This might have just as easily been put in "Clock," but it subtends a rather thorny mess--or two of them, actually.

Firstly, the adoption of a 10-hour dozenally reckoned day implies, among other things, that the world will be divided not into the current degrees, minutes, and seconds of longitude and lattitude, as at present, but into a system of wholly dozenal parts.

(Historical note: the current system of navigational coordinates was imposed by British imperialism, and ran the Prime Meridian through the grounds of the Greenwich Observatory. Never mind that that put the International Date Line in a position where it undergoes radical deformation, owing to passing over much national territory.)

Were things done without national bias, the International Dateline should have been located 11 degrees east of its present location, and I would promose this as the Dateline for a dozenal system which, instuad of using degrees of east and west longitude, will, instead, use a system similar to that used in current astronomy: Right Ascension and Declination.

Right Ascension goes eastward and by the clock; declination goes by dozenal fraction of any line of longitude from the Equater to either Pole (+ for North, - for South). So, my condo in Irvine, CA, 117° 48' 37.98" W., 33° 39' 59.97"N, would, in the new system have the coordinates R.A. 1;8A70A4B385B8, Dec. +0;45A49702138A, where the first number is in terms of dozenal hours and the second is in terms of fraction of a quarter-circle.

Now, to convert to circular measure: it is optimal to allow a gross degrees for a quarter-circle, so the Declination listed above should be +45;A49702138A°.

Having disposed of the necessary topic of circular measure and a dozenal system of terrestrial coordinates, we may now tackle the issue of the canlendar. I am in favor of the following definitions:

The world should be divided into four dozen time-zones.

The day should began at civil sunset on the (new) International Dateline, as observed at the Equator.

The year should begin at the first sunset on the International Dateline at the Equator following the transit of the sun across the celestial Equator on its apparent journey south (Autumnal Equinox).

All intercalations necessary to bring the count of years back to the modulus of the day should be made at the end of the year.

All months of the year save the third, sixth, ninth, and dozenth should consist of 30 days. The third, sixth, and ninth months should consist of 31 days, and the dozenth month should consist of as many days as is necessary to bring the count of days back to the beginning of the year as defined above.

The custom of naming days of the week and months of the year after gods or man should be abolished; it is prefereable to use ordinal-numeral forms (cf. Portuguese and Greek for examples of how this is done with the days of the week).

If the custom of dating years A.D./C.E. is to be kept, then, at the very least, the error of Dionysius Exiguus should be corrected by adding six to the count of years. But I believe it is also preferable to change from an ordinal count of the years (as is now done) to a cardinal count based on elapsed time.

I go by the new moon closest to the vernal equinox. Today is the third day of the new year by my reckoning.

That would require the last month to have 32 or 33 days. Are you sure you want that?

In my calendar proposal, I had the months be alternating 30 and 31 days, except that the last month would have only 30 days in non-leap years. Did you ever read that thread?

I've seen calendar-reform proposals with that arrangment--odd-mi,bered months having an odd number of days, even-numbered months having an even number of days, except for the last month of the year. It works, but it doesn't have quarter-years of equal length, as some businesses seem to prefer.

Alternatively, my arrangement could have the first, fourth, seventh, and tenth monts with 31 days each, and the rest (except the last month) with 30 days each. The last month would have 31 or 32 days.

Equal-length quarters are impossible due to the number of days in a year not being divisible by 4. My arrangement has quarters of 91, 92, 91, and 91/92 days, which is as close to equal as you can get. Both of your proposals have quarters of 91, 91, 91, and 92/93 days, which are as equal as mine in non-leap years and less equal in leap years.

If you want to have a quarter-based calendar, a better structure would be that of the World Calendar.

Q1: 31+30+30 = 91 days
Q2: 31+30+30/31 = 91/92 days
Q3: 31+30+30 = 91 days
Q4: 31+30+31 = 92 days

I like the World Calendar system, but I think the version I know differs slightly with each quarter having 30, 30, 30, 31 days.
This leaves one day to make up the 365 (New Year Day) and in leap years Leap Year Day would be at the end of the second quarter.
Each quarter divides by 7, keeping the same day name structure for each.
The New Year and Leap Year Days would not have one of the regular day names.
Somewhere I had a note of a rule I made up (long ago) to cover leap years; but I can't remembre it now. We'll need one, - can't base it on centuries!

That would be closer to a third than to a quarter. I guess you meant 30, 30, 31. But I've always seen it as 31, 30, 30.

I don't like having days "outside" the calendar like that. We'd probably write the two extra days as 06-31 and 12-31, like I did in my post. But we'd still have an irregular quarter, and also introduce the problem of having an irregular week.

If only there were 364 days in a year instead of 365.242, calendar reform would be so much simpler!

The best terminating-dozenal approximations of the length of the year are:

• *265.3 (365 1/4) days, with an error of 11 min 15 s per year
• *265.2Χ8 (365 13/54) days, with an error of 2 min 5 s per year
• *265.2ΧΧ6 (365 31/128) days, with an error of only 130 ms per year.

I will stop there, because any larger denominator would have the problem that the length of the leap-year cycle would be longer than the period for which it is the most accurate.

So, we could achieve great accuracy with the rule that "every 4 years is a leap year with the exception of years divisible by 128 or (*Χ8)."

Unfortunately, while this would be ideal for hex or octal, *Χ8 is a rather awkward number in dozenal, compared to *100 or *60 or even *80.

The 54-year cycle would be even more awkward, as it's not even divisible by 4.

A more practical leap-year rule would have the form:

• Every year divisible by N is a leap year (like the Julian calendar), or
• Every year divisible by N is a leap year, except for years divisible by NM which aren't, or
• Every year divisible by N is a leap year, except for years divisible by NM which aren't, except for years divisible by NML which are (like the Gregorian calendar), or
• Every year divisible by N is a leap year, except for years divisible by NM which aren't, except for years divisible by NML which are, except for years divisible by NMLK which aren't.

The average length of a year is 365+1/N-1/NM+1/NML-1/NMLK, taking M, L, and K to be ∞ for the simpler rules.

For the sake of ease of arithmetic, I will require each of the products N, NM, NML, NMLK to be FS(1) numbers up to *10000. And for the sake of accuracy, require the rule to be at least as accurate as a 35/144 leap year rule.

That gives us 28 combinations to choose from:

• N=4, M=36, L=∞, K=∞
• N=4, M=24, L=3, K=∞
• N=1, M=1, L=4, K=36
• N=2, M=1, L=2, K=36
• N=3, M=2, L=2, K=12
• N=3, M=3, L=4, K=4
• N=4, M=1, L=1, K=36
• N=4, M=3, L=1, K=12
• N=4, M=6, L=1, K=6
• N=4, M=12, L=1, K=3
• N=4, M=24, L=2, K=3
• N=4, M=24, L=3, K=3
• N=4, M=24, L=3, K=6
• N=4, M=24, L=3, K=12
• N=4, M=24, L=3, K=24
• N=4, M=24, L=3, K=36
• N=4, M=24, L=3, K=72
• N=4, M=36, L=1, K=1
• N=4, M=36, L=2, K=1
• N=4, M=36, L=3, K=1
• N=4, M=36, L=4, K=1
• N=4, M=36, L=6, K=1
• N=4, M=36, L=12, K=1
• N=4, M=36, L=24, K=1
• N=4, M=36, L=36, K=1
• N=4, M=36, L=48, K=1
• N=4, M=36, L=72, K=1
• N=4, M=36, L=144, K=1

To narrow this down, add the further requirement that the four-term rules have to be more accurate than the best 3-term rule. This gives us 9 possibilities:

• N=4, M=36, L=inf, K=inf
• N=4, M=24, L=3, K=inf
• N=3, M=2, L=2, K=12
• N=4, M=24, L=3, K=3
• N=4, M=24, L=3, K=6
• N=4, M=24, L=3, K=12
• N=4, M=24, L=3, K=24
• N=4, M=24, L=3, K=36
• N=4, M=24, L=3, K=72

To be continued...

The above calculations are based on a tropical year of 365.24218900632 days. However, the number of days in a year is decreasing. Based on the formula from that article, a year was 365.24229531288 days *1000 (Julian) years ago and will be 365.24208269976 days *1000 years from now.

Using the same methodology as in the above post, the possible leap year rules *1000 years ago were:

• N=4, M=36, L=∞, K=∞
• N=4, M=24, L=3, K=∞
• N=4, M=24, L=4, K=∞

and *1000 years from now, it will be:

• N=4, M=36, L=∞, K=∞
• N=4, M=24, L=3, K=∞
• N=4, M=24, L=4, K=∞
• N=4, M=24, L=6, K=∞
• N=4, M=24, L=2, K=2
• N=4, M=24, L=4, K=24
• N=4, M=24, L=4, K=36
• N=4, M=24, L=4, K=48

The only two rules that are common to both of these lists are:

• N=4, M=36, L=∞, K=∞
• N=4, M=24, L=3, K=∞

In other words,

• Potential rule #1: A year is a leap year if it is divisible by 4 and NOT divisible by *100. The average length of a year is *265.2Ɛ years, which is 74.87 seconds too long, or accurate to one day in 1154 (*802) years.
• Potential rule #2: A year is a leap year if it is divisible by 4 and is either NOT divisible by *80 or is divisible by *200. The average length of a year is *265.2Ɛ

So these turn out to be the same approximation of the year: They both omit 2 Julian leap days every *200 years, just with different choices of which ones to admit. So, we should choose the simpler rule:

A year is a leap year if it is divisible by 4 and NOT divisible by *100.

Too many months in my quarter ...

Well I eventually found my note about leap years, dated somewhere in the sixties, and append it here for interest.

Interesting rule, but the problem with a *1000000-year leap-year cycle is that the approximation of the year will be inaccurate by the time the cycle is over.

That's not a good leap year rule. Much too inexact. It is far inferior to the Gregorian leap year rule. It is fairly easy, however, so let's list it as the Dozenal Leap Day rule, Gregorian-style number 1:

Rule DDG1 or dozenal 100 year rule:
* every forth year is a leap year, except years divisible by 100
* year length: *265.2E days (decimal 365.243056 or 365d5h50min)

It gets much better if you omit the leap year in years divisible by *800, i.e.:

Rule DDG2 or dozenal 800 year rule:
* approximates the current tropical year
* every forth year is a leap year, except:
* years that are divisible by *80, but not *200, are no leap years, and:
* years divisible by *800 are no leap years, too.
* year length: *265.2TT6 days, which is pretty exact right now (will be exact in 2023 or so)
* In Dan's list, it would mean N=4, M=24, L=3, K=4 (that's not in his list, however...).

Since the length of the tropical year gets shorter by about *0.00048 days per *1000 years (including the fact that mean solar days also get longer due to the slowing of the earth's rotation because of tidal effects - which Dan's numbers do not take into account), a smaller year length should be considered:

Rule DDG3 or dozenal 600 year rule:
* every forth year is a leap year, except:
* years that are divisible by *90, but not *600, are no leap years
* year length: *265.2TT days (decimal: 365.2418981). That will be exact around the year 4000.
* In Dan's list, it would mean N=4, M=27, L=8, which he doesn't consider.


There may be still better rules. A very good site (perhaps a must-read for leap-rule-maker) is this: http://individual.utoronto.ca/kalendis/leap/index.htm
This guy is in favor of using the Northern Equinox as the definition of the year, because it is currently the most stable definition, i.e. with which the year length doesn't differ much in the next 2000 years.

The northern equinox year is about 365.24237 or *265.2TT9T years long. There are two possibilities to approximate this year length:

Rule DDG4 or dozenal 1000 year rule:
* every forth year is a leap year, except:
* years that are divisible by *90, but not *300, are no leap years
* years divisible by *1000 are no leap years, too.
* year length: *265.2TE days. It is only slightly better than the Gregorian year (365.242477 days instead of 365.2425).
* Dan's list: N=4, M=27, L=4, K=4

Rule DDG5 or dozenal 900 year rule:
* every forth year is a leap year, except:
* years that are divisible by *90, but not *460, are no leap years
* (equivantly, every *3000 years, the following years are exeptional leap years: 0, *460, *900, *1160, *1600, *1T60, *2300, *2760)
* year length: *265,2TT8 days. Better than dozenal 1000 rule, but a little more calculation required.
* Dan's list: N=4, M=27, L=6

You could also align the year with the Northern Solstice. This year length is not as stable as the northern equinox year, but it will stay approximately the same length for a much longer period, around 10000 years. This year is about 365.24163 days long (dozenal *265.2T967 days). Rule DDG3 (600 year rule) does not well enough for this, but consider DDG6:

Rule DDG6 or dozenal 2000 year rule:
* approximates the current tropical year
* every forth year is a leap year, except:
* years that are divisible by *90, but not *600, are no leap years, and:
* years divisible by *2000 are leap years.
* year length: *265.2T96 days. The *2000 year cycle is quite long, though.
* Dan's list: N=4, M=27, L=8, K=3

These are just Gregorian-style rules, by which I mean rules which fit into Dan's NMLK list. I prefer rules like in the website mentioned, because they keep the corresponding astronomical event (Northern Equinox or Solstice) more closely to a certain date. For this, you don't leave out leap certain years, instead you insert some leap days only after five years, in certain intervals (which are defined by the precise rule). An example is the *27/T8-rule: Every *T8 years, the following *27 years are leap years:
3, 7, *E, *13, *17, *1E, *23, *27, *30, *34, *38, *40, *44, 48, *50, *54, *59, *61, *65, *69, *71, *75, *79, *82, *86, *8T, *92, *96, *9T, *T2, *T6
Something like that. There are also arithmetical rules.

Then, you could also consider leap weaks, which retain the religously important, but very un-dozenal 7-day week cycle whilst providing a perpetual calendar. Also, other week lengths like 6-day-weeks or 4-day-weeks or the likes have corresponding leap week rules.

Well, they are possible, if you insert 4 days every leap year. Non-leap years would have 364 days (4 quarters à 91 days), leap years would have 368 days (4 quarters à 92 days). You would require a 4-day "leap period" roughly every 3 years.

In retrospect, I should have extended my list to include FS(2) instead of just FS(1): 8's and 9's aren't hard to work with in dozenal, and it appears that they do give better rules. I'll think I'll try that tomorrow (not now because I want to break my habit of staying up past midnight).

True, the leap interval doesn't have to be just one day. You could even have a 28-day leap month (every 22-23 years) so that every year could have both a whole number of weeks and equal sized quarters.

Speaking of religious significance, someone has pointed out that a 12×30 day calendar achieves Julian-level accuracy by having 7 leap months every 40 years.