mikestar13 wrote: ↑
Sat Jul 30, 2022 9:16 am
I've been rethinking the exact curve. as it stands, TMSA, calculates the percentage this way for squares and trines: x =cos(8 * orb), then p = 2 *( x -.5) but not less than zero. I actually like the way the numbers look with the formula p =cos(12 * x) from ISR, which gives closely similar results at two degrees but fades faster near the maximum orb. Seems a better fit for the observation that class 3 aspects aren't worth bothering with unless there are few/no closer aspects.
Let's see.. reconstructing... reviewing what you probably already know... Cosine runs from +1 at 0° to -1 at 180°, then back to +1, so the entire 360° loop has to be fit into 30° for trines, squares, and sextiles, meaning it has to be multiplied by 12. It then runs from +1 at 0° to -1 at 15°, which automatically drops the "below 50%" into a negative number (after 7.5°). For "manifested effect," it runs from 0% at 7°30' to 100% at 0°00'. A 3° orb is the convenient and interesting 80% level, it's at 90% until just past 2°, and - as you say - starts to taper very fast after that. It crosses the pivotal 50% mark at exactly 5°00' and is nearly gone by the time it hits 7°.
The point I'd theoretically attach to "partile" is the point at which the score rounds to 100% (0.995 or higher) of the whole
+1 to -1 length, i.e., the above value +1 then /2 is > 0.995. That's about 0°41'. Another way to say this is that the value first calculated above (cos(12x)) has a value of 0.990 or higher.
As an idle musing, not as necessarily germane to this post: I prefer the 12x form but where the +1 down to -1 is rescaled to +1 down to 0. This explains to me what's actually happening: 7.5° getting a score of 50%, meaning an "event" is at least 50% likely to happen (i.e., more likely to happen than not - an utterly meaningful "outside orb"). This gives numbers that feel more intuitively right, e.g., 1° is still 99%, 3° is 90°, 5° is 75% (half-way from baseline to perfect), etc. It's what I think of as the real curve. But I don't think this works well for TMSA's purposes because people aren't apt to intuitively understand when an aspect at the very outer fringes of acceptability has a score of 51%. It communicates better if that score has reached 0%.
So, with the 7.5° = 0 to 0°00' = 100 scale, I can see (effective) "partile" at a glance by noting 100% and 99% aspects, Class 1 is roughly (not precisely; doesn't have to be since it's still user-flexible) 80%, Class 2 ends at 50% if I set at 5° or 30% if I set at 6°, etc.
Conjuctions and oppositions use a smaller coefficient (9 in ISR, 6, in TMSA 0.4)
Yes. Now that I have a little temporary Excel file to replicate all of these, it's easy to confirm that 9x widens the curve base to 10°00' for 0%. (Did I really give these in ISR? I'd completely forgotten that I got that precise. But the ideas and use of the curve did come earlier than that.)
[quopte]while octiles use a larger one, inversely proportionate to maximum orb for the aspect.[/quote]
Ah, are you actually taking the outside Class 3 boundary for this? (Or, if it's blank, the Class 2 boundary, etc.?) That makes sense, though knowing it forces me to rethink whether to have a Class 3 octile value (for completeness, even though I don't think it's operative) or leave it blank. (I just looked: I see that I don't have one set.)
So, if outside orb of an octile is 2°, that makes it 1/5 the conjunction orb so you multiply the conjunction multiplier by 5 - using 45, coincidentally - right? It makes the 99% level just outside 0°10', the 80% (approx Class 1 drop-off) a little less than 0°50', the 50% slice at 1°20', and fades entirely at 2°. - I can't say that I can confirm this is exactly right, but the basic feel is good: It matches the idea that "partile" is only a few minutes orb, there's a critical drop no later than 1°, etc.
It does seem to drop more steeply at the very end since these aspects seem quite solid nearly to 2°, so I probably have to add a Class 3 orb just to make the whole curve run all the way out to the end. If I understood you correctly, it means that if I set a 3° Class 3 boundary (0.3 of the conjunction Class 3 boundary), you would then multiply the conjunction multiplier (9) by 3.33, or 30. This gives 99% down to just past 0°15', 80% (Class 1 approx boundary) about 1°16', 50% at 2°, etc. Again, that isn't offensive to intuition.
I have decided we don't need special marks for partility.
I think that's right. With pen and paper, I do mark it - for years I used a flare pen for partile aspects in Class 1 column, and an ordinary pen for the rest. But I think if we train ourselves and others just to look for the biggest numbers - especially to focus on 100% and 99% aspects - we get the same effect.
of course, is ultimately arbitrary. The word means "exact," astrologers began using this long ago to mean "exact to the degree," and the 1° boundary seems so close to the critical drop-off in practice that it fits well. But (aside from the fact that we probably wouldn't treat 1°02' all that different from 1°00'), if this threshold has any real, meaningful significance it is because it shows us a threshold in aspect strength. The goal, then, is to find the best expression of the tapering aspect strength curves. I do think we are so close at the 12x and 9x models above that any perception of error in it borders on tenuous fantasy. It's a very close fit to observation.
I think you should use the straight cos(12x) and cos(9x) equations. (I can't remember the reason for moving from them.)
If I make any change to calculating the strength of conjuctions/oppositions it will be along the lines used for massaging the angularily % for minor angles: alter the input value, not the formula. For conjuctions/opposition this would be orb = (orb - 3°) * 9/14 + 3° for orbs over 3° and then apply the square formula, resulting in identical strengths to 3° but slower fade out (the 3° and 9/14 would be adjusted for different maximum orbs, 9/14 assumes the default 10° for conjuntions and 7.5° for squares -- and assumes a 3° class one orb for both).
Just to be clear, while I think (for major angles) that the foreground zone is probably an exact match for the conjunction curve. - It probably IS the conjunction curve (over the years, these two kept looking increasingly like each other until I looked up one day and saw they'd converged in my head). "Foreground" probably isn't a zoning (as thought most of a century ago) so much as something singled out by the real version of "conjunct an angle." (It behaves like that.) The simple cos(12x) formula
And yes, the minor angles have been a pain. Getting them to feel
right. Maybe if you treat them the same as octiles? That means that the outside orb (3° by default) would provide a scaler - the default (for 3°) being 30x which gives 80% at 1°16' ("something more than 1°" being what feels right), 50% at 2°, and down from there?
I want to see some numbers before I decide, so some serious number crunching this weekend is in my future. I will share numbers in this thread.
My thought is that simpler is better: the straight cosine curve on a multiplied orb. But you've recently had your head deeper in this than I have, so you may be seeing something I'm missing entirely.
If I'm not missing something determinative, it seems that cos(12x) for trines, sextiles, and squares, cos (9x) for conjunctions, oppositions, and major angles, and the same with a weighted multiplier for octiles, inconjuncts, and minor angles is the way to go.