Solunars by Aldebaran Sidereal Academy

I'm writing this thread to address recent questions about the relationship of TMSA's aspect and angularity strength measurements vs. the convenient (admittedly arbitrary) aspect and angularity classes. Since the only software that gives these numbers is TMSA, I'm placing the thread in this sub-forum.

This information likely exists one or more places on the forum already. Since I couldn't easily find it, I'm sure most of the rest of you can't easily find it - so I'll write it again in one place.

Working on the simple, single guidance that "closer is stronger," I encourage the use of a simple trick Garth Allen recommended for dealing with orbs. Here is how I wrote about it in the current draft of the CSA chapter on aspects:

...I strongly recommend a trick Garth Allen (Donald A. Bradley) suggested (1957) that has been basic to my practice for 50 years: Tabulate a chart’s aspects in three columns of close, wider, and widest orbs. Specifically, list major aspects for a given chart in three columns, those with orbs 0° to 3°, 3° to 6°, and 6° to 9°.

I call these orb ranges Class 1, Class 2, and Class 3, respectively... I also assign to the three classes standard “plain English” adjectives that astrologers often use without clear definition: When I call an aspect “close,” I mean it is Class 1. By “moderate” or “wider,” I mean Class 2. If I call it “wide,” I mean Class 3.

These groupings – and this way of thinking about aspect orbs – will make your astrological life much easier and more fruitful.

Within Class 1, we need one more distinction that astrologers (especially Sidereal astrologers) use frequently: Aspects within 1° of exact are called partile, which literally means “exact.” One could argue that “partile” should be used only for an aspect with 0°00' orb, but that is not how astrologers use it. “Partile,” by convention, means within 1° because aspects this close are mostly indistinguishable from those with a 0° orb. They are functionally exact.

Always notice partile aspects.

Therefore, we have four distinctions of aspect strength based on orb: exact (partile, within 1° of precise aspect), close (Class 1: within 3°), moderate or wider (Class 2: 3-6°), and wide (Class 3: 6-9°).

Partile aspects seem magical with their vivid strength. Class 1 aspects clearly stand out as important and essential, with a sharp drop-off in perceptible importance soon after 3°. At about 6°, the drop-off becomes steep.

The most important message from this is that orbs are fluid. There is nothing absolute about the cut-off points. They are a convenient way to organize the gradually increasing and decreasing strength of aspects and angularities. The second point to understand is that boundaries of 3°, 6°, and 9° are reasonable cut-offs especially for beginners - nothing complicated, easy to remember, and work fine enough. The third point (not actually stated above) is that we can refine these boundaries with more precise measurement to calculate more narrowly when steeper drop-offs occur. A final point is that, while such boundary refinement is indeed useful, it's not a huge deal - aspect strength is fluid and gradual. No matter where you set the Class boundaries for yourself, you need make a big deal if an aspect is a little wider or a little closer.

This thread will layout the thinking behind the aspect and angularity scores in TMSA. I will break the discussion into four phases: (1) Trines, Squares, and Sextiles (2) Conjunctions & Oppositions (3) Major Angles, and (4) Minor Aspects & Angles.

Early in my study of astrology, I became persuaded that tapering aspect strength follows the pattern of a sinusoidal wave - specifically, a cosine curve - as Bradley first hypothesized around 1950 in his book Stock Market Prediction. Although Bradley changed his mind about this at the end of his life, his revised view (that aspects and angularity follow a cycloidal wave) are not persuasive to me. The pace of aspect strength degradation follows patterns marked by significant thresholds in the cosine curve's behavior.

In this first post, I need to lay out the basic theory of matching a cosine curve to aspects. This will apply to the other sections below as well and won't need to be repeated in the next three posts.

The simple premise is that, as the number of degrees two planets are separated moves from 0° to 360°, they move in and out of aspect many times, moving from exactly aspected to (between two aspects) no discernible connection at all, then back toward exact aspect, etc. That is, their aspectivity, or connection, rises and falls many times around the circle. The maximum connection or interaction is at the points we identify as aspects. The minimum connection or interaction is when their separation is somewhere between two exact aspects - in most cases, probably exactly halfway between two adjacent aspects.

Since sextiles, squares, and trines occur at 30° intervals, the widest orb they could have is plus-or-minus 15°. For example, as two planets move away from an exact sextile (60°), their connection (aspectivity) gradually fades until a point halfway between a sextile and square - at 75° separation. It then increases gradually until, at a 90° separation, their connection is again maximum at the square. It then drops off from the square to a minimum half-way between the square and trine (at 105°) and climbs back up to maximum at 120°.

If you do not already have this gradual rise and fall curve in your mind, pause a bit to form the picture. Each of these aspects has a tapering slope something like this picture (or, if the cosine curve is the correct way of measuring, it is exactly like this curve).
The cosine curve runs from +1.0 to -1.0. We can rescale it, and just call the +1 to -1 range 100% down to 0%. (This is just a convenience for us. It doesn't change the shape of anything. In the picture, this is shown as +100% down to -100%. It's the same curve.) Planets are always connected to each other, always interacting, but, for example, at an orb of, say, 13° their interaction is only at 4% of its maximum strength. In practice, this looks like "they are out of aspect" or "they have no connection."

One obvious place on the picture above is where the line crosses 0% at the middle. This is 50% of the aspect's maximum strength (halfway between -100% and +100%). I take this as the maximum effective orb for the aspect. Another interesting threshold is when it crosses the 50% line (75% of the way from -100% to +100%). This looks (on the picture) to be about 5°.

Let me pause and say how I think about these numbers. Specifically, the percentages are all fine and dandy, but percentages of what exactly? One can say "aspect strength," though I think of it a little differently, which may only be a convenient way for me to think about it. I think of the aspect score as odds that something will happen or result from the aspect. At 100% score, there is a 100% chance the aspect will express itself. At a 0% score, there is 0% chance. In between, there are greater and lesser chances. At 50% from the lowest to the highest - where it crosses the zero line in the picture above - we cross the line where it is more likely than not that the aspect will express (50%-50%). Those still aren't very good odds. A closer aspect has about a 75% chance of expressing. The closer we get to 0°, the closer we get to 100% certainty the aspect will express.

This, at least, is a useful mental device for thinking about the aspect's curve.

If scale this curve for sextiles, squares, and trines from 0% at the bottom to 100% at the top (instead of -100% to +100% as in the picture), the arbitrary 3° / 6° / 9° boundaries for the classes are crude approximations. They are equal (respectively) to strength scores of 90.5% (not a bad cut-off), 65.5% (about 2/3 strength), and 34.7% (about 1/3 strength, and surely too low to take seriously).


From the beginning, I've taken the 50% (7°30') as the full cut-off for these aspects. (Think of this as the range within which manifest expression is likely.) The next practical threshold is at 75% (more than half of the manifest strength above 50%), which is also where the drop-off curve begins its steep decline (look at the slope of the curve): This score is reached exactly (exactly!) at 5°00'. Another threshold that seems meaningful is at 90% overall strength, which is reached at 3°09' (call it 3°: there is no reason to fuss with setting a personal breakpoint at 3°09'). Finally, the score at the seemingly magical 1°00' partile threshold is 98.9%, crossing to 99.0% at 0°58' (call that 1°), and passes 99.5% (i.e., rounds to 100%) at 0°41' (again, about a degree).

Therefore, in practice, I set my personal boundaries not at 3° / 6° / 9° but at 3° / 5° / 7.5°. Partile is 1° by convention, and this fits the cosine theory.

In principle, this is exactly what TMSA is showing. However, there is one more conversion we applied to make what seemed a more usable program. It's based on a cosine curve reaching 15° either side of the exact aspect (i.e., on a 30° base).

While the above numbers represent aspect strength from a 0% at a 15° orb up to 100% at a 0° orb, we thought this might not be convenient in TMSA. A table of aspects probably would be more meaningful if it reached 0% when it dropped off the chart. Therefore, we rescaled this again so that the scores show the percentage of strength above 50% (above manifestation level). This is a simple math trick: Add 1 to the score and divide by two. This is how the final TMSA numbers are calculated:

Conjunctions and oppositions are treated differently, getting larger orbs.

Think about it: Of major (Ptolemaic) aspects, the distance from a conjunction to the nearest aspect (the sextile) or from an opposition to the nearest aspect (the trine) is double the distance between sextiles, squares, and trines. The orbs, however, do not seem to be double (in practice), though they DO seem to be larger. (This may be part of the reason conjunctions and oppositions seem stronger: at the same orb, they are stronger than, say, a square.)

Based on experience (originally sorted out in the '70s and experience seeming to confirm it in practice ever since), the Class 3 drop-off ("maximum orb") seems to be 10° instead of 7.5°. This means that the "base" of the cosine curve is 40° wide instead of 30° wide.

Widening the curve, we find that the arbitrary 3° / 6° / 9° boundaries for the classes aren't bad. They're actually better than for the smaller aspects. 3° is 95%, 6° is 79%, and 9°is 58%. Not bad, but we can do better: With a 50% threshold at 10°00', the seemingly meaningful 75% mark (more than 50% of the manifest strength above 50%) is reached at 6°40'. (I round this to 7°, which always felt like about the right day-to-day orb anyway. The 90% point is 4°05', and I tend to use 4° as a "close" category for conjunctions and oppositions based on experience. The partile level still hovers around 1°, with 99% crossed at 1°17' and 99.5% ("rounds to 100%) at 0°54'.

Therefore, in practice, I set my personal boundaries for conjunctions and oppositions at 4° / 7° / 10°. Partile is 1° by convention, and this fits the cosine theory.

In principle, this is exactly what TMSA shows. However, as above, there is one more conversion TMSA applies to make the program easier to use.[/quote]

Rescaling the curve to run from 0% at 10° (so that the scores show the percentage of strength above 50% or above manifestation level) gives the following numbers:

In principle, contacts with major angles (horizon and meridian) - meaning, the foreground zones - follows the same equations as for conjunctions and oppositions. That is, the outside boundary is 10°. With 50% of maximum strength of the foreground curve falling at 10°00', the seemingly meaningful 75% mark is reached at 6°40' (which I round to a 7° cutoff for Class 2 angularity). The 90% point is 4°05', though I tend to use 3° as a "close" category for angularity based on experience (it's the 95% level).

In practice, I set boundaries for conjunctions with Asc, MC, Dsc, and IC (in prime vertical longitude) at 3° / 7° / 10°.

So far, this sounds like the aspect rules for conjunctions and oppositions, yes? But for final display, it gets massaged so that it looks like a 4x difference in strength for the same orbs. Here is why: Both conjunction/opposition curve and the angularity curve are a cosine that reaches 50% at 10°00'. However, for aspects we thought the most meaningful display would be the percentage above 50% (meaning: the relative strength of a manifest aspect, above 50% strength). This is accomplished by adding 1 and dividing by 2.

For angularity, we want something different, though: We have not only a foreground curve (running from it's own 0-100%) but also a background curve (running from MINUS 0-100%). While aspect scores in TMSA are based on the range 50-100% (50 points spread out over 100 points), angularity is based on the range -100% to +100% (or 200 points spread out over 100 points). This makes the final numbers look different.

For aspects, anything "out of orb" doesn't appear at all. However, for angularity we want not only "out of orb" foreground (which is part of middleground) but also the whole descending background curve. This is a different puppy! The goal is to have the foreground-middleground boundary be 75% and the background-middleground boundary be 25%. (BG if under 25, FG if over 75.)

The scores for specific orbs - following the same curve - come out like this:

Minor aspects and minor angles work on their own highly customized formulae for the program. Some of these are experimental (for example, Mike and I concluded the heavily kludged minor angle curve innovation didn't work well and Mike intended to revert to the earlier, simpler formula). I'm not going to go into detail on this, but just to say enough to make the basic point of what we expect for these aspects.

From observation, minor angles have orbs that are fully active within 2° and extend to 3°: These seem to be (approximately) the viable Class 2 and Class 3 boundaries except that I can't validate ecliptical squares to MC past 2°. Nonetheless, TMSA makes no distinction between which minor angle gets the orb, so occasionally you may have to exclude a wider square to MC that TMSA detects.

Minor aspects - by which we primarily mean semi-squares and sesqui-squares - also (from observation) are quite effective to 2° and more vivid within about 1°. In theory, I don't object to the idea that there are wider (Class 3) octiles out to 3° that haven't been detected just because they are so weak. (One rarely relies on Class 3 aspects anyway.)

A cosine curve that drops below 50% at 3° hits 0% at 6°. This gives a 12° base to the curve.

Scaled to 0% at 3°, the Class 2 threshold (75%, or 50% of the value above 50%) is exactly 2°00' - just what we'd expect. The Class 1 threshold (80% on this scale) is 1°14', supporting the observation that octiles beyond 1° still feel strong. The partile level (either 99% on original curve, or 99.5% so that it rounds to 100%) is at 0°16-0°23'.

This all feels quite right compared to experience, seeming to confirm that the curve itself is correct. The 12° based curve (giving a 3° maximum orb on aspect or angularity) gives the following table.

I just skimmed the thread, but if I'm understanding it correctly... it seems like a planet square (say) Asc at exactly 1* is equally strong as a planet conjunct (say) Asc at exactly 3*? and the same for 2* square and 7* PVL conjunction?
(General astrology question, not TMSA specifically)

That's the working model, yes. See the CSA section chapter sample on Finding Aspects (Ch 12) in the Expert section for working out these things minutely.
https://solunars.com/viewtopic.php?f=70&t=7862#p55769