Aspect Orbs and Classes

[mikestar13]

This discussion started in the noviens thread, but it is not specific to noviens. We hypothize that the maximum orb for squares, trines. and sextiles is +-7°30', or 15° total (three pentades), with class boundaries of 3° and 5° and 1° being partile. We further hypotheize that generally accepted that the maximum orb for conjunctions and oppositions is +-10° or 20° total (four pentades). The class bounaries are 4° and 7°, and 1° is still considered partile. class orb

I suggest that this is mathematically inconsistent. The maximum orbs an the class 1 boundaries agree the the orb for an opposition is 4/3 the orb for a square, but applying the same ratio, the class 2 boundary should be 6°40' and partility should be 1°.

How about octiles? We have a class one boundary of 1° (which is indistinguishable from partile) and a class of 2 boundary of 2°, and a maximum orb not usually set but TMSA calculates on the basis of +-2°30'or a total of 5° (one pentade) at default orb, so again we are inconsistent. So it seems that the numbers should be one third of the numbers for a square, thus we have partiality at 20', the class 1 boundary at 1°', the class 2 boundary at 1°40.' and a maximum orb of 2°30'.

Were this adopted we would have identical sine curves except for maximum orb, and completely consistent partiality and class boundaries, and no weird edge cases where the same %strength can fall in different classes, depending on the aspect.

The curves for angularity (both major and minor) are sinusoidal but not true sine curves and are scaled differently, so the numbers are not comparable.

[Jim Eshelman]

I suggest that this is mathematically inconsistent. The maximum orbs an the class 1 boundaries agree the the orb for an opposition is 4/3 the orb for a square, but applying the same ratio, the class 2 boundary should be 6°40' and partility should be 1°.

My opinion: Except where we have had, highly precise cut-offs like the clear 7.5° line for sx/sq/tr, I favor whole degrees. People think in those terms better. Computers aside, I still think in terms of people eye-balling a chart even though I rarely do that anymore. So, I see 6°40' and go, "Yeah, about 7°."

The Big Book, of course, will tell people to use 3/6/9 boundaries for all five major aspects (even though I'll somewhere say what I use personally), partly because it's easy, partly because it's a convenient starting point, and partly because it encourages the idea that none of these are hard walls. (Even the "end of the line" 7.5°/10° lines are more of, "It's run out of steam before now anyway, right?")

How about octiles? We have a class one boundary of 1° (which is indistinguishable from partile) and a class of 2 boundary of 2°, and a maximum orb not usually set but TMSA calculates on the basis of +-2°30'or a total of 5° (one pentade) at default orb, so again we are inconsistent. So it seems that the numbers should be one third of the numbers for a square, thus we have partiality at 20', the class 1 boundary at 1°', the class 2 boundary at 1°40.' and a maximum orb of 2°30'.

These are tricky. I often find myself not wanting to think in terms of class. I see them run out to 2° and disappear (at least, where there is any clear perception - but, then, isn't though how Class 2 to Class 3 transitions feel anyway?). I've thought about putting neat 1°/2°/3° in place on the theory, "You're going to ignore Class 3 anyway, right? So why not ignore octiles past 2° the same way." Usually, I just leave Class 3 empty.

They seem really clear to 2°, and then pretty invisible after that, so 2° seems the Class 2 drop-off. All I can really say is that I'm glad I'm not the guy that has to figure out how the curve looks . (Without making the decision to do this, I end up being pretty linear instead or curvy for octiles.)

I guess all this means I had something to say but no real pretense of an answer for you. The Big Book probably will say use to 2° and give some preference if within a degree or so. I wouldn't put Class 2 tighter than 2°.

Were this adopted we would have identical sine curves except for maximum orb, and completely consistent partiality and class boundaries, and no weird edge cases where the same %strength can fall in different classes, depending on the aspect.

The shape of the semisquare curve raises related questions of minor angle curves, of course, where the same thing is true but a little more tapering: They seem solid out to 2° and mostly can be seen running out of steam down to 3°. It's still hard for me to decide if that means 2° marks Class 1 or Class 2: If Class 1, then there is no Class 3 at all; and if Class 2,then there is the aberration that in mundane charts it acts like partile!

The curves for angularity (both major and minor) are sinusoidal but not true sine curves and are scaled differently, so the numbers are not comparable.

I'm unclear why you say this about major angles (unless you're using the composite equation I recently eureka'd). I suppose there is a problem because of curve combination?

[Jim Eshelman]

Another way to say this is that I really don't have any clarity on what an octile curve would look like. I'm sure they're solid to 2°. I have no confidence in them at all beyond that (but, then, that's the way Class 3 works anyway, right?) They do seem to become even more certain within a degree. So I've been comfortable with 1°/2°/3° (or leave off the Class 3) personally.

Or I could collate a whole pile of unknowns Do I, contrary to any view I've had of my chart historically, really have a partile mundane octile Moon-Saturn at my birthplace (and is there also a 2°49' mundane Moon-Mercury octile, or is Moon-Mercury just the Novien, or is there no Moon-Mercury and it's just that I'm a Virgo while Aquarius Moon is kinda sorta like Mercury and maybe should have been its exaltation or.... my head is spinning.

I have no ecliptical octiles so I have no "feeling them from the inside. My wife has a partile one that is quite credible. My most important ex has a partile one that isn't objectionable, a 1°+ octile that is quite fitting, and a 2°+ octile that I'm not so sure fits. My second most important ex had a partile Moon octile that was exceptionally fitting. Steve's Scorpio Moon on Antares has a half-degree octile to Mars, which I think is the thing behind his certainty that has Mars has an extraordinary importance in his chart (and I don't know how I can distinguish Moon in Scorpio vs. on Antares vs. in hard aspect to Mars, but I accept that its there). He also has a degree and a half octile that matches much of what he has said about his childhood, and a degree-plus Jupiter-Neptune octile that screams aloud about his success in the theater business (and a 2 1/2° Sun-Pluto that isn't unreasonable). You have a totally credible Mercury-Mars at half a degree (though Moon is in Aries, so...), an utterly believable Sun-Pluto at a degree and a half, and Moon octiles to Mars and Saturn in the 2-3° range that I don't know how to filter out of the rest of the chart. And so on.

[mikestar13]

The thought occurs to me that major and minor aspects may have different shaped curves, as major and minor angles do. For time being TMSA will not change how it works, but I will consider various equations and run anything interesting by you. The fact the all curves are sinusoidal need not imply the exact same shape. Indeed I will explore the question of how to fit a sinusoidal curve to three points, so that the curve can fit arbitrary points (within reason).

[Jim Eshelman]

Idle thought without running the math...

I wonder if the minor aspects and a goes, seeming to rise acutely and drop off so swiftly, might be sinusoidal.

I might have to work that up. I probably have a spreadsheet with most of it done.

[mikestar13]

One change I will make is adopting your formula which is more accurate than mine IMHO, and has nicer numbers at key points.

m = 90 / maxorb
p1 = (cos(m * orb) + 1) / 2
p2 = (p1 - .5) * 2

Where orb is the absolute value of the difference between the actual position and the exact aspect, and maxorb is the highest difference that is to be considered to be in aspect (7.5 for squares, 10 for oppositions)

p1 gives the familiar 99% partility, 90% class 1, 75% class 2, and 50% class 3 (= maxorb).
p2 is the re-scaling function for display, and will remain the same regardless of how we tweak the p1 formula, so all the mathematical gyrations will focus on p1.

We can tweak p1 to interpolate between those key points by:

1. Varying the orb we use for the formula from the actual orb by a f(orb), thus if we deem the boundaries are 3/6/9 and one degree is still partility, we get 99% partility, 90% for 3 degrees, 75% for 6 degrees, and of course 50% for 9 degrees.
2. Varying m by g(orb).
3. A combination of both.

Where f and g are smooth piecewise functions not yet derived at this point in time. If research in this handling of aspects bears fruit, we can let the user specify any three non-insane class boundaries (<= 15 degrees, class 1 < class2 < class 3). TMSA already rejects insane values. I then will consider the implications for angularity. Let's say someone decides "Jim E. is wrong, the foreground is 15 degrees wide like Fagan and Bradley were working with." For this user, they would set the class boundaries for major angles to 5/10/15, and a planet at edge of foreground would always have 75% strength, no matter what non-insane class boundaries.

I'm inclined to always treat one degree as partile rather than make it user selectable. Though I could allow user to specify, obviously with a default of 1.

I'm also wondering in a change in our display philosophy might be considered. Is 50% meaning equally likely to manifest or not really that unintuitive, as opposed to 0% meaning that? If we had angularity run from 100% to -100% does this better show the expressiveness/repressiveness concept (in this case edge of foreground would be +50%, edge of background would be -50% and center of middleground would be 0%)?. Of course the two possibilities are independent of one another. BTW, this is how TMSA handles angularity internally. Thoughts?

[Jim Eshelman]

Idle thought without running the math...

I wonder if the minor aspects and a goes, seeming to rise acutely and drop off so swiftly, might be sinusoidal.

I might have to work that up. I probably have a spreadsheet with most of it done.

I just worked that up. It sucks. Totally wrong.

However, to my surprise, the sinusoidal is pretty good for octiles. Let's presume for sake of discussion that what the "nothing past 2" I'm seeing for octiles is a Class 2 drop-off and the curve runs to 3 degrees. Set a multiplier of 30 (maxorb = 3) and you get a curve with 50% at 3.0 exactly, 75% at 2.0 exactly, 90% at just a hair before 1 15' (something like 73.5') - all "feels like" 1/2/3 degrees, with 1.0 being 93%.

The fine points sort themselves out from there. The main thing is that 1/2/3 is very close to the truth, and 1.25/2/3 is mostly precise to the minute of arc, which is much better than I expect. Stunningly, 99.5% ("rounds to 100%") is exactly 30'! (I get that at 28'-30', so it probably runs there exactly at 29')

A straight sinusoid on a maxorb = 3.0 seems (tentatively) to exactly describe both minor aspects and minor angles (except for that irregularity of 2 degree orbs being the dormancy drop-off in mundane charts, though I think I understand the difference of those and natal/return charts.)

[mikestar13]

So it look like the formula with the multiplier 90 / max orb is a enough good approximation for reasonable class values, though the class values being inexact may lead to slight anomalies. Close enough for government work.

[mikestar13]

Maybe minor angles don't work precisely the same way as octiles, at least for ingresses. In a nativities and solunars, do you consider a planet 2 degrees from a minor angle the same strength as it would be at 3 degrees from a major angle?

[Jim Eshelman]

Breaking this one into two parts for response...

We can tweak p1 to interpolate between those key points by:

1. Varying the orb we use for the formula from the actual orb by a f(orb), thus if we deem the boundaries are 3/6/9 and one degree is still partility, we get 99% partility, 90% for 3 degrees, 75% for 6 degrees, and of course 50% for 9 degrees.
2. Varying m by g(orb).
3. A combination of both.

Where f and g are smooth piecewise functions not yet derived at this point in time. If research in this handling of aspects bears fruit, we can let the user specify any three non-insane class boundaries (<= 15 degrees, class 1 < class2 < class 3). TMSA already rejects insane values.

I think you are trying to force alignment (or at least allow forced alignment) of percentages and class boundaries - right? (I'm not sure. Rereading slowly didn't help.)

If so, I don't have a firm opinion on the idea. My historic, not-rethought opinion is that I'm not uncomfortable letting these two be different. - That is, the arbitrary bounds we use to sort aspects in whole degrees need not (I think) precisely align with % even when they are really close (and I think people will think in whole degrees for a long time, and only a minority of us are likely to look too closely at the percentages).

What I think is most important (I could be wrong) is that the sine curve be undistorted in shape. That is, I think it shows us what's happening.

Of course, in the spirit of flexibility, somebody could think I'm wrong about that and want to force exact bounds and % aligned, and I'd have nothing against them having a way to do this.

Or maybe I'm not answering what you actually were saying (under this Neptune 0 degree angular lunar, I'm letting my default assumption be that I'm missing the obvious).

[mikestar13]

I need a few more sanity checks for class boundaries. 4/5/6 would produce very weird % numbers for the class boundaries, but in practice the user will be setting the class one boundary at around 40% of max orb, and the class 2 boundary at around 2/3 of maxorb (with whatever inexactness is induced by choosing a round number).

[mikestar13]

Yes I am trying to force alignment, but that doesn't seem necessary. The minor angle curve remains a question. The major angle curve is the same as the aspect curve with respect to the foreground (middleground and background are more complicated).

[Jim Eshelman]

Maybe minor angles don't work precisely the same way octiles, at least for ingresses. In a nativities and solunars, do you consider a planet 2 degrees from a minor angle the same strength as it would be at 3 degrees from a major angle?

The phenomenon of dormancy is unique to ingresses. The concept doesn't exist for a natal or solunar. Therefore, there isn't really a conflict in the sense that "dormancy" is defined consistently the one place it's used. (There are also the conditions unique to ingresses that need a MUCH longer explanation but probably don't apply here anyway so I'll skip it.)

Aside from the dormancy issue, minor angle orbs seem to work identically in natals, returns, and ingresses in the sense that 2 degrees is the palatable boundary after which the drop-off is fast, effects can be seen past that point (with at least one statistical result showing 3 degrees as the end of the road, which matches observation), and a real difference between about 1 degree vs. about 2 degrees. In other words, 2 degrees looks across the board like it is a Class 2 marker. The rest falls in place from there.

If that's true, you're still right - minor angles MIGHT not work the same as octiles - but there is nothing in observation or theory I see that suggests that they are all that divergent. Nothing to require we force them to behave differently. - However, WE may behave differently in the sense that I almost always ignore Class 3 aspects and yet nearly always take Class 3 angularities seriously. (Probably the number of factors involved, not sure. 10 possibilities in combination instead of 45.)

Oh, I almost didn't answer your explicit question:

In a nativities and solunars, do you consider a planet 2 degrees from a minor angle the same strength as it would be at 3 degrees from a major angle?

No, I consider it the same strength as a major angle at about 7. - Same in ingresses EXCEPT for the dormancy question which then would seem not to be a Class distinction but rather its own kind of thing. (I could come up with at least a couple of theories about the why for this, but they'd just be theories. The gist is that geography is the bigger determinator for dormancy with ingresses.)

[Jim Eshelman]

The major angle curve is the same as the aspect curve with respect to the foreground (middleground and background are more complicated).

Got it. That's actually what I was thinking is so. - My consolidated single equation distorts this a little, especially with respect to bilateral values; but not (if we're honest with ourselves) within the bounds of what we really can tell from observation.

I'd be comfortable having the single equation used for % on angularity and then hard-set boundaries without stressing that these don't agree perfectly bilaterally.

[Jim Eshelman]

Let's say someone decides "Jim E. is wrong, the foreground is 15 degrees wide like Fagan and Bradley were working with." For this user, they would set the class boundaries for major angles to 5/10/15, and a planet at edge of foreground would always have 75% strength, no matter what non-insane class boundaries.

This would be a reason to spread the angularity curve, agreed. I do wonder whether such a person would give a dang about the % scores (meaning, it's not critical they match) but I suspect at least some of them might. OTOH, if you don't rescale it, then the last 5 degrees could still be set as boundaries AND the scores would be below 50%, which actually isn't absurd. These would all change if the person also adopted the mid-quadrant bottom.

I'm inclined to always treat one degree as partile rather than make it user selectable. Though I could allow user to specify, obviously with a default of 1.

By convention, it means 1 degree, though there might be people or circumstances where somebody wants a similar threshold to have a different value. For example, when we get to directions, some people on this forum think what is normally a 60' default boundary should be set at 15' or some other value. The flexibility to adjust the boundary checked for this "super-close" threshold might have value.

I don't think there is a rush for that, though - it could always go on the wish list. For anything created so far, I can't think of any spot where partile is used as a distinction.

[mikestar13]

In a nativities and solunars, do you consider a planet 2 degrees from a minor angle the same strength as it would be at 3 degrees from a major angle?

No, I consider it the same strength as a major angle at about 7. - Same in ingresses EXCEPT for the dormancy question which then would seem not to be a Class distinction but rather its own kind of thing. (I could come up with at least a couple of theories about the why for this, but they'd just be theories. The gist is that geography is the bigger determinator for dormancy with ingresses.)

This aligns with my thoughts: non-dormancy is not equal to class boundary or %angularity strength. The octile formula will work for minor angles.

[Jim Eshelman]

I'm also wondering in a change in our display philosophy might be considered. Is 50% meaning equally likely to manifest or not really that unintuitive, as opposed to 0% meaning that? If we had angularity run from 100% to -100% does this better show the expressiveness/repressiveness concept (in this case edge of foreground would be +50%, edge of background would be -50% and center of middleground would be 0%)?. Of course the two possibilities are independent of one another. BTW, this is how TMSA handles angularity internally. Thoughts?

Definitely a couple of ways to look at this.

At present, my vote is not to make this change, at least for aspects. I'm thinking from the point of view of a new user (perhaps someone who hasn't been in on the Solunars.com discussion of things and is simply looking at the output for the first time). The intuitive idea will be that aspects are running down to 0% then quitting, not running down to 50% (or higher) and quitting.

But you specifically mentioned angularity, not aspects. On this, my opinion is less clearcut. Yes, +100% to -100% would show the expressiveness-repressiveness more pointedly. I don't know that this helps or hurts anyone in practice. My only objection to it is that I like seeing the immediate background numbers drop to single digits - partly my eyes and partly the way my brain works, I look at the shapes of things rather than the exact characters - I scan to see single-digit numbers on angularity rather than actually read the numbers, just as I kook for the three-digit 100 before I actually read the two-digit numbers. I can see all the characters without effort, I've just learned not to actually concentrate on them when I can get my information from shape and position. This may be utterly personal and more or less unique to me.

[Jim Eshelman]

The octile formula will work for minor angles.

As I read this, I realize this is a non-trivial realization. I intuit that it has other implications not yet thought of - perhaps suggesting an answer where the question hasn't sharpened yet, or perhaps suggesting something about the underlying structure of things. (Not sure. Intuition.)

In any case... this is a non-trivial stabilization of thought on something.

[mikestar13]

One case for defining partilty comes to mind: non-foreground partile aspects. If we are going to use them, TMSA needs to know what the boundary of partility is. The idea of changing the way angularity is displayed is not that important, single digits for the remote background is eye-catching and does seem to obviate the need for "b" marks.

[mikestar13]

Eureka. Instead of the checking a box, those chart types using a separate listing of non-foreground partile aspects will have a orb of partility in minutes (0 or blank meaning don't list these), default 60'.

[Patrick Machado]

My only objection to it is that I like seeing the immediate background numbers drop to single digits - partly my eyes and partly the way my brain works, I look at the shapes of things rather than the exact characters - I scan to see single-digit numbers on angularity rather than actually read the numbers, just as I kook for the three-digit 100 before I actually read the two-digit numbers. I can see all the characters without effort, I've just learned not to actually concentrate on them when I can get my information from shape and position. This may be utterly personal and more or less unique to me.

I operate very similarly. The way angularity scores are displayed, at least, I find to be close to perfect.

One case for defining partilty comes to mind: non-foreground partile aspects. If we are going to use them, TMSA needs to know what the boundary of partility is.

I'm not skilled at usefully taking non-foreground partile aspects into account in Solunars, and only really notice them if they're very close to 0°00' (at least under 0°10' or so, I think). If defining partility were an option, I'd certainly be tempted to narrow the boundary for this purpose, at least experimentally.

[Jim Eshelman]

One case for defining partilty comes to mind: non-foreground partile aspects. If we are going to use them, TMSA needs to know what the boundary of partility is.

You're right. A hard-set value is needed, which raises the question of whether it should be user-variable

[mikestar13]

I have changed the foreground portion of the main angularity curve to your more accurate aspect formula and restored minor angularity curve to the aspect formula with a three degree max orb. Still at work adapting the MG and BG portions of the MAC. So in principle, everything is computed by the aspect formula.

[Jim Eshelman]

This is getting interesting.

[mikestar13]

Background finished. When the middle ground between angle and cadent cusp is easy, the middleground at the succedent cusp is trickier, is the center at the of the middleground at the cusp? I have already allowed for the background extending further into the succedent house than into the cadent house. Is the the middleground correspondingly foreshortened in succedent houses or is the center of the middleground at about 25 degrees of the angular houses? The latter assumption is simpler, since middleground doesn't seem to be a thing in itself but rather the comparative absence of expressiveness and repressiveness. I will post some figures later today.

[Jim Eshelman]

I take it you decided not to use the single equation for the whole circle? You certainly don't avoid hard work <g>.

[mikestar13]

Yeah I checked that in the new quanitication of angularity thread, and I see we want center of middleground at the succedent cusp. Implying the succedent cusp portion of the middleground is a bit foreshortened.

[Jim Eshelman]

I don't know that "center of middleground" is the correct idea - in my idealized picture, the middleground is two-thirds of an angular house and a third of a succedent house - but rather than the succedent cusp is the 50% strength point.

https://solunars.com/viewtopic.php?f=15&t=173#p51107

But, of course, it has the caveats we both talked about in that thread.

[mikestar13]

0 -> 1
10 -> .75
30 -> .5
40 -> .25
60 -> 0
70 -> .25
75 -> .5
80 -> .75
90 -> 1

Gives all the key points (where -> is "maps to"). A non linear 9 point curve fitting of those values gives a smooth single curve. Bessel won't help me here, it requires uniform intervals on the left side. Lagrange or Newton or cubic spines or ? will be needed.

[mikestar13]

The Python package scipy has everything I need. Downloading now, it will take 2 or 3 hours to learn to use it.

[mikestar13]

Oop, scipy had a build error. I'll have to do it the hard way with numpy (which doesn't require building).

[mikestar13]

Not so hard. Here are the angularity values for each 1 degree increment in the quadrant, Lagrange interpolated. The key points are right, but the foregrounds are grossly asymmetrical. Trying different interpolation methods.

0.0 100%
1.0 84%
2.0 73%
3.0 67%
4.0 64%
5.0 63%
6.0 64%
7.0 66%
8.0 69%
9.0 72%
10.0 75%
11.0 78%
12.0 80%
13.0 82%
14.0 84%
15.0 84%
16.0 85%
17.0 84%
18.0 83%
19.0 82%
20.0 80%
21.0 78%
22.0 75%
23.0 72%
24.0 69%
25.0 66%
26.0 63%
27.0 60%
28.0 56%
29.0 53%
30.0 50%
31.0 47%
32.0 44%
33.0 41%
34.0 39%
35.0 36%
36.0 34%
37.0 31%
38.0 29%
39.0 27%
40.0 25%
41.0 23%
42.0 21%
43.0 19%
44.0 17%
45.0 16%
46.0 14%
47.0 12%
48.0 11%
49.0 9%
50.0 7%
51.0 6%
52.0 4%
53.0 3%
54.0 2%
55.0 1%
56.0 0%
57.0 0%
58.0 0%
59.0 0%
60.0 0%
61.0 1%
62.0 2%
63.0 3%
64.0 5%
65.0 7%
66.0 10%
67.0 13%
68.0 17%
69.0 21%
70.0 25%
71.0 30%
72.0 34%
73.0 39%
74.0 45%
75.0 50%
76.0 55%
77.0 61%
78.0 66%
79.0 70%
80.0 75%
81.0 79%
82.0 83%
83.0 86%
84.0 89%
85.0 91%
86.0 93%
87.0 95%
88.0 97%
89.0 98%
90.0 100%

[mikestar13]

Now the same values for piecewise Lagrange interpolation, much more symmetrical and more natural than my previous piecewise linear interpolation, though the numbers aren't grossly different from piecewise linear which I had been using:

0.0 100%
1.0 98%
2.0 95%
3.0 92%
4.0 90%
5.0 88%
6.0 85%
7.0 82%
8.0 80%
9.0 78%
10.0 75%
11.0 74%
12.0 73%
13.0 71%
14.0 70%
15.0 69%
16.0 68%
17.0 66%
18.0 65%
19.0 64%
20.0 62%
21.0 61%
22.0 60%
23.0 59%
24.0 57%
25.0 56%
26.0 55%
27.0 54%
28.0 52%
29.0 51%
30.0 50%
31.0 48%
32.0 45%
33.0 42%
34.0 40%
35.0 38%
36.0 35%
37.0 32%
38.0 30%
39.0 28%
40.0 25%
41.0 24%
42.0 22%
43.0 21%
44.0 20%
45.0 19%
46.0 18%
47.0 16%
48.0 15%
49.0 14%
50.0 12%
51.0 11%
52.0 10%
53.0 9%
54.0 8%
55.0 6%
56.0 5%
57.0 4%
58.0 2%
59.0 1%
60.0 0%
61.0 2%
62.0 5%
63.0 8%
64.0 10%
65.0 12%
66.0 15%
67.0 18%
68.0 20%
69.0 22%
70.0 25%
71.0 30%
72.0 35%
73.0 40%
74.0 45%
75.0 50%
76.0 55%
77.0 60%
78.0 65%
79.0 70%
80.0 75%
81.0 78%
82.0 80%
83.0 82%
84.0 85%
85.0 88%
86.0 90%
87.0 92%
88.0 95%
89.0 98%
90.0 100%

[Jim Eshelman]

I wanted to compare this to what my single formula generates. It's interesting how they move apart and back together. I've bolded anything more than 3% points different (which is most of it).

0.0 100% 100.0%
1.0 98% 99.8%
2.0 95% 99.1%
3.0 92% 97.9%
4.0 90% 96.4%
5.0 88% 94.5%
6.0 85% 92.3%
7.0 82% 89.8%
8.0 80% 87.1%
9.0 78% 84.3%
10.0 75% 81.3%
11.0 74% 78.3%
12.0 73% 75.3%
13.0 71% 72.3%
14.0 70% 69.4%
15.0 69% 66.7%
16.0 68% 64.1%
17.0 66% 61.7%
18.0 65% 59.5%
19.0 64% 57.5%
20.0 62% 55.8%
21.0 61% 54.3%
22.0 60% 53.1%
23.0 59% 52.1%
24.0 57% 51.4%
25.0 56% 50.8%
26.0 55% 50.4%
27.0 54% 50.2%
28.0 52% 50.1%
29.0 51% 50.0%
30.0 50% 50.0%
31.0 48% 50.0%
32.0 45% 49.9%
33.0 42% 49.8%
34.0 40% 49.6%
35.0 38% 49.2%
36.0 35% 48.6%
37.0 32% 47.9%
38.0 30% 46.9%
39.0 28% 45.7%
40.0 25% 44.2%
41.0 24% 42.5%
42.0 22% 40.5%
43.0 21% 38.3%
44.0 20% 35.9%
45.0 19% 33.3%
46.0 18% 30.6%
47.0 16% 27.7%
48.0 15% 24.7%
49.0 14% 21.7%
50.0 12% 18.7%
51.0 11% 15.7%
52.0 10% 12.9%
53.0 9% 10.2%
54.0 8% 7.7%
55.0 6% 5.5%
56.0 5% 3.6%
57.0 4% 2.1%
58.0 2% 0.9%
59.0 1% 0.2%
60.0 0% 0.0%
61.0 2% 0.2%
62.0 5% 1.0%
63.0 8% 2.3%
64.0 10% 4.0%
65.0 12% 6.3%
66.0 15% 9.1%
67.0 18% 12.3%
68.0 20% 16.0%
69.0 22% 20.0%
70.0 25% 24.5%
71.0 30% 29.2%
72.0 35% 34.2%
73.0 40% 39.3%
74.0 45% 44.6%
75.0 50% 50.0%
76.0 55% 55.4%
77.0 60% 60.7%
78.0 65% 65.8%
79.0 70% 70.8%
80.0 75% 75.5%
81.0 78% 80.0%
82.0 80% 84.0%
83.0 82% 87.7%
84.0 85% 90.9%
85.0 88% 93.7%
86.0 90% 96.0%
87.0 92% 97.7%
88.0 95% 99.0%
89.0 98% 99.8%
90.0 100% 100.0%

[mikestar13]

The difference in the foreground (the most important area) is attributable largely but not entirely due to the edge of foreground scaling to ~80% rather than my 75%.

[Jim Eshelman]

Also the background not extending do far mto midquadrant.