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Cayley-Klein “Distance”

Here is an account of this “distance” from Lawrence Edwards.


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We see that the logarithm is there simply for its ability to ‘convert’ multiplication into addition, the better to satisfy our intuition about distance!
That is to say, it is there because

log s a.b = log s a + log s b,

where s is the base of logarithms (if x = sy, then y = log s x).

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This may be less contrived and artificial than it might at first seem to be, as
we have at least one sense that works logarithmically. This is our sense of musical pitch.
At each multiplication-by-two of audio frequency, we perceive an additional octave.

If f1 and f2 are frequencies, and

log s (f2 / f1) = x, where (f2 / f1) = 2 and x = 1, then

s1 = 2, which implies that s = 2.

Division of a perceived octave into a scale of twelve intervals perceived to be equal (semitones), gives a pitch-difference per semitone of

p = (log 2 2) / 12 = log 2 21/12,

which implies that the ratio of frequencies corresponding to pitches perceived to be a semitone apart is the twelfth root of two.

One could be tempted by this to think that human perception of distance in perspective also involves an inherent logarithmic action—but it does not.

It is intriguing that reference is almost invariably made to such things as intuition and consciousness in discussions like the foregoing, because these things are faculties, not geometric elements, or numbers—yet here we have a consciousness (that of a ‘being’) said to be capable of making things (numerically) equal!

Dare I say it? It is all most unprojective, and surely anthropogenic.

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