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TECHNICAL^SUPPIY DEPARTMENT
William BraidLWtiteXeehnical Editor
The Scale Drawing—The Important
Subject of Octave and Semitone Ratios
All the Factors Which Enter Into the Problem of String-Tone Production Are Interdependent,
a Matter Which Is Often Neglected—Determination and Proof of Ratio—Shortening
and Loading—Arriving at the Striking Point of the Hammers
HIS matter of octave and semitone ratios
is far more important than is usually sup-
posed, because upon its proper settlement
depend a great many other factors. One of the
points in the treatment of strings, which is
often if not usually overlooked, is that all the
factors which enter into any problem of string-
tone production are interdependent. One can-
not touch one without disturbing all the others.
If lengths are altered for any reason, then stiff-
ness, flexibility, weight and all other possible
elements of the whole must be altered accord-
ingly if the result is to be the same as it was
before. It is like an algebraic equation. If it
be true, for instance, that x + y — z = 0, then
plainly if x changes in value, either or both y
and z must change too. And if we regard our
unknown quantities as functions of each other,
then whatever changes one must change both
the others.
Now in the case of the pianoforte string we
are dealing with a large set of variables, each
of which is a function of all the others, either
one by one or all together. If one changes,
the others must change in proportion. Or, as
one may say, if h = f (x -f- y -f- z), then any
change in any one of the three values on the
right-hand side of the equation will change
the meaning of the left-hand side. Or, con-
versely, if any one of the right-hand variables
changes while the left hand remains the same,
then the other two variables must change pro-
portionally.
Hence, if we tamper with the octave ratio
of pitch (1 : 2), then we must understand that
in so doing we shall also be tampering with
the weight and the stiffness of our strings at
the same time.
As I said two weeks ago, we could, of course,
manage easily enough in stringing with a 1 :2
ratio, if it were not for the excessive length we
should develop in the bass and the equally
excessive flexibility, which would produce con-
ditions of slackness fatal to the sort of sound
desired from those strings. We must, there-
fore, modify the octave ratio for the purpose of
cutting down both excessive length and exces-
sive flexibility.
Determination and Proof of Ratio
Flexibility varies directy as length, other
things being equal. If, therefore, we made our
octave ratio 1 : 1.750, we should be taking care
of the flexibility matter very nicely, provided
that also we allowed greater weight as the
lengths increased. But this greater weight
must also be only just enough to balance the
loss in length due to the abandonment of the
full octave ratio, 1 : 2. Therefore, considering
the relation which changes in weight bear to
pitch, we compromise further and make our
octave ratio 1 : 1.875. If now, along with thus
scaling our lengths, we increase the unit weight
accordingly, neither too much nor too little, as
we go down, we shall find that we may arrive
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throughout the greater part of the instrument's
compass at something very like an equality of
the one remaining factor which of course, is a
component of all the others. This factor is
tension.
It would also be possible to prove the cor-
rectness of this octave ratio, 1 : 1.875, in another
way. We might simply calculate the mass
(length xx V cross-section area) of each string,
and then multiply this in each case by the fre-
quency of vibration (pitch number). The one
factor of course goes up as the other goes
down. That is, the string becomes heavier as
the pitch lowers. Now if the weighting were
done rightly, then the product would be almost
the same throughout from each of the multipli-
cations.
The proof by equality of tension is perhaps
as good as any, however, for it is hard to be-
lieve that any one can misunderstand the true
meaning of the phrase. There has been a fear-
ful lot of ignorant talk on the subject, and in
the article of last week I said something about
that which I need not repeat here, especially
as I am giving a full treatment of the points
involved in succeeding articles. At present I
shall take it for granted that readers will agree
with me upon the proposition that it is better,
from the viewpoint of construction and of
standing in tune, that a pianoforte should have
to bear virtually equal stresses over equal areas
of its surface.
I shall then consider this proof by tension.
Without stopping to make the calculation,
which has been explained in this department
many times, I merely point out that if one takes
a string of No. 88 unison and scales it at two-
inch length, with No. 13 wire (America Steel &
Wire Co.'s gage), it will give out the note C 7
(4,185 v.p.s) when stretched to a tension of
very nearly 154 pounds. Assuming for the mo-
ment that this figure represents something de-
sirable in the way of a level of tension we shall
find that if we continue to scale at the same
octave ratio, taking Nos. 87, 86, and so on in
turn with No. 88 fixed at two inches the tension
will gradually drop a pound or two at a time
until at No. 83 it has dropped to 146 pounds.
The average of the tensions from No. 88 to 83,
inclusive, on this basis will be found to be very
William Braid White
Associate, American Society of Mechanical
Engineers; Chairman, Wood Industries
Division, A. S. M. E.; Member, American
Physical Society; Member, National Piano
Technicians' Association.
Consulting Engineer to
the Piano Industry
Tonally and Mechanically Correct Scales
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nearly 150 pounds.
If we now change at the next unison to No.
13J^ gage of wire, which is .001 inch thicker
than No. 13, the tension will jump up again,
always supposing that the length scale remains
the same (1 : 1.875 for the octave). At the
next unison lower it will once more begin to
descend and will reach 147 pounds at No. 77.
The average, however, again will be found to be
150 pounds. Changing then to No. 14 wire, which
again is about .001 inch thicker than No. 13j^,
we find that the tension level once more comes
up; and the process is repeated.
In other words, so long as we can scale our
lengths steadily in the proportion of 1 : 1.875
for each octave distance, or (what is the same)
at 1 : 1.059 for each semitone, we shall find that
a change of one-half number of wire in the
middle of each octave will give us the equality
or virtual equality of stress over small areas,
which, I am assuming, we desire.
Those who desire to use higher tensions can,
of course, begin if they wish with No. 13J4 and
proceed downward accordingy, with propor-
tionate results.
I shall not, however, pursue this subject of
tensions and wire gages any further from this
point until we come to it again in due course.
It has been used so far simply as a proof of the
correctness of the choice of octave ratio.
Another point to be noted in passing is that
if we scaled out our strings upon the octave
ratio and with the changes .of wire gage recom-
mended above, the proof by multiplication of
mass with frequency would have given a virtual
constant.
Shortening and Loading
I take it then that we may consider our length
ratio as proved. Considering that we have a
grand piano to design of not more than five
feet two inches over all, and that we have
twenty-six unisons in the overstrung section, it
must be evident that the understrung strings
near to the bass will become too long for the
size of the piano. They, therefore, will have to
be shortened to whatever extent may be neces-
sary. We shall be guided, however, in doing
this by the facts already before us. It will be-
come a question of balancing the shortening by
increase of weight. If there were no question
of change in stiffness to be considered; if, in
other words, the efficiency of the string as a
vibrating body were not interfered with by any
increase in weight, then we might do almost
anything we desired in the way of shortening
and loading. In fact, the very small grand piano
would then be a much better musical instrument
than it commonly is. Unfortunately the ques-
tion of shortening and thickening (loading) is
bound up with the question of stiffness. The
string may become too stiff. If, and when, this
happens, the vibration of that string becomes
rod-like, which means that it vibrates almost
entirely in segments and hardly at all in its
full length. The resuting tone is harsh, short,
and disagreeable, and, moreover, such a string
can hardly at all be tuned correctly. We must
then take great care in dealing with this matter,
as we shall see in due course.
Meanwhile it is only necessary to say that,
since we have adopted the octave ratio 1 : 1.875,
this means that if C 7 is two inches in length,
C 6 will be 2" X 1.875 = 3.75" in length. It also
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