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I had to revisit Betza notation for the purpose of making a Chu Shogi rule page on my website. of course Chu Shogi, with its multi-step Lion, poses some unique problems not yet dealt with. In arriving here I also saw Joe's appeal for a revamping of this notation. Let me start with some general remarks after reading the page and previous discussion:

• I do support using C and Z in stead of L and J.
• The page suggests that leaps larger than 3 should be written as combinations of shorter leaps in brackets. Like [WC] for the Giraffe's (4,1) jump. This is an ambiguous prescription for encoding, however; it could just as well have been written as [DN] or [HF].
• The same notation is useful to indicate an intermediate square when this really matters (rather than just making up for the lack of letter definitions for larger jumps). The case of [WF] for the Mao was discussed.
• I think the logic of the system dictates the 'move modality' (as opposed to directional) modifiers to be outside the brackets, i.e. n[WF], as [WF] is a single leap, similar to N. But nN is ambiguous as to where it could be blocked. I think it would be undesirable to allow modifiers inside the brackets. The hypothetical n[WA] case should IMO decribe a move that can ONLY be blocked on the W square. If the A move should also be blockable, it can be written as n[WF2]. So a path specification should mention all squares that are to be considered in the path of an oblique leap, that are unambiguously implied by linear leaps such as D, H, A or G.
• Modifiers usually apply to all these intermediate squares. We also have no notation (as yet) for a H leap that can only be made when the W square is occupied and the D square is empty, and trying to create it with the bracket notation seems to overload the latter.
• The inward-outward problem for such combined leaps is not yet solved. It stands to reason that the F part of [WF] should by default only mean outward. In fact, composite leaps can always be decomposed that all components fall in the same 'octant', and each atom has only one move falling in that octant. If it would be needed to move outside the octant to probe conditions on an intermediate square, the path would be perceived as bent. To describe bend paths, we could adopt the convention that the first step is imagined to lie in the forward direction to define the frame of reference. Any subsequent leap that is not in that exact same direction can then be prefixed with a modifier. E.g. n[DsW] for a hypothetical version of the Mao, that can be blocked on the D squares (but jumps over the W squares, or it would have been n[W2sW]).
• This creates a problem when you want to refer an oblique leap to an earlier one of the same kind (as in the Rose, which makes all Knight jumps in ever changing directions). Because in stead of a pair of most-forward moves, the frame of reference is rotated such that there now is a single one. So notations like ff and fh are no longer applicable. This can be easily fixed by using f, fr, r, br, b, bl, l, fl for the eight directions, though. And fs could then mean fr + fl etc.
• I would propose to resolve the ambiguity in encoding longer jumps by using non-functional intermediate squares by the requirement that only the final jump can be oblique (i.e. use [WC] rather than [CW] for the Giraffe), and that the first step should be as short as possible ([WC] rather than [DN] or [HF]). Orthogonal leaps count as smaller than diagonal leaps of the same number of squares (i.e. use [DA] rather than [AD]). Now some new stuff:
• I do not like the p (Pao) and g (Grashopper) prefixes very much. They seem to be very specialized. I Fairy-Max I use a much more general system to describe hoppers. It assumes they have a primary and secondary move, the latter becoming active after the hop. This is very flexible. The Betza system allows something similar: mRcj[RR] = Canon j[RW]j[BF] = j[QK] = Grasshopper This is an extension of the bracket notation to allow the brackets to contain slider/rider moves in case it has a j prefix. Normally the use of a sliding first step would lead to an undefined intermediate square, but the j prefix could make it to mean the location of the first obstacle. So the Canon slides to the platform with a R move, then switches to the secondary move, which is also Rook-like (and by default in the same direction), but the over-all move can only capture. The Grasshopper could do only a single step from the platform, i.e. its secondary move is W or F (but it can both capture and non-capture with the over-all move). Bent hoppers could change direction at the platform. E.g. j[RF] would be a 'Bifurcating Grasshopper', landing on two squares diagonally behind the platform. Mats' bifurcators do not bifurcate at the platform, but on the square before it. A new modifier could be introduced for this (and 'y' suggests itself): y[RB] would be a bifurcator that moves upto the obstacle as a Rook, and then continues as a Bishop in either direction. And y[QfsQ] = y[RB]y[BR] Now the Lion problem: This is related to other leaps that involve intermediate squares. What makes it fall outside the existing system is that it is a 'capture and continue' square, which is exactly what makes the Lion and related pieces special. In Chess it is sort of implicit that your move ends as soon as you capture. There could be a modifier other than m or c to encode such a capture, and 'e' suggests itself. So eD would mean a move that captures en-passant on the implied intermediate W square (and could both move to and capture on the D square). A triple jump only capturing on the second square and completely ignoring the first would be e[DW] in the systematics we used before. fne[FF] = Checker (e.p. capture on the F square, A square must be empty) Another thing that I sorely missed is a symbol for arbitrary bending of the path. If [KK] would by default mean a second K step in the direction of the first, the notation for the Lion move would become very cumbersome, as you would have to write out all possible bends separately. Like e[WF]e[WsW]e[WbF]e[WvW]e[FfF]e[FvW]e[FsW]e[FbF] If 'a' would mean any/all/arbitrary, you could simply write e[KaK] meaning a K step in any direction after the initial K step to the e.p. square. So we would have KNADe[KaK] = Lion RbBfe[FvF] = Soaring Eagle BsbRf[WvW] = Horned Falcon