Angularity
Posted: Thu May 05, 2022 5:33 am
Currently, TMSA calculates angularity according to one of two fixed models, mid -quadrant or cadent cusp according to where the trough of planetary expressiveness is taken (the peak is always at the angles, and neither peak nor trough will change).
Let me describe how the current curve works in some detail as a basis for the changes coming (to simplify this discussion we will ignore minor angles but they will be handled in an analogous fashion over a narrower range of degrees).
First, lets establish some ideas. TMSA expresses strength as a percentage ranging from 0 to 100. Key points are: angles = 100% edge of foreground = 75%. center of middleground = 50% , edge of the background = 25%, center of background = 0%. These numbers are not calculated directly, but strength is internally expressed as a number ranging from -1 to +1 which is then rescaled to a 0% to 100% range for printing.
Let's make a naive attempt at calculating some numbers. First of all, we take the longitude difference in degrees between the midheaven and the planet, call this d. Then x = d mod 90.
Let's get a few values of the function y = cos(4 * x):
x y
0 +1
15 +0.5
22.5 0
30 -0.5
45 -1
60 -0.5
67.5 0
75 +0.5
90 1
This is almost what we want for the mid-quadrant model, but not quite (it makes the foreground too wide). So we use a series of statements in the form of if x >= a and x < b, then x = (x- a) * m/n + c, where m, n and c are constants for that range of values. For example if x >= 0 and x < 10, x = (x -0) *3/2 + 0. We then calculate y = cos(4 *x), so that 10 degrees and 80 degrees both yield +0.5, the edge of the foreground, which we want for the mid-quadrant model. 45 degrees remains 45 and yields -1.
Now for the cadent cusp model, we do the same for the foreground area, but for other ranges of degrees, 60 (the cadent cusp) is converted to 45.
Now the equation y = cos(4 *x) is the preferred method of calculating strength based on modified quadrant position and will be the TMSA default (it's the method we use in TMSA 0.4, but we could equally use a cycloid equation (though this would change some of the modifications needed).
But what if we don't want the foreground to be ten degrees wide? What if we want fifteen degrees like in early Siderealist usage? What if we wanted five degrees like cosmobiologists most likely would if they regarded angularity? Never fear, the degree ranges and the corresponding constants can be calculated at run time! Ten degrees is Jim's preferred foreground width and mine and will always be the TMSA default.
So you will always see final strength values ranging from 0% at the center of the background to 100% at the angles, with 75% at the edge of foreground, 25% at the edge of the background, and 50% at the center of the middleground, regardless of where within reason those edges are located. TMSA does all the mathematical heavy lifting in a microsecond or two.
Let me describe how the current curve works in some detail as a basis for the changes coming (to simplify this discussion we will ignore minor angles but they will be handled in an analogous fashion over a narrower range of degrees).
First, lets establish some ideas. TMSA expresses strength as a percentage ranging from 0 to 100. Key points are: angles = 100% edge of foreground = 75%. center of middleground = 50% , edge of the background = 25%, center of background = 0%. These numbers are not calculated directly, but strength is internally expressed as a number ranging from -1 to +1 which is then rescaled to a 0% to 100% range for printing.
Let's make a naive attempt at calculating some numbers. First of all, we take the longitude difference in degrees between the midheaven and the planet, call this d. Then x = d mod 90.
Let's get a few values of the function y = cos(4 * x):
x y
0 +1
15 +0.5
22.5 0
30 -0.5
45 -1
60 -0.5
67.5 0
75 +0.5
90 1
This is almost what we want for the mid-quadrant model, but not quite (it makes the foreground too wide). So we use a series of statements in the form of if x >= a and x < b, then x = (x- a) * m/n + c, where m, n and c are constants for that range of values. For example if x >= 0 and x < 10, x = (x -0) *3/2 + 0. We then calculate y = cos(4 *x), so that 10 degrees and 80 degrees both yield +0.5, the edge of the foreground, which we want for the mid-quadrant model. 45 degrees remains 45 and yields -1.
Now for the cadent cusp model, we do the same for the foreground area, but for other ranges of degrees, 60 (the cadent cusp) is converted to 45.
Now the equation y = cos(4 *x) is the preferred method of calculating strength based on modified quadrant position and will be the TMSA default (it's the method we use in TMSA 0.4, but we could equally use a cycloid equation (though this would change some of the modifications needed).
But what if we don't want the foreground to be ten degrees wide? What if we want fifteen degrees like in early Siderealist usage? What if we wanted five degrees like cosmobiologists most likely would if they regarded angularity? Never fear, the degree ranges and the corresponding constants can be calculated at run time! Ten degrees is Jim's preferred foreground width and mine and will always be the TMSA default.
So you will always see final strength values ranging from 0% at the center of the background to 100% at the angles, with 75% at the edge of foreground, 25% at the edge of the background, and 50% at the center of the middleground, regardless of where within reason those edges are located. TMSA does all the mathematical heavy lifting in a microsecond or two.