13
the three A's first, we can establish a double octave and two octaves that fit as well as is possible
both with each other and with the scale of the given piano.
Next we subdivide this wide interval into six equal parts by tuning six contiguous major thirds that
fit between the three A's perfectly. Finally, with every fourth note already tuned, we fit the three
missing notes within each major third primarily by tuning a pattern of thirds and fourths.
Tuning wide intervals first and then subdividing them has important advantages over the usual
methods of building up wide intervals by tuning a succession of narrow ones. In the first place, it
guarantees that the wide intervals will be as harmonious as possible, and that the narrow intervals
will be adjusted or forced to be compatible with them. Secondly, small errors in tuning narrow
intervals cannot add up to become large errors in the wide intervals, no matter how difficult the
scale of the piano. This not only leads to greater accuracy on well-scaled pianos, but also greatly
reduces the number of problems associated with tuning poorly scaled pianos.
Direct-Interval Tuning
Direct-interval tuning is a way of using the Sight-O-Tuner II that exactly simulates the way fine aural
tuners tune by ear. Each interval is tuned by setting its width to a specified number of cents, which
is verified by a direct measurement. Hence the term "direct-interval tuning". The sequence of
intervals followed is circular, just as in aural tuning, and this makes it quite easy for the tuner to
check each interval aurally as it is tuned. (If you are not familiar with measuring the width of
intervals in cents, refer to the section entitled "Measuring the Width of Musical Intervals").
First, two single octaves and the double octave are tuned using direct-interval measurements.
Second, the double octave is subdivided in six equal parts with a set of contiguous major thirds that
mathematically fit this span perfectly, as determined by direct interval measurements. This leaves
every fourth note tuned, and the three untuned notes within each major third are then tuned with
fourths and thirds, again by direct interval measurements. Follow the step-by-step procedure below,
and be sure to check all intervals aurally as you tune:
Step 1.
Tune A4 to zero cents, and A3 from A4 as a 2-4 octave 1 cent wide. (Refer to Appendix H
if you are not familiar with the different kinds of octaves.)
Step 2.
Tune A2 from A3 as a 3-6 octave 1 cent wide. Check the A2-A4 double octave, and if it is
more than 4 cents wide, divide the excess by three and narrow both octaves by this amount. (E.g.,
if double octave is 5.5 cents wide, 5.5-4 is 1.5, divide by 3, and narrow both octaves .5 cent.)
Step 3.
Tune three major thirds of equal cents width between A2 and A3. You must first guess
how wide to tune them and then see how the guess works out and revise it if necessary. A good
first guess is 13.5 cents. So tune F3 from A3 13.5 cents wide, then C#3 from F3 13.5 cents wide.
Measure the width of the A2-C#3 third. If it is also 13.5 cents wide, you were lucky and these
three thirds are the correct width on the first guess. If you were not so lucky, average the three
thirds (two of which were 13.5 cents wide), and tune all three to this average value by retouching
C#3 and F3.
Step 4.
Now tune C#4 from A3, and F4 from C#4, as thirds of this same value. You have now
tuned five contiguous thirds all the same width, a width that fits exactly the A2-A3 octave. To see
whether this width fits the A3=A4 octave, measure the width of the last third, F4-A4. If this
agrees with the other thirds, you were lucky again and these six thirds are all tuned. If you weren't
lucky, take the discrepancy of the last F4-A4 third, divide it by 3, and move F4 by this amount in
the direction that will reduce the discrepancy. (E.g., if you had five thirds at 13.5 cents, and the last
was 12 cents the discrepancy is 1.5 cents. Take one-third of this, 0.5 cents, and move F4 flat by
this amount. This leaves you with four thirds at 13.5, one at 13, and one at 12.5. This is quite
reasonable on that "inharmonic instrument," the piano).