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tuned when 4 beats of A2-C#3 occur at the same tempo as 5 beats of C#3-F3, and in addition, 4
beats of C#3-F3 occur at the same tempo as 5 beats of F3-A3.
Step 5.
Now tune C#4 and F4 to divide the A3-A4 octave into three equal parts with thirds. You
may have to taper the width of the thirds downwards slightly in the upper octave on account of the
inharmonicity of the piano.
Step 6.
Check that the three major tenths formed on the seven notes tuned so far also in the ratio
of 4 to 5. Also check the C# and F octaves with both the third-tenth and minor-third-sixth tests.
Scale problems will show up at this stage, and it may be necessary to compromise slightly the
perfectly rising thirds to get satisfactory octaves and tenths.
Step 7.
Fill in the six untuned notes between F3 and C#4 to get a nine-note mini-temperament, but
be sure not to change already tuned notes. 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. Then 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. Check the G3-B3 third, which is the
test interval for this tuning. If it is too small, you must expand your fourths, and vice versa. With
just nine notes to worry about, it is always possible to get five perfectly rising thirds and four
matched fourths no matter how poorly scaled the piano may be. The beat rates may not be very
close to theoretical, but they will be right for the given piano and its inharmonicity characteristics.
So tune the piano, and let the beat rates fall where they may!
Step 8.
Tune down to A2 and up to A4, and use the contiguous third test to place each note
initially. Check each note with the fourth and fifth, and then the major sixth and octave as they
become available. The final result should be two octaves tuned with rising thirds all the way, with
all fourths quite even and acceptable, and with all fifths nearly pure.
APPENDIX F
Contiguous-Interval Tuning Tests for Electronic Piano Tuners
By Dr. A.E. Sanderson
Two contiguous musical intervals are intervals that touch each other, in other words, share the note
in the middle. Tests that use contiguous intervals are easy to learn and use, and tell the tuner
explicitly which notes are at fault and what to do to correct them.
Contiguous major thirds will beat in the ratio of four to five because the major third itself consists of
two notes whose frequencies are in the ratio of four to five. Displacing any interval up the
keyboard will speed it up theoretically in the ratio of the frequencies of the two root notes involved.
Therefore two contiguous major thirds should beat in the ratio of four to five, two contiguous minor
thirds in the ratio of five to six. Similarly, two contiguous fourths should beat in the ratio of three
to four and two contiguous fifths in the ratio of two to three. However, on the piano this
theoretical relationship holds well only for the major and minor thirds. The fourths and fifths are so
strongly affected by inharmonicity that these contiguous intervals beat at almost the same speeds.
Using the above facts, we can develop a test for one note of the piano at a time. Take C4 for
example. Play down a third and up a third G#3-C4 and C4-E4, keeping time at the rate of four
beats of the lower one, and then at five beats of the upper one. Think of it as four beats to the
measure, followed by five beats to the measure. The tempo of the two kinds of measures should
agree. If the upper beat rate is too fast, it indicates that C4 may be flat, and vice versa.
Before moving C4, we need more evidence. Play down a fourth and up a fourth, G3-C4 and C4-F4,
and listen for near equality of the beat rates, or an upper beat rate just slightly faster than the
lower. If C4 is flat the upper fourth will be faster than the lower, and vice versa. If both the fourth