harmonic entropy, neutral zones

A while back I posted on my concept of harmonic entropy. In February
1997 I ran a computer program to compute the harmonic entropy of all
intervals within the octave in 1-cent increments, based on the
assumption that our brain can ideally recognize ratios with numerator up
to N but our hearing of frequencies is blurred in the form of a normal
distribution with standard deviation 1% (based on Goldstein's work). I
hadn't looked at the results yet, so as a preliminary study I listed the
local minima and maxima below.

Note that the minima appear to approach the just values as N increases,
but the number of minima remains approximately constant. Note also that
there is a definite maximum at around 348 cents. This means that
harmonically, the brain interprets the neutral third with a variety of
ratios, none of which is predominant enough to allow the brain to make a
decision. As Johnny Reinhard said, a sort of neutral zone. Other neutral
zones appear to be stabilizing for N=80 at around 285 cents, 423 cents
(giving the 9/7 a very narrow range of acceptable flattening!), 457
cents, and 537 cents.

The local minima and maxima were as follows (maxima denoted with *):

N=80:

*57
264 (7/6=267)
*285
316 (6/5=316)
*348
387 (5/4=386)
*423
437 (9/7=435)
*457
498 (4/3=498)
*537
581 (7/5=583)
*615
620 (10/7=617)
*656
702 (3/2=702)
*746
814 (8/5=814)
*845
885 (5/3=884)
*924
970 (7/4=969)
*999
1021 (9/5=1018)
*1041
1051 (11/6=1049)
*1145

N=40:

*72
219 (8/7=231)
*242
272 (7/6=267)
*286
314 (6/5=316)
*348
386 (5/4=386)
*426
433 (9/7=435)
*454
498 (4/3=498)
*543
586 (7/5=587)
*654
703 (3/2=702)
*752
811 (8/5=814)
*843
884 (5/3=884)
*923
968 (7/4=969)
*996
1021 (9/5=1018)
*1130

N=20:

*110
171 (11/10=165)
*197
255 (7/6=267)
*287
319 (6/5=316)
*346
384 (5/4=386)
*421
439 (9/7=435)
*450
497 (4/3=498)
*545
585 (7/5=583)
*643
701 (3/2=702)
*761
818 (8/5=814)
*844
885 (5/3=884)
*933
972 (7/4=969)
*1042
1057 (11/6=1049)
*1096

N=10:

*201
270 (7/6=267)
*285
318 (6/5=316)
*347
382 (5/4=386)
*428
436 (9/7=435)
*444
503 (4/3=498)
*552
577 (7/5=583)
*619
710 (3/2=702)
*783
812 (8/5=814)
*840
887 (5/3=884)
*933
965 (7/4=969)
*997
1023 (9/5=1018)
*1049

(remember that for N=10, ratios of 11 aren't even considered)

I wrote,

>> Paul H., the nonlinear processing that the ear does actually
increases
>> the frequency resolution over what the standard uncertainty relation
>> would give for an analyzer with the ear's temporal resolution, thus
>> decreasing (rather than creating) the incidence of beats.

Paul H. wrote,

>?? I must be misunderstanding something. If you analyze the signal
>using a process which decomposes it linearly, such as a Fourier
>transform, no energy is detected at the beat frequency. What is a
>decrease from nothing? I think this is what Gary Morrison was talking
>about when he said (talking about bram's beat-canceling idea):

You are correct only with an infinitely long sampling window. The
temporal resolution of the ear is far better than that, on the order of
several milliseconds. Standard signal processing theory says that the
temporal resolution times the frequency resolution is greater than 1 or
2*pi or some such constant. The ear beats this by performing several
layers of processing, which is what I refer to as "nonlinear".

Paul H. wrote,

>?? I must be misunderstanding something. If you analyze the signal
>using a process which decomposes it linearly, such as a Fourier
>transform, no energy is detected at the beat frequency. What is a
>decrease from nothing? I think this is what Gary Morrison was talking
>about when he said (talking about bram's beat-canceling idea):

Reading this again, I note an even more important point: the ear does
_not_ detect energy at the beat frequency; even beat rates in the
audible range do not necessarily produce audible tones (they only can be
said to do so if the conditions are right for difference tones: loud
enough for the real nonlinearities of the ear to be inmportant), as
Helmholtz observed. The beat is only a variation in the energy at an
average frequency of the original waves.

On Thu, 8 Oct 1998, Paul H. Erlich wrote:
> the ear does
> _not_ detect energy at the beat frequency [snip]

This is pretty much the point I've been trying to make all along. Not
very successfully, apparently.

--pH http://library.wustl.edu/~manynote
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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