New quantification of angularity

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New quantification of angularity

Post by Jim Eshelman » Mon May 08, 2017 11:52 pm

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 / succeedent / cadent houses, but wityh really big cusp orbs. I don't count this in what follows, but it's no farther 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 centerted 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 succeedent cusps, and BACKGROUND being the remaining third of space that is centered around the cadent cusps.

This is well-supported by Bradley's earlty 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 succeedent 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 succeedent and cadent cusps as of equal strength, which conflicts with my perception (and is unsupported by the particular Bradley statistics I cited above). Nonetheless, there was always a sense that midquadrant itself was pretty weak - as if the background zone stretched deeper into the succeedent 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-qudrant.

The result is the following:
New Grounds 2.jpg
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 succeedent house. This is 30° out of every quadrant.

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


That's how I see it now - a slight tweak of what I've taught previously.
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Re: New quantification of angularity

Post by Jim Eshelman » Mon May 08, 2017 11:53 pm

Venus_Daily wrote: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.
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Re: New quantification of angularity

Post by Jim Eshelman » Mon May 08, 2017 11:54 pm

DDonovanKinsolving wrote:
Zenith wrote: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

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Re: New quantification of angularity

Post by Jim Eshelman » Mon May 08, 2017 11:55 pm

Zenith wrote: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.
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Re: New quantification of angularity

Post by Jim Eshelman » Mon May 08, 2017 11:56 pm

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.

Image

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.

Image

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.

Image
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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:03 am

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.)
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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:04 am

Arena wrote: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.

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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:04 am

Arena wrote: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.
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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:06 am

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

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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:07 am

Arena wrote: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.
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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:07 am

Jupiter Sets At Dawn wrote: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?

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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:08 am

Arena wrote: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.

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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:09 am

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.
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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:09 am

Jupiter Sets At Dawn wrote:
Arena wrote: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.

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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:10 am

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.
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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:10 am

Arena wrote: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.

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Re: New quantification of angularity

Post by Jim Eshelman » Tue May 09, 2017 12:11 am

Jupiter Sets At Dawn wrote: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?

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Re: New quantification of angularity

Post by mikestar13 » Thu Jul 20, 2017 3:32 pm

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.
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Re: New quantification of angularity

Post by Jim Eshelman » Thu Jul 20, 2017 3:40 pm

mikestar13 wrote:
Thu Jul 20, 2017 3:32 pm
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.
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Re: New quantification of angularity

Post by mikestar13 » Thu Jul 20, 2017 3:56 pm

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.
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Re: New quantification of angularity

Post by mikestar13 » Thu Jul 20, 2017 10:14 pm

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.
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Re: New quantification of angularity

Post by mikestar13 » Sat Jul 29, 2017 5:40 pm

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
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Re: New quantification of angularity

Post by Jim Eshelman » Sun Jul 30, 2017 9:35 am

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!
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