I spent the last evening experimenting with sound and colours.

Listen to this audio first. If you are unable to play the audio in the
browser, use the download link below it to download and play it on your
system.

You will hear two pairs of tones separated by 2 seconds of silence. Each
tone is 2 seconds long. The first tone has a frequency of 100 Hz. I
won't tell you the frequency of the second tone right now. But you
should notice that it is a little greater than the frequency of the
first tone. You'll need good earphones to listen to this. Most laptop
speakers can not play these low frequency tones well.

The third tone after the silence has a frequency of 400 Hz and you'll
have to wait a little to know the frequency of the fourth tone.

Now, try to answer this question. Is the difference in frequencies in
the first pair of tones equal to that of the second pair of tones? If
you feel they don't sound equal, which difference sounds greater?

I hope you find the first change in frequency greater than the second
one because that's how a normal human brain is supposed to perceive
these differences in frequencies. If you do, you might be surprised to
know the frequencies of the second and the fourth tone. The frequency of
the second tone is 110 Hz and that of the fourth tone is 410 Hz. Yes,
there is a difference of 10 Hz in both cases. Why then does the first
difference sound greater than the second one? In fact, the first
difference would sound roughly four times greater than the second
difference. Why?

We sense pitch in a logarithmic fashion. Let me explain what this means.
Let us consider a strange company which pays you one gold coin for every
2 days you work. Your income is directly proportional to the number of
days you work. The number of gold coins is always half the number of
days you work. You earn the gold coins in a linear fashion. The graph
looks like this:

Now consider another company which pays one gold coin if you work for 4
days, two gold coins if you work for 16 days, 3 gold coins if you work
for 64 days and so on. In other words, you earn n gold coins if you work
for 2

^{2n} days where n is a positive integer. The graph looks
like this:

Note that the graph is no longer a straight line. That's why, in this
case, the income is not said to grow linearly with the number of days of
work. But note that the number of gold coins earned is half that of the
exponent to which 2 is raised to equal the number of days. So, if we
plot a graph of the gold coins earned versus these exponents, we should
get a graph similar to the one we got for the first case. Have a look:

Note that the x axis is log

_{2}(number of days of work).
log

_{2}(n) is the exponent to which 2 has been raised to equal
n. So, we see that in this case, the number of gold coins earned is
directly proportional to the logarithm of the number of days of work.
Hence, the income here is said to be in a logarithmic fashion.

Similarly, the perception of pitch is directly proportional to the
logarithm of the frequency of the sound.

There is another interesting thing to note in the second example. You
earn one gold coin for 4 days of work. After 4 days, to earn an extra
gold coin, 12 extra days of work is needed. The number of extra days
required to earn an extra gold coin is 3 times the number of days of
work done so far. This pattern holds throughout the graph. For example,
after earning three gold coins with 64 days of work, 192 extra days of
work is required to earn the fourth gold coin. This pattern holds true
if you want to analyze how much extra effort is required to increase
your income by 2 gold coins or any number of gold coins you want to
consider. Try it.

So, at any point, to earn a certain number of gold coins, the number of
extra days of work required is directly proportional to the number of
days spent in work so far.

Our perception of pitch of sound is similar to the number of gold coins
earned in the second case. The change in frequency (frequency of the
second tone - frequency of the first tone) required to cause a certain
perceived change in pitch is directly proportional to the frequency of
the sound one is hearing. So, At 400 Hz, we need a change of 40 Hz to
create the same effect as a change of 10 Hz at 100 Hz. The following
audio demonstrates this. The fourth tone in this audio has a frequency
of 440 Hz.

Let us see why such a relation between perceived change in frequency,
actual frequency and actual change in frequency implies that our
perception of pitch is logarithmic.

Let dp be the perceived change in frequency. Let df be the actual change
in frequency. Let f be the frequency at which the change occurs. As per
the discussion above,

dp ∝ df / f
⇒ dp = k · df / f, where k is a constant

Let us integrate both sides of the equation.

**∫**dp = **∫**k · df / f
⇒ p = k · ln f + c, where c is a constant

So, here we can see that the perceived pitch is directly proportional to
the logarithm of the actual frequency. Now if we represent the first
actual frequency as f_{1st}, the second one as f_{2nd}
and the corresponding perceived changes in pitch as p_{1st} and
p_{2nd}, the perceived change in pitch is:

Δp = p_{2nd} - p_{1st}
= (k · ln f_{2nd} + c) - (k · ln f_{1st} + c)
= k · ln f_{2nd} - k · ln f_{1st}
= k · ln (f_{2nd}/f_{1st})

So, when you hear a change of 10 Hz at 100 Hz, you perceive a change in
pitch of k · ln(110/100) = 0.095k. However, when you hear the same
change in frequency at 400 Hz, you perceive a change in pitch of only k
· ln(410/400) = 0.024k.

I tried to do a similar experiment with colours: http://susam.in/downloads/files/weber-fechner/color.html.
Just like the sound experiment, here the intensity of blue increases
from 20 to 40 in the first change and 200 to 220 in the second change.
However, I don't think the perceived change in brightness in the first
instance appears to be greater than that of the second one. I find both
the changes to be almost equal. So, does Weber-Fechner law not work
here? The Wikipedia
article on Weber-Fechner law mentions that *the eye senses
brightness approximately logarithmically*. Is there a flaw in my
experiment?

For a moment, I thought that perhaps, the intensity of blue I am
specifying in the RGB code is not directly proportional to the
brightness of the color. But the formula GIMP uses to calculate the grayscale
brightness from a color image implies that the brightness is directly
proportional to the intensity of blue we specify in the RGB code. Or is
it that the intensity we specify in the RGB code is already directly
proportional to the logarithm of the brightness?

A related post: Listening
to superimposed waves.