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12
THE
MUSIC TRADE
REVIEW
OuTTECHNIGAL DEPARTMENT
THE PERFECT SCALE.
(Continued.)
The first step towards the determination of any
problem cannot be taken until we have satisfac-
torily isolated, each and all, the various factor;
which go to make it up. So-, in the case of piano
scale drafting, we are totally unable to know, even,
whether an approximately perfect scale is possible
until we have discovered and accurately stated all
the factors in the case. As was remarked last
week, exactly the failure to do this is responsible
for nearly all the bad practice and worse theory
which we continually encounter.
In the present case, then, what are the necessary
factors? First of all, undoubtedly, comes that of
the strings, and of their behavior under all possible
conditions. In fact, the whole question of scale
design becomes very largely indeed a matter of
knowing exactly what strings will do in given con-
ditions. Once this is known, the rest becomes
comparatively simple. What, then, are the con-
trolling conditions in which the strings of a piano
are compelled to perform their functions?
They are, to put the matter quite simply, the
conditions which arise from the lengths of strings,
from their weights, the net tensions imposed on
them, and the pitch at which they vibrate respect-
ively. If we are to design a scale, then, it is plain
that the first question is to determine all about
these conditions of length, weight, pitch, tension,
etc. And, when we have entirely done this, and
entirely settled how these factors shall interact
upon each other, then we shall have gone a long
way towards settling all our problems. For, in
the end, all depends upon the strings.
Now, the most important matter to be considered
at first is that of pitch. For everything finally
comes down to the number of vibrations which
each string shall perform in a given time; in
short, to the precise pitch at which it shall sound.
When we consider the piano, we must first remem-
ber that it is an instrument tuned in equal tem-
perament. This means that the proportions of
pitch relation between the various members of the
scale are not only fixed, but uniform. In other
words, each octave of tones tuned in equal tem-
perament consists of 12 equidistant semitones.
Hence, naturally, the octave proportion of pitch
being 1:2, it follows that the semitone proportion
is 1:12th root of 2 (1:1.059). This being the case,
should not the string proportions also be arranged
in exactly the same way, after the same propor-
tions, but in a reverse direction?
If piano strings were without weight, perfectly
uniform and perfectly elastic, this would be all
right. But they are nothing of the kind. On the con-
trary, they possess the attribute of weight, which
increases as length increases, while lacking the other
attributes above mentioned. Hence, the same
proportions cannot be applied to the string lengths,
for the very simple reason that tones need be con-
sidered only in the light of their one property
(vibratory speed), while strings must be consid-
ered as differing in lengths, and differing propor-
tionately in weight and tension with every change
in length.
At the same time, however—and this is a very
important point—the various factors which enter
into the design of an octave of piano strings must
be just as rigidly calculated as if only the pitch
were being considered. Because you cannot make
string lengths for an octave in the proportion of
1:2, you must not suppose that there is no definite
proportion. On the contrary, there is such a pro-
portion, and it is just as fixed and rigid as is that
of the pitch proportion in equal temperament. It
corresponds to the pitch proportion of equal tem-
perament. In fact, if it did not correspond thereto,
it would be useless.
This is a point little recognized by "practical"
designers. They forget that the piano is an equal
temperament instrument and that you cannot pos-
sibly get it to stand in tune or give its true tone
quality unless its strings are proportioned in a man-
ner corresponding with the pitch proportions in
which it is tuned.
Now, the great consideration is this: That it
does not so much matter what octave proportion
you choose in string lengths, so long as the semi-
tone proportion be based upon the octave propor-
tion in exactly the same way as would be the case
if we were dealing with pitch in equal tempera-
ment. That is the big secret.
Since to increase the length of a string is to
double its weight and also halve its tension, and
since the pitch of a string varies inversely as its
weight and directly as the square root of its ten-
sion, it follows that a deduction of one-eighth
from the doubled length of the string will, at the
pitch distance of an octave, compensate for the
weight and tension factors. But since this shorten-
ing of the doubled length decreases the weight, and
by the decrease in length adds to the tension, we
must make one more compensation. This we do
by adding again one-sixteenth of the doubled
length. So our octave proportion comes out at
1:1.875; i. e., 1:[2-(1-16 of 2)]. So that, just,as
the pitch proportion in equal temperament is 1:2,
so is the string length proportion 1:1%, which is
the same thing as 1:1.875.
This being the case, it is plain that the semitone
proportion must, following the same plan, be 1:12th
root of 1.875. When this is worked out it comes
to 1:1.0538. And this factor constitutes the key
to all good scale design.
Now, if we choose a good length for the highest
string in our scale (C7) and proceed to build all
other string lengths descending upon this first one,
by multiplying at each semitone by 1.0538, we
shall have a perfectly proportioned scale from the
top of the piano down to the region where the
overstringing begins. The place of this latter is
determined by the space limitations of the piano
itself. The matter of bass string proportioning
will be considered later.
The method which I here suggest has a further
advantage. It is possible to know what the tension
on a given string will be when certain other things
are known. These things are (1) the pitch in
vibrations per second, (2) the speaking length in
inches, (3) the weight in grains. The pitch of any
tone in equal temperament should be known. Every
scale designer ought to make his plan with the un-
derstanding that the piano is to be tuned to a cer-
tain pitch. Naturally, the International pitch should
be used, since that is the only real standard we
have. Given the pitch of any tone in equal tem-
perament (as A 435), and we can get the pitch
of the whole chromatic scale therefrom by using
the semitone factor of equal temperament pitch,
as described above, namely, 1.059. Multiply by this
ascending and divide by it descending. Thus the
pitch of all tones in the scale at equal temperament
may easily be obtained. I give here the pitch scale
of one octave, from middle C (C3) up to C4. In-
asmuch as the pitches double in every octave
ascending and halve in every octave descending, it
will be an easy matter for readers to expand this
table into one covering the whole scale. The
pitches of the semitones in the octave mentioned
are (ascending) :
C 3
258.65
C sharp
274.03
D
.
290.3
D sharp
307.5
E
325.8
F
345.2
F sharp
365.7
G
387.5
G sharp
410.5
A
435.
A sharp
.
460.8
B
488.2
C 4
517.3
By doubling each of these at each octave ascend-
ing and halving at each octave descending, the
whole scale can readily be worked out.
Now, when this has been done, the designer
should obtain a table showing the weight per 100
feet of each kind of steel wire used for piano
strings. By means of this, combined with the
lengths of his proposed strings, which he will
have already worked out according to the formula
above described, and the pitches of the various
tones to which the strings correspond, he will be
able to find out in advance just what will be the
tension on any proposed string with any given
number of wire. He may do this by squaring the
pitch, as expressed in vibrations per second, mul-
tiplying this by the weight in grains and again by
the length in inches and dividing by 675,000. This
will give the tension in pounds. In this manner, by
carefully choosing the wire, the tension throughout
the whole of the treble or plain wire sections may
be arranged to suit the requirements of the scale.
This eliminates guesswork entirely.
To give one instance: It is found by taking the
weight of 100 feet of No. 13 wire that its weight
per inch is 1.52 grains. The length of string C 7
(top C) is, let us say, 2 inches. Its pitch on the
International basis will be 4,138 vibrations per sec-
ond. Now the tension on a string of that makeup
will be
4,138 x 4,138 x (1.52x2) x2
675,000
that is to say, 154 pounds and a fraction. This is
just about right. Taking No. 14 wire, which weighs
1.7 grains per inch, we have
4,138 x 4,138 x ( 1 . 7 x 2 ) x 2
675,000
that is to say, 172.5 pounds, which is too much.
So we see that No. 13 wire is the wire to use in
this particular place.
(To be continued.)
Communications for this department should be
addressed to the Editor, Technical Department,
The Music Trade Review. W.B.W.
Hammer Head
and
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