By J. Coleman, Sr.
A piano string has a series of partials (sometimes erroneously called harmonics) which are approximately whole number
multiples of the fundamental frequency (first partial). For example the 3rd A on a piano (counting from A0, A1, A2) has a
theoretical frequency of 110 cycles per second (or Hertz). If it is multiplied by 2, you have 220 Hz (2nd partial). If one
places his finger lightly on the middle of a string, he can force it to vibrate at its 2nd partial. If A2 is lightly touched at a
distance of 1/3 the length from one end after the note is played, the string will be forced to vibrate at its 3rd partial
(approximately 330 Hz which is 3 times the fundamental pitch). One can continue to divide the string by 1/4, 1/5, 1/6, 1/7,
1/8, etc. This will cause the string to sound at its 4th, 5th, 6th, 7th, and 8th partials respectively.
In order to further clarify, let me say that when a string is forced as above to vibrate in three parts by touching it at the 1/3
point, we say that this is the 3rd partial because one can see the string breakup into 3 parts with 2 nodal points in
between. At the same time one notices that the pitch jumps one octave plus a 5th (19 half-steps above).
One should learn the note location for the partial series for each note of the chromatic scale. Here are the notes that
correspond to the locations of the first 12 partials of the note Middle C.
C4 C5 G5 C6 E6 G6 Bb6 C7 D7 E7 F#7 G7
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. Inharmoni city 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
