Historical temperamentals

[Richard Moody]

: A440A@AOL.COM
> To: pianotech@ptg.org
> Subject: Re: Historical temperamentals
> Date: Sunday, August 08, 1999 12:31 PM
> 
> I am sorry all,  My last line of the preceding post read:
> >precision found in the square root of two. 
> 
> Culpa all my meas, it should have said, of course, the 12th root of two.  
> Please excuse my sloppiness. 
> Ed Foote
> 


No slopiness at all. ET can be derived from the square root of two.  The square root
ot two is the ratio of the tritone in ET.
(F#,Gb).
  
BTW for those who are curious as to how the "ancients" figured out something as
complicated and abstract as the 12th root of 2, before log tables, slide rules or
calculators, it might have been done thus......The mean or the square root of the
octave is the square root of 2. 
This is the tritone. If two minor thirds makes up a tritone,  the mean or square
root of the tritone should yield a minor third. If three semitones makes a minor
third, the cube root of a minor third should get one of those semitones.   yup
checks out on my calculator......
       

proof  Given that for spread sheets the square root of two is entered as....2^(1/2).
 (This follows from the fact that "two squared" is written as 2^2)

 2^(1/2)^(1/2)^(1/3) = 2^(1/4)^(1/3) ...... 2^(1/4)^(1/3) = 2^(1/12) = twelth root
of two. 

Now how the ancients realized 18/17 approximates this, (98.95 cents) I haven't found
out yet. This is the proportion some used to set the frets for lutes.   ric