Adjusting the Relative Percentages of a Whole

[Thetom]

Hi, how can I work out the relationship between percentages of a whole, when one of those percentages changes? I'm rubbish at maths and can't really explain myself properly, but I'll do my best. Here's the problem:

I want to be able to workout the percentage of component colours in any mix of colour, and then change the amount of one component but keep the ratio between the other colours the same. I need to do this a hundred times, but here is just one as an example:

I have a Purple (P) that I have mixed using...

30% (W)hite
55% (C)yan
11% (M)agenta
4% (Y)ellow

I've tried to express this mathematically (and probably unconventionally) as...

P = 30%W + 55%C + 11%M + 4%Y

Now I need to mix a new colour (D) by decreasing the amount of white by half while keeping the relation between the other colours the same. I then need to find the percentages of these component colours to allow me to mix the new colour. Sooo... (30%W - 50% = 15%W)

D = 15%W + ?%C + ?%M + ?%Y

C = ?%
M = ?%
Y = ?%

How can I find out the percentages of CMY once I have decrease W by a certain factor (in this case 50%)? It's important that the ratio between the other colours doesn't change, just their ratio to white.
I've tried to explain my problem as clearly as I can, but I'm rubbish at maths. If it's not clear what I mean, please let me know. I can go into more detail, or include some diagrams which will allow me to explain it more clearly.

Any help with this practical problem would be very, very... helpful :)

 

[DaveC426913]

Very strange colour palette you're using. CMYW? That's a new one.

Anyway,

30% (W)hite 
55% (C)yan 
11% (M)agenta 
4% (Y)ellow

If W=15, then

C = 55 + 55/70*15
M = 11 + 11/70*15
Y = 4  + 4/70*15

More generally,

Given, A + B + C + D = 100

If A' = A-x
then
B' = B + B/(B+C+D)*(x)
C' = C + C/(B+C+D)*(x)
D' = D + D/(B+C+D)*(x)

Example:

W=35,C=27,M=22,Y=16
x  = 25

W' = 10
C' = 27 + 27/(65)*25
M' = 22 + 22/(65)*25
Y' = 16 + 16/(65)*25

That could probably be boiled down further...

 

[Thetom]

Very strange colour palette you're using. CMYW? That's a new one.

Black and white simply adjust the value of a colour. Value is how light or dark a colour is. Adding white increases the value. Printers (for example) use CMYB, and the black decreases the value of the colour to make it darker. A printer doesn't use white ink though, it just uses the white from the paper. Really the palette is CMY + BW. I'm eventually going to be using a pallet of CMY & RBG & BW . It's for mixing polymer clays.

Here's an example of the colours I've mixed using CMYW if your interested. The transition of colour at the bottom of the photo is the sum of the component colours at any given point along the event line . I'm trying to formulate a new way of blending clays. I can explain in more detail of your interested? I'm finding it hard to work out the maths part though.

http://a.imageshack.us/img839/5738/dsc00106z.jpg

If W=15, then

C = 55 + 55/70*15
M = 11 + 11/70*15
Y = 4 + 4/70*15

Brilliant! Thanks, I just need to work that out, and make sure I understand what has happened so I can do it myself. I think I get the relationship...

Each component (CMY) has added to it... its own value divided by 70 (the old remaining percentage) times 15 (the new percentage of W). I think that explains it!?

I find it hard to express this stuff in English, let alone mathematically. The only way I've been able to do it so far is visually, by drawing out a graph on squared paper and counting the squares when I changed a value. Takes far to long doing that.

Thanks again. I'll try applying it to other problems to see if I got it.

 

[Thetom]

More generally,

Given, A + B + C + D = 100

If A' = A-x
then
B' = B + B/(B+C+D)*(x)
C' = C + C/(B+C+D)*(x)
D' = D + D/(B+C+D)*(x)

Yes! I think I understand that. I'll try applying it and see what happens...

 

[DaveC426913]

Each component (CMY) has added to it... its own value divided by 70 (the old remaining percentage) times 15 (the new percentage of W). I think that explains it!?

No, times W minus 15. See my example, where I made a change that is not 50%.

Do a few examples on paper, using round numbers to ensure it's working. I'd hate for you to apply it to the clay and have it come out all wrong.

 

[DaveC426913]

Here's an example of the colours I've mixed using CMYW if your interested. The transition of colour at the bottom of the photo is the sum of the component colours at any given point along the event line . I'm trying to formulate a new way of blending clays. I can explain in more detail of your interested? I'm finding it hard to work out the maths part though.

http://a.imageshack.us/img839/5738/dsc00106z.jpg

This is fascinating. Can you walk me through it?

 

[Thetom]

This is fascinating. Can you walk me through it?

I would love too :) Give me a minuet to get some examlpes uploaded that explain what is going on.

I think I'm abit confused with this equation, actually.

I'll run through what I just did..

A = 30
B = 55
C = 11
D = 4

X = 50%
-----------------------------
A' = A-x

B' = B + B/(B+C+D) x (x)

B' = 55 + 55/ (55 + 11 + 4) x 50

B' = 55 + 55 / 70 x 50

B' = 55 + 55 / 3500

B' = 55 + 0.015714285714285714285714285714286

B' = 55.02

lol. I'm deffinatly doing somthing wrong. I need to think about this

 

[Thetom]

No, times W minus 15. See my example, where I made a change that is not 50%.

Ok I did that now (i think) and my new answer for B' is 66.8.

 

[Thetom]

Ok I think I got it...

A' = 15
B' = 66.8
C' = 13.4
D' = 4.9

100.1 Total (due to a bit of rounding)
I see where I went wrong now. I'll try some more examples on paper to test...

 

[DaveC426913]

B' = 55 + 55 / 70 x 50

B' = 55 + 55 / 3500

You have done the order of operations wrong here. Order of operations is left to right, i.e.:

= ...+ 55 / 70 x 50
= ...+ 7.857 x 50
= ...+ 39.28

 

[Thetom]

Am I right in thinking that X in this equation isn't the percentage that W was reduced by, but is the result of W being reduced by a certain percent?

I have tried to resolve the example you gave in your first post. I think I understand the order of operations. I have included some brackets in my workings out to help show the order I did things.

W=35,C=27,M=22,Y=16
x = 25

W' = 10

C' = 27 + (27 / 65) x 25
C' = 27 + 10.4
C' = 37.4

M' = 22 + (22 / 65) x 25
M' = 22 + 8.5
M' = 30.5

Y' = 16 + (16 / 65) x 25
Y' = 16 + 6.2
Y' = 22.2

Totals 100.1. This looks promasing :)

 

[DaveC426913]

I'm working on a calculator but haven't got it working yet.

http://www.davesbrain.ca/science/balancerizer.html

Ironically, the calculation is working fne, it's the field entry giving me grief.

I'll have to away to bed and finish it tomorrow.

 

[Thetom]

Ok' I am still totally stuck. I just don't get it :(
Here's a new equation... I'll just show the workings for C. I'm not sure of the correct usage of brackets. I have just put brackets around parts that were done first. What am I doing wrong?..

C = 58, M = 10, Y = 5, W = 27
---------------------------
W' = W - 29%
W' = 27 - 7.83
W' = 19.17
--------------------------
C' = C + C/ (C+M+Y) x W'
C' = 58 + 58/(58+10+5) x 19.17
C' = 58 + (58/73) x 19.17
C' = 58 + (0.79 x 19.17)
C' = 58 + 15.23
C' = 73.28

The problem is that when I do it for Y and M they all total greater than 100. Where am I going wrong. I'm totaly lost in the process and can't see what is happening.

 

[Thetom]

I'm working on a calculator but haven't got it working yet.

http://www.davesbrain.ca/science/balancerizer.html

Ironically, the calculation is working fne, it's the field entry giving me grief.

I'll have to away to bed and finish it tomorrow.

Oohhh, that looks amazing! If you can make it work I'll buy it off you. That would be really, really helpful. Thanks for even trying it! :D.

I'll write up a little explination of what I'm doing for your reading pleasure...

 

[Thetom]

Here's a quick description of what I'm upto...

Polymer Clay is a plastic clay that can be 'fired' in an oven to set it. There are a quite few brands of clay; FIMO, Sculpey and others (i only really use Fimo).

It is possible to mix two different coloured clays together to make a new colour. Mixing blue and yellow will make a green, for example. A Skinner Blend, devised by a lady called Judith Skinner, is a method of mixing coloured polymer clays so that there is a gradual transition from one colour to another. This is a very clever method which has advanced the quality of art allot in the past decade. A Skinner Blend works like this:

Two or more triangles of clay are arranged to make a 'pattern sheet'. Like this...
[PLAIN]http://a.imageshack.us/img185/6987/rbsk20copy.jpg

This pattern sheet it then folded and passed through a pasta machine to squish the colours together. The process of folding and squashing is repeated several times (20+) to get the clays to fully mix into a graduation. Like this...
[PLAIN]http://a.imageshack.us/img689/5034/rbsk27copy.jpg

(Thanks to PoLeigh for the above images). Here's one I made earlier...
[PLAIN]http://a.imageshack.us/img688/9063/56979160.jpg

As you can see the final blend is a result of adding the different colours, at any vertical instance, to create the final colour.
A more complete description of skinner blends can be found here, if you didn't get it from my example. http://www.desiredcreations.com/howto_cabasicskinnerblend.htm

I have been doing a study on these Skinner Blends, with the hope of identifying the different variables to devise a new (and hopefully more accurate) method. To do this I have had to find names for some of the different elements of the Skinner Blend, so that I can address them. One of these elements is what I call the event line, shown below. The event line is just the horizontal line from left to right...
[PLAIN]http://a.imageshack.us/img828/1945/bgeo1.jpg

It occurred to me that any point (or event) on the event line could be broken down into the component colours, allowing me to plot a matrix of planned events and then produce a pattern sheet that will produce those events. I'll explain that with some examples...
Here are 5 different colors that I have mixed together and found the ratio of component colours for...
[PLAIN]http://a.imageshack.us/img833/2319/blend2mixedcolours.jpg

I have then arranged these values in a matrix, just as a simple and clear way of presenting them...
[PLAIN]http://a.imageshack.us/img541/6323/blend2mixedmatrix.jpg

Using these colours I then designed a graduation of colours to produce, so the colours blend from one to another. I can identify the events I want to happen from this. So I know that half way along the event line, for example, I need a certain ratio of colours...
[PLAIN]http://a.imageshack.us/img836/9899/blend2eventline.jpg


I've then plotted each event, using the values from the mix matrix, into a pattern sheet. Each event should then be comprised of the exactly the right amount of different component colours. Here's what the pattern ended up looking like...

[PLAIN]http://a.imageshack.us/img821/70/blend2patterndesignandm.jpg

You can see that at event 1 (numbered at the bottom) there is 43% white, 35% yellow, 18% magenta and 4% cyan. Which all comes together to make a brownish orange. Event two is mainly yellow with a bit of cyan which makes a lime green. Hence the events of the blend have been broken down into their constituent parts to be mixed together in the pasta machine. Here's the pattern sheet made out of clay...

[PLAIN]http://a.imageshack.us/img6/1020/blend2pattern.jpg

Next the whole thing is folded and flattened, folded and flattened until the colours have blended together. Here's some pic's of the process, ending in the final blend...

[PLAIN]http://a.imageshack.us/img824/4836/mid1.jpg
[PLAIN]http://a.imageshack.us/img265/3185/mid2.jpg
[PLAIN]http://a.imageshack.us/img21/1859/mid3d.jpg

[PLAIN]http://a.imageshack.us/img826/351/blend2c.jpg

So, that is the basics of what I have done. I'm now trying to create a blend that progressively increases/decreases the value (lightness) of the colours as the transition progresses. This is what I need these equations for. I havn't managed to do it yet so can't show you examples of what i mean.
Basically, I want the colours to blend from one to another at a certain rate... but have the value change at a different rate. Or to put it another way, I want the amount of white to change in a non-linear fashion. So I need to be able to work out the ratios of CMY at different events on the event line, and then progressively adjust the amount of white, without disturbing the ratios of CMY. As soon as I have achieved this, I'll post a pic to show you what I mean.
I'm working on writing up my findings to present to the Polymer Clay community, and so this post was a small practice for when I do that.

 

[Thetom]

Here's a fuller description of the problem I'm having. Below shows a linear transition from blue to white. At event 1 white is 0%, at event 2 white is 50% and event 3 white is 100%.
[PLAIN]http://a.imageshack.us/img16/7476/32509735.jpg

And here is what I mean by a nonlinear transition. The amount of white increases exponentially (it may not be exponential, nonlinear may be a more accurate description).
[PLAIN]http://a.imageshack.us/img27/1041/66966317.jpg
(note: i just made those values up, they may not correspond to the curve.)

Now, finding the percentage of the component colours, when there is only two colours is easy. I can just subtract the percentage of white and the remainder is the percentage of blue. But when there are three or more colours it is much harder (for me).

A liner transition of blue white and yellow (Fig.3)...
[PLAIN]http://a.imageshack.us/img697/1421/43814554.jpg

But if I do the same for a nonlinear transition, it doesn't work...
[PLAIN]http://a.imageshack.us/img833/1316/37733789.jpg

The curve of the white has increased the amount of yellow compared with the amount of blue. So I need a way of increasing one of the component colours while maintaining the relationship between the other colours.

So referring back to fig.3 I can see that at event 2 there is 50% blue 25% yellow and 25% white. or there is twice as much blue than yellow. I know, from plotting the nonlinear curve of white, that at event 2 I want there to be only 15% white (as an example). So how to now adjust the amount of blue and yellow accordingly. This is my problem. I still haven't managed to get my head around the maths of it yet.

------------------------------------------------------------------------------

So using the original values I gave in the OP, event one was:

W = 30%
C = 55%
M = 11%
Y = 4%

The next event saw a decrease in white by 50% and me stumped..

W = 15%
C = ?
M = ?
Y = ?

There will be a whole series of events like this. Ideally, it should be possible to not only adjust the white in a nonlinear fashion, but also adjust the other colours, relative to each other. So the amount of white may increase at one rate, the amount of cyan increases at another rate, but the amount of yellow and magenta both increase at the same rate. I would like to be able to do this to any number of colours.

Hopefully that has illustrated how all this is being applied practically - so you can see exactly what I'm doing with these numbers.

 

[DaveC426913]

I have corrected your example:

C = 58, M = 10, Y = 5, W = 27
---------------------------
W' = W - 29%
W' = 27 - 7.83
W' = 19.17
--------------------------
C' = C + C/ (C+M+Y) x W'
C' = 58 + 58/(58+10+5) x 7.83
C' = 58 + (58/73) x 7.83
C' = 58 + (0.79 x 7.83)
C' = 58 + 6.19
C' = 64.19

Conceptually, what you are doing is taking 7.83 grams of clay out of white (So: 27-7.83). And adding that amount of clay (7.83g) back in, splitting it proportionally between Cyan, Magenta and Yellow (each will be a fraction of that 7.83g).

 

[DaveC426913]

Hopefully that has illustrated how all this is being applied practically - so you can see exactly what I'm doing with these numbers.

This is fascinating.

You know, in theory, this whole calculation and layout process could be taken off your hands and handled by a program. You'd simply input your desired colours at your desired event locations, press a button, and it would print out a template for your final layout of colours.

 

[Thetom]

Conceptually, what you are doing is taking 7.83 grams of clay out of white (So: 27-7.83). And adding that amount of clay (7.83g) back in, splitting it proportionally between Cyan, Magenta and Yellow (each will be a fraction of that 7.83g).

Right, i understand this. I had thought to use that value (the 7.83) but didn't actually try it. I think it's important for me to understand what is happening conceptually. To have control of the process one must understand the process. Thanks for your help with this DaveC426913. I'm off out for the afternoon but when I'm back later i'll have a good go at applying this and see if I can get the right numbers in a few test runs

This is fascinating.

Agreed , I'm pretty new to this art form, but it's the process that has me hooked. I've only scratched the surface by explaining the Skinner Blend.

You know, in theory, this whole calculation and layout process could be taken off your hands and handled by a program. You'd simply input your desired colours at your desired event locations, press a button, and it would print out a template for your final layout of colours.

Yeah I know, but I have no idea how to produce such a thing, or how easy it would be for a programmer to do. Do you happen to know anywhere I could at least ask to see if that would be an easy project for someone to do? A programming forum maybe? I thought perhaps excel could atleast give me the values. I'll ask my brother-in-law later. He's pretty hand with computers.

Anyway, thanks again. I never would have worked that out on my own (was trying for about 5 hours last night). I'll get practicing with my calc later...

 

[Dickfore]

This thread makes my head hurt. Kudos to DaveC for the patience.

 

[Thetom]

Yeah, sorry about that. I probably could of explained it much more simply. If I read this from somone else I would struggle to decode it. Thanks again Dave :)

 

[Thetom]

But, to be fair, most of the threads on this forum make my head hurt. So that isn't saying much^

 

[Dickfore]

[PLAIN]http://a.imageshack.us/img21/1859/mid3d.jpg

[PLAIN]http://a.imageshack.us/img826/351/blend2c.jpg

On the other hand, these look great. Well done.

 

[Thetom]

On the other hand, these look great. Well done.

Thanks. You guys are the first people I've shown. You saw it here first... and quite possibly last. But I really think I'm onto something with this. Just need to get it all worked out. Thanks for the kind words.

 

[DaveC426913]

OK, the balancerizer is working. Not very user-friendly ... but it's working.

http://www.davesbrain.ca/science/balancerizer.html

 

[DaveC426913]

I'm curious about something.

This Skinner Blend is a method for transitioning linearly from one colour to another.
Your method improves on that incrementally by allowing non-linear gradients.

It seems to me though, that the very nature of the mechanical blending is only so accurate. Even with the very best of skills with Skinner Blend, one will get only a rough result compared to what one envisioned - you simply can't control the blending down to the millimeter.

It seems to me, that inherent approximateness is more than enough to swamp the subtleties of your non-linear blends.

 

[DaveC426913]

Yeah, sorry about that. I probably could of explained it much more simply. If I read this from somone else I would struggle to decode it. Thanks again Dave :)

No, this stuff is right up my alley. Colour spaces, blending, algorithms, sculpture. Application of science, math and programming to art. Love it.

 

[Thetom]

I'm curious about something.

This Skinner Blend is a method for transitioning linearly from one colour to another.
Your method improves on that incrementally by allowing non-linear gradients.

It seems to me though, that the very nature of the mechanical blending is only so accurate. Even with the very best of skills with Skinner Blend, one will get only a rough result compared to what one envisioned - you simply can't control the blending down to the millimeter.

It seems to me, that inherent approximateness is more than enough to swamp the subtleties of your non-linear blends.

Ahh, down to the nitty-gritty. I think you may be right about that. The main benefit of this new method is the non-linear gradients. And perhaps that is the only benefit, making this laborious process un-desirable.
And there is an inherrent inaccuracy. One which I am trying to minamize. 1mm - no, probably not. But the blend in those pics accuratly produced a 5mm band. I want to see how far I can get that down. When you create the blend it tends to spread horizontally. Making the blend the exact same width as the pasta machine limits this spreading, and so it spreads out vertically. You can see this in the bulges at the top of the sheets in the pics.

I'd like to give a few more thoughts on this, but must dash now... I'll post again later...