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Step 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.
Step 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.
Step 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.
Step 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).
Step 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.
Aural Tuning
The two-octave "A" temperament may be tuned aurally with exactly the same note-tuning
sequence. This makes it easy to check an electronic tuning aurally at every step--very helpful in
avoiding errors. Aural tuners can study and possibly improve their tuning by using the SOT II to
measure the width of tuned intervals after setting this two-octave temperament very carefully by
ear.
Step 1.
Tune A4 to 440 Hz. Use F2 as test note, 17th to fork should beat same as 17th to A4.
Tune A3 from A4 as a 2-4 octave, 1\2 beat wide. That is, the tenth should beat 1\2 beat per
second (bps) faster than the third.
Step 2.
Tune A2 from A3 as a 3-6 octave 1\2 beat wide. That is, the major sixth should beat 1\2
bps faster than the minor third.
Step 3.
Check the double octave, a 1-4 interval to be less than 1 beat wide. That is, the 17th beat
should be less than 1 bps faster than the third. If the double octave is too wide, compromise both
octaves slightly to get and acceptable double octave.
Step 4.
Divide the A2-A3 octave into three equal parts by tuning C#3 and F3. These thirds can be
tested very accurately with the contiguous thirds test. This test states that two contiguous thirds
must have relative beat rates in the ratio of 4 to 5, that is 4 beats of the lower one require the same
amount of time to complete as 5 beats of the upper one. This test then does not require knowledge
of beats per second, only a good sense of rhythm or tempo. In this case, C#3 and F3 are correctly
