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I am currently reading a very nice book on scales in physics. There is a discussion on the different physical scales which are based on the effect of the corresponding phenomena, the given examples are the Beaufort scale, the Richter scale , star luminosity scale, the decibel scale and the diatonic musical scale. All these scales have the common feature that they relate logarithmically to the intensity of the corresponding effect,

$$ S = A \, \ln(I) + B \, .$$

$S$ is typically a number, $A$ and $B$ are system-specific quantities that are use to set the scale of one chosen object. For example, if the keys of a piano are labeled from 1 to 12 (starting at middle C), the frequency $F$, and the numbers $N$, are related by

$$ N = \frac{12}{\ln(2)} \ln(F) - \frac{12}{\ln(2)} \ln(440) + 8 \, . $$

Here the fact that A is 440 Hz and the doubling of the frequency every 12 keys constrain $A$ and $B$.

My question is the following: Why are such log scales so common in nature? They seem to emerge in extremely diverse situations. Is there a common mechanism for this to happen independently of the particular physics involved?

Note that the Mohs scale of rock hardness seems to be a counter-example of this.

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    $\begingroup$ These scales aren't "common in nature". It's just that the quantities we choose log scales for typically range over many orders of magnitude, and a scale of $1-10$ is more convenient for humans than a scale of, say, $1-10^7$. $\endgroup$
    – ACuriousMind
    Mar 30, 2015 at 14:02
  • $\begingroup$ Comment about the piano formula you have: 440 Hz is typically the tuning standard A$_4$, not G. Also, this is just a convenient re-arrangement of how to split an octave into 12 equal pieces. $\endgroup$
    – Bill N
    Mar 30, 2015 at 14:43
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    $\begingroup$ While Why is the decibel scale logarithmic? has a more specific title at least some of the answers actually address this more general question. $\endgroup$ Mar 30, 2015 at 14:50
  • $\begingroup$ Lying with statistics might be another reason. As ACuriousMind stated you can emphasize effects on a log scale which would not be visible on a linear scale. $\endgroup$
    – Randy Welt
    Mar 30, 2015 at 14:55
  • $\begingroup$ @AcuriousMind, I agree that log scales are very practical because they change huge numbers into usefull ones. However a scale of the form $N = A \ln(\ln(I))+B$ or $N = A\ln(I)/I+B$ does the trick as well. Why is it the simple log scale that is most used? And, more importantly, why is the same scale applicable to so many phenomena? $\endgroup$ Mar 30, 2015 at 15:06

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Exponentials come as solutions of this differential equation:

dx/dt=c*x

where c is a constant and x and t variables. The solution is of the form:

x(t)=e^(c*t)

A great number of measurements and observations we make can be approximated by this equation.

Once the solution is exponential it is logical to take the log since the numbers become large fast and will not fit optionally in a plot.

So the question becomes why this simple approximation, which leads to the differential equation is so prevalent in nature

delta(x)/delta(t)~c*x

particularly for time measurements.

Note that this is a first order equation. Suppose that the real solutions obey complicated differential equations. The first order solution , will be this lowest form for many cases of physical observations, particularly where there is no symmetry, as with time measurements. Thus it acquires the prominence you describe.

It is a bit similar to the role of the harmonic oscillator for potentials, because it is the lowest order in an expansion for a symmetric potential. That is why the harmonic oscillator has such a prominent role in approximations.

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At the sensory/neuropsychological level, humans (and probably other creatures) detect and respond on a compressed level versus the actual intensity detected by the eyes, ears, nerve endings, etc.

Perceived brightness and loudness changes don't correspond linearly to changes in physical intensity (power/area). Also, sound pitch perception depends on consecutive multiples (a doubling of frequency produces a matching sounding musical pitch).

Examples of these perceptions: 1) Hipparchus divided stars into magnitudes, the 1st magnitude being the brightest, 2nd magnitude the next, on down through 6th. When we eventually got around to cardinalizing his ordinal system, (relating luminosity to his magnitudes), we ended up with a logarithmic compression.

2) Testing of human hearing perception on with large populations indicates that loudness follows a logarithmic behavior compared to physical intensity.

3) If you ask male and female casual singers to match a pitch spontaneously, they will most often sing an octave apart, indicating they hear an octave as being the same note even if the frequency is doubled. The octave is split by multipliers, traditionally into 12 steps (that's an totally different story).

4) The cochlea is the organ which translates vibrations into nerve signals. Current studies indicate that frequency placement along the (unfolded) length of the cochlea is logarithmic with frequency.

I would say that human biophysics of sensory receptors pushes us to explain relationships in a logarithmic fashion.

There are also specific things in nature which are non-linear and must be mathematically described as either logarithmic or exponential. Two that I can think of right now are 1) radioactive decay populations 2) voltages in cylindrical geometries.

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  • $\begingroup$ thanks for your answer. I agree that there might even be an evolutionary reason for our senses to work in such a way: It makes us sensitive to a huge range of stiumly. However, what about richter or beaufort scales? They have nothing to do with biology and still work just like decibels. $\endgroup$ Mar 30, 2015 at 15:09
  • $\begingroup$ Moreover, how come ears and eyes work in the same way? There is no reason for hearing not to work with a $\ln(\ln(I))$ law and sight working with $\ln(I)/I^2$, for example. Instead we have one single simple law for all theses phenomena. $\endgroup$ Mar 30, 2015 at 15:17
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Your question appears to confuse the map for the territory. The scales you mentioned don't occur in nature; they occur in humans' descriptions of nature.

Phenomena - actual earthquakes, radiation from stars, etc. in their full complexity are what occur in nature. We often make measurements based on these phenomena and get results such as the total energy released by the earthquake or the total energy received from the star. Finally, we often convert the results of the measurements using a log scale. This says very little about nature or the unity of the phenomena. It is simply a choice of how to communicate and think about the results of the measurements.

We are free to make log scales for anything. We could report our heights on a log scale, if we wanted. In practice, we usually do this when we expect the quantities to vary over many orders of magnitude. We simply report the order of magnitude directly; doing so is making a log scale.

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