relative vs. absolute

[Mark Mandel]

On Fri, Jun 5, 2009 at 1:25 PM, James A. Landau
<JJJRLandau at netscape.com> <JJJRLandau at netscape.com> wrote:
>
> Quite a good answer, but two comments:

Thanks.

> 1.      "100 degrees Fahrenheit isn't twice as hot as 50" is correct, but 100 degrees Rankine *is* twice as hot as 50 degrees Rankine.  That’s because the Rankine scale uses Fahrenheit-sized degrees but its zero is set at absolute zero, thereby making it a Ratio rather than an Interval scale.  Similarly, as you point out, degrees Kelvin (degrees Celsius with zero at absolute zero) is a ratio scale.  See below my signoff.

Good point. That's part of why I specified Fahrenheit, but the
Fahrenheit and Kelvin scales differ on two dimensions (size of
interval and zero point), which damages the comparison.

> 2.      There is a fifth kind of scale, namely logarithmic.  Decibels are a logarithmic scale.  53 decibels is twice as loud (or twice as strong) as 50 decibels, and 47 decibels is half as loud/strong as 50 decibels.  E.g. electrical types speak of a “3-decibel drop” when they talk about splitting a current into two, as when one wire comes in and two go out.  Other logarithmic scales you might be familiar with include the Richter scale for earthquakes, the pH scale for acid/base strength in chemistry, Malthus (who said food production can be measured on a ratio scale and population on a logarithmic), and in music octaves.

Nice. I knew that, of course, but I never connected them with the
four-part classification... which is itself an ordinal scale :-). I
would draw a distinction, though, with the musical example:
Acoustically (in terms of waveform: frequency) the musical scale --
the octave and its divisions -- is a logarithmic scale, but auditorily
(in terms of perception: pitch) it is an interval scale: from middle C
to G is a fifth, just the same as from C' to G' (in the octave above
middle C) and from C, to G, (in the octave below). Similarly for
decibels with amplitude vs. loudness.

> It is possible to extend this list beyond logarithmic, but with the exception of power towers (e.g. one, googol, googolplex…) I can’t recall ever having seen a power tower scale in use, and even in advanced mathematics power towers are quite rare.

Stopping to think about it -- a bad idea! -- the log and power tower
types of scale seem to veer off in a different direction than the
nominal - ordinal - interval - ratio sequence. Each type in N-O-I-R
allows more kinds of operation than the ones before it, but a log
scale is just a different way of looking at a ratio scale, more
convenient to us because it compresses the larger orders of magnitude.
What operations are possible on a log scale that aren't possible on a
ratio scale?

In fact, can you do anything with a log scale *except* compare levels
and intervals? All these examples have a reference value, an arbitrary
zero point like the Greenwich meridian for longitude:
 - pure water at some temperature for pH
 - "one micrometre on a seismograph recorded using a Wood-Anderson
torsion seismometer 100 kilometres (62 mi) from the earthquake
epicenter" for the Richter scale (WP)
 - middle C for pitch
 - somehow determined threshold of perception for decibels
I'm not counting Malthus here, as he didn't actually propose a scale,
and zero food production very quickly means zero population.

--
Mark Mandel

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