with time owing to the change in Earth’s rotation rate and other changes. So a vary complicated rule with a varying mean year would be need to minimise the drift. Irv has suggested such rules. Furthermore, only the drift for one ecliptic longitude would be minimised. This is because precession causes the time intervals between the solstices, equinoxes and other fixed ecliptic longitudes to change slowly over time.

As explained in a previous note, if a calendar with a fixed mean year  is sufficiently accurate, the drift varies in direction over  ecliptic longitudes and there are two ecliptic longitudes that have no drift at all at any one time. These have been called the calendar seasons. Irv has found that one of these calendar seasons (the one that occurs after perihelion and before aphelion) is stable.
 
Just one definition (and I'll probablly not find out about any replies till mid-July):
 
Here's how I've been defining "drift":
 
I've been using "drift" to mean the difference between the solar ecliptic longitude at a particular time and calendar date, and the _desired_ ecliptic longitude for that time and date.

We (Victor and I at least) prefer to separate this “drift”  into drift and jitter. This separation is valid only for a calendar with a fixed mean year.

That's what Minimum Drift is intended to minimize.

But Mike has already said that it actually minimises something else. That something else is the jitter. The drift can be kept low for a few thousand years by an appropriate choice of mean year.

Karl

 

10(10(11

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[Karl Palmen]

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Dear Victor and Calendar People

 

 

 

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Victor Engel
Sent: 03 July 2009 00:06
To: CALNDR-L@...
Subject: Re: Drift and Jitter RE: June-September season

 

Drift vs. jitter:

 

Mike,

 

Here is how drift and jitter are used on this forum.

 

Jitter is the measure of how much the calendar date misses the target

date during a calendar cycle.

Drift is the measure of how much this error changes over subsequent

calendar cycles.

 

KARL SAYS: This presupposes that the calendar has a cycle. It is also not precise enough to measure.

 

I prefer to define the jitter as the movement of the calendar year against the mean calendar year, particularly the maximum range of this movement. This works for any calendar that has a mean year, not just those that have a cycle.

Over short periods of time over which the drift is negligible, the jitter is approximately the movement of the calendar year moves with respect to the equinoxes, solstices and other ecliptic longitudes.

 

The jitter that is minimised to the maximum range of the jitter movement (latest minus earliest). If a calendar has a rational mean year, minimising this jitter causes it to have a cycle. Also all calendars that have a cycle also have a finite jitter.

 

 

Take the Gregorian and Julian calendars, for example. There is jitter

each time an adjustment is made. There are more complex adjustments to

the Gregorian calendar than to the Julian calendar. The Julian

calendar consequently has very little jitter when compared to the

Gregorian calendar. The mean year Julian year, however, is too long.

Over subsequent cycles, the calendar date drifts away from the target

date.

 

The jitter of the Julian calendar is 0.75 days, which is the difference between the number of leap years in a period of 1 leap year and the mean number of leap years in 1 year.

 

The jitter of the Gregorian calendar can be measured thus: Find the longest period of time that begins and ends with a leap year and has a leap year every four years. Such a period is 1904 to 2096 inclusive. It has 193 years of which 49 are leap years. The average number of leap years for 193 years is 193*97/400 = 46.8025. The difference (49-46.8025) is the jitter, which is 2.1975 days.

 

Now consider the same calendar, but with years whose number is divisible by 4000 made into common years (mean year 365.24225 days). Find the longest period that behaves like the Gregorian calendar and also begins and ends with a 193-year period as defined above. Such a period is the 3393 years of 304 to 3696. It has 8*(97)+49=825 leap years compared with an average of 0.24225*3393 = 821.95425. This leads to a jitter of 3.04575 days.

 

Karl

 

10(10(11

 

 

Victor

 

 

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[Karl Palmen]

Dear Mark, Irv and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Mark J. Reed
Sent: 27 May 2009 13:13
To: CALNDR-L@...
Subject: Re: Leap Week Calendars RE: Sorry Brij--I confused you with someone else

As mentioned earlier, the ISO 8601 week-numbering system for the
Gregorian calendar effectively defines a leap-week calendar, in which
most years have 52 weeks but 71 out of 400 years have 53 weeks. Since
the rule is based on the weekday on which the corresponding Gregorian
January 1 falls, the pattern is not obvious when considered in
isolation. It is almost the ideal (irregular) 5,6,5,6,6 pattern that
Karl mentioned, but there is one seven-year gap (between years 296 and
303 modulo 400).

KARL SAYS: Here I show you the closest matches of the ideal pattern in which the leap years spread as evenly as possible.
Firstly I state that the leap years of an ideal pattern are those years Y such that

(( 71*Y + K ) mod 400 ) < 71

for a constant K.
Furthermore, the interval pattern I gave (of 6,6,5, 6,6,5, etc. in (2) below ) begins at
the year whose accumulator (( 71*Y + K ) mod 400) is equal to 0.

K=178 would place the mean new year closest to the mean new year of the Gregorian calendar, but it is not the closest match.
Below I list the ISO leap week years that are not leap week years in an ideal pattern of a given value of K along with the accumulator in (). Those years that have small accumulator have the ideal leap week year one year earlier and those with a large accumulator have the ideal leap week year one year later. I use the 400 year interval 1801-2200 in these examples.

K=178:
1807(075), 1824(082), 1896(394), 1903(091), 1914(072),
1920(098), 1931(079), 1948(086), 1976(074), 2020(398),
2048(386), 2065(393), 2076(374), 2093(381), 2172(390),
2189(397).
16 leap years disagree.

In the order of there accumulators they are
1914(072), 1976(074), 1807(075), 1931(079), 1824(082), 1948(086), 1903(091), 1920(098)
2076(374), 2093(381), 2048(386), 2172(390), 2065(393), 1896(394), 2189(397), 2020(398)
This shows that there are an equal number that are one year early and one year late compared to this ideal pattern.


However, K=173, K=181 and K=185 have only 15 leap years disagreeing.

K=173:
1824(077), 1896(389), 1903(086), 1920(093), 1931(074),
1948(081), 2020(393), 2048(381), 2065(388), 2076(369),
2082(395), 2093(376), 2144(397), 2172(385), 2189(392).

K=181:
1807(078), 1824(085), 1852(073), 1896(397), 1903(094),
1914(075), 1920(101), 1931(082), 1948(089), 1976(077),
2048(389), 2065(396), 2076(377), 2093(384), 2172(393).

K=185:
1807(082), 1824(089), 1852(077), 1903(098), 1914(079),
1920(105), 1931(086), 1948(093), 1959(074), 1976(081),
2048(393), 2076(381), 2093(388), 2182(073), 2172(397).

In the order of their accumulators they are

K=173:
1931, 1824, 1948, 1903, 1920
2076, 2093, 2048, 2172, 2065, 1896, 2189, 2020, 2082, 2144.

K=181
1852, 1914, 1976, 1807, 1931, 1824, 1948, 1903, 1920
2076, 2093, 2048, 2172, 2065, 1896.

K=185:
2128, 1959, 1852, 1914, 1976, 1807, 1931, 1824, 1948, 1903, 1920
2076, 2093, 2048, 2172.

In each case the first row of years has the ideal leap week year one year earlier and the second row has the ideal leap week year one year later.


Karl

10(10(17



On 5/27/09, Palmen, KEV (Karl) <karl.palmen@...> wrote:

> Dear Mike and Calendar People
>
>
>
> Brij has suggested several leap week calendars, where a week is
> intercalated to a year of exactly 52 weeks = 364 days.  See
> http://www.hermetic.ch/cal_stud/palmen/lweek1.htm .
>
>
>
> All of Brij's proposed leap week rules make all years whose number is
> divisible by six have a leap week and some other years, which Brij calls
> Kepler's leap week years have a leap week.
>
>
>
> Other leap week rules have been suggested. See
> http://www.hermetic.ch/cal_stud/palmen/lweek1.htm#rules .
>
>
>
> There is a trade-off between simplicity and minimising the jitter of the
> year with respect to the seasons (or mean year). Extreme examples (for
> the 400-year cycle of 71 leap weeks) are
>
> (1)    5:40:400 in which a year has a leap week if its number is
> divisible by five, but not divisible by 40, unless also divisible by 400
> (similar to Gregorian 4:100:400), which varies about two and half weeks
> against the mean year and
>
> (2)    one whose leap weeks occur on years five or six years apart in
> intervals:
>
> 6,6,5, 6,6,5, 6,6,5 6,5
>
> 6,6,5, 6,6,5, 6,6,5 6,5
>
> 6,6,5, 6,6,5, 6,6,5 6,5
>
> 6,6,5, 6,6,5, 6,5
>
> 6,6,5, 6,6,5, 6,6,5 6,5
>
> 6,6,5, 6,6,5, 6,6,5 6,5
>
> 6,6,5, 6,6,5, 6,5
>
> which gives the minimum jitter of one week against the mean year. Irv
> has suggested a similar rule for a 293-year cycle of 523 leap weeks.
>
>
>
> Karl
>
>
>
> 10(09(03 till noon
>

[MIKE OSSIPOFF]

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Victor,
 
Yes, then jitter was what I meant. The leapyear rule I proposed shouldn't have any drift at all, except insofar as the number of days per mean tropical year gradually changes.
 
Maybe I should just call my leapyear rule proposal "Minimum Distance", where distance refers to the distance along the ecliptic between the actual and desired positions of midnight UT, NorthI/1...the actual and desired solar ecliptic longitude at midnight UT, NorthI/1.
 
This is a rare opportunity for me to get on the computer during my family's visit.
 
Mike Ossipoff
 
 
 
> Drift vs. jitter:

>
> Mike,
>
> Here is how drift and jitter are used on this forum.
>
> Jitter is the measure of how much the calendar date misses the target
> date during a calendar cycle.
> Drift is the measure of how much this error changes over subsequent
> calendar cycles.
>
> Take the Gregorian and Julian calendars, for example. There is jitter
> each time an adjustment is made. There are more complex adjustments to
> the Gregorian calendar than to the Julian calendar. The Julian
> calendar consequently has very little jitter when compared to the
> Gregorian calendar. The mean year Julian year, however, is too long.
> Over subsequent cycles, the calendar date drifts away from the target
> date.
>
> Victor
>

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[MIKE OSSIPOFF]

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From what others have said, maybe I oversimplified a bit when I said that "Minimum Distance" has no drift at all, except for that caused by the change in days per tropical year. But, for practical purposes, my statement was probably ok.
 
Mike Ossipoff
 

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[MIKE OSSIPOFF]

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X-OriginalArrivalTime: 27 Jul 2009 23:56:50.0637 (UTC) FILETIME=[E8B4BFD0:01CA0F15]



Calendarists=2C
=20
I've posted the details of my fixed terrestrial-seasonal calendar proposal.=
 I'd also like to post similar details for a fixed declination-calendar=2C =
and then for non-fixed terrestrial-seasonal and declination calendars. Four=
 proposals=2C four postings. Then I'll repost them all together in one post=
ing=2C so=2C then=2C my proposals will be posted all in one place=2C approp=
riately-titled.
=20
I've chosen for the fixed declination-calendar to be next=2C for this posti=
ng=2C because I feel that people here prefer a fixed calendar=2C even thoug=
h its requirement for a new leapyear rule=2C a leapweek rule=2C could compl=
icate the proposal. And I know that some here don't like the idea of a terr=
estrial-seasonal calendar=2C and so=2C for those two reasons=2C I'm posting=
=2C here=2C the details of my proposal for a fixed declination-calendar.
=20
By "declination-calendar"=2C I mean a calendar whose largest year-divisions=
=2C the seasonal-divisions=2C are based directly on the amount of solar dec=
lination. Not on equinoxes and solstices and not directly ultimately based =
on solar ecliptic longitude. And not with seaston starting-dates intended t=
o approximiate the perceived terrestrial-seasons' starting dates.
=20
Goals:
=20
There are two extreme-declination seasonal periods=2C North and South=2C in=
tended to be the periods when (as nearly as possible=2C given other constra=
ints) the north or south solar declination is in the top half of its range.
=20
Because this is a fixed calendar=2C and because a fixed calendar offers max=
imum convenience only if all of the year divisions consist of whole numbers=
 of weeks=2C this need places a constraint on the lengths of North and Sout=
h--but they still are periods when the north or south solar declination is =
very nearly in the top half of its range.
=20
As a result of these goals and constraints=2C  32.5 degrees is chosen as th=
e solar ecliptic longitude at which for North to begin.
=20
Year-divisions:
=20
Four declination-seasonal year-divisions:
=20
South=2C Northward=2C North=2C and Southward.
=20
North and South each have 17 weeks
=20
Northward and Southward each have 9 weeks=2C for a total of 52 weeks.
=20
Each of those four year-divisions is divided into months=2C of 4 or 5 weeks=
.
=20
The seasonal year-divisions have these month-lengths (in weeks)=2C in this =
order:
=20
North and South: 5=2C4=2C4=2C4
=20
Northward and Southward: 5=2C4
=20
(I call those "Mike-months". Karl has suggested a different ordering of the=
 5 & 4 week months=2C to distribute them more uniformely=2C even if less si=
mply. I have no objection to Karl-months)
=20
Months are named by number. For example=2C NorthI=2C NorthII=2C NorthIII an=
d NorthIV.
=20
Leapyear rule:
=20
Leapweeks are used=2C with the intent to minimize the distance=2C on the ec=
liptic=2C between 32.5 degrees and the solar ecliptic longitude at midnight=
=2C UT=2C NorthI/1=2C with this 364-day-year calendar.
=20
Definition of "initial d"=2C for the purpose of specifying starting date an=
d leapyear rule:
=20
"Initial d" is defined for the new calendar's first 365 days in use. For th=
e new calendar's first 365 days in use:=20
=20
d2 =3D 32.5 minus the solar ecliptic longitude at midnight=2C UT=2C NorthI/=
1.
=20
=20
If the solar ecliptic longitude =3D 32.5 degrees after midnight=2C UT=2C No=
rthI/1=2C then the "Outside Date" is the midnight=2C UT directly after the =
time when the solar ecliptic longitude equals 32.5 degrees.
=20
If the solar ecliptic longitude =3D 32.5 degrees before midnight=2C UT=2C N=
orthI/1=2C then the "Outside date" is the midnight=2C UT=2C directly before=
 the time when the solar ecliptic longitude equals 32.5 degrees.
=20
Divide (days between midnight=2C UT=2C NorthI/1 and the Outside Date) by (3=
2.5 degrees minus the solar ecliptic longitude on the Outside Date). Take t=
he absolute value of the result of that division.
=20
That gives "days per degree".
=20
Multiply d2 by days per degree. That gives the initial d. The starting valu=
e of d=2C used in the leapyear rule and the starting-date rule.
=20
Starting date:
=20
The new calendar comes into use on the beginning (midnight) of the day afte=
r the old calendar's December 31.
=20
Every year and every year-division begins on a Monday. The new calendar's f=
irst year in use starts on a Monday. That Monday can be the first day in wh=
ich the new calendar comes into use (the day after the old December 31)=2C =
if that is a Monday=2C or it can be any Monday during the 365 days previous=
 to that.
=20
The Monday on which the new calendar starts is chosen so as to minimize the=
 magnitude of initial d.
=20
Leapyear rule:
=20
The initial value of d is the "initial d" determined as described above.
=20
If=2C for any year=2C d> 7/2 - 1.24219=2C then that year is a leapyear.
=20
(I've intended to write=2C here=2C the better implementatin described by Mi=
chael Deckers. If this is different from what Michael Deckers wrote=2C then=
 Deckers' rule is what I intend).
=20
Immediately when a non-leapyear ends=2C d + 1.24219 becomes the new value o=
f d.
=20
Immediately when a leapyear ends=2C d + 1.24219 - 7 becomes the new valule =
of d.
=20
Mike Ossipoff
=20
=20
=20
=20
=20
=20
=20
=20
_________________________________________________________________
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