11
way that they were all equally out of tune. The difficulty with this approach was that it took 12
steps before one could tell if he was doing alright.
More recent approaches to equal temperament involve one or two test intervals for each step taken
in the tempering system. More care is being given to insure that each interval (such as the minor
3rds, major 3rds, 4ths, 5ths, minor 6ths, major 6ths) is equally tempered and is compatible with all
its parallel similar intervals. Some of the more popular recent temperament systems are by George
Defebaugh, Bill Stegeman, Dr. Al Sanderson (2 octave A-A temp) James Coleman (F-A temp) and
Mark Peele (10th temp). These may be seen demonstrated at various Piano Technician's Guild
Institutes, Conferences and Seminars.
Since the Stretch Calculator tunings involve the accurate tuning of the 4th partial of each note in
the F-F temperament, all the intervals which involve the 4th partials will be beautifully tempered
(such as major 3rds, 4ths). Other intervals such as minor 3rds, major and minor 6ths, which are
involved with higher order partials, will also be beautifully tempered. Since the greatest irregularity
in partial alignments occurs in the 1st, 2nd, and 3rd partials, there may be some slight unevenness
heard in the octave and 5ths.
Now, by the simple practice of playing certain test intervals while tuning with the Stretch Calculator
mode, one can have a double check on his aural test as well as assuring that the visual judgments
are more accurate. If one tunes the stretch tuning system from top to bottom or at least from C5
down, when the 4th 1/2 step down is achieved, a major 3rd aural test is available without the upper
note interfering with the LED display. In arriving at the seventh 1/2 step down, a 5th is available for
aural judgment without LED interference from the upper note of the interval. This 5th interval can
be followed on down to C3. One may notice a slight variation in the sound of the 5ths especially as
one approaches the lower area of the scale. Sometimes this is merely due to the slipping or
instability of the previously tuned upper note of the 5th. But, with the shorter scaled pianos,
irregularity of the lower partials may cause the beat frequency to be greater than expected. At this
point one may make a decision to alter the lower note to smooth out of the 5th interval (which of
course may change the beat rate in other intervals based on this lower note). By doing this
judiciously one can have a better tempering than can be had with either aural or visual methods
alone.
APPENDIX D
What are Partials and Beats?
By James W. Coleman, Sr.
A piano string has a series of partials (sometimes erroneously called harmonics) which are approxi-
mately 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