Inharmonicity in strings

[Richard Moody]

----------
> From: Jim Coleman, Sr. <pianotoo@IMAP2.ASU.EDU>
> To: Billbrpt@aol.com
> Cc: pianotech@ptg.org
> Subject: Re: Inharmonicity in strings
> Date: Saturday, June 13, 1998 10:33 PM
> 
> Hi again Bill:
> 
> I stand corrected by Kent Swafford. You can hear the 8:3 relationship.
> 
> Jim Coleman, Sr

Actually I don't think you can HEAR the 8:3.  If you can, what's the first
thing that strikes you about the beats?  You can hear an 11th, but to
split a hair, I don;'t think the beats that are supposed to be there (from
the 8 and 3 partials) can actually be heard. 

	To be truthful, I can hear the 8:3 on a synthesizer, and yes I did have
to check the math to make certain the rate was what I thought.  This was
starting at a -- d" (a3--d5). Still they are pretty damn weak, and given
the difficulty (comparative) of hearing 4:3 fourths on the piano, I doubt
they could be heard at all there.    

 I believe the original question was about sine waves having no partials
execpt the fundamental, and the implications of that in tuning.   

But to address the postulation... no coincedent partials--no beats...
consider the following.... 

	These tests were conducted on an analog synthesizer, harpsichord type
patch,(tweeked to give clearest beats)  with unlimited sustain. 

It can be demonstrated, that clear audible beats disappear out side the
simple ratios, (non doubled)(lowest numerators)  of the intervals. 

For instance, a third, 5:4 with root at c# (3) beats 5.5 /sec.  the tenth
at 5:2 with the F now at f' (4) beats the same.  at  17th  the ratio is
5:1 the beat is the same. However at the 24th or c# - f'' (F 6) no beats
are readily descernable. In fact they seem vanish.   Now there should be a
10:2 but can they be heard enough to be usefull, if at all? There was
always a doubt if I was hearing any beats  there. 

Consider the 5:3 ratio, the 6th.  Observe the ratio is composed of 2 prime
numbers. Take f--d'  (f 3 -- d4)  Pert'near 8 bps, and very distinct.  
What about f--d"  (f3--d5) ? NO BEATS ! !   The nearest coincedent
partials would be 10--3. The beats should be doubled, but they are not
nearly (if at all)  audible to hear that.

Richard Moody