Ptolemaic Aspects - toward a unified theory

[Jim Eshelman]

On the way home today, I had a seed idea that I thought would give a unified theory of the five Ptolemaic aspects - why these and not others, etc. It struck me as a series of nested sine curves. Maybe someone else would like to play with this?

Here's the basic thing: The five aspects 0°, 60°, 90°, 120°, and 180° are actually exact event fourths of the scaling of a sine wave. That is, measured on the slope of a sine wave (or, turned inside out, a cosine wave, actually), the half-way spot between 0° and 90° isn't 45° - it's 60°! Same with the trine as the "midpoint" of the square and opposition. Here are the cosines of each of those angles.

0° 1.0
60° 0.5
90° 0
120° -0.5
180° -1.0

They are even intervals. First of all, this defines the importance of those angles and the way they form a numerical set. Second, I thought if I could generate another "strength" curve that used the intervals of the cosine curve as its argument, I'd have nested sine waves describing a complete aspect model.

In theory, it's simple; however, getting the bugs out of it is not simple. You can't, for example, multiply the number of degrees before taking the function; rather, you have to use the multiplied cosine itself as the argument of a different formula. But the slope coming off the conjunction and opposition are so slow that it gives gigantic orbs to those two aspects, even though the sextile, trine, and square have quite reasonable orbs that approximate observation.

Here's how far I've gotten: Calculate the cosine of 2 x the angular separation for values 0 through 180. (You could do 0 to 360, but one doesn't need to do that.) This will be a value ranging from -1 to +1. I then want to multiply this by 360 so that the scaling of the sine curve is transferred to a full circle, from which I will again calculate a "strength" score. The formula for all of this is: (COS C +1)/2

You get a graph like the one below. Notice that the five aspects are precisely defined, their strength scaled from 0 to 1 (call it 0% to 100% if you like). For the three central aspects, the orbs are reasonable: a square drops below 99% at about 1°, 90% at about 3°, 75% a little before 5°, and 50% at about 5°. The trine and sextile are similar.

But the conjunction and opposition … whoo! They stay at 99% out to 10°, 90% at 18°(!!!), then drops more rapidly to 75% at 23° and 50% at 31°. This (ahem!) does not match observation!

Anyway, I thought I'd share how far I got in case it catches anyone else's imagination.

[SteveS]

Not sure I understand where you are going with this thread. But your post here reminds me of your writings in 'Interpreting Solar Returns' (ISR) about 'Aspect Orbs'. I have forgotten, have you posted your ISR 'Aspect Orbs' on this forum? I do know this, when I first read your writings on aspect orbs in ISR, it offered me much better clarity in determining how I finalized my personal use for aspect orbs in determining potency for chart reading.

[Jim Eshelman]

Yes, they're here: viewtopic.php?f=20&t=233

Where I'm going is the possibility that there is a single formula that not only measures relative strength of aspects by orbs but also defines the existence of the aspect in the first place. Possibly such a thing doesn't exist, but I suddenly got this idea yesterday that maybe it does exist.

Across the whole length of astrology's history, I'm not sure there has ever been a clear answer to the question of why these five angularities specifically are the primary aspects. I do think that their distinction comes from the fact that they 0°, 60°, 90°, 120°, and 180° are the exact quarters and octaves of the cosine curve, something that exists in nature. Since aspect strength based on orbs also seems to change along the same curve (they don't taper linearly, they taper curvilinearly), I suddenly got this concept that there is a way in which these non-linear curves are nested inside of each other.

So far, though, it's not obvious to me how the final pieces would fit for that to be true - how to generate the conceptual single curve that not only describes orbs but actually defines the aspects themselves.

[SteveS]

Jim wrote:

Across the whole length of astrology's history, I'm not sure there has ever been a clear answer to the question of why these five angularities specifically are the primary aspects. I do think that their distinction comes from the fact that they 0°, 60°, 90°, 120°, and 180° are the exact quarters and octaves of the cosine curve, something that exists in nature.

Ah, now I understand where you are coming from.

[mikestar13]

My personal theory of "why" these aspects. Let's assume aspects are always measured on the shorter arc, so only separations of 0 to 180 degrees need be considered. For the first harmonic we get 0/1 = 0 degrees for the conjunction, which is unique, starting our set of Ptolemaic aspects. For the second harmonic 0/2 = 0 degrees, which is the already defined conjunction again (so ignore it) and 1/2 = 180 degrees which is the opposition.The opposition is unique, add it to our set. For the third harmonic, we get 1/3 which is the unique trine, add it. Similarly the fourth yields the unique square. OTOH, for the fifth harmonic the are two new aspects, the similar but not quite identical quintile and biquintile, so not unique, don't add anything to the set. But for the sixth harmonic, a unique new aspect, 1/6 is a sextile. 0/6 is a conjuction, 2/6 is a trine and 3/6 is an opposition. So our Ptolemaic set of aspects is complete. Each higher harmonic will produce at least two new aspects as the fifth does. I'm intrigued by Jim's idea of cosine curves.

[Jim Eshelman]

Interesting demonstration.

May i suggest... The 1st harmonic doesnt exactly exist because aspects aren't between points but between great circles... And the conjunction and opposition are on the same great circle, the same aspect.

[mikestar13]

Jim, I can't quite grasp the formula you are using. Are you using an Excel spreadsheet to calculate? If you could share it, so I can understand the formula you are currently using, I think I can tweak it to give more reasonable conjunction/opposition orbs.

[Jim Eshelman]

Jim, I can't quite grasp the formula you are using. Are you using an Excel spreadsheet to calculate? If you could share it, so I can understand the formula you are currently using, I think I can tweak it to give more reasonable conjunction/opposition orbs.

https://1drv.ms/x/s!Ansnmu2xbktamelp2umTw4h6CwBjeA