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Scott

http://www.nature.com/news/2008/080529/full/news.2008.866.html
And
http://web.mit.edu/newsoffice/2012/thinking-logarithmically-1005.html

emre

http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law

GroundTruthTrek

I think that the case where logarithmic scales are "better" than linear scales is if you want to emphasize proportional difference rather than absolute difference. For example, many people prefer to know what % difference there is between the price of things, rather than the absolute difference. If you wanted to capture prices on a graph and express the difference in this way, you'd want to plot log(price) rather than just price. Alternately you could plot the % difference on a linear scale, but that then doesn't give the reader access to information on the absolute prices, which may be of interest as well. Obviously in many cases the simple fact that readers wouldn't know how to read the log axis might incur a greater cost than the more appropriate representation of the data, but I think that decision should be considered on a case-by-case basis.

In the case of electromagnetic radiation, it doesn't make much sense to me at all to plot on a linear scale... Such a scale would suggest that the difference between visible light and gamma rays is irrelevant in comparison to the difference between the microwaves in your oven and those coming out of your internet router. This case is a bit different than the more subtle case I outlined above. In this system the number (whether frequency or wavelength) is not really a very direct expression of the nature of what is measured... if we had a name for log(wavelength) like we have a name for log(earthquake energy) [moment magnitude] then we'd rather use that and conceal the fact that any log math was used at all.

I used a log axis on a plot for public consumption recently. I did it partly because I felt there was some intrinsic value in challenging the audience to read it... In the blog post including this plot, I included a sidebar explanation of log axes: http://www.groundtruthtrekking.org/blog/?p=2651

Gary

Isn't sound perceived logarithmically? I.e., an octave represents a doubling of frequency. So the piano keyboard is a logarithmic scale....

Yep

Human audio perception is definitely logarithmic, which is why we use log-log plots in the audio field, for both amplitude (dB are logarithmic) and frequency.

Audio circuit noise is often plotted on a linear axis, for instance, with equal energy per linear bandwidth. This is not what humans perceive, and it's misleading to show it that way. We hear with equal energy per *percent* bandwidth, which completely changes the plot and emphasizes high frequency hiss.

Human perception of brightness is also logarithmic, as are many other senses: https://en.wikipedia.org/wiki/Stevens%27_power_law

Ben Lien

The sound scale covers frequencies that aren't generally thought of as sounds, so what stimuli are we deaf to? Is it meaningful to say we can't hear the ocean going up and down with the tides? (10E-4.6 Hz) That may be about the same frequency of pressure change from a low-pressure weather system passing through, which some people can sense, although mostly in the joints. My inner ear can detect pressure changes from altitude changes at fractional Hertz. At some point, especially on the low end, it becomes meaningless to think of vibration frequencies as "sound." I would agree, however, that ultrasound is a notable gap in human perception, especially when compared with bats and porpoises.

Regarding the electromagnetic spectrum, my skin can detect some frequencies of infrared, although frequency differentiation is not possible, and direction of arrival is difficult to determine. I suppose that's pretty weak, so I'm forced to concede significant blindness.

The question of what we are missing, though, hasn't been addressed. I suspect the only ultrasound I am routinely subject to comes from switching power supplies, which I'm happy to miss, although it might be fun to hear bats at night. For the electromagnetic spectrum, I'm not convinced there is much information coming to me from natural sources, except in near-IR and UV, which might be nice to see. Sensing pulses from astronomical events doesn't sound useful, but probably happens across the spectrum.

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