Inside Beats

[A440A@AOL.COM]

Virgil is quoted as writing: 
"IF A3 has a fundamental frequency of 220hz., then F3 sounding at
176 hz. would produce a pure major 3rd at an exact 5:4 ratio.
True so far, it would then seem that, in order to expand the
F3-A3 3rd by 7bps, F3 should be lowered by 7 beats to 169 hz."

And then Richard writes:
<<Ok... so I may be the one who misunderstands... I will certainly
re-read this article a few times to see if I can figure out what
he is getting at....but for the moment I must admit I am clearly
miffed...grin.. so if anyone can shed some light on this I would
be greatfull.>>

Greetings, 
   I think you got it right.  It may help if we define at what frequency the 
"beats" are being counted, since there are several rates going on 
simultaneously in any interval.
   In Virgil's example, the   F3, with a fundmental of 176 Hz, has its fifth 
partial at 880 Hz, as does the A beginning with a fundamental of 220.  (We 
are leaving inharmonicity out of this comparison, as it doesn't change the 
outcome appreciatably). 
    If you drop the F by 7 Hz to 169, its fifth partial becomes 845 Hz,  so 
you create a beat rate at the 5:4 location if 35 Hz.  which we don't even 
hear as a beat rate.  
However,  Virgil has written that you drop the F by 7 *BEATS*, not cycles.  
By using the word "beats", a requirement is created for something to compare, 
since beats come from two differing frequencies.  Does he mean that the F3 
would be dropped until it beats at 7 bps with its original pitch?  
     There is only 9.8 Hz diffence between F3 and E3, so dropping F by 7 Hz 
means dropping it over a 1/4 step.   Dropping the F3 by 7 cents means that it 
is only going down approx  .7 bps.  
Vague and ambiguous writing, I believe.
Ed  Foote