The two-octave "A" temperament may be tuned as follows.
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 1 beat or less
wide. That is, the 17th beat should be 1 bps or less faster than the third.
If the double octave is
too wide, compromise both octaves slightly narrower to get an acceptable wider double octave.
Step 4. Divide the A2-A3 octave into three equal parts by tuning C3
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 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
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 ninenote
mini-temperament, but be sure not to change already tuned notes. Tune up
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.