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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:
S tep 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.)
S tep 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.)
S tep 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.
S tep 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).
S tep 5 . Now every fourth note is tuned from A2 to A4. Fill in the missing notes between F3 and
C#4 to get a nine-note mini-temperament. You must now take a guess at the width of the fourths,
try it on the mini-temperament, and revise it as necessary to make the fourths fit with the thirds,
whose width you already know. A good first guess for fourths is 2.5 cents. Tune up a fourth from
F3 to A#3 down a third from A#3 to F#3, up a fourth from F#3 to B3 and stop. Now tune down a
fourth from C#4 to G#3, up a third from G#3 to C4, down a fourth from C4 to G3 and stop. You
have filled in the six missing notes of the mini-temperament, now you can tell whether your
guess on the width of the fourths was correct from the width of the G3-B3 third, which is the check
interval.
S tep 6 . Measure the width of the G3-B3 third and compare it to what it should be. If it is smaller
than the other thirds, your fourth guess was also too small, and vice versa. The size of the error
in the fourths equal one-quarter the error in the G3-B3 check interval because four fourths were
tuned to get to the check interval.
S tep 7. Retune the mini-temperament as in Step 5 with the new correct value for the widths of
the fourths. Check it aurally to see that you have five perfectly rising thirds, and four equally
good fourths. This result can always be achieved even on the most poorly scaled piano, since
we have not been asked to make any compromises up to this point.
S tep 8 . Tune outwards from the mini-temperament with major thirds down to A2 and up to A3.
The thirds tuned downwards are constant width but the thirds tuned upwards may have to be
calculated if their width varies. Use straight-line interpolation based on the three known thirds in
the A3-A4 octave, to get numbers for the missing thirds. (E.g., in Step 4 these three thirds, A3-
C#4, C#4-F4 and F4-A4 were 13.5, 13, and 12.5 cents, respectively. Straight-line interpolation
gives values of 13.4, 13.3, and 13.1 for A#3-D4, B3-D#4 and C4-E4, and 12.9, 12.8, and 12.6 for
D4-F#4, D#4-G4 and E4-G#4, respectively).
S tep 9. The electronic tuning of the two octaves A2-A4 is now complete. A very careful aural
check is advisable at this point. Pay particular attention to the fourths, which are wider than
theoretical owing to the effects of inharmonicity. The fifths are purer than theoretical for the
same reason, and rarely cause any trouble.
