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To all interested and capable in this type of thing:
The following are intructions I have written for my apprentice who is trying
to learn to tune a Thomas Young or other representative 18th Century style
Well Tempered Tuning (WT). In my opinion, the best temperament scheme for
virtually any type of temperament is best accomplished from an A fork in the
F3-F4 octave. Working on that premise, I figured out what I believe to be a
good, representative 18th Century style WT, perhaps identical to the Thomas
Young.
I'd like for those who know how to analyze verbal instructions and create
their mathematical equivalents to take a look at this and either confirm or
correct what I have written.
Most specifically, I have made only one specification as to beats per second.
In Owen Jorgensen's book, Tuning, the F3-A3 3rd beats theoretically at 3.8
beats per second. I give the specification, 4 beats per second, which can
easily be estimated or confirmed by affirming 2 beats per tick of a metronome
set at 120.
One important question I have is how the very last interval, C4-E4 (which
historically, in many WT's, is usually the very *first* interval to be tuned)
should end up theoretically, according to the instructions I have written.
Jorgensen says the Thomas Young C4-E4 3rd should beat theoretically at 4.1
Beats Per Second (BPS), nearly indistinguishable from the F3-A3, 3.8 BPS.
Our attempts at this temperament end up with the C3-C4 3rd beating 2-3 BPS.
Also, I think my G3-D4 5th beats a little faster than the Thomas Young
theoretical 1.4 BPS.
The purpose of writing instructions this way is to provide the very least
amount of subjectivity. A good tuner should be able to come up with the same
results each time, matching the consistency which ETD users get. The
temperament sequence lies easily within the reach of the left hand in the
part of the piano which is usually considered the optimum area for
temperament construction.
Of special note are steps 11, 12 & 17 which tell you to "temporarily tune" a
pure interval. These steps set up the Equal Beating (EB) process which
create the end result. During fine tuning, it is not necessary to repeat
this whole process. As long as the desired relationships are established, it
is better to skip through these elementary steps the same way one skips
through them when doing Algebra
Any and all comments (but not flames) are welcome.
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Tuning a Representative 18th Century Well Tempered Tuning
from an A Fork within the F3-F4 Octave
1. Tune A4 to A-440 pitch source.
2. Tune A3 from A4, a 6:3 octave.
Test for 6:3 octave: play A3 and C4, then C4 and A4. When both
intervals beat
exactly the same, the octave is correct.
3. Tune F3, from A3 a widened 3rd to beat at 4 beats per second.
4. Tune C4 from F4, a pure 5th. The A3-C4 minor 3rd will beat twice as fast
as
the F3-A3 Major 3rd (about 8 beats per second).
5. Tune F4 from C4, a pure 4th. This will form a perfect 4:2 octave, F3-F4.
6. Tune Bb3(A#3), a pure 5th from F4. This will also make a pure 4th,
F3-Bb(A#3).
7. Tune Eb4(D#4), a pure 4th from Bb3(A#3).
8. Tune Ab3(G#3), a pure 5th from Eb4(D#4).
9. Tune Db4(C#4), a pure 4th from Ab3(G#3).
10. Tune Gb3(F#3), a pure 5th from Db4(C#4).
11. Temporarily tune D4 to A4, a pure 4th.
12. Temporarily tune G3 to D4, a pure 5th.
13. Listen to the G3-C4 4th and notice the strong beating.
14. Sharpen G3 until the G3-C4 4th beats exactly the same as the G3-D4 5th.
15. Sharpen D4 until the A3-D4 4th and the G3-D4 5th beat exactly the same.
16. Tune B3 from G3 so that the G3-B3 3rd beats exactly the same as the
F3-A3 3rd (4 beats per second).
17. Temporarily tune E4, a pure 5th from A3.
18. Notice the resultant beat between E4 and B3.
19. Flatten E4 until the A3-E4 4th and the B3-E4 4th beat exactly the same.
20. The C4-E4 3rd should beat gently, 2-3 beats per second.
Bill Bremmer RPT
Madison, Wisconsin
<A HREF="http://www.billbremmer.com/">Click here: -=w w w . b i l l b r e m m e r . c o m =-</A>
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