well tempered temperaments

[A440A@aol.com]

<< > Greetings,

> << Just wondering if anyone can point a web link or a book on tuning well

> temperaments. >>

Umm,  Ok,  you were asking HOW to tune a well temperament...??  There is a 
lot on the web, but here is one that you may be interested in trying, 
compliments of Paul Bailey RPT.


                                Bearing Plan
                                   for the
                        P.B.  '93   Well Temperament    

1) TUNE C to a pitch source.

2)TEMPER F a 1/12 comma narrow fifth below C.
        (This could be done with no estimating by going one round of the
E.T 
thru Just Intonation bearing plan of John Farey,1807;[see O.J.'s TUNING, 
SEC.85-86.] 
      Very careful testing of all the fifths and fourths  and the
just third would be required for accuracy, and any working tuner will almost
surely use direct tempering with one or two tests instead. It is very 
interesting
to me that if one goes the long way 'round for the 1/12 comma fifth, this
becomes a complete just intonation and/or exact copy of beat rate 
temperament, with
 absolutely no estimate-guestimate tempering.)

3) TUNE Bb a just fifth (or fourth) from F; keep going by just fifths or 
fourths to Eb, Ab, Db and Gb. Spread this around the keyboard enough so
that 
you can-

4)TEMPER B a wide fourth above F#(Gb)  so that the third B-D# beats at the 
same rate as the third  F#-A# a fourth lower.

5) TUNE E just to B.

6) TEMPER  A a wide fourth above E so that A-C# beats at the same rate
asE-G# 
a fourth lower.

7) TEMPER  D between  Bb and  F#  so that two conjunct equal beating thirds

are formed.

8)  TEMPER  G  so that  Eb-G beats at the same rate as Bb-D a fifth higher.

                        Cents Deviation from Equal Temperament

C=+6.2  C#=-1.6         D=+1.4  D#=+2.3         E=-2.1          F=+6.2  

F#=-3.6 G=+49   G#=+0.3         A=0.0           A#=+4.2         B=-0.1
 
Now, for a more traditional approach,  this is a plan for the Kirnberger, 
compliments of Carey Beebe: 

Here's how you go about setting Kirnberger III:
1. Tune your c'' (that's the c one octave above middle c'!) to 
a tuning fork, and tune middle c' in absolute perfect tune 
below it.
2. Now we want to determine the e' a perfect third above 
middle c': This will be a revelation if you have never heard 
a perfect third before--Your violinist friends are going to 
find it unbearably flat, but we want it in perfect tune. (Hint-
-if you split your interval over the octave, you will find it 
easier to hear because the speed of the beats is halved.)
3. Next, the hard part. Make all the fifths around the circle 
between C and E equally narrow and rough. We are 
dividing the comma among these four fifths, instead of the 
twelve of equal temperament, so they are going to be three 
times as out of tune as the equal tempered fifths we know 
and love on the piano. To tune a narrow fifth, first tune it 
perfectly, then lower the top note (or raise the bottom note) 
until you can't bear it any more. That's probably about a quarter of a comma! 
All your fifths C-G, G-D, D-A and A-E should sound equally rough. Don't move 
your c' or e'--you've already tuned them, and must keep that interval as the 
beautiful perfect third. (Actually, it wasn't that hard, was it?)
4. Tune all the fifths from the flat side of C around the circle of keys 
absolutely perfect. Stop about the Eb, and begin again working around the sharp 
side of E, tuning all those fifths absolutely perfect. (In theory, you'll 
end up with one fifth a little narrow, in fact very close to an equal 
tempered fifth, but in practice, they should all sound perfect.)
            When you are happy with your middle octave, you must bring the 
rest of your instrument into tune with what you have done. Tune down in octaves, 
and up in octaves.

When you feel you've mastered Kirnberger, it's an easy 
matter to change it into something else, so now it's time 
for Werkmeister III. He proposed his third temperament 
in 1691 (Those Darwinists reading this will, I trust, forgive 
me working backwards in history for ease of tuning--in 
musical terms, what came later was not always better!), 
and this is how you can accomplish it on your i
nstrument:
1. From your nicely tuned Kirnberger, sharpen that e' that 
your violinist friend so much despised, so that instead of 
the beautiful perfect third with middle c', it makes a 
perfect fifth with a.
2. Tune down the octave to e, and raise the b so it is a 
perfect fifth with the e.
Congratulations--You're done! You've tucked that last 
narrowed fifth a little bit further around the circle, 
improving the sharp keys. And you will be forever loved 
by your string-playing colleagues because they needn't detune the top string 
on their fiddles.
 


Ed Foote RPT 
http://www.uk-piano.org/edfoote/index.html
www.uk-piano.org/edfoote/well_tempered_piano.html