Barleywine
However Ibn Ezra uses a very convoluted system, which looks at the degrees of the Sun, Moon, Ascendant, Part of Fortune and pre-natal szygy. For each one of these five degrees we award points for rulership, exaltation, triplicity, terms and face - so planets who rule these degrees or who have them as a dignity begin to accumulate points. The planetary hour ruler and the planetary day ruler get scores as well, then each planet gets a score for its house placement,etc. At the end of the point scoring we tot up the points for each planet and the winner (Almuten) is the Almuten of the Chart. This process is not identical to the essential and accidental dignities and includes additional considerations and ignores others.
Morinus does not use Ibn Ezra's method, though this is by far the most common method quoted.
As always, your thoroughness is appreciated. I don't necessarily want to DO it by hand (I'm delighted to leave that to the computer), I just want to know WHAT it's doing so I can make sense of the result. If it means getting familiar with more convoluted methods, so be it. Can you point me to a reliable source to get started?
I see what you're saying about the dispositor chain being more likely to complete with fewer variables in the chart. Let me try my chart to see what happens:
Jupiter in Sagittarius (in it's sign of rulership but dispositing nothing)
Saturn in Leo dispositing Moon in Capricorn
Moon in Capricorn dispositing Sun, Mercury and Venus in Cancer
Mercury in Cancer dispositing Mars in Virgo
Sun in Cancer dispositing Saturn in Leo
This one is a bit of a "closed loop" (what the software coders used to call a "do-loop" that has Saturn doubling back on itself) with Jupiter left out in the cold. It could also start with Sun or Moon and create the same kind of loop excluding Jupiter.
Here's a random thought, based on nothing but idle conjecture since I've never seen anything written on it. I was wondering whether a planet's sign of exaltation might bring it into the dispositor mix. For example, Jupiter is dignified (exalted) in Cancer, which would give it sway over Sun, Mercury and Venus in Cancer, Sun covers Saturn in Leo, Saturn covers Moon in Capricorn, and Mercury covers Mars in Virgo. You mentioned earlier that Jupiter is more important in my chart that it seems at first. Saturn and Moon are in detriment, Mercury and Venus are combust and Mars (although it rules the Ascendant) is peregrine, leaving Jupiter in a relatively positive position. However, Morinus gives chart almuten to Mars with Venus second, followed by Moon. Jupiter and Saturn are about even with Sun and Mercury completely out of the running. It must be all of that convolution you described .
EDIT: Here's some input from the "other" traditional astrologer I communicate with.
"A planet which disposits, disposes of, or rules another because the other is in one of its areas of dignity. For example, when Jupiter is in Taurus, Venus, the planetary ruler of Taurus, disposits Jupiter. Planets can also be disposed by the lesser dignities, so the dispositor of Jupiter by exaltation is the Moon which is naturally exalted in Taurus.
The last sentence seems to answer my question about whether Jupiter can "disposit by exaltation." If it can, that ties it in to everything else through its exaltation in Cancer.