Egregious Tone

Over at Beyond Necessity Ed Ockham has a post on the Shepard Tone, a mysterious chord sequence which seems to rise (or fall) steadily without actually 'getting anywhere'.  Earlier, I had suggested that the Shepard tone (and also the Waterfall Illusion) were examples of 'perceptual contradictions' and possibly of relevance to Ed's ongoing discussion as to whether the principle of non-contradiction is learned through experience.

Ed seems to think that the Shepard tone presents no difficulties.  He says

It is a tone which ascends in pitch but does not ultimately ascend in pitch, just as an orbit is a movement which changes place but which does not ultimately change in place. What's the problem?

 and

Why should we find the 'paradox' any more paradoxical than angular movement or modulo change?  If you keep on turning around long enough, you will be facing the same direction again.

I think Ed is making a slippery distinction here between ascends and ultimately ascends which I altogether fail to grasp.  And I don't see the analogy with circular motion, such as the hands of a clock, either.  But no matter.  Here is a suggestion.

One possibility we should consider, which Ed may be hinting at with his clockface analogy, is that the relation is at a lower pitch than, which I denote by '<', is not transitive.  Restricted to pure tones < is transitive and this is our usual musical experience and expectation.  The Shepard tone presents us with a sequence in which successive chords are in this relation.  From p1<p2, p2<p3, p3<p4, etc, we expect p1<pn, and it's disconcerting when this seems to fail.  So it would be interesting to compare non-adjacent chords in the Shepard sequence.  Is it true that we judge pi<pj for i<j?  Looking at the diagram in the first section of the WP article, reproduced here, suggests that the answer might be No.

Compare the first chord (p1) with the seventh (p7).  The notes in each chord are separated by octaves and each note in p7, apart from the highest (which is low in intensity), is one tone lower than the corresponding note in p1.  So I would hazard that we would say that p7<p1 and certainly ~p1<p7.  But if this is right (anyone with a piano or synthesiser who can try it?) then it means that < is not transitive over the Shepard chords.  And this in turn means that we shouldn't expect that the Shepard sequence 'goes anywhere'.  We shouldn't expect that pn, for large n, is pitched way above p1.  But we do, presumably because the transitivity of < does hold for pure tones and for most of the chords we are exposed to in music.

Added later:  And of course p8 is equal to p1 so it's highly unlikely that anyone would judge that p1<p8.  In conclusion, it seems there are two concepts of ascent in pitch that we can apply to a chord sequence.  The first is a short-range or local ascent:  we listen to a short segment of notes, p(i), p(i+1), p(i+2), say, and judge that p(i)<p(i+1) and p(i+1)<p(i+2).  The second is a long-range or global ascent: we judge that p(i)<p(j) for any i<j.  My local ascent is Ed's unqualified ascent and my global ascent is Ed's ultimate ascent, perhaps.

How does this connect with the law of non-contradiction?  Well, suppose we are so benighted as to hold that there is just one notion of ascent in pitch for a chord sequence.  Listening to the Shepard sequence for a long while together with background knowledge of the range of pitch we can hear tells us that the sequence does not ascend.  But listening to fragments tells us that it does ascend.  So we do seem to be brought to embrace a contradiction.   However, we are so committed to the principle of non-contradiction that in such situations we look for a way out.  In this case we reconceptualise our auditory perceptions:  local ascent does not imply global ascent.