# Colour Chords

Colour "Chords"

Hi,

couldn't think of a better title, but it turns out to be rather fitting for what I'm about to describe, I think...

I thought of this shortly after being introduced to my first lab involving a spectrometer. We perceive different wavelengths of light in the visible spectrum as different colours, each wavelength associated with a distinct corresponding colour, just as we perceive different wavelengths of sound as distinct pitches.

In sound, however, waves can be combined to form, say, chords, or other combinations of multiple pitches, which are instantly recognisable as such. Very rarely in nature do we come across a perfect sinusoidal sound wave with only one pitch.

If this analogy holds true for light, it would seem to imply that there exist colour 'chords', so to speak, that are combinations of light at different wavelengths, and thus do not correspond to one single wavelength. This confuses me, because I've always taken for granted that any colour the eye can see can be expressed using a single wavelength.

So my question is this: am I right in assuming that these colour 'chords' exist, or does a combination of colours have a single wavelength that can somehow be derived from its component wavelengths? (E.g. say you mix yellow and blue light of equal intensity. Does the resulting green light have a wavelength that is the average of the red and blue lights' wavelengths, or can it only be expressed as a combination of both?)

I hope I've expressed myself clearly enough. Any help is appreciated.

Cheers,
-q

Thanks, that helps!

I would add that the human mind adds a "trick" layer; letting us experience colours which cannot occur (reddish green, yellowish blue) in reality. That isn't the physical interpreation of what you're talking about, but it's interesting. *shrug*

collinsmark
Homework Helper
Gold Member

Hello quozzy,

So my question is this: am I right in assuming that these colour 'chords' exist, or does a combination of colours have a single wavelength that can somehow be derived from its component wavelengths? (E.g. say you mix yellow and blue light of equal intensity. Does the resulting green light have a wavelength that is the average of the red and blue lights' wavelengths, or can it only be expressed as a combination of both?)

Yes, light does form such chords! But before going into detail, take note of Frame Dragger's comment, because its important to this topic:

I would add that the human mind adds a "trick" layer; letting us experience colours which cannot occur (reddish green, yellowish blue) in reality. That isn't the physical interpreation of what you're talking about, but it's interesting. *shrug*

It is interesting, and relevant too. Your computer screen is only capable of producing red, green and blue light. When you're screen emulates a yellow light, it is actually transmitting red and green light at the same time.

If you were to take a prism, and examine the spectrum of this yellow light coming from the computer screen, you wouldn't see much yellow light, if any, in the spectrum. Instead you would see a peak at green, and another peak at red.

Not all yellow light has this characteristic though. Shine sunlight through yellow cellophane, and pass that through a prism, you will see yellow.

So there is a difference between natural yellow light, and the "yellow" light created by a computer monitor. It's just that our human eyes (and brain) can't tell the difference. The color cones in our human retinas are similar the computer screen in the respect that they only pick up red, green and blue light. Yellow light excites both the green and red cones somewhat, and the brain interprets that circumstance as yellow.

There is some math behind this too. It relies on some trigonometry:

$$cos(a) + cos(b) = 2cos \left(\frac{a+b}{2}\right) cos\left(\frac{a-b}{2} \right)$$

and in terms of a time-based signal,

$$cos(\omega _1 t) + cos(\omega _2 t) = 2cos \left(\frac{\omega _1 + \omega _2}{2}t \right) cos\left(\frac{\omega _1 - \omega _2}{2}t \right)$$

Let's look at that in a bit more detail.

$$\left(\frac{\omega _1 + \omega _2}{2} \right)$$

is the average frequency of $$\omega_1$$ and $$\omega _2$$.

So if you are adding two tones together, say at 50 Hz and 70 Hz, that term corresponds to average of the two, in this case 60 Hz.

The other part,

$$\left(\frac{\omega _1 - \omega _2}{2} \right)$$

is the difference, divided by 2, between the two frequencies. This term modulates the other term at a lower frequency. This is exactly why you hear beats when you tune a musical instrument to the pitch of another musical instrument. It is called the beat frequency. As the two notes approach the same frequency, the beat frequency decreases. When you hear no more beats, they are perfectly in tune.

Okay, back to our light example. Suppose we want to add red and green light (I'm going to use approximate number here for simplicity).

Red: 430 THz
Green: 545 THz

$$cos \left( (430 \ \mbox{THz}) t \right) + cos \left( (545 \ \mbox{THz}) t \right) = 2cos \left( (488 \ \mbox{THz})t \right) cos \left( (58 \ THz)t \right)$$

Note that 488 THz is roughly the frequency of yellow light. Our eyes are not capable of detecting the 85 THz beat frequency, so we just interpret the sum of red and green as yellow light.

Although human eyes cannot detect the beat frequency of mixing red and green light, our human eyes can detect the difference between the combination of red and blue light (such as magenta) compared to a single color with a frequency in between (such as green)! But the explanation might not be considered as interesting. It's just simply that the combination of red and blue light excites the red and blue cones in the retina, without exciting the green cones so much. And that has to do with spectral energy and corresponding physiology.

Hello quozzy,

Yes, light does form such chords! But before going into detail, take note of Frame Dragger's comment, because its important to this topic:

It is interesting, and relevant too. Your computer screen is only capable of producing red, green and blue light. When you're screen emulates a yellow light, it is actually transmitting red and green light at the same time.

If you were to take a prism, and examine the spectrum of this yellow light coming from the computer screen, you wouldn't see much yellow light, if any, in the spectrum. Instead you would see a peak at green, and another peak at red.

Not all yellow light has this characteristic though. Shine sunlight through yellow cellophane, and pass that through a prism, you will see yellow.

So there is a difference between natural yellow light, and the "yellow" light created by a computer monitor. It's just that our human eyes (and brain) can't tell the difference. The color cones in our human retinas are similar the computer screen in the respect that they only pick up red, green and blue light. Yellow light excites both the green and red cones somewhat, and the brain interprets that circumstance as yellow.

There is some math behind this too. It relies on some trigonometry:

$$cos(a) + cos(b) = 2cos \left(\frac{a+b}{2}\right) cos\left(\frac{a-b}{2} \right)$$

and in terms of a time-based signal,

$$cos(\omega _1 t) + cos(\omega _2 t) = 2cos \left(\frac{\omega _1 + \omega _2}{2}t \right) cos\left(\frac{\omega _1 - \omega _2}{2}t \right)$$

Let's look at that in a bit more detail.

$$\left(\frac{\omega _1 + \omega _2}{2} \right)$$

is the average frequency of $$\omega_1$$ and $$\omega _2$$.

So if you are adding two tones together, say at 50 Hz and 70 Hz, that term corresponds to average of the two, in this case 60 Hz.

The other part,

$$\left(\frac{\omega _1 - \omega _2}{2} \right)$$

is the difference, divided by 2, between the two frequencies. This term modulates the other term at a lower frequency. This is exactly why you hear beats when you tune a musical instrument to the pitch of another musical instrument. It is called the beat frequency. As the two notes approach the same frequency, the beat frequency decreases. When you hear no more beats, they are perfectly in tune.

Okay, back to our light example. Suppose we want to add red and green light (I'm going to use approximate number here for simplicity).

Red: 430 THz
Green: 545 THz

$$cos \left( (430 \ \mbox{THz}) t \right) + cos \left( (545 \ \mbox{THz}) t \right) = 2cos \left( (488 \ \mbox{THz})t \right) cos \left( (58 \ THz)t \right)$$

Note that 488 THz is roughly the frequency of yellow light. Our eyes are not capable of detecting the 85 THz beat frequency, so we just interpret the sum of red and green as yellow light.

Although human eyes cannot detect the beat frequency of mixing red and green light, our human eyes can detect the difference between the combination of red and blue light (such as magenta) compared to a single color with a frequency in between (such as green)! But the explanation might not be considered as interesting. It's just simply that the combination of red and blue light excites the red and blue cones in the retina, without exciting the green cones so much. And that has to do with spectral energy and corresponding physiology.

Ahhh... someone with a fine knowledge of animal biology and physics. Too bad you're not a woman living near me. :rofl: Anyway, I would add that my example was of those colours we percieve which can't be created through combination of Red and Green and Blue. If you take magic "physics paint, both red and green, and mix it, your computer will tell you that what you have created, is a shade of brown. There is NO SUCH COLOUR as "greenish red". It has been shown in a recent study (or rather confirmed) that humans (at least) believe we see such pseudo-colours. We DO see "reddissh green", and can be made to distinguish between that, and brown, or red, and green.

Why? See Collinsmark's post, and then add that our entire visual perception of the world is a very fine filter that MOSTLY lets out brains operate autistically, with minimal outside input. After all, if you're an animal, there might be an advantage to percieving shadings that don't strictly exist... perhaps as a means of telling the ripeness of fruits and vegetables.