From: jseboldt@pnet51.orb.mn.org (John Seboldt)
Newsgroups: rec.music.classical
Subject: Re: Temperments
Date: 26 Sep 92 04:25:02 GMT
Organization: People-Net [pnet51], Minneapolis, MN.

egras@ohstpy.mps.ohio-state.edu rants and raves:
>        I had some quesions about the temperments used in establishing
>the arranged pitches.  I was told that the standard modern way in the
>12th root of 2 way.  This comes from the 12 pitch classes.  So if we
>decide that the >A should 440Hz, then an A# is 2^(1/12)*440Hz, a B is
>2^(2/12)*440Hz and so on. Is this the equal-tempered scale?  I don't
>think so since the pithes are equally spaced.  I was also told that the
>method in practice today is the equal-tempered temperment which involves
>dividing an octave interval into 100 cents and all the pitches having an
>equal number of cents between them.
 
Yes, in equal temperament pitches ARE equally spaced, logarithmically
speaking. The "cents" measurement is another way of putting it.
 
However, acoustically, only the octave is a just, acoustically "pure"
interval in equal temperament.  The equal spacing of the 12 notes means
that every key is equally OUT OF TUNE!  Amazing what we have learned to
tolerate, but true.
 
The facts of life in tuning a keyboard instrument are:  not every
interval can be pure with 12 semitones. (fifth, freq. ratio of 3/2;
fourth, ratio of 4/3; major third, ratio of 5/4).  If you tune a series
of fifths (c to g, g to d, d to a, etc), you will NOT end up with
another C, but a B-double-sharp, slightly sharper than C. That
difference has to be folded back somehow if you want the octaves to be
pure.  This series of fifths is the basis of the Pythagorean temperament
-- I'm not too clear how useful this was, perhaps only for melodic
instruments.
 
Thus the different ways to deal with the situation.  Mean-tone
temperament tended toward mostly pure thirds and small fifths -- most
keyboard music from 1500 to about 1650 (???).  You can't play in all
keys; and sometimes you have to pick whether to tune an E-flat or a
D-sharp, for instance.  On a harpsichord, you just retune for the piece;
on an organ, there were often separate keys (front and back halves of a
"black" key).  Some intervals were intolerable: f-sharp to b-flat could
NOT be used as f-sharp to a-sharp in an F-sharp major triad!
 
As harmonic language moved further afield, temperaments evolved that
were more usable in "distant" keys.  USABLE being a matter of your ear
adjusting; they still favored the more common triads, so more common
keys were still purer than equal temperament.  Systems by Werckmeister
and Kirnberger around 1700 were the sort of tuning Bach used (he did NOT
use entirely equal temperament).  True equal temperament didn't take
over till 1800-1850, depending on the country you're talking about.
 
As a harpsichordist/organist, I find these  tuning differences very
discernible. As the piano (dominant 19th century keyboard instrument)
became fuller in sound, the duller harmonic structure made equal
temperament easier to take.  With all the brilliance of the
harpsichord's upper partials, equal temperament is much more obviously
out of tune, and a more "consonant" temperament (depending on the piece
and period) makes all the difference.  Nothing like a pure third to put
your mind at ease.
 
Discerning players of wind and string instruments will tune differently
in different circumstances -- with a keyboard, one way, with other winds
or strings, another, depending on their ears, attentiveness, and how
they feel that day!
 
Sincerely, Your Friend <Pfeifenwerk>
 
:  John Seboldt...Minneapolis, MN...                            :
:  As a ham, K0JD...as a human...well, you judge!               :
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