Hi Ed, Thanks for the instructions, I was indeed asking how to tune a well temperament for an aural tuner, though all that didn't seem to come out :) A few questions if you don't mind: - What is 1/12 of a comma? - When doing step 3 in the P.B. '93 well temperament (and step 3 of Kirnberger) where tuning the Eb, Ab, etc. via just fifths I presume you are to cover over 2 octaves? And when it says "Spread this around the keyboard" I presume that is tuning those notes in different octaves around the keyboard? - I am particularly interested in the Young 1799 temperament, which seems to be highly regarded. Does anyone have a details that could help an aural tuner? Thanks! - John > 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
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