Points of Maximal Beating

Paul,

Is there a way to mathematically calculate the cents values for what
would be the most maximally beating points between any two primary
ratios? For instance, what would be the maximally beating point
between 6/5 and 5/4, or between 4/3 and 3/2? Is there, or could there
be, a number of them between certain intervals?

I realize this would somewhat be dependent upon the timbre being
used, but let's say for ease of discussion, it's a harmonic timbre
under consideration. Or perhaps it's better to consider only a sine
wave?

Thanks,

Jacky

>Is there a way to mathematically calculate the cents values for what
>would be the most maximally beating points between any two primary
>ratios? For instance, what would be the maximally beating point
>between 6/5 and 5/4, or between 4/3 and 3/2? Is there, or could there
>be, a number of them between certain intervals?

>I realize this would somewhat be dependent upon the timbre being
>used, but let's say for ease of discussion, it's a harmonic timbre
>under consideration. Or perhaps it's better to consider only a sine
>wave?

Of course, for a sine wave the question is meaningless -- no beats will
occur at all, unless the dyad is played in a low register and the tones
actually beat against one another, in which case the beating will simply
decrease smoothly as one moves from 6/5 to 5/4. For a harmonic timbre,
though, the answer depends on both the amplitudes of the partials, and on
how you define "maximally beating" (presumably each instance of beating is
to be weighted by both its amplitude and rate, but the details of the
calculation are pretty subjective, though a proper treatment will have to
take into account the critical band of the human ear, which varies according
to register).

You may recall the "golden mediant" Margo and Dave K. described some time
ago:

http://www.egroups.com/message/tuning/12915

"As a solution of the above quadratic one finds that Phi is (sqrt(5)+1)/2 or
approximately 1.61803398874989484820459.

Thus we can find the Golden Mediant for the region of maximum complexity
between two simpler intervals by setting x=1, y=Phi. For two such interval
ratios i:j and m:n where i:j is the simpler ratio:

(i + Phi * m)
GoldenMediant(i/j, m/n) = -------------
(j + Phi * n) "

That could be a useful catch-all in many situations -- in other words it
usually (but not always) will be close to the right answer regardless of
specifics.

So, for your 6/5 - 5/4 example, the golden mediant is about 339.3439 cents,
and for your 4/3 - 3/2 example, the golden mediant is about 560.0666 cents.

Good question Jacky.

Paul, thanks for pointing folks to Margo's and my post.

Just a reminder that Margo and I later agreed that it should be called
the "noble mediant". Note that the most complex of the two ratios is
always the one that gets "phi-weighted". And as Paul mentioned, it
doesn't always work. Notably it fails near the unison, octave and
perfect fifth.

The noble mediant formula of course gives a frequency ratio, not cents
or a fraction of an octave, so to get cents you still have to apply
ln(x)/ln(2)*1200 to the result. Any decimal places in such cent values
are totally irrelevant, as are most of the decimal places we gave for
phi.

Regards,