12
Oct 5th 4th 3rd 3rd 3rd 2nd 2nd 2nd 2nd half
When one is listening to various tempered intervals, there is at least one particular area where one
can hear the beat phenomenon between coincident partials of the two notes. For example: when
listening to the 5th (F3-C4), these 2 tones have partials which occur in close proximity to the note
C5 (the 3rd partial of F and the 2nd partial of C). These are called the first of lowest coincident
partials. If there is a slight difference in the pitch of the two coincident partials, one can hear a
slight waver in the tone. This is called the beat phenomenon at the pitch of C5. It is the difference
of frequency or Hz of these two partials. One can calculate the beat speed if one knows the cents
reading of each of these two partials. The following formula is helpful to convert cents difference
to beats per second!
Beats=ref. note Hz x 2 raised to (upper cents deviation/1200) minus ref. HZ x 2 raised to (lower
cents dev./1200).
The reference note frequency can be found in charts, but it is so easy to calculate using the 1/12
root of 2, which is the half step ratio. If we need to know the Hz of C5 we merely multiply A440
times 1.0594631 three times to get 523.2251. In the process we find A# at 466.164, B 493.883.
If we wish to find Hz below A440 we divide by 1.0594631. G# equals 415.305, G=391.995 etc.
Now let's use the formula above to find the beat rate of the interval F3-A3 (Major 3rd). The 5th
partial of F3 is at the note location of A5. The 4th partial of A3 is also near A5. When properly
tuned, on most pianos they will create a beat rate of approximately 7 beats per second. Let's say
that the 4th partial of A3 reads +3.7 cents and the 5th partial of F3 reads -10 cents at the
reference note of A5 (880 Hz). At A5, -10 cents is the same as G# +90 cents since we have 100
cents per half step. Now, taking the higher reading first, we have
Ref Hz 880 x (2 raised to (3.7/1200)) = 881.883
Ref Hz 830.61 x (2 to (90/1200) power) = 874.932
This leaves us with a difference of 6.95 Hz, which is the beat frequency of the lowest coincident
partials of these two notes.
With the aid of a scientific calculator one can easily compute the beat rate of any interval. Dr.
Sanderson has an excellent set of notes on how to tune a beautiful 2 Octave Temperament by
carefully measured intervals using either aural principles or machine techniques. You may find it
interesting to measure your resulting interval widths after setting a careful machine tuning or vice
versa, you may find it more interesting to measure interval widths after very carefully tuning by ear.
(See Appendix E)
APPENDIX E
The Two-Octave "A" Temperament
By Dr. A.E. Sanderson
The two-octave A temperament is probably the first temperament designed to take into account the
inharmonicity of pianos strings. Inharmonicity not only changes the beat rates from their theoretical
values for all intervals on a piano, it also creates impossible tuning conflicts as well. The simple
octave splits up into different kinds of octaves, depending upon which pair of coincident partials are
tuned to zero beat. Even the single, double, and triple octaves are incompatible intervals on a piano,
and can only be tuned to sound "as good as possible," not perfect, because inharmonicity makes
perfection literally unattainable.
The two-octave A temperament is tuned from the "outside in." That is, the wide intervals, two
octaves and the double octave, are tuned first. This is done so that octave tuning problems with a
piano will show up at the earliest possible stage when they are relatively easy to correct with small
compromises. Many pianos have, unfortunately, incompatible tuning requirements, and by tuning