## Thursday, May 03, 2012

### The Shepard tone

Someone commented (see here and here) about the enigmatic Shepard tone, suggesting it was a case of choosing between the apparently inviolable principle of contradiction, and a phenomenon that apparently contradicted it. I wonder about that.

Look at the Wikipedia definition, which is actually quite good. It says (I paraphrase slightly) it is a tone that continually ascends in pitch, yet which ultimately seems to get no higher. There are two interesting words in that definition, namely 'seems' and 'ultimately', and applying or removing them gives four possible combinations.
(1) The Shepard tone ascends in pitch but does not ascend in pitch

(2) The Shepard tone ascends in pitch but does not seem to ascend in pitch

(3) The Shepard tone ascends in pitch but does not ultimately ascend in pitch

(4) The Shepard tone ascends in pitch but does not ultimately seem to have ascended in pitch
Definition (1) is clearly absurd.  A definition needs to give us a way of distinguishing one thing from another, but this gives us nothing, since it includes pitches which do ascend and those which do not ascend, i.e. includes every kind of pitch whatsoever.  Definition (2) is better, but is it correct?  Surely not. If we take the first two or three tones as they occur in order, they clearly are ascending, and it is not that they just seem to ascend, at least in the initial phases of the sequence.  Both definitions (3) and (4) incorporate the term 'ultimately', and here we are getting somewhere.  What seems paradoxical about the tone is the way that after a full octave has been ascended, we seem to be  back where we were.  It's like one of those Sisyphean nightmares where we seem to be climbing forever, and find ourselves back in the same place.  Or the Blair Witch Project (for those who remember that).

Yet, in this case at least, is there really any contradiction between appearance and reality?  The whole point of the octave interval is the strong resemblance between the two tones of the interval (say, middle C and top C).  And where there is resemblance or similarity there is (formal) identity.  So it is no paradox to say we are ultimately back in the same place. All the Shepard tone does is to eliminate the respect in which the tones of the interval are different, i.e. eliminates the respect in which C and C' are different, while retaining the similarity.

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.  It is midnight, then time passes for each successive hour until it is midnight again.  Definition (3) captures the Shepard tone best. 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?

David Brightly said...

The problem with (3), I think, is that if we interpret 'ascends in pitch' as p1<p2 & p2<p3 & p3<p4 &..., or in other words, ∀n. p(n)<p(n+1), and we interpret 'does not ultimately ascend in pitch' as ∃n. ~p(n)<p(n+1), then we do indeed have a contradiction. Or is there another interpretation of 'ultimately'?

Edward Ockham said...

Surely it's another interpretation of "<"? Think of rotation, or a clock. It's 1 o'clock. The movement from 1 to 2 is plus 1 hour. And likewise 2 to 3, 3 to 4 etc. Then 12 to 1, gets us back to 1, yet we have still added an hour.

David Brightly said...

I'm utterly baffled by your last two paragraphs :-(

Is it that 'ultimately ascending in pitch' does not imply 'ascending in pitch'? I don't understand how you see 'ultimately' qualifying 'ascending'.

As to the clock metaphor, how does the hand 'ultimately not change place'? Surely it's changing place all the time even though it perennially returns to places it has visited before.

Edward Ockham said...

>>As to the clock metaphor, how does the hand 'ultimately not change place'? Surely it's changing place all the time even though it perennially returns to places it has visited before.
<<

Well ultimately (say 12 hours later) it is in the same place. If you are in the same place, you haven't changed places. So, ultimately you haven't changed place.

David Brightly said...

I can equally well say

Well ultimately (say 13 hours later) it is in a different place.

But that would equally abuse 'ultimately', which, for me, presumes a final steady state. There is no such state in the Shepard tone or in the rotating clock hand.

Anthony said...

"It says (I paraphrase slightly) it is a tone that continually ascends in pitch, yet which ultimately seems to get no higher."

You paraphrase horribly. The tone does not continually ascend in pitch. The tone creates the illusion that it continually ascends in pitch.

David Brightly said...

I have an analysis here.

Anthony said...

I agree that the Shepard tone is not an example of a perceptual contradiction.

We perceive the tone as continually ascending in pitch. We do not perceive it as staying the same in pitch. When we recall a previous pitch and compare it to the current pitch, we judge that they are the same, but this involves recall, so it is not perception.