Mathematical roots of Major Aspects

[Jim Eshelman]

Major aspects are the five Ptolemaic aspects: Conjunction, Opposition, Square, Trine, and Sextile.

Questions often arise that confuse their "majority" with the issue of aspect intensity; e.g., people ask why Trine and Sextile a "major" when semi-square and sesqui-square are more dynamic. The simple answer (which needs elaboration) is that the trine is a much more significant aspect than the semi-square, but it has a different (softer, or more static) nature. The "cheap and easy" alternative answer that I can get away with sometimes is that major aspects have wider orbs, minor aspects have narrower orbs.

However, there is something structurally significant in te mathematics of these five major aspects that sets them apart from all other aspects.

First, I think the major aspects are inherently connected to a sine wave. As you may know, I think the tapering strength of aspect orbs follows a sine curve (or, specifically, a cosine curve). But, furthermore, there is an even spacing of these aspects in terms of the sine/cosine curve:

cos 0° = +1.0
cos 60° = +0.5
cos 90° = 0
cos 120° = -0.5
cos 180° = -1.0

Measured along a cosine curve, the trine and sextile are the half-way points between a square and the conjunction or opposition!

There is another way to show this. If my basic trigonometry were better, I could likely explain that this is the same thing. Again, it demonstrates basic ways in which not only is a square half-way between a conjunction and opposition, but - measured along the curve - the sextile is half-way between a square and a conjunction, the trine half-way between the square and opposition.

See the diagram below:

Major Aspects Circle.png

I have cut the diameter of this circle into even fourths. (Well, once you make allowances for my eye-balling it.) Notice that when you draw line segments perpendicular to the diameter at one-fourth, one-half, and three-fourths of the way along the line, these cross the circle not at 45 / 90 / 135 but at 60°, 90°, and 120°.

The major aspects seem to have hard mathematical reasons they are locked in as being of the greatest importance overall.

[mikestar13]

Very convincing rationale, and it dovetails perfectly with the inferences from harmonic theory. Ptolemy was wrong about quite a few things (choice of zodiac not least) but he was right about the major aspects. Not that minors are to be neglected. But a beginner who reads a nativity using Ptolemaic aspects only to a 5 degree orb (as I did for many years) will miss some things, but won't get anything wrong.

[Jim Eshelman]

Agreed

[Jim Eshelman]

Unfortunately, the effort to create a single mathematical curve that both necessarily defines these aspects and shows their strength, does not fall easily from these numbers.

I found a way to create an argument for the aspectivity function based on the turning points in the cosine curve, and then to use this nonlinear argument to create an aspect strength curve. It does indeed uniquely define these five aspects. It also gives orbs (and orb-strength drop-offs) that match experience for trines, sextiles, and squares. However, if makes the width of conjunctions and oppositions gigantic, totally unreasonable.

For the curious (and those who might be able to improve upon this effort to define a single function), here is what I did:

Where X is the elongation between two bodies in degrees, calculate the first value as:
MOD(((COS(X)+1)*2),1)*360
Notice that this gives a 0°-360° value to each degree of elongation, which cycles back to 360° (i.e., completes the circle) at 0°, 60°, 90°, 120°, and 180°. Then, where Z is the value just calculated, calculate aspect amplitude with
(COS(Z)+1)/2

Here is what you get:

Sinusoidal.png