Solunars by Aldebaran Sidereal Academy

PERSPECTIVES ON ASPECTS
by Garth Allen in American Astrology (from "Your Powwow Corner," AA 9/57, later reprinted & somewhat expanded in AA 4/74)

Every astrological teacher is asked one particular question more often than any other and with a maddeningly dependable frequency: "What size orbs do you allow" Any answer, even if culled from textbooks of any school or historical vintage, is an expression of opinion - unavoidably so in the absence of a full-bodied statistical report on the subject. The newer student of astrology should not be discouraged by the apparently contradictory range of answers he may have collected from a series of interviews or from combing standard books. There is a logical solution to the problem of orbs if you stop assuming that there must be a rigidly drawn boundary like an invisible retaining wall with vibrations on one side and nothing on the other.

In fact, apart from the mechanics of the subject, there is no real problem at all where orbs are concerned if you correlate the answers you have collected with the sources of your answers. As a fairly reliable rule, you will find that the more technically-minded the source happens to be, the smaller will be the orb proffered. If the teacher you approach tends to be rather loose and liberal in his or her delineations, you can expect a king-size orb for an answer - and little or no distinction between exact, close and wide aspects. For example, should the teacher unblinkingly blame a sprained ankle on a transit of Uranus through the 12th solar house, you can bet that teacher has a no-holds-barred, anything goes approach to delineation as a whole, and will have no qualms about using a nine-degree departure in chart comparison studies to explain a case of love at first sight. (Leave by the nearest door when you run across this kind of cookie.)

At the other extreme is the phrenetic hairsplitter who uses not only every funny little minor aspect ever proposed, but some that may not even have been invented yet. The orb limit you'll hear him insisting upon as valid will be in keeping with this mathematical agoraphobia. However and fortunately, the vast majority of practicing astrologers use a sensible, medium-sized, somewhat elastic orb, evaluating a combination of planets in terms of the importance of the components. (The "astrodyne" concept of evaluating chart components represents an effort to weigh each factor in terms of relative impact in a reasonable way.)

[Emphasis added. - JAE]
The Sun and Moon, in this logical view, usually enjoy a wider reach in their effectiveness. A wide aspect between planets in angular houses is unquestionably more important than a closer aspect between planets in weak places. Then there are wide "straddling" aspects, often found in triple conjunctions where the center planet is six degrees from one planet, say, and four degrees from another in the other direction. The two outer planets of the trio are not close enough by themselves to command respect as a conjunction, yet the three together have to be considered conjoined since they fence in a crucial zone of the chart and cannot rightly be interpreted as free agents. The old phrase "translation of light" can be used with a variety of scattered wide aspects which become knit together in this way through a chain of imaginary orb circles on the celestial sphere.

The strength of an aspect is called its amplitude, and while it would seem natural that the power of an aspect should be directly related to its closeness to exact phase, it is quite unscientific to assume that this relationship exists. Why? Because physicists know of no forces or energic phenomena in the universe having linear variation of intensity. Thoe of you with a flair for science know what this means. If astrological forces are forms of natural energy, albeit not recognized as such yet, then an aspect 1°20' from partile cannot simply be classed as "twice as strong" as an aspect 2°40' from partile or three times stronger than an aspect 4°00' out of exact phase.

There is probably a rough correlation of this sort, but only a very rough one and it may be a long time before the actual state of affairs is clarified through research in the future. The one certainty that emerges from our present understanding, though, is that the power of an aspect tapers off from its peak in a curvilinear fashion until it becomes so weak at a fair distance from partile, as to be unappreciable. [Emphasis added.] One thinks inevitably of the law of gravity in this connection: every movement of your finger gravitationally affects the most distant quasar ("Every particle in the universe attracts every other particle with a force... blah, blah"), even if "unappreciable." Having visualized the curving slope-off, you will find it easy to see why we personally are suspicious of all fixed-orb theories. That is, we can't agree to any pushbutton scheme wherein there is a point where an aspect turns on and off, in operation one moment and then out of commission when the fictitious line is stepped across.

In reply to endless inquiries, the writer's personal approach, whether to the study of of a single chart or to the analysis of a series of charts with some particular goal in mind, is to tabulate the aspects first in easily spotted categories. The average chart contains 5 or 6 major (classical) aspects within 3°00' and these I call first order aspects or aspects of the first order, since they are undoubtedly the strongest. I have run across many cases where a chart contained only one or two first-order aspects and recall at least one instance where there were 16 aspects closer than three degrees,so the average of 5 1/2 per chart is just that - an average.
[NB - "Major aspects" is a technical term specifically meaning the five Ptolemaic aspects. That's what he is referencing here. The average is obtained by considering that these five aspects touch eight points around the zodiac; 3° on either side is 6° x 8 aspects = 48° out of a possible 360° that are in aspect to a specific planet; 48/360 = one time in 7.5; multiply this by 45 possible aspect planet-pairs, and get 6.]

Aspects between 3°00' and 6°00' I call second-order aspects, and those from 6°00' to 9°00', third-order aspects. There is no special reason for using these terms, these orders, they being only convenient in making sure I pay first attention to the closest couplings before bothering about the wider ones. Whether others adopt this nomenclature or not is not too important but this is my personal hangup until someone offers a better mousetrap.

[NB - For economy of syllables, I call them Class 1, Class 2, and Class 3 aspects, or close, moderate, and wide. - JAE]

As for the three-degree divisions, they too are somewhat arbitrary, although they were found to provide the most practical dividing lines for our previously published statistical studies on the charts of athletes, artists and confirmed bachelors. We found that when an aspect pair of planets stood out conspicuously in charts with a common denominator, it was the first-order category that did the standing out. Among professional athletes, for instance, the test ratio reached its peak at 2°50' from partile, and among the bachelors the significant formations attained their height of incidence within 3°35'. The drop-off in significance beyond four degrees is startlingly steep, and in the statistical studies of specific events there is invariably a glaring absence of cases past the four-degree boundary, justifying the general real that "if it doesn't happen within three degrees, it ain't agonna happen!" In the collective analysis of suicides, the margin was even narrower.

A safe remedy for the confusion you feel about orbs is to dispense with even the foregoing scientifically suggested borderlines, because the statistics show only where aggregate significance ends and not where an actual influence may cease to be perceptible in the individual case. A fruitful viewpoint to adopt is to think of the strength of an aspect in terms of its uniqueness in everyday experience. How many people in your community were born under such and such a formation? The ratios for what we call the full-circuit planetary pairs [aspects involving at least one planet that moves faster than Jupiter] are easily calculated. Full circuit pairs are those 32 planetary pairs which complete the full cycle of major aspects, from conjunction back to conjunction, in less than 2 1/2 years. Incidentally, the rarest of all aspects in astrology is a Sun-Mars opposition. (Why this should be so will become readily clear if you inspect your ephemerides and note that the opposition always occurs with Mars retrograding at its fastest pace, hence zipping through the aspect in a comparatively brief time.)

Where transiting orbs are concerned, if life were simple and astrological principles simpler yet, almost every event would take place according to the partile timetable. Events tend to coincide with partile transits even in this complicated cosmos for the simple reason that the sheer intensity of close aspects mathematically singles out a person from the population for a particular experience. They tend to, that is. But the supreme fact of astrology is that the mundane structure of any horoscope has a modifying effect on the intensity of all planetary forces [emphasis added]. The hampering pressures are greatest in those zones of the chart we call succedent and cadent. It is only in the zones centering around the three basic great circles (horizon, meridian and prime vertical) that the inhibiting pressures are relieved or removed [emphasis added]. Therefore, it is no mystery why so many major events in life take place before or after the time of partile aspect.

A promised or indicated event can more easily come to pass when the aspect is quite platic but in a strong position in the horoscope than when it is exactly partile and weakly situated. It is when the two conditions, closeness and angularity, occur simultaneously that a crisis is virtually inevitable. [emphasis added]

[This is where the heavily edited reprint diverges from the original. Both still have a lot more, but they are mostly the same until this point. - JAE]

A supplement, in fairness and explanation...

The article above is identical with the way it first appeared in American Astrology in the 1960s. Soon before his death, though, Garth Allen reprinted it with a new section at the end. I have left this off.

Notice that he speaks of aspect strength tapering in a curvilinear fashion. Under his own name, Donald A. Bradley, the author of the above wrote a book called Stock Market Prediction in 1950 or '51, in which he introduced the idea that a sine curve is the proper way to measure varying aspect strength by orb. (I still think this is the correct approach.)

In the last year or two of his life, Bradley changed his mind. Gary Duncan came back from his last in-person visit with Bradley in Tucson, and told me, "He now thinks aspect strength curves look like this big splash." He was referring to a different wave form, a cycloidal wave, which, drawn out on a graph, looks more like a big water splash, with sharp, acute peaks instead of the round-topped crests of sine waves. In Bradley's reprint of the above article, he added this at the end, with a formula for calculating the theoretical amplitude of an aspect for any amount of separation between two planets.

I believe he moved in this direction after seeing one statistical study after another produce a graph that had the same sort of sharp, peaked distribution, rather than a rounded-top curve on the statistical graph. I disagree with his conclusion; I think the graphed appearance is a consequence of the nature of the statistical methods (a consequence of the math), and does not represent the operation of the aspect in nature.

I especially appreciate the sinusoidal wave because it has significant thresholds and drop-off points at the same places we observe them to be in real life. For example, there are significant numerical thresholds at (within a few minutes) of the 1°, 3°, and 7.5° thresholds.

FWIW...

BTW., further discussion on the above article is here: viewtopic.php?f=16&t=175

A little more on the sinusoidal wave model...

Draw a sine wave that peaks at the exact location of an aspect (say, a square) and tapers bilaterally to a zero point 15° on either side. (At that point, it starts rising again from 0° to be a sextile or a trine).

Normalize the curve so that it is running from 0% at 15° away to 100% at a 0°00' orb. Judge these percentages to mean the likelihood that the effects of the aspect will be felt. Notice that this doesn't give any absolute orb, because even at, say, a 12° orb there would still be almost a 10% chance that the aspect would manifest - it just isn't very likely.

There are some interesting thresholds in this curve.

First, we ask ourselves: At what point is it more likely we will have an effect than not? This translates to: Where is the chance of an effect 50%, such that any lesser orb is more than 50% chance of a manifestation? This threshold is at 7.5°, which is where I put my most extreme, outermost practical orb for sextiles, trines, and squares.

Is there anything in the 5-6° range which seems, most of the time, the practical outermost threshold for these aspects? Yes, at 5°00; the score is 75%.

Well, how bout the magical 3° cut off point, which easily marks Class 1 aspects (and is both the aggregate or statistically-visible threshold in every study done, and the aggregate or collective impact level usually seen as practical in mundane astrology)? Yes, this is well marked. A score of 90% is reached exactly at 3°09'

How about 99%? Where does the likelihood of a manifestation first round to 99%? This, it turns out, marks the partile threshold: 0°58' to be exact. It then rounds to 100% (i.e., reaches 99.5%) at 0°41'.

These all make sense in terms of what we actually see happening in astrology.


But, of the major aspects, the gap between the conjunction or opposition and its nearest other aspect (conjunction to a sextile, or opposition to a trine) is not 30° away but 60°; and experience suggests that the curve is, in fact, broader around the conjunction and opposition. Not twice as wide, but wider.

So, for these aspects, I scale to a 20° reach for 0%, and get the following thresholds:

50% at 10°00' (the farthest I go in extremis for a conjunction or opposition)
75% at 6°40' (I use 7° as my practical stop-point for these aspects)
90% at 4°05' (I use 4° as my cut-off for Class 1 oppositions or conjunctions)
99% at 1°17' (essentially the partile spot)
100% at 0°54' (essentially partile also)

Something I know Bradley worked on, on-and-off, was a way of quantifying the combination of aspect partiality and angularity. I don't think he ever came to a conclusion that fully satisfied him. I have a crude way of doing it that is too tedious to apply all the time (I don't really need to quantify it), but that I've applied to my chart and a few others. It starts with taking the orb strength curve just discussed, and then modifying this for the distance planets are from the angles. I calculate a "strength" factor for each of the two planets, average them, and multiply it by the aspect's strength score. It's probably too simplistic, but it's been something to play with occasionally.

[Interrupting myself (5/21/22): Bradey proposed slightly different model, where the two angularity scores and the aspect score are multiplied together. His is A x B x C whereas the one with which I experimented is (A+B)/2 x C, which downplays the individual angularities a bit. Suppose Mercury were foreground at 100% angularity, Saturn middleground at 45% angularity, and Mercury square Saturn close at 96% strength. His model gives 1.00 x .45 x .96 = 0.432 (43%), mine above gives (1.00 + .45)/2 x .96 = 0.696 (70% strength). I think this is a better result, because a Class 1 aspect with one planet exactly angular and the other not weak makes more sense at 70% strength than 43% strength. This also alleviates the problem of a single planet exactly on a cadent cusp wiping out the aspect altogether with a 0.00 multiplier.]

My closest aspect is Venus square Pluto, 0°13' orb. With no consideration of angularity, it has a strength of 99.9%.

But for birthplace it is in the immediate background, and for location it is in the immediate foreground. Thus, after adjusting for angularity, for birthplace its overall score is 22%; for locale, where I have lived for four decades, its score is 98%. Big difference!

For birthplace, this closest of aspects - after everything is adjusted for proximity to the angles - ranks behind the following aspects (some of which have orbs much wider than I usually bother with):

63.9% Moon-Mars sextile
48.6% Moon-Venus trine
38.5% Moon-Uranus trine
36.5% Moon-Jupiter trine
30.5% Mars-Neptune square
28.4% Sun-Mars square
26.2% Neptune-Pluto sextile
25.4% Venus-Mars sextile
24.1% Uranus-Neptune square
23.4% Jupiter-Neptune square
21.8% Venus-Pluto square

That's pretty far down the list for a 0°13' aspect, the closest in my chart! But it makes sense given its cadency. (My Moon-Jupiter and Moon-Uranus aspects are made much stronger - even at 6° orb - by Moon's angularity and the middleground placement of Jupiter and Uranus. But the 0°17' Jupiter-Uranus conjunction itself only gets a 17% score.)

TheScales_BothWays wrote:Care to show how you calculated your aspects' strengths, like how you did it at your last post?

The aspect strength itself is a straight sine curve, as described in detail above. The angularity weighting is complex, but boils down to finding an expressiveness value for each planet in a layered set of curves, based on its placement in a quadrant, averaging the two planets' weights (not sure that's a valid way to do it, but might be a good-enough approximation),and multiplying the result of this by the strength value for the aspect itself.

NOTE 5/21/22: TMSA now takes care of calculating this angularity weighting. It's no longer complicated for the astrologer.

When this was reworked and then reprinted in 1974, Bradley appended formulae for calculating relative strength of angularity and of aspects as a cycloidal wave. I probably should append his formulae here in case others want to work with them. As a reminder, I still think the wave is sinusoidal instead; but, obviously, his presentation deserves the respect owed his decades of work on the matter.

He wrote:

Garth Allen wrote:Is there a not-too-complicated way to cope with these conditions both quantitatively and qualitatively in any horoscope - that is, accurately but not in such a technical setting as to repel the incipient student? There is, indeed, and arriving at it was eye-opening in more ways than the one intended at the outset.

For years, nay, even decades, yours truly had labored under the assumption that horoscopic intensities were basically sinusoidal or rounded wavelike in nature, in the familiar manner of sine waves. They aren't, though, and we should have realized this much earlier than was the case. Astrological amplitudes, we now know, are cycloidal in action - they build up increasingly to their peak, which is sharply pointed and not domed. In mathematics, by the way, a perfect cycloidal curve is etched in space by a point on the rim of a rotating wheel; a good demonstration of this effect would be to roll a wheel bearing a tiny point of light across the floor of a dark room - the movement of the light is cycloidal. Both aspect and mundane-position amplitudes behave in this same fashion, reaching their crests as points in space and time. It is for this reason that solar and lunar returns, say, are possible; the charts in question are cast for the peaks of the cycloidal waves whereas sinusoidal action would negate the specialness of these moments in time.

We have devised formulas, based on cycloidal action, for the relative evaluations of both aspects between two bodies or points and the mundane importance of their positions in the chart. The intensity formulas for mundane position are independent of which house system or circular or spherical concept is adopted by the individual student. Table I [not reproduced], for instance, measures intensity of mundane position around any basic symmetrically divided circle used in astrology. The strongest points (such as the Ascendant or any other angular cusp, by spherical system or just plain equal-house, it not mattering which) are given a value of 3.00, the weakest a value of 1.00.

Table 2 [not reproduced] has for its argument the distance in arc between any two planets or points, i.e., their longitudinal differences being anywhere between 0°00' and 180°00'. In this layout the maximum value is also 3.00, the weakest being 0.00 (in theory but not in actual practice, as a study of the lowest tabular value, 0.25, will make plain). The multiplication of the two values obtained for each planetary pair, with a maximum of 27.00, will provide an index figure which gives a scientifically reasonable rating of the importance of the configuration being dealt with. All 45 possible planetary pairs can be handled with relative ease and rapidity using these tables as printed in their simplified form. The keen student will, of course, want to calculate (and furnish to his colleagues) his own more-detailed version of these tables.

Begging the indulgence of the nontechnical readers who don't give a hang about realities, we reproduce the formulas for the benefit of the wider awake. Table I is based on the formula
Intensity = 1 + 2(1-sin 2D)
where D = distance from fiducial point (Midheaven, Ascendant, East Point, or whatever).

Table 2 I based on the formula
Intensity = (1-sin 2D) + (1-sin 3D) + (1-sin 6D)
where D = difference between two longitudes. (Of those to whom simple trig is a challenge, we ask forgiveness for our "sins".)

These data are offered for what they may be worth; at least they put astrological research and practice on a nonblurry, exacting, no-nonsense basis that allows accurate determinations to be made and, thereby, obviously superior chartwork to be achieved. Putting astrology into terms understandable to educated people is the least the astrological field can do toward furthering the best interests of its subject.

These equations have some small errors. A negative sign in the sine value flips them at the wrong point, but taking the absolute value of the sine solves that problem. Also, for a better fit of observation, I suggest in the last equation changing 6D to 4D.