Here is another contribution which might be relevant to this thread topic (mast balance of JR).

Conventional thinking, and basic geometry, dictates a relationship between mast balance and yard angle (for a given halyard attachment point on the yard). 

In other words, if we make a decision as to the sling point on the yard (usually just a little aft of the midpoint of the yard) then on a conventional modern junk rig the ideal relationship between the yard angle and the mast balance is a matter of simple geometry. I thought that this was most elegantly expressed by Paul Th in a post many months ago, when he gave us this diagram:

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A good diagram is worth a page of words, and that's a good diagram.

The significance of the word “ideal” is that if we depart too far from the yard-angle/mast-balance relationship shown in Paul’s diagram, then more effort is required to make the sail drape correctly (all these hauling parrels to drag the battens this way and that). And also, since harmonising the mast-balance and the yard-angle has the effect of reducing to a minimum the “halyard angle” (a useful concept coined by Arne) – this also means less effort required to hoist the last metre or so of the full sail to its highest point.

Different planforms (such as fully fanned, Reddish types) might require a somewhat different diagram, but in the end, gravity and geometry will dictate the “ideal” proportions and relationships. The closer we can get to the ideal geometry, the less requirement for hauling forces in the horizontal direction to make the sail drape correctly, and the less effort required to hoist that last metre or two of sail.

There are a lot more things to consider in the design of a good sail, a business which is better left to people with a proven track record – but this geometric consideration is one of the basic considerations in the design of a modern, conventional junk sail, if it is to behave conveniently.

But here’s the thing: there is always going to be someone who thinks outside the box.

First, I must quote again from Arne, in an earlier post on this thread: “My conclusion is to not be too categorical in the question of running parrels.  My hard rule is that there are no hard rules...”

All of the above is based on the assumption that batten parrels must be soft, and forces on the sail and battens must be balanced out as much as possible, in order for the battens and sail to sit where they should on the mast.

But what if the batten parrels were to be rigid, and to allow no fore-and-aft movement?

The geometric “rules” could then go out the window.

Steve D has done just that with the rig on his Serenity. (see Boat of the Month September 2023 – you can go to https://www.junkrigassociation.org/page-1858517 and scroll down to Sept 2023). Due to the elimination of soft batten parrels, replaced by his rigid “egg-formers” (which owe quite a lot to Paul McKay’s aerojunk variant D-formers) Steve has made a sail which breaks the geometric relationship between mast balance and yard angle .

It is geometrically “all wrong” – yet it appears to be highly successful in almost every respect – and no hauling parrels of any description are required. This is a sail which needs only a halyard and a sheeting system (though when it first appeared I thought it would be improved by the addition of just a standing boom downhaul). The simplest-to-operate junk rig yet seen. Whether the mechanical detail of that rigid batten parrel would be reliable if scaled to a larger size I do not know, and I personally prefer my spanned running batten parrel downhauls – but the thing is, it works almost perfectly on this boat despite breaking the rules of geometry. (I say “almost” because if the mast were a little too short, then the halyard angle might be too large, and then perhaps the sail might require a lot of effort to fully hoist. Perhaps Steve can comment on this).

One last comment – referring to the post in the other thread,  by David D:

“When one reads of THPs and YHPs and running parrels and changing mast balance and the associated adjustments for windward or downwind sailing or reefing, in order to get the sail to set well, it seems the simple uncomplicated junk rig is no longer such a simple uncomplicated thing.”

I understand what David means, and don't disagree. There are enough ropes to pull as it is. However, the junk rig was never “simple uncomplicated”. I believe camber in the sails has complicated things further, but simpler versions are evolving now. In any case, simplicity was never the junk rig's merit (and nor were Western windjammer rigs simple either). The merit of these early rigs was that they could be built with third-world materials, and they could be operated reliably by minimum man-power. A certain amount of complexity is a worthwhile price to pay in order to get a rig which is simple to operate (and requiring less effort). 

Able to be d.i.y.-built with low tech, inexpensive materials, very simple to operate, and with performance at least as good as its contemporaries. That was always, and still is, I believe, the merit of the junk rig.