Angularity

[mikestar13]

Currently, TMSA calculates angularity according to one of two fixed models, mid -quadrant or cadent cusp according to where the trough of planetary expressiveness is taken (the peak is always at the angles, and neither peak nor trough will change).

Let me describe how the current curve works in some detail as a basis for the changes coming (to simplify this discussion we will ignore minor angles but they will be handled in an analogous fashion over a narrower range of degrees).

First, lets establish some ideas. TMSA expresses strength as a percentage ranging from 0 to 100. Key points are: angles = 100% edge of foreground = 75%. center of middleground = 50% , edge of the background = 25%, center of background = 0%. These numbers are not calculated directly, but strength is internally expressed as a number ranging from -1 to +1 which is then rescaled to a 0% to 100% range for printing.

Let's make a naive attempt at calculating some numbers. First of all, we take the longitude difference in degrees between the midheaven and the planet, call this d. Then x = d mod 90.

Let's get a few values of the function y = cos(4 * x):

x y
0 +1
15 +0.5
22.5 0
30 -0.5
45 -1
60 -0.5
67.5 0
75 +0.5
90 1

This is almost what we want for the mid-quadrant model, but not quite (it makes the foreground too wide). So we use a series of statements in the form of if x >= a and x < b, then x = (x- a) * m/n + c, where m, n and c are constants for that range of values. For example if x >= 0 and x < 10, x = (x -0) *3/2 + 0. We then calculate y = cos(4 *x), so that 10 degrees and 80 degrees both yield +0.5, the edge of the foreground, which we want for the mid-quadrant model. 45 degrees remains 45 and yields -1.

Now for the cadent cusp model, we do the same for the foreground area, but for other ranges of degrees, 60 (the cadent cusp) is converted to 45.

Now the equation y = cos(4 *x) is the preferred method of calculating strength based on modified quadrant position and will be the TMSA default (it's the method we use in TMSA 0.4, but we could equally use a cycloid equation (though this would change some of the modifications needed).

But what if we don't want the foreground to be ten degrees wide? What if we want fifteen degrees like in early Siderealist usage? What if we wanted five degrees like cosmobiologists most likely would if they regarded angularity? Never fear, the degree ranges and the corresponding constants can be calculated at run time! Ten degrees is Jim's preferred foreground width and mine and will always be the TMSA default.

So you will always see final strength values ranging from 0% at the center of the background to 100% at the angles, with 75% at the edge of foreground, 25% at the edge of the background, and 50% at the center of the middleground, regardless of where within reason those edges are located. TMSA does all the mathematical heavy lifting in a microsecond or two.

[mikestar13]

TMSA 0.5.0.0 may not be released until about the forth of July. I'm being very painstaking with this and other revisions in order to get them exactly right. Then with all the scaffolding built, releases adding new features (noviens, transits quotadians ...) will come quicker.

[Jim Eshelman]

This is an excellent and important post, Mike. While I trusted that you had the math right, I never fully understand which tricks you pulled off to accomplish it (I didn't really need to), and this is quite interesting. (Most users, of course, won't care, but having it documented like this is still important and some version of this should be in the documentation.)

As for 4th of July, I have (as you probably) suspect a Caduceus of emotions intertwining: On one hand, there are some improvements I "can't wait" to get my hands on. (The one that comes up almost daily is the angle labels on the foreground table; but there are more, including accurate Q2 rates, wanting to see how much they're really off in practice.) OTOH, it's really exciting to see how conscientiously you're reviewing the whole program and refining its operations, contents, and look. This will be a really BIG DEAL version!

[dflow]

Hi Mike,
I'm implementing the angularity curve in my ruby code as per TMSA 0.4.9.2 source, but I have one discrepancy...
I get all the same percentages as below but for some strange reason Sun yields 32% instead of 89%. Any ideas?

Z means that is close to the Zenith, does that mean it is 89% angular to the Zenith instead of 32% to AC/MC ?
What does G in the column header stand for?

From TMSA (8 May 1979 10:57:00 AHDT 61N12'39" 149W49'42")

Code: Select all

Pl Longitude   Lat   Speed    RA    Decl    Azi     Alt     PVL    Ang G
Mo 14Vi20'50"  2N 0 +12°24' 188°52'  1S39   7°38' -30°14'  77° 9'  61%   
Su 23Ar10'37"  0N 0 +58' 1"  45°10' 17N 5 141° 4' +41° 6' 305°46'  89% Z 
Me  2Ar 4' 9"  2S32 + 1°38'  25°31'  7N52 167°38' +36° 8' 286°21'  65%   
Ve 24Pi26'28"  1S42 + 1°12'  18° 5'  5N49 176°51' +34°34' 274°34'  94% M 
Ma 29Pi59'52"  0S45 +45'29"  22°55'  8N46 170°44' +37°16' 281°57'  72%   
Ju  7Cn23'59"  0N40 + 7'20" 124°16' 20N24  64° 1' + 9°13' 349°46'  74%   
Sa 12Le37'35"  1N58 - 0' 5" 159°32' 10N45  36°14' -12°42'  20°52'  57%   
Ur 24Li35'46"  0N21 - 2'32" 226°41' 17S 9 319°13' -40°41' 127°13'  29%   
Ne 25Sc29'55"  1N27 - 1'18" 259°11' 21S37 282°25' -31°24' 147°59'   0%   
Pl 22Vi35'30" 17N38 - 1'23" 202°28'  9N36 352°42' -18°58' 110°16'  58%   
Er 20Pi11'53" 19S13 + 0'34"  20°58' 11S57 174°23' +16°42' 288° 3'  62%

BTW what is the difference between show.py and show2.py ?

thanks in advance!

[Jim Eshelman]

Mike may have more to say, but I can answer part of this before I get back to work (taking down building power tonight and soft-landing network systems first.

I get all the same percentages as below but for some strange reason Sun yields 32% instead of 89%. Any ideas? Z means that is close to the Zenith, does that mean it is 89% angular to the Zenith instead of 32% to AC/MC ?

Minor angles have their own independent curves. Whichever possible angle contact is stronger shows. This Zenith contact (upper square to Asc) is on a 32° base IIRC instead of a conjunction' 10° base.

What does G in the column header stand for?

"Ground" as in Foreground, Middleground, Background.

[mikestar13]

The portion of the piecewise main angularity curve for bodies near the main angle is cos(6*x) where x = distance from nearest angle while for minor angles it is cos(20*x) TMSA 1.0 is shifting to a very similar but slightly different more complex formula that yields very similar but "more satisfying" numbers.

In the original formulas the range of the function is of course -1 (exactly on a cadent cusp) to +1 (exactly on an angle), this is the converted for display purposes to a range of 0% to 100%, with 25% being the edge of the background and 75% being the edge of the foreground.

Note that based on discussions with Jim E, the formula for minor angles was modified to make the drop off slower from 0 to 2 degrees and faster from 2 to 3 degrees, based on the idea that an ingress is dormant unless a planet is within 3 degrees of a major angle or 2 degrees of a minor angle, therefor these limits must have the same strength, an assumption Jim and I have concluded is erroneous. 1.0 will be using a more sophisticated version of cos(20*x) and dropping this idea. My SSR was calculated by my own copy of 0.4.9, which is modified to use the original formula, thus yielding Jupiter at 99% rather than the 100% Jim quoted based on calculating my SSR on his publicly released copy of 0.4.9. Hope this clears everything up.