my modest proposal

[Richard Halstead]

Eight day week calendar:

200 cycles of 3 years each
year 1,2 368 days, year 3 360 days
on the 28th cycle, years 82, 164 etc. 360 days
on the 200th cycle, year 601 begins the next 28 year cycle so no "leaps"
until 682
12 months
30 days in month except month twelve with 38 days for years 1,2,4,5,7,8 etc

Corrections\critiques welcome.

Just throwing it out there to folks who've given it much more careful
thought, I'm sure.

[Irv Bromberg]

On 2012 Jan 12, at 18:46 , Richard Halstead wrote:

Eight day week calendar:

200 cycles of 3 years each
year 1,2 368 days, year 3 360 days
on the 28th cycle, years 82, 164 etc. 360 days
on the 200th cycle, year 601 begins the next 28 year cycle so no "leaps" until 682
12 months
30 days in month except month twelve with 38 days for years 1,2,4,5,7,8 etc

There has been a lot of emphasis here on conserving the traditional 7-day sabbatical cycle, so it really seems futile to propose an 8-day week.  Anyhow...

Richard, you didn't state what is your calendar mean year, and which equinox or solstice is your intended target.

It seems that you have tried to implement an 8-day leap week calendar, but your cycle is not symmetrical, nor is it smoothly spread.

From your description I can't tell how many years there are in total in your cycle, and how many of them are long.

I call them "long" rather than "leap" because for 8-day leap week cycles it is the long years that are the common years, the short years are exceptional, or at least less common.

For the northward equinoctial year your best cycle choice is of course the 293-year cycle with 192 long years per cycle.

Mean year 365+71/293 days = 365d 5h 48m 56+152/293s = about 365.242320819112628 days.

To obtain a symmetrical smoothly spread cycle, it is a long year if (192 * Year + 146) MOD 293 < 192.

Or if you prefer, it is a short year if (101 * Year + 494) MOD 989 < 101.

For the north solstitial year your choices are extremely limited due to the nearby 29-year short mean year mixer cycle with 19 long per cycle, which prevents any reasonably accurate short cycle from existing in the range of the north solstitial mean year.  Allowing denominators up to 1000 years, the most accurate choice is the 989-year cycle with 648 long years per cycle, having a mean year of 365+239/989 days = 365d 5h 47m 59+269/989s = about 365.241658240647118 days.

To obtain a symmetrical smoothly spread cycle, it is a long year if (648 * Year + 494) MOD 989 < 648.

Or if you prefer, it is a short year if (341 * Year + 494) MOD 989 < 341.

There is another cycle whose mean year is slightly longer than the present era north solstitial year but it is interesting:

It has a calendar mean year of 365+29/120 days so it has the same mean year and day count as 8 repeats of a 120-year leap day cycle.

It is a 960-year cycle with 629 long years per cycle.

The mean year in time units is interesting because it is a whole number of minutes:  365d 5h 48m 0s.

It can't be made perfectly symmetrical because it has an even number of years per cycle, but it could be made almost symmetrical:

It is a long year if (629 * Year + 480) MOD 960 < 629.

Or if you prefer, it is a short year if (331 * Year + 494) MOD 989 < 331.

Although its mean year is slightly too long today, the north solstitial mean year will be getting slightly longer for a while, so this cycle is an excellent choice for quite a few millennia to come.

For more information about Symmetrical Leap Cycles please see that topic at <http://www.sym454.org/leap/>.

On that same web page you'll also find available my freeware "Ford Circles of Leap Cycles" and the "Fixed Leap Cycle Finder" spreadsheets with VBA macros, which will allow you to explore for other leap cycles.

-- Dr. Irv Bromberg, University of Toronto, Canada

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

[Karl Palmen]

Dear Richard sand Calendar People

The wording of the proposal is not very clear. I make a guess at what it means.

It seems to be a 3-year cycle in which two years have 368 days and one year has 360 days with some exceptions which have 360 days instead of 368 days. It seems that Richard is proposing a 600-year cycle with the exceptions on the 82nd, 166th, 250th, 334th, 418th, 502nd and 586th year of each 600-year cycle (assuming his "164 etc" is wrong rather than his wording). This makes seven exceptions, so making 207 short years in each 600-year cycle. Mean year is thus 368 - 8*207/600 = 365.24 days exactly, which is too short to be accurate.

Let's look for accurate cycles.
From http://the-light.com/cal/kp_NdaySolarCyc.html I see cycles

N  Days  Years LongYrs  YrLength
---------------------------------
8  10592    29      19  365.24138
8  85832   235     154  365.24255
8  75240   206     135  365.24272
----------------------------------

Every cycle with mean year between any two of those shown can be made up by adding whole numbers of those two cycles.

Calendar people may note that 235 + 29 = 264, which is equivalent to eight 33-year cycles (mean year 365.2424242... days). Add another 29 to get the 293-year mentioned by Irv. Add more 29s to get Irv's other suggestions.

I note that a rule with 3-year cycles with exceptions would produce a jitter well in excess of the minimum jitter of 8 days. For example take any of the 7 exception years and the year before it and you get 2 consecutive years of 360 days totalling 720 days this is 10.48 days shorter than two mean years of 365.24 days, so producing a jitter of 10.48 days. The worst interval the 81st year to the 586th year inclusive, which is 506 years in total of which 176 years are short (169 from 81, 84, 87, ... 585 + all 7 exceptions). The average number of short years in 506 years is 506*(207/600) = 174.57. The difference is 1.43, which indicates a jitter of 8*1.43 = 11.44 days over these 506 years.
 
A minimum jitter cycle would consist mainly of 29-year cycles in which the every 3rd year starting with either the 1st or 2nd year has 360 days and the other years have 368 days. The exceptions of the 29-year cycles would be formed by extending a 29-year cycle by 3 years to 32 years.

Accurate cycles with just one 32 have 235, 264, 293, 322, 351, 380, 409, 438, 467, 496, 525, ... (adding 29 each time) years.


Karl

12(07(20

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Richard Halstead
Sent: 12 January 2012 23:46
To: CALNDR-L@...
Subject: my modest proposal

Eight day week calendar:

200 cycles of 3 years each
year 1,2 368 days, year 3 360 days
on the 28th cycle, years 82, 164 etc. 360 days
on the 200th cycle, year 601 begins the next 28 year cycle so no "leaps"
until 682
12 months
30 days in month except month twelve with 38 days for years 1,2,4,5,7,8 etc

Corrections\critiques welcome.

Just throwing it out there to folks who've given it much more careful
thought, I'm sure.
--
Scanned by iCritical.

[Karl Palmen]

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

Dear Irv and Calendar People

 

Irv’s suggestions ensure the minimum jitter of 8-days and consist of periods of 29 and 32 years as described in my previous note.

The symmetry requires that a 29-year or 32-year cycle split by the start of the symmetrical cycle.

 

The 29-year cycle can be defined symmetrically, but less regularly as having it’s 2^nd, 5^th, 8^th, 11^th, 14^th, 16^th 19^th, 22^nd, 25^th and 28^th years short.

All of Irv’s symmetrical suggestions can be formed by symmetrically placing these 29-year cycles along with 3-year cycles of which the 2^nd year is short.

 

I’ve mentioned a cutting algorithm that cuts symmetrical cycles into two types of symmetrical parts, which are sequenced symmetrically. In this case the 1^st level (or iteration) on the cutting algorithm cuts into 3s of ‘LsL’ and 5s of ‘LsLsL’, where ‘L’ is a long year of 368 days and ‘s’ is a short year of 360 days. At the 2^nd level it cuts into 29s (= 3+3+3+5+3+3+3) and 61s (=29+3+29), where ‘+’ is concatenation (sequencing).

The 293-year cycle cuts into (29+29+29+29+61+29+29+29+29)  and the 989-year cycle has twice as many 29s each side of the 61. The 960-year cycle is the same as the 989-year cycle but with one 29 removed from the start or end, so making an almost symmetrical sequence of symmetrical 29s and 61s.

 

Also, in my previous note I was wrong about the worst period for jitter in my guess of Richard’s proposal. Add two years to the end (one of which is short), extending it to be 81^st year to 588^th year. It has 508 years of which 177 years are short.

 

Karl

 

12(07(20

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 13 January 2012 03:03
To: CALNDR-L@...
Subject: Re: my modest proposal

 

On 2012 Jan 12, at 18:46 , Richard Halstead wrote:

Eight day week calendar:

200 cycles of 3 years each
year 1,2 368 days, year 3 360 days
on the 28th cycle, years 82, 164 etc. 360 days
on the 200th cycle, year 601 begins the next 28 year cycle so no "leaps" until 682
12 months
30 days in month except month twelve with 38 days for years 1,2,4,5,7,8 etc

 

There has been a lot of emphasis here on conserving the traditional 7-day sabbatical cycle, so it really seems futile to propose an 8-day week.  Anyhow...

 

Richard, you didn't state what is your calendar mean year, and which equinox or solstice is your intended target.

It seems that you have tried to implement an 8-day leap week calendar, but your cycle is not symmetrical, nor is it smoothly spread.

From your description I can't tell how many years there are in total in your cycle, and how many of them are long.

I call them "long" rather than "leap" because for 8-day leap week cycles it is the long years that are the common years, the short years are exceptional, or at least less common.

 

For the northward equinoctial year your best cycle choice is of course the 293-year cycle with 192 long years per cycle.

Mean year 365+71/293 days = 365d 5h 48m 56+152/293s = about 365.242320819112628 days.

To obtain a symmetrical smoothly spread cycle, it is a long year if (192 * Year + 146) MOD 293 < 192.

Or if you prefer, it is a short year if (101 * Year + 494) MOD 989 < 101.

 

For the north solstitial year your choices are extremely limited due to the nearby 29-year short mean year mixer cycle with 19 long per cycle, which prevents any reasonably accurate short cycle from existing in the range of the north solstitial mean year.  Allowing denominators up to 1000 years, the most accurate choice is the 989-year cycle with 648 long years per cycle, having a mean year of 365+239/989 days = 365d 5h 47m 59+269/989s = about 365.241658240647118 days.

To obtain a symmetrical smoothly spread cycle, it is a long year if (648 * Year + 494) MOD 989 < 648.

Or if you prefer, it is a short year if (341 * Year + 494) MOD 989 < 341.

 

There is another cycle whose mean year is slightly longer than the present era north solstitial year but it is interesting:

It has a calendar mean year of 365+29/120 days so it has the same mean year and day count as 8 repeats of a 120-year leap day cycle.

It is a 960-year cycle with 629 long years per cycle.

The mean year in time units is interesting because it is a whole number of minutes:  365d 5h 48m 0s.

It can't be made perfectly symmetrical because it has an even number of years per cycle, but it could be made almost symmetrical:

It is a long year if (629 * Year + 480) MOD 960 < 629.

Or if you prefer, it is a short year if (331 * Year + 494) MOD 989 < 331.

Although its mean year is slightly too long today, the north solstitial mean year will be getting slightly longer for a while, so this cycle is an excellent choice for quite a few millennia to come.

 

For more information about Symmetrical Leap Cycles please see that topic at <http://www.sym454.org/leap/>.

 

On that same web page you'll also find available my freeware "Ford Circles of Leap Cycles" and the "Fixed Leap Cycle Finder" spreadsheets with VBA macros, which will allow you to explore for other leap cycles.

 

 

-- Dr. Irv Bromberg, University of Toronto, Canada

 

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

--
Scanned by iCritical.

[Richard Halstead]

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

Okay, you guys are too smart for me, but yes, you get the gist of my proposal and have already improved upon it.  Please feel free to modify and distribute without attribution, I hereby grant all recipients a perpetual license under the GPL or whatever lawyers say... ;)

Yes, I understand that anything other than a seven day week hasn't got a snowball's chance, but it's fun to kick the ideas around, better than sudoku anyway.

On the solstice question, I can't see how to avoid having the northern winter solstice jumping all around December with any variation on my OP I can imagine.

At least it has the advantage of each month always beginning on the same day of the week.  But in addition to numerous other shortcoming, the business folks would not tolerate adding nine percent or so to the fourth quarter on the 46 week years, unless they extended the first quarter to April 2 etc....  I was only really shooting for novelty...

On 1/13/2012 10:55 AM, Karl Palmen wrote:

Dear Irv and Calendar People

 

Irv’s suggestions ensure the minimum jitter of 8-days and consist of periods of 29 and 32 years as described in my previous note.

The symmetry requires that a 29-year or 32-year cycle split by the start of the symmetrical cycle.

 

The 29-year cycle can be defined symmetrically, but less regularly as having it’s 2^nd, 5^th, 8^th, 11^th, 14^th, 16^th 19^th, 22^nd, 25^th and 28^th years short.

All of Irv’s symmetrical suggestions can be formed by symmetrically placing these 29-year cycles along with 3-year cycles of which the 2^nd year is short.

 

I’ve mentioned a cutting algorithm that cuts symmetrical cycles into two types of symmetrical parts, which are sequenced symmetrically. In this case the 1^st level (or iteration) on the cutting algorithm cuts into 3s of ‘LsL’ and 5s of ‘LsLsL’, where ‘L’ is a long year of 368 days and ‘s’ is a short year of 360 days. At the 2^nd level it cuts into 29s (= 3+3+3+5+3+3+3) and 61s (=29+3+29), where ‘+’ is concatenation (sequencing).

The 293-year cycle cuts into (29+29+29+29+61+29+29+29+29)  and the 989-year cycle has twice as many 29s each side of the 61. The 960-year cycle is the same as the 989-year cycle but with one 29 removed from the start or end, so making an almost symmetrical sequence of symmetrical 29s and 61s.

 

Also, in my previous note I was wrong about the worst period for jitter in my guess of Richard’s proposal. Add two years to the end (one of which is short), extending it to be 81^st year to 588^th year. It has 508 years of which 177 years are short.

 

Karl

 

12(07(20

 

From: East Carolina University Calendar discussion List [CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 13 January 2012 03:03
To: CALNDR-L@...
Subject: Re: my modest proposal

 

On 2012 Jan 12, at 18:46 , Richard Halstead wrote:

Eight day week calendar:

200 cycles of 3 years each
year 1,2 368 days, year 3 360 days
on the 28th cycle, years 82, 164 etc. 360 days
on the 200th cycle, year 601 begins the next 28 year cycle so no "leaps" until 682
12 months
30 days in month except month twelve with 38 days for years 1,2,4,5,7,8 etc

 

There has been a lot of emphasis here on conserving the traditional 7-day sabbatical cycle, so it really seems futile to propose an 8-day week.  Anyhow...

 

Richard, you didn't state what is your calendar mean year, and which equinox or solstice is your intended target.

It seems that you have tried to implement an 8-day leap week calendar, but your cycle is not symmetrical, nor is it smoothly spread.

From your description I can't tell how many years there are in total in your cycle, and how many of them are long.

I call them "long" rather than "leap" because for 8-day leap week cycles it is the long years that are the common years, the short years are exceptional, or at least less common.

 

For the northward equinoctial year your best cycle choice is of course the 293-year cycle with 192 long years per cycle.

Mean year 365+71/293 days = 365d 5h 48m 56+152/293s = about 365.242320819112628 days.

To obtain a symmetrical smoothly spread cycle, it is a long year if (192 * Year + 146) MOD 293 < 192.

Or if you prefer, it is a short year if (101 * Year + 494) MOD 989 < 101.

 

For the north solstitial year your choices are extremely limited due to the nearby 29-year short mean year mixer cycle with 19 long per cycle, which prevents any reasonably accurate short cycle from existing in the range of the north solstitial mean year.  Allowing denominators up to 1000 years, the most accurate choice is the 989-year cycle with 648 long years per cycle, having a mean year of 365+239/989 days = 365d 5h 47m 59+269/989s = about 365.241658240647118 days.

To obtain a symmetrical smoothly spread cycle, it is a long year if (648 * Year + 494) MOD 989 < 648.

Or if you prefer, it is a short year if (341 * Year + 494) MOD 989 < 341.

 

There is another cycle whose mean year is slightly longer than the present era north solstitial year but it is interesting:

It has a calendar mean year of 365+29/120 days so it has the same mean year and day count as 8 repeats of a 120-year leap day cycle.

It is a 960-year cycle with 629 long years per cycle.

The mean year in time units is interesting because it is a whole number of minutes:  365d 5h 48m 0s.

It can't be made perfectly symmetrical because it has an even number of years per cycle, but it could be made almost symmetrical:

It is a long year if (629 * Year + 480) MOD 960 < 629.

Or if you prefer, it is a short year if (331 * Year + 494) MOD 989 < 331.

Although its mean year is slightly too long today, the north solstitial mean year will be getting slightly longer for a while, so this cycle is an excellent choice for quite a few millennia to come.

 

For more information about Symmetrical Leap Cycles please see that topic at <http://www.sym454.org/leap/>.

 

On that same web page you'll also find available my freeware "Ford Circles of Leap Cycles" and the "Fixed Leap Cycle Finder" spreadsheets with VBA macros, which will allow you to explore for other leap cycles.

 

 

-- Dr. Irv Bromberg, University of Toronto, Canada

 

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

--
Scanned by iCritical.