# relative vs. absolute (UNCLASSIFIED)

Mullins, Bill AMRDEC Bill.Mullins at US.ARMY.MIL
Fri Jun 5 20:16:17 UTC 2009

```Classification: UNCLASSIFIED
Caveats: NONE

> 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.

That pitch can be described in intervals is not indicative of the scale,
so much as it is of human perception.  A fifth interval is a frequency
(and wavelength) ratio, not a difference.  The to get from the frequency
of middle C to G, you multiply by a unitless constant, not add a certain
number of cycles per second.

>
> 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?

The fact that it is easier to slide reference points around on a log
scale is a difference in utility, if not capability, I believe.  As an
engineer, I can easily speak of a signal being 30 dB (for example) above
threshold, in ways that are certainly more convenient than "1000 times
greater" would be.  The expressions "above" and "below" are more precise
than the analogous "times greater" and "times less than" expressions
would be (although, if pressed, I'd say "one one-thousandth of" rather
than "1000 times less than").  It's not that you can't do these on a
ratio scale, it's just that they are done better on a log scale.

>
> In fact, can you do anything with a log scale *except* compare levels
> and intervals?

No.  But that is a feature, not a bug.

> 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

And astronomical star magnitudes, and f/stops in photographic exposure.

Classification: UNCLASSIFIED
Caveats: NONE

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