Re: [tuning] harmonic entropy continuing

In Pauls list of intervals with big troughs (4/3, 3/2) lesser troughs
(thirds and sixths) he got to the "non trough" regions in the minor
seconds where "maximal ambiguity" is occurring. There is a
similar hump of ambiguity between the 4/3 and 3/2 which, I
wonder, is it centered on SQRT(2), the 12tet tritone?

Bob Valentine

>>Mediants define the conceptual boundaries of the ratios and since >simpler
>ratios are farther away from more complex ones and complex ones >cluster
>together, there are more possible actual intervals (rational or >otherwise)
>that can be conceived as simpler ones than for more complex >ones.

Joe Pehrson wrote,

>Wazzat?? Here's where I start to get lost (it had to happen). Is this
>because of the bell curve??

Simpler ratios are hermits while more complex ratios are more sociable. Plot
all the ratios up to a specified integer limit (say, 20) on a number line
and see this for yourself. So in a sense the simpler ratios occupy more of
the space on the number line. Hopefully this addresses your confusion
regarding those three paragraphs, and others by Monz. Let me know if it
doesn't. BTW, I discovered a detailed mathematical derivation of this fact
in 1993, which I could share with you. The analagous observation for chords
of 3 notes is shown by the Voronoi plots I was spewing out a few months ago
(remember?).

>Could someone possibly construct such synthesized examples??

I thought we already had some on the net. If not, I'll produce some to put
on Carl's website.

Bob Valentine wrote,

>In Pauls list of intervals with big troughs (4/3, 3/2) lesser troughs
>(thirds and sixths) he got to the "non trough" regions in the minor
>seconds where "maximal ambiguity" is occurring. There is a
>similar hump of ambiguity between the 4/3 and 3/2 which, I
>wonder, is it centered on SQRT(2), the 12tet tritone?

Bob, why don't you look at the graph? If you do, you'll see that there are
two roughly equal maximal ambiguities between 4/3 and 3/2 (using the
assumption that the listener's frequency resolution is 1%) and they occur at
around 538¢ and 656¢. There is a large trough at 7/5, and a really small one
at 10/7, but the area between these ratios (including sqrt(2)) is lower in
ambiguity than the 538¢ and 656¢.