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Old 02-12-2012, 04:15 AM
DevKin DevKin is offline
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Geometric white to black color progression

Hello.

I've been searching far and wide for an answer to this question, but I to no avail. Because of that, I registered on your forum. Anyway...

I have this task at my art school. I have to make 3 white to black progression scales with paint. Each scale has 10 segments of equal size.

The first scale is based on arithmetic progression, the second - geometric progression and the third is a scale that doesn't have the problems connected with the previous two scales.

While I had no problem making the first arithmetic scale (just add one more drop of black paint in each segment than the previous segment had).

0, 1, 2, 3, 4, 5 black drops and so on.

Alas, I have no idea how to progress geometrically. If I start with two drops on the second segment I go up to hundreds of drops in the last.

TL;DR: Using paint (not the MS variant) how do I make a 10 segment scale from white to black progressing geometrically?

P.S. I apologize in advance for any art terms that I didn't use properly, I am not studying in English.
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Old 02-12-2012, 08:22 AM
DevKin DevKin is offline
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Re: Geometric white to black color progression

Perhaps I posted in the wrong section of the forum, can someone re-redirct me please?
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Old 02-12-2012, 08:43 AM
marecdl marecdl is offline
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Re: Geometric white to black color progression

Hello Devkin,

I would quite know, but, if you converted the scale in two triangles?
For example: you draw a line form the upper left corner of the "bar" where you make the segments in, to the lower left corner.
One of those triangles you paint black, the other white, and then you blend the colours?
I think it would make a nice scale...

Marc.
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Old 02-12-2012, 12:15 PM
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Marigold Marigold is offline
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Re: Geometric white to black color progression

Hi DevKin,

Quote:
Alas, I have no idea how to progress geometrically. If I start with two drops on the second segment I go up to hundreds of drops in the last.

I'd understand your instructions as follows:

Black
Mixture 1 = Black + 1 unit white
Mixture 2 = Mixture 1 + 1 unit white
Mixture 3 = Mixture 2 + 1 unit white
...

This will not add white in equal increments, but will create each new step starting from a mixture that does already contain increasing amounts of white - so the color will be lightenend much quicker.

Susanne
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Old 02-20-2012, 11:21 AM
frogparty frogparty is offline
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Use to denote nudity/mature subject matter Re: Geometric white to black color progression

Mathematically an arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant e.g. 2, 4, 6, 8, 10...
A geometric progression is one where each term is in a common ratio to the one preceeding it e.g 2, 4, 8, 16, 32, 64 ... is a geometric progression where 2/4 = 4/8 = 8/16 = 16/32 ... = 0.5 is the common ratio.

How can one apply this maths to painting a grey scale? The obvious way is to apply the progression to the amount of black paint in the mix. However, if you had the expertise you could attempt to apply the progression to the value (luminance) of the mixes instead. Maybe this is what is required for the third scale, rather than progress by arithmetic of geometric addition of amounts of black paint, paint a scale with even steps in value.

Say we are going to attempt 10 step progressions with different amounts of black paint (effectively exploring the tinting power of the black).

An arithmetic (linear) approach would be, listing the parts of white to black in each mix -
White:Black
10 : 0
9 : 1
8 : 2
7 : 3
...
1 : 9
0 : 10

A geometric approach (amount of black increasing geometrically) would be to take an amount A of white and add a unit of black, then take amount A of the resulting mix1 and add a unit of black to make mix 2, then take amount A of mix2 and add a unit of black etc. To be truly geometric it is important to keep ratio of the amount of base mix to black added each time the same. Marigold didn't mention that detail, and without it the results will be slightly different. The common ratio between mixtures is then a constant A/(A+1)

The resulting scale depends on the relationship between amount "A" and the "unit" of black. Black paint is usually strong, so with A of 1 unit the white would be overwhelmed within a couple of steps and the mix look black.

Quote:
Alas, I have no idea how to progress geometrically. If I start with two drops on the second segment I go up to hundreds of drops in the last.
Actually that is the same as taking A = 1 unit, mix 1 is half and half black and white, mix 2 is 1/4 white, mix 3 is 1/8 etc. by 1/64 (1.56%) white, if not before, it looks black. But progressive mixing, in this case half and half each time, is easier way to think about it than counting out hundreds of drops

Just dawned on me to ask what medium are you using, you talk of "drops" so some kind of liquid colour I guess. I have been thinking in mixing dollups of acrylic or oils, but what I have said about amounts would also work for making dilute solutions of a liquid black.

Anyway say we do some progressive mixing with A of 5 units, mix 1 is 5/6 (83.3%) white, mix 2 is 25/36 (69.4%) white etc. The fractions get a bit boggling, but if I list the percentage whites for each mix using different amounts A and adding a unit of black each time it might show something of how it works.

Mix A = 2 A = 5 A = 10
1 66.6% 83.3% 90.9%
2 44.4% 69.4% 82.6%
3 29.6% 57.8% 75.1%
4 19.8% 48.2% 68.3%
5 13.1% 40.2% 62.1%
6 8.8% 33.5% 56.5%
7 5.8% 27.9% 51.3%
8 3.9% 23.3% 44.7%
9 2.9% 19.3% 42.4%


Starting with 10 times as much white as black (A = 10 units), after 9 mixing steps you would still have a good amount of white in the mix, maybe too much.

Quote:
Using paint (not the MS variant) how do I make a 10 segment scale from white to black progressing geometrically?

There are many different geometric series that go from somewhere above 0 to somewhere below 100 in 9 steps. An arbitrary example would be
28.7, 33.0, 37.9, 43.5, 50.0, 57.4, 66.0, 75.8, 87.1 (common ratio 1.148). In your paint programme set the value/luminance colour co-ordinate to these values. Not sure how meaningful that would be, other than to give a feel for values being in common ratio rather than linear - something different from what we have been doing mixing amounts of paint that way. I suspect that mixing in a geometric progression of amounts may teach us something about tinting power, not sure that seeing a geometric progression of values does.

But curiosity got the better of me, so here is a comparison of (first row) a linear variation of CIE Lab L values in steps of 10, along side two arbitrary geometric progressions of L. Both have too big a leap to black at the end.



But perhaps the bottom one is a bit like the top? If a progression is chosen that covers the dark greys nearer to black then there are big steps in the light grey area instead.

Hope this is of some help.
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