relative vs. absolute (UNCLASSIFIED)

[Mark Mandel]

On Fri, Jun 5, 2009 at 4:16 PM, Mullins, Bill AMRDEC <
Bill.Mullins at us.army.mil> wrote:

> 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.
>

I think you're missing my point: *Pitch is a phenomenon of human perception.
Frequency is a phenomenon of physics. They are not the same. *Describing
pitch with an interval scale makes sense, because we're describing our
perceptions.

Let me return to the definitions and examples I gave earlier, which I think
we agree on:

Interval: The values are ordered and can be subtracted, but not added,
multiplied or divided. The interval between Sept. 12, 2000 and Sept.
12, 2009 is 9 years, but "Sept. 12, 2000 + Sept. 12, 2009" is
meaningless. Similarly,   Jan. 1, 2000 isn't twice as (anything) as
Jan. 1, 1000,   100 degrees Fahrenheit isn't twice as hot as 50, and
40 degrees west longitude isn't four times as west as 10 degrees west,
except as measured from their scales' arbitrary zeros.

Ratio: The values can be treated as numbers: meaningfully subtracted,
added, multiplied or divided. $6 is twice as much as $3,   300 degrees
Kelvin is three times as hot as 100, and a person of 60 is twice as
old as a person of 30.   40 degrees north latitude is twice as far
north as 20 degrees north (and the equator, unlike the Greenwich
meridian, is not arbitrary).

Pitch has intervals -- a fifth from C to G, a semitone from A to Bb. But it
doesn't have ratios. What is 3 * C, or even 3 * middle C, or middle C / 2?
To a musician, the question is meaningless. Pitch has no zero point. Middle
C is a point to measure from, like the Greenwich meridian or the freezing
point of water, and equally arbitrary. You can call it zero if you like, and
describe the other notes as so-and-so many semitones above or below it, but
you still can't can't add middle C to G-above-middle-C, or divide middle C
by 2, any more than you can do it with longitudes or Celsius temperatures.

Pitch does have a ratio aspect in its cyclical, or rather helical, nature.
Go up one octave and you've climbed in one dimension, but in another you've
come back to where you were. All C's share an identity relation for which I
can think of no analogue in temperature or earthquakes or dB or any of the
other phenomena we've discussed in this thread.

Which puzzles me, now that I think of it. Perhaps we should consider the
musical scale as something different from these others, eligible for that
distinction because it is describing perception rather than physics.


> > 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.


Yes. If I left any impression that I thought log scales were useless, I was
writing poorly. I meant only that in the NOIR classification, they should be
considered a subtype of ratio scale rather than a fifth type.

Regards,
Mark Mandel

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