Just, Pythagorean and Meantone on the Diatonic

[Richard Moody]

>> Richard writes:
> > I would have guessed that keys and key signatures along with the
> >major and minor scales besides the "modes" would have been known at least
> >since the appearance of the keyboard which I have heard can be traced
back to the 13th century.  The music staff must have had an influence also.
>
> Greetings,
>     The earliest keyboards  dated from at least 200 B.C.  By 1000 A.D. we
> know of organs with several octaves, but no more than 7 notes per.
> Pythagorean tuning is strongly suggested for these instruments by a lot of
> things.
>     From 1200 to 1400 or so,  the tuning world seems to have tried a lot
>of things, from Pythagorean to Just Intonation, but seems to have settled
on
> what we call a meantone by the mid 1400's, later quantified by Aaron in
>1513.  Keyboards took several forms, but the 7/5 12 has always seemed to be
>the most user friendly.
> Ed Foote


    7 notes made the diatonic scale of course. (or as some say, two tetra
chords) An amusing exercise is to tune those first 7 (the white keys) in
just intonation.  That would be the 5th, 4th and 3rd all pure from C.   A
pure 3rd from F makes A and a good 4th from E also.  B would be a 3rd from
G, and it does make a pure sounding 5th with E. It has no 4th.   Now we are
left with D.  Poor D, tuned as a 5th from A it sounds OK, but the fourth it
makes  with G sounds terrible.  It sounds even worse as a m3rd with B.  (if
you tune 2 octaves).  Part of the reason is that D can have a 9/8 ratio with
C or a 10/9 ratio depending on where it is comming from.  So we are faced
with the first problem of tempering a diatonic scale.  You
can sharpen A and tune D ---equal beating perhaps---to G and A.

     This is the emperical approach.  The first theoretical approach
is to tune all the fifths pure according to the school of Pythagorus as Ed
mentioned.
Since the fourth is an inverted fifth we can easily tune "up a fifth down a
fourth"  to get all 7 notes.
    Or you can do what the Medieval theorists did and propose D as the mean
between C and E  or the mean of the ratio 5/4.  (C--E is a major 3rd.  the
ratio of a major 3rd is 5/4)   That mean would be the square root of 5/4
which is another problem they had to figure out.  Do you suppose the mean of
the two ratios of D , 9/8 and 10/9 is the same as the square root of 5/4?
(yeppers) Anyhow, this  is where the name Meantone comes from. The fact that
this D would also match the D formed by a series of 5ths (starting from C)
flattened by one fourth of the comma of Didumus (syntonic)  must have been
amazing then as it is now.
    It is just as amazing  for the tuner to guess right and tune a series of
narrow  5ths and have the 3rds come out pure.  But since that rarely happens
a little fudging helps, like tuning  E  pure and tune 3 5ths to it, and F
when tuned pure to A should make a narrow fifth with C. If you don't like it
you can adjust A to E and try again.  An example of the theoretical
real-ized
by the emperical.
    If you play old tunes in these 3 tunings of the diatonic, like "Amen"
or "Praise God from Whom all Blessings Flow"  I think you will praise
Meantone the most.
If you try the ancient carols like "Green Sleeves"  or "What Child is this"
you quickly find these 7 notes aren't enough in any mode.    ---.ric