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 test and the third test indicate the C4 is flat, this is very strong
evidence that C4 should be moved. But to nail down your decision, you can add a contiguous fifth test as well.
To check contiguous fifths, play down a fifth and up a fifth, F3-C4 and C4-G4. If C4 is flat, the lower fifth will beat faster
than the upper.
In our example, we have now used three tests, and six other notes to check up on one note. If all the tests indicate that
C4 is flat, then it is a good idea to move C4. If some tests say flat and some say sharp, then leave C4 where it is and go
on to test other notes. Eventually you will find the main culprit or culprits, the notes for which all three tests say the same
thing. Move these notes without hesitation. Your temperament will improve steadily as you find and correct each note
that fails all three contiguous tests.
The range of this contiguous-interval test is at least from C3 to C5, a two-octave span. After tuning the whole piano,
unisons and all, start applying this test at C3. Move up one semitone at a time, and correct any note that fails all three
tests before moving on. Go up to C5 this way. If you like, you may make a second pass from C3 to C5 and polish your
tuning even more. Eventually you will reach the point where no notes can be improved upon, and at that point you will
have an extremely fine tuning. A supertuning if you will!
APPENDIX G
Octave Tuning
By R. Baldassin
Tuning octaves with the Accu-Tuner can be directly related to tuning octaves aurally with specific interval tests. These
interval tests and electronic setting instructions have been included here for three primary reasons: 1) So that aural tuners
will know which SAT settings correspond to the interval tests they have been using. 2) So that SAT users may expand
their aural tuning abilities by checking aurally with interval tests, and 3) To raise in general the level of knowledge relating
to octave tests, electronic setting instructions, and their use in piano tuning.
Knowing that there are several types of octaves, aural tests and electronic setting instructions are necessary to insure that
the appropriate type is being tuned in a given area of the piano. Since only one type is in tune at a time, and so rare is the
exception, the tests and setting instructions for one type only should be used at a time. The exception, of course, would
be in a transitional area, changing from one type to another.
Two different aural tests along with the electronic setting instructions have been provided so that the findings may be
double checked. Since the aural tests employ the use of intervals for comparison, both expanded and contracted, and
either the upper note or lower may be the reference to which we are tuning, four classifications of aural tests result. Be
sure to note the Classification for each test so as to correctly interpret the findings and make the proper adjustments in
tuning the octave.
CLASS A:
Lower note is the reference note. If the beat rate between the test note and the upper note is
too slow
as
compared to the beat rate of the test note and the reference note,
raise the upper note
. If the beat rate with the upper
note is
too fast, lower the upper note
.
CLASS B:
Upper note is the reference note. If the beat rate between the test note and the lower note is
too slow
as
compared to the beat rate of the test note and the reference note,
raise the lower note
. If the beat rate with the lower note
is
too fast, lower the lower note
.
