temperament interval widths?

[Robert Scott]

Ron Koval asked:

>Can someone post the width of intervals from pure in equal
>temperament?  I know major 3rds are 13.7 cents wide, 5ths 
>are about 2 cents wide, but how about the 4ths, 6ths, and 
>minor thirds?  I'm trying to make an easily understood graph
>to "pre-listen" to the alternate temperaments.

(I know you mean "narrow" on the fifths, right Ron?)

I don't know if staring at a graph is going to be as satisfying
as actually hearing various temperaments, but let me give you
some tools to answer your questions yourself.  The nominal pitch
of the fundamental in non-stretched equal temperament is
given by starting with A4 = 440 Hz and then multiplying or
dividing by 1.059463094 as you go up or down from there in half-steps.
Once you have all the fundamental frequencies you can compute the 
deviaiton from pure (called "just") intervals by knowing what whole-
numbered ratioes define each just interval.  Octaves are 1:2.
Fifths are 2:3.  Fourths are 3:4.  Major thirds are 4:5.
Minor thirds are 5:6.  Other intervals can be dervied by combining
the ones here.  For example, a 10th is an octave and a major third,
so the ratio is 4:10 = 2:5.  Once you know the ratio for a just
interval, you can compare it with the ratio of the equal-tempered
interval.  For example, the ET major third is 1:1.25992 while the
ratio for a just major third is 4:5 = 1:1.25.  The ratio of these
two ratioes is 1.007936.  You can convert this approximately to
cents by subtracting 1 and dividing by .00057779.  (I know, to be
precise you have to use logarithms, but in this range the approximation
is very good.)  In this case you get .007936/.00057779 = 13.7 cents.
Now you can do the same with any other just interval.

Robert Scott
Detroit-Windsor Chapter