ramblin' (tuning by pure 5ths)

[Richard Moody]

----------
> From: pianoman <pianoman@inlink.com>
> To: pianotech@ptg.org
> Subject: Re: ramblin' (tuning by pure 5ths)
> Date: Saturday, June 28, 1997 6:57 AM
> 
> 
> 
> > I will post the fifth beat rates from C3 to A5, seeing is
believing. 
> > rm
> > Dear Richard,
> Yesterday I tuned an "L" and used Jim Colemans stretch numbers for
his "L"
> and measured several intervals for several thirds and several 5ths
for the
> normal FAC and altered FAC.  I will send them this weekend when I
can get
> some free time.  I am interested in seeing the beat rates on the SS
you
> talk about. 
>  This scientific research eats up tuning time!
> 
> James Grebe from St. Louis
> pianoman@inlink.com

James and LIst, 

	The beat rates for fifths for Jim Coleman's WTO, (Well Tempered
Octave(s) follow.  Keep in mind these numbers are from a spread sheet
(SS) that computed the beat rates for 5ths, (among other intervals)
at an octave stretched to 2.004, as used in the formula for computing
the positive adjacent note of an equal temepred octave. or
multiplying the freq, times the twelth root of two.  Since our octave
is widened a little, the number used is  freq. times the twelth root
of 2.004.  The spread sheet needs to see this as   F*(2.004^1/12). So
if the frequency is A440 and you multiply by (2.004^1/12) you get the
freq of Bb.  To get the frequency going down, you multiply by
(2.004^-1/12).  If your eyes are blury, you might miss the - right
after the ^ .   

	So.... starting with fifths from A220 (A3) that is the fifth A3-E4up
we have.... 
	A3-E4 = .024, .035, .027, .028, .030, etc, etc, to the A440-E5 fifth
which is .048,  which is a little less than one beat in 20 seconds.  
That's as they say in South Dakota, is purt'near pure.  For contrast
in Equal Temperament the same fifth is 1.488 bps.  

	Now Jim says the fourths beat faster.  Here is the rate...3.456 bps.
  But that high on the keyboard you don't notice it as 3 beats per
second. (A440 -D5). In ET this fourth beats a little less than 2
beats per second, you don't notice that either.  

	Now for the octave beat rate, .878 for the A220 to A440 octave.  or
nearly one beat per second. This beat you will hear and be able to
count. But the amazing thing is in a chord this beat becomes "lost" 
ie you simply don't hear it. 

Richard Moody