I've come up with some improvements of his scheme. Firstly I reduce jitter, by having the intercalation more evently spaced and secondly I correct the Metonic cycle by truncating at the beginning, so losing a saltus lunae, therby having a one day correction at truncation instead of a two day correction.

Tom found 16 possible places to add a leap day to the tree calendar at or next to a cancelled sacrifice. The cancelling of the sacrifice provides space for the leap day. I found a 17th such place.

Tom classified them into 4 types:

- Type 1: leap day and cancelled sacrifice 0u+ of tree month
- Type 2: leap day 0f and cancelled sacrifice 1u+ of tree month
- Type 3: leap day and cancelled sacrifice 29u+ of tree month
- Type 4: leap day 29f and cancelled sacrifice 28u- of tree month

GN Month Date Rune Type Pass 1 B 0 U 1 12 2 T 0 U 1 1 4 F 0 f 2 2 5 M 29 f 4 3 7 F 29 U 3 4 7 H 0 f 2 4 8 R 0 f 2 5 9 B 0 f 2 5 10 H 29 U 3 6 10 T 0 f 2 6 11 R 29 U 3 7 13 T 29 U 3 8 13 M 0 f 2 8 15 F 0 U 1 9 16 M 29 U 3 10 18 H 0 U 1 11 19 R 0 U 1 12I decide that only one cancellable sacrifice is needed per pass. Using two would add more complication for little gain through jitter loss.

So I select one cancellable sacrifice per pass by
preferring type 1 and 3 to type 2 or 4
and preferring Elder (R) to Birch (B).
In all cases the cancellable sacrifice of each pass is selected
when there are several.

Also eliminating Birch removes the need for that long note at
the end of Tom's version.

The remaining 12 leap days are

GN Month Date Rune Type Pass 2 T 0 U 1 1 4 F 0 f 2 2 5 M 29 f 4 3 7 F 29 U 3 4 8 R 0 f 2 5 10 H 29 U 3 6 11 R 29 U 3 7 13 T 29 U 3 8 15 F 0 U 1 9 16 M 29 U 3 10 18 H 0 U 1 11 19 R 0 U 1 12All but three are either type 1 or type 3, where the leap day coincides with the cancellable sacrifice. The other three are at passes 2,3 and 5.

The 12 passes are almost evenly spaced, so are these 12 selected possible leap days. Each comes 20, 21 or 22 tree months after the previous one.

Tom uses a 57-year cycle of 14 leap days, which is eventually corrected (after 6 57-year cycles) by a truncate Metonic cycle. Each 57-year cycle is made up of 3 Metonic cycles A, B and C, where A and B have 5 leap days each and C has just 4 leap days. My improvement sticks to this idea, but spaces the leap days more evenly, then Tom's.

In particular Tom has a leap day at the start of his B cycle
which is just *one* pass after
the last leap day of the preceeding A cycle,
which in turn is just two passes after the previous leap day.
This would cause considerable jitter.

I suggest having a 57-year cycle of leap days with the following passes:

3 6 8 11 1 4 6 9 12 2 5 7 10 12They form 19-year cycles as follows:

B 2 5 7 10 12 C 3 6 8 11 A 1 4 6 9 12The resulting leap days over the A,B and C cycles are

Cyc/GN Month Date Rune Type Pass B 4 F 0 f 2 2 B 8 R 0 f 2 5 B 11 R 29 U 3 7 B 16 M 29 U 3 10 B 19 R 0 U 1 12 C 5 M 29 f 4 3 C 10 H 29 U 3 6 C 13 T 29 U 3 8 C 18 H 0 U 1 11 A 2 T 0 U 1 1 A 7 F 29 U 3 4 A 10 H 29 U 3 6 A 15 F 0 U 1 9 A 19 R 0 U 1 12Only three of the 14 leap days that occur in the 57 years do not occur on the cancelled sacrifice. If the trucated (Deficient) cycle were formed from the last 11 years of the C cycle, it would have 3 leap days. We only want 2 leap days. Therefore I take out the 8 years with Golden numbers 4 to 11 inclusive. The effect of this is as though the saltus lunae in year 4 were put into reverse.

The leap days of the D cycle and neighbour leap days are thus:

Cyc/GN Month Date Rune Type Pass C 18 H 0 U 1 11 [ D 4-11 skipped ]--------- [ 5 passes 2-6 skipped ] D 13 T 29 U 3 8 D 18 H 0 U 1 11 B 4 F 0 f 2 2The first leap day of D comes 4 passes after the previous leap day. This large gap is a correction of the 57-year cycle, where intercalation is a little too frequent (giving mean year of 365.2456... days).

The complete 334-year cycle is made up of Metonic cycles thus:

BCA BCA BCA BCA BCA BCDFor the complete arithmetic of the 334-year cycle. See Tom's article

*Karl Palmen* November 2003