New quantification of angularity

[Jim Eshelman]

On and off for, oh, 40 years, I've thought that we needed a slightly better descriptive model of relative angularity - what are historically called "the grounds."

Historically, among Sidereal astrologers, there have been two primary models that seem to contradict each other. (At the very beginning, there was a third model - which boiled down to classic angular / succedent / cadent houses, but with really big cusp orbs. I don't count this in what follows, but it's not far from what I regard as the truth, as you'll see below.)

MODEL 1. "Three grounds" being 30° of prime vertical longitude centered on the traditional house cusps. The FOREGROUND was the 30° of PV centered on the angular cusps (i.e., about half a house on either side of an angle), MIDDLEGROUND as the 30° of PV centered around the succedent cusps, and BACKGROUND being the remaining third of space that is centered around the cadent cusps.

This is well-supported by Bradley's early published studies of Sidereal Lunar Return planets and natal horoscopes of murderers. It is very close to what I have long perceived as the real working of things. That is, the weakest (least expressive) part of a quadrant does appear to be the cadent cusps themselves. Also, the areas right around the succedent cusps never have seemed excessively weak/inexpressive - very much feeling "middling." The weaknesses of this model are, for example, that a planet rising through the 12th house doesn't suddenly jump from "foreground" to "background" half way in between. In a gradual curve, there has to be a middleground area in there somewhere.

MODEL 2 is much easier to graph, but I don't think it's as accurate. It says that the angles are the strongest points, and the middle of each quadrant is the weakest - a simple rising and falling of strength or expressiveness of a planet. Fagan was sold on this during most periods. - As mentioned above, I don't find that it's quite right. For example, it would mark the succedent and cadent cusps as of equal strength, which conflicts with my perception (and is unsupported by the Bradley statistics I cited above). Nonetheless, there was always a sense that mid-quadrant itself was pretty weak - as if the background zone stretched deeper into the succedent house than the simple "half house" model.


Over the last few months, I've been clarifying and cross-checking perceptions, and drawing out of myself the most direct and simple statements of what appeared to be true about angularity. When I looked at these outside of either of the prior models, and without any prior consideration that the zones had to be equal size, I found that there is a simple pattern. I present this below.

First, I got honest with myself that I've never really seriously thought that a foreground zone reached a full 15° either side of the angles. It certainly doesn't on the cadent side (the drop-off is faster); and it really doesn't on the angular side, either. That was just the fruit of thinking in overly neat "15° either side of the angle" models. I could always live with this acceptably because I always practiced and taught that planets much closer, especially within 10° or 7° or 2°, were way stronger. In truth, I think the foreground zone only extends 10° of PV longitude either side of the angles.

Second, I knew a middleground zone had to lie between the high point of an angular cusp and the low point of a cadent cusp. (The drop-off is gradual.) That had to be considered.

Third, I had tended to think of the "immediate background" as the 10° or so on either side of a cadent cusp. I knew this reached further into the mid-quadrant though, and probably included Fagan's "dumb notes" at mid-quadrant.

The result is the following:

Grounds areas SML.png

THE FOREGROUND is the area 10° of PV longitude either side of the angles, or 20° out of every quadrant.

THE BACKGROUND is the area from 10° of PV longitude on the cadent side of a cadent cusp clockwise to 20° into the succedent house. This is 30° out of every quadrant.

THE MIDDLEGROUND is everything else - two separate zones in each quadrant, and totaling about 40° out of every 90°.


That's how I see it now - a slight tweak of what I've taught previously.

[Jim Eshelman]

in your theory does the foreground still widen orbs?

No. Orbs are orbs.

However, there may be times when we pay less attention to all but the strongest aspects if they aren't foreground, especially outside of natal charts.

[Jim Eshelman]

Would the number of degrees on either side of a cusp increase or decrease when the quadrant isn't a perfect 90 degrees?

We measure angularity in the Campanus mundoscope. If you do that conversion (which almost every astrology software program can do), the true status of foreground/middleground/background can be seen directly.

To answer your question, yes the zodiacal degrees included in the angular zones do increase or decrease according to Sidereal Time and geographic latitude (while a planet's declination must also be considered). It's useful to think about it three-dimensionally, not just in terms of a flat drawing.

-Derek

[Jim Eshelman]

Would the number of degrees on either side of a cusp increase or decrease when the quadrant isn't a perfect 90 degrees? In my chart, for example, the quadrants are either 67 degrees or 113 degrees.

If you are measuring it on longitude, then yes... and you'll still be off on planets with high latitude.

The correct way to measure it is in Prime Vertical longitude instead of celestial longitude. The quadrants in PV are always exactly 90°. It's only the elliptic that is stretched across them evenly.

If you want to have a way to estimate just from a zodiac-based horoscope, think of "10 degrees" as meaning "One-third of a Campanus house." However, there will still be distortion unless you actually look at it in PV longitude.

[Jim Eshelman]

Elsewhere today, I talked about what I see as two relatively independent curves existing, one peaking at angular cusps for expressiveness, one peaking at cadent cusps for repressiveness. Here are some pictures of what these would look like.

First, here is the Angularity / Expressiveness curve. It peaks at the angles, and troughs out half way in between, in the mid-succeedent houses. More or less, after it drops below the baseline, it isn't doing anything much.



Next, here is the Cadency / Inexpressiveness curve. It "peaks" (is at its strongest negative value) at the cadent cusps, and troughs out (highest on the graph) half way in between, in the mid-angular houses. More or less, after it rises above the baseline, it isn't doing anything much.



Here is the complicated one. This one more or less lets each of the curves have their complete and entire say when they are at their strongest, and when they are both rather tepid in strength it blends them. Notice how a "middleground" area is defined as rough, irregular, no certainty in the lines, etc. in between more distinctive foreground and background zones, especially in the succeedent cusp (classic middleground) range, and is a much more acute shift in the cadent areas.

[Jim Eshelman]

Some interesting "natural thresholds" are shown in these numbers. In particular, one might wonder why foreground, though active out to about 10°, seems to have such a sharp drop-off after 7°. Well, the repressiveness curve (though at the least of its strength) becomes slightly stronger than the expressiveness curve at about 7.5°.

Angularity drops below 50% of its positive score at 15°, which I used to take as the foreground boundaries; but, in practice, that seems too much. Well, half of that - a 75% score on the angularity curve - falls at a distance of 10°-11° (bilaterally) from the angles (10° is 77% strength, 11° is 72%).

Similar distances apply to repressiveness and the cadent cusps.

There is a zone 7.5° either side of the succeedent cusps where both the angularity curve and cadency curve are negative values, i.e., both are relatively inactive. This 15° band is the most naturally defined middleground zone because it's in a kind of limbo with neither angularity nor cadency strongly operative. (Implicitly, there is a more generous middleground zone as an area that is neither acutely foreground nor acutely background - I take this in practice as the 40° zone more than 10° from both angles and cadents.)

In the cadent houses, though, this transition is more acute. There is no zone where both angular and cadent curves cease to function. Angularity drops below the Zero line at 67.5° (22.5° cadent from the angles), and cadency only crosses the zero line at 82.5°, or 7.5° cadent from the angles, though it is at half-strength at 75°, or 15° cadent from the angles. In between, a reasonable transition curve is formed by blending the two scores.

It's a model. It might make it easier for some of you envision what I feel is happening at different microparts of the quadrants. Its biggest weakness is how strong the curve stays through the angular houses, and I'm not sure it persists that strongly. (But, of course, I could be wrong about that.)

[Jim Eshelman]

By what criteria do you measure? How does astrological "frequency" or strength get measured? I can see the graphs, they look cool... but I don't know how a planet makes it into the 25% strength or the 75% strength or 99%.

F.ex. if you have a natal chart and you see angular Sun. How do you measure it's strength in the angularity - by what criteria? You would need to know the person to judge if the Sun was very strong, or just average or rather weak.

[Jim Eshelman]

I mean, by what criteria do you measure? How does astrological "frequency" or strength get measured? I can see the graphs, they look cool... but I don't know how a planet makes it into the 25% strength or the 75% strength or 99%.

Oh, I see. - These graphs are showing the 90° of any quadrant (measured along the prime vertical). The 0° spot at far left is the exact angle (either Asc, MC, Dsc, or IC). The points 30° and 60° are, respectively, the succeedent and cadent cusps. Sorry that wasn't clear.

F.ex. if you have a natal chart and you see angular Sun. How do you measure it's strength in the angularity - by what criteria? You would need to know the person to judge if the Sun was very strong, or just average or rather weak.

It's a purely astronomical measurement off the mundoscope (the basis for determining angularity). For example, my Sun (as shown in the mundoscope) is 19°19' into the first house (19°19' below Ascendant), so it would appear at the 19° mark on the graphs.

Sorry, I thought that would be obvious from the original discussion at the top of the thread.

Notice that the blue curve, designated as expressiveness and angularity, peaks at the 0° point and troughs halfway between as described. The red curve, designated as repressiveness and cadency, peaks at the 60° point (the cadent cusps) and troughs halfway between as described.

[Jim Eshelman]

ok. I see, these are just pure numbers, just degrees from an angle... but not an attempt to measure real strength of manifestation.

[Jim Eshelman]

ok. I see, these are just pure numbers, just degrees from an angle... but not an attempt to measure real strength of manifestation.

They are an attempt for quantify the strength of a planet a particular distance from the angles.

[Jim Eshelman]

Arena, my opinion is generally a planet that's conjunct an angle within less than a degree is very strong, up to 3° is strong, up to 7° noticable if nothing else is stronger, and up to 10°notable at least in natal work. These orbs are based on observation.
I don't call anything more than 3° from an angle "angular." The word I use for more than 3° up to 10° from a major angle is "foreground" and I call this "ease of expression" not "strength" because I think "strength" is a misunderstanding of what "angular" means even in tropical astrology. The major angles are MC, IC, ASC and DSC. Everything else, I would use up to a 1° orb.

Is that what you are asking about here?

[Jim Eshelman]

Yes this was all clearly understood by my mind a long time ago. I was not asking about numbers, nor angles.

No the question was more about what actual measurements in manifestation would have been used to say a planet is strongly manifested, as f. ex. being within 3 degr. from an angle we would expect it to act out "strongly" or manifest somehow obviously in the personality. But what would be used to measure it in real life samples. I know Gauqelin did have some certain traits in his research - but I am guessing that is the only real measurements that have been done on natal charts.

[Jim Eshelman]

No, actually some Bradley studies in the early '50s identified the "three grounds" as the framework for natal angularity. The appear to match results he published in the same general period for Sidereal Lunar Returns.

You can see some of these results graphically reproduced in one of the introductory chapters of my Interpreting Solar Returns.

[Jim Eshelman]

Yes this was all clearly understood by my mind a long time ago. I was not asking about numbers, nor angles.

No the question was more about what actual measurements in manifestation would have been used to say a planet is strongly manifested, as f. ex. being within 3 degr. from an angle we would expect it to act out "strongly" or manifest somehow obviously in the personality

I can't figure out what you're asking. What do you mean "actual measurements in manifestation"? Either you're asking about actual measurements (which is what Jim and I have both give you) or you're asking what it looks like when someone has a planet angular.

But what would be used to measure it in real life samples. I know Gauqelin did have some certain traits in his research - but I am guessing that is the only real measurements that have been done on natal charts.

From this, I think you're asking about how we know what having a particular planet angular looks like. From your previous messages today, I suspect you want to know specifically how we know what it looks like when someone has either the Sun or Neptune angular, and how we know that because you don't see those traits in your partner or yourself.

The answer is observation, and the answer is also statistics. And statistics, like in SMA, (yet) don't explain everything. Someone has an angular sun, but that person doesn't just have an angular sun. He has an angular SUn in a particular constellation, and it's aspecting one or more other planets, some more strongly than others, and he has other aspects as well.

If you want to do your own studies, maybe get a pile of psychological assessments matched with birthdata and see what statistical corelations you can come up with.

[Jim Eshelman]

Jupe, she didn't understand what the curve was (I didn't explain it well). She thought it was a data distribution, I think, rather than an angularity modelling.

[Jim Eshelman]

Yes JSAD I was thinking more in those lines, not particularly about me or my partner, but more in general on how the statistics on strength were gathered. By what measurement, which keywords etc.
I do not know of any other such data than Gauqelin data and now Jim has added that Bradley did some data gathering.

I do of course realize that we need to look at more things than only angularity, like which planets are they aspecting and how are the luminaries placed/aspected and so on. But angularity is important, it is very important.
Surely we do expect closely angular planets to show in the personality since we emphasize angularity so much in the sidereal world of astrology.

If f.ex. my partner, has an angular Sun by measurement of the recorded birth time and place, and there is not even a single word or behavior to be seen in him that corresponds with angular Sun description, and he even acts opposite to that description, then I am one who does doubt his chart. And if I see this happen more times, I start asking questions and I do ask myself if it is indeed possible then that his birth time is not correctly recorded, or that the program or usage of houses is somehow not right. I ask myself to learn in order to understand this better. It is not because I am doubting the methods or astrology, because I strongly believe there is so much truth to it.

[Jim Eshelman]

I also don't think I've ever seen a description of Sun angular that's accurate or useful. As an angular Sun person who is sure my time of birth is accurate, I question the descriptions, not the chart.

I don't think Fagan's description of angular Sun people is all that. Jim pointed out a few things in Fagan's descriptions of other planets he feels are off, so it's not just me, although I think Fagan's descriptions are pretty unflattering, and he'd met a couple of Sun angular people he wasn't fond of.

Just for instance, everything I read about angular Sun says I should be an extrovert. I'm not. I'm a classic introvert, meaning not that I hate and fear groups, but that I find other people draining, and prefer my own company (and that of my critters.) I don't get lonely. I am self-contained and self-directed, although not particularly self-centered or self-ish. While I often end up as leader in groups, it's not because I worked and politiked and strived to be the leader. It's because the leader is the person other people choose to follow.

Does any of that sound like your partner?

[mikestar13]

James,

Would you be willing to share the equations used to calculate the "complicated one"? I find the graph pretty clear and convincing, but I at a loss how to quantify the "entire say" concept. I am writing some astrology software based on the Swiss Ephemeris and would prefer a more accurate estimate of angularity than method 2.

[Jim Eshelman]

Would you be willing to share the equations used to calculate the "complicated one"?

There isn't a single equation. There is one equation applied to two different places, added in some places and left stand-alone in others, editing each output cell one at a time to pick where the peak curve had sole ownership, other areas where the trough curve owned it, and other places where I just let their total have play.

Totally crafted picture, nothing that can reduce to a single valuation.

I'll see if I can find it. I have one spreadsheet with about 23 variations, and I may not have even kept this one since it's a visual kludge.

[mikestar13]

Jim,

Thank you for your prompt response and your willingness to help. Meanwhile, I'll keep hacking at it. A google search led me to the concept of phase modulation, which has some equations for irregular sine curves (peaks and troughs not equidistant), which looks rather like what we're looking at. I will gladly share anything I come up with.

[mikestar13]

The following technique produces somewhat similar results to your complicated curve:

1. Measure the planet's distance in degrees counterclockwise along the prime vertical from the previous angle (ascendant for eastern hemisphere, below horizon planets, etc.). Call this A.

2. If A<=60 then set A to 3/4*A.

3. If A>60 then set A to 3/2*A-45.

4. Calculate cos(4*A).

This procedure yields a value of +1 at the angles, -1 at the cadent cusps, 0 at the succedent cusps and at 75 degrees

Not quite your idea, but not that much harder to program than method 2. IIRC, this is a very good approximation of the graph in Interpreting Solar Returns.

[mikestar13]

I've been doing some further refinements of the curve produced by the equations. The revised procedure:

1. Measure the planet's distance in degrees counterclockwise along the prime vertical from the previous angle (ascendant for eastern hemisphere, below horizon planets, etc.). Call this a, the result will be p.

2. If a<=10 then p=cos(6*a).

3. If a>=10 and a<=50 then p=cos(3*a)/sqrt(3)

4. If a>=50 the p=-cos(6*a).

Notice at exactly 10 or 50 degrees, both of the two applicable formulas produce identical results. The region where p>0.5 is the foreground, where p<-0.5 is the background, between -0.5 and 0.5 is the middleground. A table of p values for each degree of a quadrant follows.

0 1.000 1 0.995 2 0.978 3 0.951 4 0.914 5 0.866 6 0.809 7 0.743 8 0.669 9 0.588
10 0.500 11 0.484 12 0.467 13 0.449 14 0.429 15 0.408 16 0.386 17 0.363 18 0.339 19 0.314
20 0.289 21 0.262 22 0.235 23 0.207 24 0.178 25 0.149 26 0.120 27 0.090 28 0.060 29 0.030
30 0.000 31 -0.030 32 -0.060 33 -0.090 34 -0.120 35 -0.149 36 -0.178 37 -0.207 38 -0.235 39 -0.262
40 -0.289 41 -0.314 42 -0.339 43 -0.363 44 -0.386 45 -0.408 46 -0.429 47 -0.449 48 -0.467 49 -0.484
50 -0.500 51 -0.588 52 -0.669 53 -0.743 54 -0.809 55 -0.866 56 -0.914 57 -0.951 58 -0.978 59 -0.995
60 -1.000 61 -0.995 62 -0.978 63 -0.951 64 -0.914 65 -0.866 66 -0.809 67 -0.743 68 -0.669 69 -0.588
70 -0.500 71 -0.407 72 -0.309 73 -0.208 74 -0.105 75 -0.000 76 0.105 77 0.208 78 0.309 79 0.407
80 0.500 81 0.588 82 0.669 83 0.743 84 0.809 85 0.866 86 0.914 87 0.951 88 0.978 89 0.995
90 1.000

[Jim Eshelman]

That's probably right. I remember my basic procedure was to plot each curve (centered on angles, centered on cadent) separately, and give each one it's say when it was dominant (stronger in its own direction), then average them in between - and maybe do a little more finessing to curve-smooth in a place or two. But what you've done here seems to replicate that. Cool!

[Jim Eshelman]

I'd never posted a graphical representation of Mike's equation above. Here's what it looks like, peaking at the angular cusps and troughing at the cadent, with a less consequential tapering between.

Angularity.jpg

[Jim Eshelman]

Mike in particular, but anyone else interested... I just came up with a new (crude, needs refinement) math model for generating the angularity curve. It's really geeky, and the curve-shaping needs some polish, but the basic principle works. For now, I'll treat this as if the changes in strength are all linear, but I know they aren't. It's just an approximation.

It's always been unclear to me what geometry would force the quadrant-minimum to be at cadent cusps (30° after an angle in the course of normal daily rotation). Today, considering the question of whether Campanus house circles make mundane trines and sextiles to angles, I thought the following three thoughts at the same time:

• Especially if houses actually exist (but not depending on their existence, I tend to think of the rhythms between as being not from angle to adjacent angle Ie.g., not from IC to Asc) but two separate rhythms, one swooping from inmost at IC to outermost at MC (and then back), and a separate one swooping from self-focused maximum at Asc to others-focused maximum at Dsc (and then back).
• In non-hard aspects, the semi-sextile and quincunx are both inconjunctions or non-aspects - almost anti-connections.
• Every intermediate cusp has a soft-aspect connection to one major angle and a no-connection to the other. This produces some fascinating patterns!

Consider:

• The 3rd, 5th, 9th, and 11th cusps make no (mundane) aspects to MC - are null points - and either trine or sextile Asc.
• The 2nd, 6th, 8th, and 12th cusps make no aspects to Asc - are null points - and either trine or sextile MC.

Taking IC-to-MC as an example, if a peak of strength is at IC, it would hit a "no aspect" (or anti-aspect) at the 3rd cusp - a perfect explanation for "all that strength went away." It them climbs through a mundane trine to MC (2nd), square (Asc), and sextile (12th) - building strength all the way until it peaks at MC. (I'm inclined to take the no-aspect at 11th cusp as "momentum carries it.") - The model would have four of these simple waves around the wheel.

Fictitiously treating this as if it's a linear phenomenon (and sorting out the details of the curve later), try the following (very simple if you draw it out, but takin a lot of lines to write):

• Give IC (angle) a score of +2.
• Give 3rd cusp (cadent) a score of -2. [These mark the extremes.]
• Give Asc a score of 0 (crossing the line).
• In between, this gives 2nd cusp (succedent, trine MC) a score of -1.
• Also give 12th cusp (sextile MC) a score of +1.
• 11th cusp gets +1.5, halfway between the +1 at 12th and +2 at MC.
• Repeat this around the other half of the circle.

After this MC-IC oscillation, do exactly the same thing between Asc and Dsc: Asdc +2, 12th -2, 11th -1, MC 0, 9th +1, 8th +1.5, Dsc +2, and repeat for the other half of the circle.

When you add the two values these two separate curves gives to each angle, you get:

• Every angular cusp (1st, 4th, 7th, 10th) has a score of 2 (or 100%).
• Every succedent cusp (2nd, 5th, 8th, 11th) has a score of 0.5 (or 50%).
• Every cadent cusp (3rd, 6th, 9th, 12th) has a score of -1 (or 0%).

It's different from earlier models and produces a desirable result. It also ties in some interesting theoretical considerations that may or may not turn out to be valid. I haven't explored what a full valuation of the curve would look like through the quadrant.

[mikestar13]

I will check this out and derive a curve. This makes a great deal of sense.

[Jim Eshelman]

I will check this out and derive a curve. This makes a great deal of sense.

It's a new though <g>. If valid, not sure if it's sinusoidal or cycloidal. I'm playing further with a decanate by decanate breakdown on a (false) linear assumption.

[Jim Eshelman]

Applying linearly in thirds of houses (just to get more of a curve) it has some extremely desirable details and undesirable ones. Good enough to maintain interest and keep looking. It's an awfully close match for the theoretical model above without being perfect, The following pattern is replicated in each quadrant. (I've drawn lines at the foreground, middleground, background boundaries as defined in the model of this thread.)

Code: Select all

ANG +2.00  100%
    +1.33   78%
---------------
     0.00   33%
CAD -1.00    0%
    -0.50   17%
     0.00   33%
---------------
SUC +0.50   50%
    +1.00   67%
---------------
    +1.50   83%
ANG +2.00  100%

These, of course, are all approximations. But look at how it breaks at a score of 0.0 exactly where the asymmetrical background boundaries are. - I do think foreground strength leans a little too far into the angular side, but this could be the curve shape, it's not bad: 10° on one side of the angle is 83%, 10° on the other side is 78%, they're pretty close as is.

[mikestar13]

I notice that this model makes the peaks at the angles twice as high as the troughs at the cadent cusps are low. I will play with the numbers a bit myself. I can derive a linear equation, then converting it to cosine or cycloid curves should be easy. In fact, TMSA could offer all three calculation methods, linear, cosine (the default) or cycloid.

[Jim Eshelman]

In principle, it currently seems to me that this model has no basis unless soft aspects to horizon and meridian actually are formed (mundanely, of course), which challenges a lot of little things. I don't know if it's true or not. Guess I'll be doing a lot of looking at people having planets within 2° of intermediate house cusps in the mundoscope. (I still think the idea of Venus sextile my Asc is absurd.)

[mikestar13]

Let me approach this a different way, for this discussion we will ignore minor angles.

In the natural rhythm of the universe,the highest ease of expression of a planet's energy is at the angles, the lowest is at mid-quadrant. Whether the rise and fall is linear, cosine, or cycloid doesn't affect this argument--presume linear for simplicity.

This is exactly how ingresses work, since they are subject to no distortion from individual human consciousness (only the collective unconscious).

For natals and solunars, this is not the case.The paramount importance of individual consciousness imposes a distortion. Human beings find it much easier make something weaker than to make it stronger. So the rise is slower than the fall.

As an example, improving the code of TMSA is difficult (though rewarding), worsening it would merely require Terry's cat to walk across my keyboard. (Thank God for backups!)

This difference is least felt when the energy is perceived as strong near the angles, most felt at mid-quadrant, where the trough is pushed a full fifteen degrees and so occurs at the cadent cusps, and the rise from cadent cusp to mid-quadrant is particularly small.

[mikestar13]

Indeed It just occurred to me, maybe the true trough for natals and solunars is neither the cadent cusp nor mid-quadrant but somewhere between-- perhaps at twenty-five degrees of the succedent houses.

[Jim Eshelman]

But that's not what the formal studies showed. The trough was at the cadent cusp, which in practice does seem the weakest spot.

[mikestar13]

The cadent cusp basis is what TMsA already does. I see there is no need to change the math just our understanding of it. The difference is I will rewrite the function to first convert the number of degr4ss CCW from the appropriate angle to the equivalent number of degrees in the ideal mid-quadrant model. Thus 0 is still 0, 60 becomes 45, 90 remains 90, taking the foreground width into account (default ten degrees, so 10 becomes 15 and 80 becomes 75). Then the equivalent degrees will be fed into a cosine, cycloid, or linear function (user choice. default cosine) to generate a strength number (internally ranging from -1 to +1, converted for printing to 0% to 100%).

I will also implement cosine, cycloid, and linear calculation for aspects (user choice, default cosine).

All choices will be per option set, and angularity and aspects need not be the same (documentation will advocate keeping them the same).

[Jim Eshelman]

I think you're only describing how to do the calculations - is that right? Otherwise, I'm confused.

I'm all for more code-efficient ways to do complicated math

PS - You seem fond of linear models (and that's what I did as a stop gap immediately above). I think the one thing we can be confident is not the real 'curve' is a straight line, because we don't find that kind of behavior in nature. Nonetheless, (1) it's your program and (2) I can see it giving some ability to estimate how other shapes might behave.

[mikestar13]

I'm thinking a serious astrologer would only use it as a basis for comparison if at all, and I may drop it entirely when I write the code.

[Jim Eshelman]

I'm thinking a serious astrologer would only use it as a basis for comparison if at all, and I may drop it entirely when I write the code.

In principle, angularity is so simple. In practice, landing exactly what it is and how it behaves (perhaps in ways that don't ultimately matter in practice, but matter much in understanding) is tough.

No wonder astrologers fell into the "angular, succedent, cadent - that's all you have to know" mode.

[mikestar13]

Been working with cycloid curves. Here is the strength for each degree of the quadrant:
Deg Cos Cyc
0 100% 100%
1 100% 95%
2 99% 90%
3 98% 84%
4 96% 79%
5 93% 74%
6 90% 69%
7 87% 64%
8 83% 59%
9 79% 55%
10 75% 50%
11 73% 48%
12 71% 46%
13 69% 45%
14 67% 43%
15 65% 41%
16 63% 40%
17 61% 38%
18 59% 36%
19 57% 35%
20 55% 33%
21 53% 32%
22 51% 30%
23 49% 29%
24 47% 27%
25 45% 26%
26 43% 24%
27 41% 23%
28 39% 22%
29 37% 20%
30 35% 19%
31 33% 18%
32 31% 17%
33 29% 16%
34 27% 14%
35 25% 13%
36 23% 12%
37 21% 11%
38 20% 10%
39 18% 10%
40 17% 9%
41 15% 8%
42 14% 7%
43 12% 6%
44 11% 6%
45 10% 5%
46 8% 4%
47 7% 4%
48 6% 3%
49 5% 3%
50 4% 2%
51 4% 2%
52 3% 1%
53 2% 1%
54 2% 1%
55 1% 1%
56 1% 0%
57 0% 0%
58 0% 0%
59 0% 0%
60 0% 0%
61 0% 0%
62 1% 1%
63 2% 1%
64 4% 2%
65 7% 3%
66 10% 5%
67 13% 7%
68 17% 9%
69 21% 11%
70 25% 13%
71 30% 16%
72 35% 19%
73 40% 22%
74 45% 26%
75 50% 29%
76 55% 33%
77 60% 37%
78 65% 41%
79 70% 46%
80 75% 50%
81 79% 55%
82 83% 59%
83 87% 64%
84 90% 69%
85 93% 74%
86 96% 79%
87 98% 84%
88 99% 90%
89 100% 95%
90 100% 100%


The cycloid curve has the same peaks and troughs as the cosine curve but some significant differences:

1. Edge of the foreground (10 and 80) 50% vs 75%
2. Center of middleground (75) 29% vs. 50%
3. Center of middleground (22) 30% vs. 51%

Cycloid curves make every strength weaker except the peak and trough--a quite noticeably faster fall off. It might be needful to modify the cycloid formula (in addition to the modifications already made to rescale Bradley's equation from 1 to 3 to 0% to 100%).

[Jim Eshelman]

I knew it was a steep drop-off, but hadn't remembered just how steep. It definitely requires getting used to some different numbers,. (There's no real middleground that makes sense.)

Any "100% strong" since of partility evaporates at once. The 75% threshold (our current working cut-off) is at 5 degrees. By 10 degrees it's at 50%. Then (due to your crafting, it mirrors on the other side. (Nice taper you pulled off.) Yes, the numbers you quote at the end.

[mikestar13]

I have some small errors in my piecewise equations (they are a bit of a b*tch to handle). These are the correct results, identical to version 0.4 for cosine curves, but calculated by a better method for adaptation to cycloid curves and/or different foreground widths. These are the results for both curve types and ten degrees foreground width:

Deg Cos Cyc
0 100% 100%
1 100% 95%
2 99% 90%
3 98% 84%
4 96% 79%
5 93% 74%
6 90% 69%
7 87% 64%
8 83% 59%
9 79% 55%
10 75% 50%
11 73% 48%
12 72% 47%
13 70% 46%
14 69% 44%
15 67% 43%
16 65% 41%
17 64% 40%
18 62% 38%
19 60% 37%
20 59% 36%
21 57% 34%
22 55% 33%
23 53% 32%
24 52% 31%
25 50% 29%
26 48% 28%
27 47% 27%
28 45% 26%
29 43% 25%
30 41% 23%
31 40% 22%
32 38% 21%
33 36% 20%
34 35% 19%
35 33% 18%
36 31% 17%
37 30% 16%
38 28% 15%
39 27% 14%
40 25% 13%
41 23% 12%
42 21% 11%
43 19% 10%
44 17% 9%
45 15% 8%
46 13% 7%
47 11% 6%
48 10% 5%
49 8% 4%
50 7% 3%
51 5% 3%
52 4% 2%
53 3% 2%
54 2% 1%
55 2% 1%
56 1% 1%
57 1% 0%
58 0% 0%
59 0% 0%
60 0% 0%
61 0% 0%
62 1% 1%
63 2% 1%
64 4% 2%
65 7% 3%
66 10% 5%
67 13% 7%
68 17% 9%
69 21% 11%
70 25% 13%
71 30% 16%
72 35% 19%
73 40% 22%
74 45% 26%
75 50% 29%
76 55% 33%
77 60% 37%
78 65% 41%
79 70% 46%
80 75% 50%
81 79% 55%
82 83% 59%
83 87% 64%
84 90% 69%
85 93% 74%
86 96% 79%
87 98% 84%
88 99% 90%
89 100% 95%
90 100% 100%

Some intuitive impression. The cosine curve gives more reasonable numbers for key points:
0 -> 100% (the angle)
10 -> 75% (edge of foreground)
25 -> 50% (center of middleground)
40 -> 25% (edge of background)
60 -> 0% (cadent cusp)
70 -> 25% (edge of background)
75 -> 50% (center of middleground)
80 -> 75% (edge of foreground)
90 -> 100% (angle)
These key points seem about right, but the cosine curve seems to stay at or near 100% too long as you move away from the angles. The cycloid curve looks rather better out to three degrees or so from the angle, but then starts dropping too fast. I'm hoping inspiration will show me a non linear curve that has the advantages of both.

[Jim Eshelman]

More or less my impressions, too. The cosine curve has always seemed most fitting to the thresholds.

As for how fast it drops off, this is partly because we are doubling to that - 100 to +180 becomes 0 to 100. Without this "display" hack, IIRC it drops below 99.5 (ie, round to 100) at approximately partile.

This matches the perception that, for most purposes, partile is "exact."

[Jim Eshelman]

I'm sympathetic to the cycloidal curve in that it was Bradley's last mature thoughts on the subject and this deserves respect and attention. I also recognize that my sensibility about the number thresholds comes from working minutely with the cosine values for almost half a century - it surely has molded my sensibilities in the matter - and yet I recall it feeling remarkably right when I first looked at it in aspects.

I think we should let people explore it. I probably wouldn't even have tried to normalize it to bottom out at cadent, but just offered it as he presented it at the end of his life. Once implemented, I will look at it for specific things and let my mind wrap around it a bit. But it's going to take a lot to make me think that the cosine curve is wrong.

(A secondary reason for this is that the sine-cosine curve is basic to the geometry of defining the five major aspects: Cosine values for 0°, 60°, 90°, 120°, and 180° are, respectively, +1.0, +0.5, 0.0, -0.5, and -1.0, shown by the lines bisecting, and then bisecting again, a radius of any circle.)

[mikestar13]

One thought would be to drop normalization and get the exact curve Bradley used, letting the background fall mid-quadrant. I will still normalize his 1.000 to 3.000 range to 0% to 100%, since to have two radically different ways of expressing strength would be too confusing. So perhaps we will have three angularity models: cadent cusp cosine, mid-quadrant cosine, and cycloid mid-quadrant. Only the first two will have adjustable foreground width. I will post revised Bradley curves in a couple of hours.

[mikestar13]

Bradley's curve, expressed as 100% to 0% rather than 3 to 0:

Deg Cyc
0 100%
1 97%
2 93%
3 90%
4 86%
5 83%
6 79%
7 76%
8 72%
9 69%
10 66%
11 63%
12 59%
13 56%
14 53%
15 50%
16 47%
17 44%
18 41%
19 38%
20 36%
21 33%
22 31%
23 28%
24 26%
25 23%
26 21%
27 19%
28 17%
29 15%
30 13%
31 12%
32 10%
33 9%
34 7%
35 6%
36 5%
37 4%
38 3%
39 2%
40 2%
41 1%
42 1%
43 0%
44 0%
45 0%
46 0%
47 0%
48 1%
49 1%
50 2%
51 2%
52 3%
53 4%
54 5%
55 6%
56 7%
57 9%
58 10%
59 12%
60 13%
61 15%
62 17%
63 19%
64 21%
65 23%
66 26%
67 28%
68 31%
69 33%
70 36%
71 38%
72 41%
73 44%
74 47%
75 50%
76 53%
77 56%
78 59%
79 63%
80 66%
81 69%
82 72%
83 76%
84 79%
85 83%
86 86%
87 90%
88 93%
89 97%
90 100%


Much nicer numbers, but the trough is inevitably at 45 degrees.

[Jim Eshelman]

Much nicer numbers indeed.

[mikestar13]

A hybrid curiosity. The curve we get from using the blending the cycloid and cosine curves together near the angles and using the cosine curve exclusively for areas of the quadrant more than ten degrees from the angle. I use all cycloid at the angle, and fade in more and more cosine as the distance from the angle increases. The fade in is linear, so if I took this seriously, I would polish this with non-linear fade in.

Deg strength
0 100%
1 97%
2 94%
3 92%
4 90%
5 88%
6 86%
7 84%
8 81%
9 78%
10 75%
11 73%
12 72%
13 70%
14 69%
15 67%
16 65%
17 64%
18 62%
19 60%
20 59%
21 57%
22 55%
23 53%
24 52%
25 50%
26 48%
27 47%
28 45%
29 43%
30 41%
31 40%
32 38%
33 36%
34 35%
35 33%
36 31%
37 30%
38 28%
39 27%
40 25%
41 23%
42 21%
43 19%
44 17%
45 15%
46 13%
47 11%
48 10%
49 8%
50 7%
51 5%
52 4%
53 3%
54 2%
55 2%
56 1%
57 1%
58 0%
59 0%
60 0%
61 0%
62 1%
63 2%
64 4%
65 7%
66 10%
67 13%
68 17%
69 21%
70 25%
71 30%
72 35%
73 40%
74 45%
75 50%
76 55%
77 60%
78 65%
79 70%
80 75%
9 78%
8 81%
7 84%
6 86%
5 88%
4 90%
3 92%
2 94%
1 97%
0 100%

[Jim Eshelman]

IOW (if I'm getting this correctly at first glance), your taking the one curve and forcing on it the most desirable characteristics of the other while making sure it doesn't get its worse features?

Even though I don't have a Moon-Mercury conjunction, I can see the satisfaction and fun of bending the universe to your will this way There is nothing known to me in nature that would behave in this blended way,, but it's very good art.

[mikestar13]

Exactly, Jim. I just can't take it seriously but it has a high level of aesthetic appeal.

[Jim Eshelman]

This one deserves... Eureka! What follows isn't perfect, but it's damn close.

I have succeeded - with a very simple concept that makes completely logical sense - in creating a single formulation that describes the desired curve in a single equation. Every primary anchor of the definition is exactly, precisely right! A few of the transition numbers look weird (are a little fuzzy), but I can live with that at this stage. I will describe in this post the original thesis this entire thread has been building on; describe the mental concept that we've been trying to fit it into; describe the simple, elegant idea that resolves it all without fudging; and give the final numbers (what the curve looks like).

Concept/Premise:
The concept that makes sense of all the statistical research Bradley published on angularity in his life PLUS Fagan's conceptual leanings PLUS what seems the truth from day-to-day practice is this: Two curves exist in nature. An EXPRESSIVENESS curve peaks exactly at the angles and falls to its minimum exactly halfway between (mid-quadrant: middle of the succedent houses). Also, a REPRESSIVENESS curve peaks (maximizes) at the cadent cusps, falling to its minimum exactly halfway between cadent cusps (which is the middle of the angular houses).

The problem is that these don't align with each other. The first important realization, then, was that each becomes less significant at its minimum. That is, if the REPRESSIVENESS curve is at its weakest, it doesn't really matter much. Same with the EXPRESSIVENESS curve, though it's bottoming out mid-quadrant closely ties into where the cadent / repressiveness curve maximizes. (It's a little awkward because we don't get the same "bleed" of effect in the angular houses.)

How we worked around all that:
The initial thought was that whichever of the two was strongest would prevail - that the other wasn't really relevant. Therefore, the expressiveness curve was the primary thing we heard from near the angles, and the repressiveness curve the main thing having a voice near the cadent cusps. In between, there were places it seemed reasonable to average the two, then a middleground area where we found a lot of clumsy noise of no particular consequence but unusual shape - easy to ignore. - Still, it always felt a little clumsy and, in TMSA, Mike solved the practical problem by force a smoothing of the curve by brute force.

The expressiveness and repressiveness curves, and their blending by the rules just summarized, are here https://www.solunars.com/viewtopic.php? ... 1107#p1104
You can see in the last graph on that post what my blended curve looked like and the awkward middleground.

Today's Eureka:
I realized I'd being trying to awkwardly make up rules that dealt with each curve fading in importance as the other grew... when there was a totally natural way to do it.

I had scaled the expressiveness curve from +1 at the angles to -1 halfway between them (mid-succedent); and the repressiveness curve from -1 at the cadent cusps to +1 halfway between them (mid-angular). I needed each of them to become gradually less important as they left their maximum score so that the other curve gradually became more important. It would be nice if this happened smoothly, naturally, without forcing the issue.

My Doh! moment this morning was this: If a curve is moving from +1 to -1 in strength, its importance can also fade at the same rate! In concept (rescaling it to +1 to 0), it amounts to not much more than using the SQUARE of the calculated score instead of the number itself. (LOTS of things in nature taper at this rate.) The mathematical rendering is more complicated than that, but the concept is EXACTLY that: The delta (rate of change) of the strength curve itself is exactly the same as the delta rate of its importance.

Consequence: Each curve becomes gradually, naturally less important where it should, and usually where the other curve is becoming gradually more important. Then, YOU JUST ADD THEM TO GET THE STRENGTH SCORE where "strength" here means how much expressiveness exceeds repressiveness.

It's... elegant. Here is what the final curve looks like it. (This required NO fudging!) After it, I give Bradley's statistical findings in the SLR studies.

Eureka.png

z-Summary.jpg

The basic parameters are intact: The score is at +1 (+100%) exactly at the angles, 0 (or 50%) exactly at the succedent cusps, -1 (0%) at the cadent cusps, and back to 0 (50%) exactly in the middle of the cadent houses. It defines the angles, the cadent cusps, and the two precise middleground junctures (succedent cusp and mid-cadent) precisely.

Where I wish it were different are certain other thresholds. For example, +0.5 that used to fall at 10° (seeming outer bound of foreground) falls at about 12°. The low-score (formal background) zone doesn't quite include the mid-quadrant, but starts between 47° and 48°. These are symmetrical: The consequence of background reaching nearly to the mid-succedent is that foreground reaches nearly to mid-angular. All the others are about right: The other background threshold (-0.5) is just past 10° from cadent, and the other foreground threshold (+0.5) is just a bit more than 10° from the angle.

[Jim Eshelman]

Here is the awkward math.

Calculate angular and cadent curves as before. In Excel, I have COS(4x) and COS(4(x-60))* (-1). (Mike will have this or his own way of doing it.)

The "faded p scale" - angularity curve (which happens to be in my column P, hence the name) - is the prior result (call it p) multiplied the same number scaled 0 to 1 instead of -1 to +1: p*((p2+1)/2)

The same effect for the repressiveness score (call it q) took a bit of head scratching to get it to work out right, but it turns out to be: q*(1-((q2+1)/2))

This looks clumsy and forced, but it's really a single simple thing: As the score changes from maximum to minimum, it is multiplied by itself to that its contribution fades from 100% to 0% on the same scale (at the same rate).

After getting these two numbers, you just add them. The sums range from +1.25 to -1.25. Divide the score, therefore, by 1.25 to get a "pretty" number.

Here is the value of each degree from the angle (each degree in the mundoscope quadrant) expressed as %.

0 100.0% [precisely foreground]
1 99.8% [1° = 100%]
2 99.1%
3 97.9% [3° = 98%]
4 96.4%
5 94.5%
6 92.3%
7 89.8% [7° = 90%]
8 87.1%
9 84.3%
10 81.3% [10° = 80%]
11 78.3%
12 75.3%
13 72.3%
14 69.4%
15 66.7%
16 64.1%
17 61.7%
18 59.5%
19 57.5%
20 55.8%
21 54.3%
22 53.1%
23 52.1%
24 51.4%
25 50.8%
26 50.4%
27 50.2%
28 50.1%
29 50.0%
30 50.0% [precisely middleground]
31 50.0%
32 49.9%
33 49.8%
34 49.6%
35 49.2%
36 48.6%
37 47.9%
38 46.9%
39 45.7%
40 44.2%
41 42.5%
42 40.5%
43 38.3%
44 35.9%
45 33.3%
46 30.6%
47 27.7%
48 24.7%
49 21.7%
50 18.7% [10-11° = 20%]
51 15.7%
52 12.9%
53 10.2% [7° = 10%]
54 7.7%
55 5.5%
56 3.6%
57 2.1%
58 0.9%
59 0.2%
60 0.0% [precisely background]
61 0.2%
62 1.0%
63 2.3%
64 4.0%
65 6.3%
66 9.1% [6-7° = 10%]
67 12.3%
68 16.0%
69 20.0% [9° = 20%]
70 24.5%
71 29.2%
72 34.2%
73 39.3%
74 44.6%
75 50.0% [precisely middleground]
76 55.4%
77 60.7%
78 65.8%
79 70.8%
80 75.5%
81 80.0% [9° = 80%]
82 84.0%
83 87.7%
84 90.9% [6° = 90%]
85 93.7%
86 96.0%
87 97.7% [3° = 98%]
88 99.0%
89 99.8% [1° = 100%]
90 100.0% [precisely foreground]