Properties of the Meyer-
Palmen Solilunar Calendar
by Karl Palmen
First published 1999-04-29 CE / 102-25-02-15 MP

### Introduction

Recently I came upon a solilunar calendar invented by Denis Elliott which uses a 1689-year cycle of which 622 years are long, just like Peter Meyer's Goddess Lunar Calendar of 1998. I suggested to Denis Elliott an enhancement of the rules for his calendar which would improve its tracking of the seasons and in my opinion make it simpler.

This rule can be generalised to any calendar that has a Y-year cycle with L long years such that

1. Y-L years have 354 days,
2. M years have 385 days and
3. L-M years have 384 days

I first told the CALNDR-L mailing list about this class of calendars on 24 February 1999 in A class of 354:384:385 Solilunar Calendars. I now refer to such calendars as YLM calendars.

In March 1999 Peter Meyer invented a YLM calendar with Y=6840, L=2519 and M=1328, which is called the Meyer-Palmen Solilunar calendar (MPSLC). It has other properties besides being a YLM calendar that make it worth discussing.

### The Rules for YLM Calendars

1. The year y years after a given year is long (i.e. has 384 or 385 days) if (y.L) mod Y < L. All other years have 354 days.
2. A long year y has 385 days if and only if ([(y.L)/Y].M) mod L < M , where [x] is x rounded down to a whole number.
Y, L and M are constants. For a solilunar YLM calendar (L/Y)+12 must be close to the mean number of synodic months in a tropical year and (354.Y+30.L+M)/(12.Y+L) must be close to the synodic month in days.

For the nth long year after the given year, the expression [(y.L)/Y] in rule 2 is simply n.

### YLM New Years and Seasons

On 1 February 1999 I sent to CALNDR-L a note Lunar Calendar New Year Drift Against Seasons in which I gave a simple method of estimating the drift of the new year of such a calendar against the seasons.

For an n-year period of l long (13-month) years in a calendar with a cycle of Y years of which L are long, the number of months a which the calendar drifts ahead of the seasons over the period is given by

a = n*L/Y - l

Because each year must have a whole number of months, the best a solilunar calendar can achieve is always to have the absolute value of a less than 1 (in practice the maximum is (Y-1)/Y if L and Y have no common divisor). A calendar that does achieve this can be considered to have optimal intercalation. All YLM calendars have optimal intercalation.

However, this concept of optimal intercalation assumes that all months are equal in length. Since this is not the case, the range of new years may be slightly more than a month, rather than slightly less. This is shown elsewhere for the MPSLC. This range could be reduced a little by changing the month lengths to some non-YLM rule.

### Year-to-Year Use

I'd expect that the MPSLC could be used as follows year-to-year, and a similar usage could be applied to any YLM calendar. Initially for a long year the two remainders would be worked out according to the two rules. Then for each year the first remainder (from rule 1) would be incremented by adding L mod Y to last year's first remainder. For each long year the second remainder (from rule 2) would be similarly incremented by adding M mod L to the last long year's second remainder.

The table below shows how it could be done to calculate the MPSLC year types and hence new years for the next 19 years.

The first remainder is the number obtained by rule (i) and the second remainder is the number obtained by rule (ii) in the definition of the MPSLC.

 ```Year    1st Remainder  2nd Remainder  Length   New Year 102-25    335 (quotient=2263)  97      385     1999-03-17 102-26        2854                     354     2000-04-05 102-27        5373                     354     2001-03-25 102-28        1052           1425      384     2002-03-14 102-29        3571                     354     2003-04-02 102-30        6090                     354     2004-03-21 102-31        1769            234      385     2005-03-10 102-32        4288                     354     2006-03-30 102-33        6807                     354     2007-03-19 102-34        2486           1562      384     2008-03-07 102-35        5005                     354     2009-03-26 102-36        0684            371      385     2010-03-15 102-37        3203                     354     2011-04-04 102-38        5722                     354     2012-03-23 102-39        1401           1699      384     2013-03-12 102-40        3920                     354     2014-03-31 102-41        6439                     354     2015-03-20 102-42        2118            508      385     2016-03-08 102-43        4637                     354     2017-03-28 102-44        0316           1836      384     2018-03-17```

### Long and Short Years of the MPSLC

The MPSLC has interesting properties in its arrangement of long and short years, not only because it's a YLM calendar but also because its complete cycle is one month short of 360 Metonic cycles. One Metonic cycle is 19 years of which 7 are long.

The MPSLC groups years into cycles of 60. I find it better to group the years into cycles of 360 (=6x60), so that cycle 0 consists of six 60-year cycles numbered 0 to 5 inclusive. This makes the year 102-25 into 17-025. Also it is useful to put the first remainder in such a form, but having units going from 0 to 359 rather than from 1 to 360. E.g., the year 17-026 has remainder 2854 = 07-334. Except in the case of year 00-001 which has remainder 06-359, the 360s part of the remainder determines whether a year is long or short. It is long (with this exception) if and only if this first part is less than 7.

It can be shown that the remainder of year y of 360-year cycle c is

(11.c+7.y mod 19) — the number of 360s
and (360-y) — the units (0 to 359)

Hence with the exception of year 00-001, a year is long, if and only if

(11.c+7.y - 1 mod 19) < 7

or noting 7x7 = 11 mod 19,

(7.(7.c+y) - 1 mod 19) < 7

or noting 11x11 = 7 mod 19,

(11.(c+11.y) - 1 mod 19) < 7

For each 360-year cycle the long (L) and short (s) years are as follows (noting the exception of year 00-001):

 ```                   1234567890123456789 360 year cycle 00: LsLssLssLsLssLssLss 360 year cycle 01: sLsLssLssLssLsLssLs 360 year cycle 02: ssLssLsLssLssLsLssL 360 year cycle 03: LssLssLsLssLssLssLs 360 year cycle 04: sLssLssLssLsLssLssL 360 year cycle 05: LssLsLssLssLsLssLss 360 year cycle 06: sLssLsLssLssLssLsLs 360 year cycle 07: ssLssLssLsLssLssLsL 360 year cycle 08: sLsLssLssLsLssLssLs 360 year cycle 09: ssLsLssLssLssLsLssL 360 year cycle 10: LssLssLsLssLssLsLss 360 year cycle 11: sLssLssLsLssLssLssL 360 year cycle 12: LsLssLssLssLsLssLss 360 year cycle 13: sLssLsLssLssLsLssLs 360 year cycle 14: ssLssLsLssLssLssLsL 360 year cycle 15: LssLssLssLsLssLssLs 360 year cycle 16: ssLsLssLssLsLssLssL 360 year cycle 17: LssLsLssLssLssLsLss 360 year cycle 18: sLssLssLsLssLssLsLs```

This repeats each era (of 6840 years), of course.

It may be seen that for each new 360-year cycle, each long year is delayed one year, except for one long year that is delayed by two. The 1-year delay is simply due to the fact that 360 is one less than a multiple of 19 (361 = 19x19). So only the delay of two is a real interruption of the Metonic cycle. The exception of year 00-001 occurs when the delay goes over the end of the 19-year cycle.

This delay occurs over the year with the biggest remainder, which at the start of the 360-year cycle is at least 6821 = 18-341. These are the 19 short years that occur 19 years after a long year. They are listed below (using the 60, 360 and 19-year cycles).

 ```    60      360      19     years until next one  1 042-08  07-008  133-01   353 s  2 048-01  08-001  151-12   372 L  3 054-13  09-013  171-04   353 s  4 060-06  10-006  189-15   372 L  5 066-18  11-018  209-07   353 s  6 072-11  12-011  227-18   353 s  7 078-04  13-004  246-10   372 L  8 084-16  14-016  266-02   353 s  9 090-09  15-009  284-13   353 s 10 096-02  16-002  303-05   372 L 11 102-14  17-014  322-16   353 s 12 108-07  18-007  341-08   372 L 13 000-19  00-019  000-19   335 s      14 006-12  01-012  019-11   353 s    15 012-05  02-005  038-03   372 L     16 018-17  03-017  057-14   353 s  17 024-10  04-010  076-06   353 s 18 030-03  05-003  094-17   372 L 19 036-15  06-015  114-09   353 s```
The pattern of intervals between these is
`sLs Ls sLs sLs Ls sLs sLs`
Starting with each such year the pattern of long and short years is identical to the pattern of long and short intervals between them, if you start with year 42-08. It also works with 36-14.

 ```   year   remainder Long/short  1 42-08     6832     s  2 42-09     2511     L  3 42-10     5030     s  4 42-11     0709     L  5 42-12     3228     s  6 42-13     5747     s  7 42-14     1426     L  8 42-15     3945     s  9 42-16     6464     s 10 42-17     2143     L 11 42-18     4662     s 12 42-19     0341     L 13 42-20     2860     s 14 42-21     5379     s 15 42-22     1058     L 16 42-23     3577     s 17 42-24     6096     s 18 42-25     1775     L 19 42-26     4294     s```
The remainders for the next 19 years are 19 less and so on until 48-01 which has a remainder of 6821 greater (in fact it's 6839). This gives the same pattern

`sLs Ls sLs sLs Ls sLs sLs`
Thus the MPSLC has a limited self-similarity.

### Length of Meton

The previous section dealt only with whether a given year is long or not and not how many days it has if long. The extra month of the long year is called "Meton". It has 30 days in a 384-day year and 31 days in a 385-day (monster) year. Rule 2 determines how long Meton is in a long year.

If it is known that this is the nth long year, then Meton has 31 days if and only if

n.M mod L < M

For the MPSLC M = 1328 and L = 2519. Because M is close to half of L (68.5 more) the long years usually alternate in the length of Meton. The value of the second remainder goes up 137 mod 2519 every pair of long years. The alternation is interrupted when adding this 137 mod 2519 causes the second remainder to go down rather than up. This happens once every 17 or 19 long years and a total of 137 times in an era of 6840 years.

The long years can be divided into 137 periods in which they alternate between a 31-day and 30-day Meton for 17 or 19 long years with the first and last having a 31-day Meton. Each period lasts 46, 47, 51 or 52 years and has a mean length of 49.927.. years. A 19-long-year period adds 42 mod 2519 to the second remainder and a 17-long-year period takes 95 mod 2519 from the second remainder. The first long year in each period has a second remainder between 0 and 136 inclusive to ensure that both it and the previous long year have a Meton of 31 days.

Starting with the last year of the era, these 137 periods have 19 or 17 long years as follows:

 ```19,19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17, 19,19,19,17, 19,19,17, 19,19,17, 19,19,17,```
Although the number 2520 is divisible by all numbers up to ten, 2519 is divisible by 11 (11x229). There are 11 rows in the above table. Each row has either 184 or 239 months in it averaging to 229 months (cf. 353, 372 and 360).

This pattern of 19,19,17 is reminiscent of my lunar Yerm Calendar. This has a pattern of 17,17,15 in a single long row. Indeed if you count the first month of the last yerm of the cycle as 0 then month m is long (has 30 nights) if

m.M mod L < M

where M=451 and L=850. All other months have 29 nights.

### A Redundant Feature of YLM Calendars

In looking for the pattern that the length of Meton follows I noticed it is totally independent of the pattern of the long years. I then realised that the L in rule 1 of the YLM calendar definition need not equal the L in rule 2. Instead we could have:
1. The year y years after a given year is long (i.e. has 384 or 385 days)
if (y.L1) mod Y < L1. All other years have 354 days.
2. A long year y has 385 days, if and only if ([(y.L2)/Y].M) mod L2 < M , where [x] is x rounded down to a whole number.
For such a calendar one can find Y', L' and M' to form a YLM calendar identical to the one described.
1. L' is the lowest common multiple of L1 and L2,
2. Y' = Y.L'/L1 and
3. M' = M.L'/L2
This means that one could adjust the MPSLC to take account of the changing rate of the Earth's rotation without always changing which years are long.

I suggest that it may be worth trying Y=353 and L1=130 with suitable values of L2 and M.