PIANOFORTE STRINGS AND THEIR PROPER DIMENSIONS.
The strings of a modern pianoforte are made of cast steel and possess
a relatively great thickness and stiffness. That is to say, they enjoy
these characteristics to a far greater degree than do the strings of
any other musical instruments that employ such agents for the purpose
of generating musical sounds. The strings of any member of the viol
family, for example, are so totally unlike those of the pianoforte that
no comparison of their respective behavior when subjected to tension
can be of interest to any save the scientist. In dealing with the
strings of the pianoforte then, we face an isolated and unusual problem
which we shall have to consider at some length. We shall investigate
the peculiar effects produced by the high tension, great thickness
and great stiffness of the strings, as well as the singular phenomena
exhibited in the case of the covered bass strings. We shall note that
the strings are responsible for many unpleasant things of which they
are seldom accused, and that their proportions as to length and tension
do not comprehend in themselves the whole problem that the scaling of
them presents to the designer. This matter of the internal nature of
the steel and other wire has not, unhappily, received that attention to
which its importance justly entitles it. No treatment of the principles
of pianoforte design could be considered complete, however, without
some discussion of the phenomena thus presented. The investigation
which we shall undertake will lead us to the development of more of
those general principles that we are now engaged in enunciating, and we
shall then be able to formulate certain rules of wide application which
may be employed in the practical consideration of the problems with
which the whole matter of pianoforte design abounds.
As is generally known, the strings that are charged with the duty of
emitting the sounds comprehended within the two lowest octaves on the
pianoforte are customarily constructed of a combination of steel wire
and some other, usually copper or iron. The latter is wound over a core
of the former wire, and this winding is graduated, as to the amount and
thickness of the material employed, according to the pitch to which it
is desired that each string shall be tuned. There is an obvious reason
for this procedure. For, as we have already shown, two strings whose
lengths are as 2:1 will, other things being equal, emit musical sounds
separated by the interval of an octave. Consequently, under perfect
mechanical conditions, the length of each string of a pianoforte should
conform to the rule thus indicated, and should be one-half or double
the length of that which produces the octave above or below it; the
absolute application of this rule, however, being subject to certain
practical modifications throughout the entire compass. These will be
discussed later.
Even in the absence of such considerations, however, this ideal
condition could not be attained. The mechanical difficulties presented
would always operate to forbid the carrying out of such an arrangement
throughout the whole compass of the instrument. For, to follow the
rule with entire consistency would necessitate a length of 256 inches
for the lowest C, on an assumed length of 2 inches for the highest
note of the same denomination. As this would imply a length or height
of the instrument of nearly 24 feet it is not difficult to see that
such construction is impossible. Furthermore, evenness of tone quality
would be seriously hindered if the lowest strings were of any such
dimensions. To secure equality of tonal result it is necessary, as
has been noted above, that we should be able to equalize, as far as
possible, the particular forms of vibration that pertain to each
string. Obviously, the nature of the blow that would produce a given
form of vibration in a string of 256 inches in length must be very
different from that which would produce similar forms in a string only
one-tenth as long. Again, to maintain such long strings at the required
tension involves mechanical problems that savor more of engineering
than of pianoforte building.
For these and cognate reasons, therefore, the practice has arisen
of artificially slowing the rate of vibration in the bass strings by
wrapping them with brass, iron or copper wire. Naturally, the form of
the vibrations excited in these wrapped strings is entirely different
from any that the plain steel wire is capable of producing. The iron
or copper wire is itself thrown into vibration both independently of
and together with the cord of steel, so that we have the phenomenon of
one string emitting two separate series of vibrations, with resultant
disarrangement of the generated upper partials and concomitant
production of beats in a more or less appreciable quantity. Now
if, in addition, the bass strings are not scaled with approximate
correctness as to their relative lengths, thicknesses, and other
dimensions, it follows that there will be two distinct and different
causes of dissonance and unevenness of tone-quality, either of which is
sufficient, in itself, to produce very unpleasant tonal results. It is
clear, then, that particular attention must be paid to the designing of
the string arrangement, if excellence of tone-quality is to be anywhere
approached.
It is, fortunately, possible to give quite precise directions for the
calculating of string dimensions. As a preliminary, we must remind the
reader of the rules that were laid down in Chapter IV, relating to the
behavior of stretched strings. It will be recalled that we had occasion
to observe that these rules would require certain modifications in
practice, as they referred only to ideal musical strings which are of
perfect flexibility and perfect uniformity, and are stretched at an
absolutely constant tension.
The first modification that appears upon investigation has reference
to the division of string-lengths. It has already been pointed out
that, in practice, we cannot obtain the octave above the fundamental
tone of a given pianoforte string by dividing it exactly in the middle.
Conversely, an exact doubling of the length does not produce the exact
octave below the given fundamental tone. This discrepancy occurs on
account of the fact that the shortening or lengthening of a given
string causes a corresponding change in the tension at which it is
maintained and in the density of adhesion of its molecules.
Now if we double the length of a string in order to obtain the octave
below its fundamental tone, we decrease its tension, and this causes
a slowing of the frequency of vibration. Then again, the increased
resiliency of the string brought about by the lengthening tends also to
decrease the frequency. The frequencies of vibration of a string vary
directly as the square root of the tension, inversely as the thickness,
and directly also as the stiffness. These axioms being admitted, we
observe that to obtain an octave lower than a given fundamental tone,
we must obtain one-half the frequency that produces the fundamental.
Therefore, as we see from above, the double length must be decreased
by one-fourth to allow for the automatic decrease of stiffness which
varies directly as the frequency. And this modification must itself be
modified to compensate for the increase in frequency produced by the
very act of shortening. Therefore we must consider the tension, and
we find that to reduce this tends again to decrease the stiffness in
exactly the same proportions as it was before increased. But frequency
of vibration varies as the square root of the tension; therefore we
take the square root of one-fourth, which was the fraction first
arrived at. This root is one-sixteenth and is the differential factor
that must be subtracted from the ideal octave lengths, in order to
obtain the practical lengths.
It will be found of course, as must be apparent to the reader, that
the differential factor here suggested does not provide a complete
solution to the problem of allowing for the exhibited differences
between theory and practice. It does, however, provide a true guide to
the lengths. There is of course a difference of produced frequency to
be allowed for yet. Fortunately, however, this is provided for by the
graduated thicknesses of pianoforte wire. By taking advantage of this
almost geometrically proportioned graduation of diameter we are able to
calculate a stringing scale that, if adhered to, will give the nearest
possible approximation to complete harmony between theory and practice.
That is to say, we can proceed with a string-length calculation based
upon the differential factor already obtained, and then by arranging
the distribution of the string thicknesses according to the diameters
that are provided by the manufacturers of music wire, we may obtain a
true estimation, not only as to the thickness of wire to be used at
each place, but also as to the lengths proper to each string. Of course
the reader will remember that the matter of pitch is of considerable
importance in all calculations of this kind. A difference in pitch
implies difference of tension when the other factors remain equal, and
we therefore have calculated the following tables on the assumption
that the pitch to be used is that known as the International or C 517.
Attention is, therefore, directed to the following
TABLE SHOWING TRUE LENGTHS OF OCTAVE STRINGS FROM THE HIGHEST C STRING
TO THE LAST C STRING THAT IS USUALLY LEFT UNWRAPPED.
C⁵ = 2.048 in. = 2-1/25 + ... Approx.
C⁴ = 2.048 × 1.9375 = 3.968 in. = 3-24/25 + ... "
C³ = 3.968 × 1.9375 = 7.688 in. = 7-17/25 + ... "
C² = 7.688 × 1.9375 = 14.875 in. = 14-7/8 + ... "
C¹ = 14.875 × 1.9375 = 28.820 in. = 28-4/5 + ... "
C = 28.820 × 1.9375 = 55.828 in. = 55-4/5 + ... "
[NOTE.--The length of the first string is chosen arbitrarily,
but as given is a very close approximation to the practice of the best
American makers. The vulgar fractions are calculated from the decimals
and the error in no case exceed about one-fiftieth of an inch. The
differential factor is, as we know, 1/16. Therefore we multiply by (2
[-] 1/16) or 1-15/16; in decimals 1.9375.]
The above table, then, affords us a reliable guide to the scaling of
the unwrapped strings. At the same time, however, it is not by any
means complete, for the reason that there is no method shown as yet for
the calculation of the other and intermediate string-lengths. We are,
however, able to accomplish this task by the aid of a very ingenious
rule proposed by the late Professor Pole, F.R.S. It is as follows:
The proper length of any string may be determined from that of any
other string, provided that the length and frequency of the second
string be known. Given these factors: Then,
(1) Take the logarithm of the length of the known string.
(2) Multiply the number .025086 by the number of semitones that the
sound to be given by the required string length is above or below the
sound produced by the given string.
(3) If the required string is below the given string, add together the
two numbers obtained; if it be above, subtract the second number from
the first; the result in both cases is the logarithm of the required
length.
For example, we have calculated already the proper length of C. In
hundredths of an inch this length is expressed as 2882. The log. of
this number is 45943. (This may be verified by any table of logarithms.)
It is required to obtain the length of the string that,
_caeteris paribus_, will produce one semitone above C.
45943 = log. of 2882
02508 (6) = .02508 (6) × the number (1) of
semitones that required string
sounds above given string
-------------
By subtraction, 43435 = log. of 2718 = length of required
string in hundredths of an inch
∴ Required length for C-sharp = 27-18/100 inches
By reversing the process described above, and adding instead of
subtracting, the proper lengths for the semitone below and all others
in descending progression may be calculated with accuracy.
Having thus settled the matter of string lengths, we may proceed to
consider the questions of diameter. But it is first of all necessary to
warn the reader that the lengths that have here been calculated refer
only to such pianofortes as are capable, by reason of their size, of
taking the ideal string-lengths. Very small uprights, for example,
cannot be brought within that classification, except as regards the
highest of their strings. In all pianofortes, no matter what their
size, the higher strings are practically identical in length; but it
will be found that shortness of height in an upright or of length
in a grand begins, towards the middle of the scale, disastrously to
affect the string proportions. As already pointed out, there are
only two ways in which these disproportions can be overcome. These
are through alterations in the tension or in the thickness. But such
alterations necessarily disturb the whole tonal balance; and here we
find a very strong reason for the poor tone that the average atrophied
grand or upright possesses. Moreover, it must not be forgotten that
disproportionate thickness or unduly slackened tension affect the
actual nature of the vibrations that are set up within the string.
And the affections are operative both as to frequency and to form.
Therefore, naturally, bad tone and inability to stand in tune. This is
not intended as an argument against the small pianoforte; but it is
desired here to show that these little instruments, whether horizontal
or vertical, must not be expected to perform impossibilities. If we are
obliged to build small instruments, we must revise our calculations
and tabulate the string-lengths according to a different basis of
apportionment. For the purpose of the present work, however, the
calculations have been made on the assumption that the standard size of
pianoforte is to be designed.
Turn we then to the consideration of string diameters. The cast
steel wire that is used for the pianoforte strings is supplied in
definitely numbered and graded thicknesses. The numbers that are used
generally run from No. 13 to No. 24. According to the tests made at
the Chicago World’s Fair by the aid of Riehle Bros.’ testing machine,
the wire of these numbers was of the following diameters and broke at
the following strains. The wire manufactured by the firm of Moritz
Poehlmann, Nuremberg, Germany, has been selected from among the various
products that were subjected to these tests, on account of its superior
durability and evenness of gradation.
Number Diameter in fractions of an inch Broke at strain of
13 .030 325 lbs.
14 .031 335 "
15 .032 350 "
16 .035 400 "
17 .037 415 "
18 .040
19 .042
20 .044
Now it is a well known fact, and, indeed, obvious from what has
already been said, that the proportional relations as to length,
tension, diameter and breaking strain do not permit any other
arrangement for the scaling of wire than that which is universally
accepted by piano makers. That is to say, the shortest wires are taken
from the thinnest numbers, and vice-versa, the whole scaling being so
arranged as to secure for each tone that its strings shall be stretched
at approximately the same tension. Experience and the observations
of the most eminent manufacturers seem to have established that the
strain upon each of the uncovered strings should be maintained, as
nearly as possible, at 160 lbs. If this be done it will be found that a
pianoforte so constructed will produce the proper pitch at each string
when the lengths are as calculated in the tables referred to. It will,
of course, be necessary to arrange with due proportion the number of
strings that are to be taken from the wire of each number. It will be
found that the best practice takes into account the half sizes not
shown here and strings the instrument with an average of five tones
to each thickness of wire, beginning at 13 or 13-1/2 and continuing
down to the end of the unwrapped strings according to the general
directions suggested. Experience and the individual ideas of the
designer, assisted by such knowledge as this work aims to impart, are
the best guides that can be followed. Empirical induction, based upon
observation and experience, provides the only possible and practical
means for arriving at the true and proper arrangements to be made for
each individual instrument. This empiricism extends with particular
force to all string arrangements and is seen nowhere so conspicuously
as in the variety of methods that are adopted by manufacturers in
determining the number of strings within the unwrapped sections of the
scale. Thus, certain makers carry the wrapping over to the beginning
of the treble strings and have two or three string-groups provided
with wrapped wire before the overstringing is begun. The idea here
is either to correct original defects of scale design or to shade
down the break in tone that so often occurs at the point where the
overstringing usually begins. From observation of the practice of the
best makers, it may be said that the tone C below middle C is usually
the first overstrung tone. Of course, when the instrument is very small
it will often be found that it is impossible to give the last unwrapped
strings their proper lengths. In this case these offending strings may
either be covered with light wrapping or may be put bodily over into
the overstrung portion of the scale, in which latter case they will be
wrapped anyway.
Supposing then that the matter of the number of overstrung strings has
been determined, we may proceed to the consideration of the dimensions,
number and covering of the strings that are to serve here. We are
obliged to confess that the problem of attaining to good tone in the
bass is, indeed, difficult. It is by no means hopeless, however, as the
success of more than one eminent maker has already demonstrated.
The simplest, most obvious, and easiest way out of the inherent
difficulties of the scaling of bass strings is to be found in the
consideration of their proper lengths. It does not require very much
thought to perceive the truth that the longer the strings the less
weight need be imposed upon them. If, in fact, we make the bass strings
to approach as far as may be to the lengths that they would require
to have if unwrapped, we shall be able to reduce proportionally
the amount of artificial control that has to be exercised over the
vibration speed. Not only this, but the greater length thus attained
implies greater tension. That is to say that, as we saw before, the
tension at which a string is stretched acts to overcome the slowness
of vibration-speed induced by its greater length, and, consequently,
tends to generate a more regular progression of the upper partials (as
experiment has demonstrated), with resultant tendency to greater purity
of tone-quality.
We may, in fact, accept it as an axiom that the bass strings should be
as long and, simultaneously, as lightly weighted as possible, and that
the weight of them should be strictly proportioned to the pitch of the
musical tone that they are desired, at a given tension, to emit. As far
as the second clause of these conditions is concerned it is well to
remind the reader that limitations of space within the body of a piano
usually determine the possible lengths of the bass strings. So much so
is this the case, indeed, that it is not often possible to make any
great difference in their respective lengths. The best makers appear to
be agreed in a method of treating the problem that is at once simple
and effective. They recognize the great advantage of scaling the bass
strings at the greatest possible length, and then they take care that
the descending increase of length is no greater than to make the lowest
bass string one-fifth longer than the highest. At the same time they so
graduate the weight of the wrapping material that the same results are
attained as would naturally follow if they were as accurately scaled,
in proportionate length, as are the plain wire strings.
This equalization is, of course, only approximate. For the forms of
vibration excited in two strings of the same pitch will be different
whenever the various factors that govern the emission of sound by
them are variable. Thus when the factor of length is varied, no
counter-adjustment of tension, thickness or density can restore to
the string so modified the exact form of vibration that it may have
originally possessed. Consequently it becomes impossible to induce from
artificially weighted strings precisely the same series of partial
tones that a plain wire filament will emit, even when the tones
generated by the two strings are of the same pitch.
The lesson of this is plain. As perfection of tonal quality can only
be attained in part, it especially behooves us to pay strict attention
to such scaling of the bass strings as will furnish a complement of
sound producing agencies that may be relied upon to induce as nearly
as possible the same successions of partials as are habitually emitted
throughout the higher sections of the piano. Thus it becomes evident
that the greatest practicable length and the least practicable weight
are the chief factors that must govern the designer in laying out the
scale for the bass strings.
The relative densities of the wrapping material employed in the
manufacture of bass strings have been the subject of considerable
study. Brass, which was the earliest object of experiment, has long
been superseded by either copper or iron. As to the relative advantages
possessed by these two materials, it can be said at once that the chief
and almost the only advantage presented by the latter lies in its
relative cheapness. Acoustically, however, copper forms by all means
the most suitable material for the winding of bass strings, and this
for the following reasons: The specific gravity of copper is 8.78,
while that of iron is but 7.78. Again, the former metal, while inferior
in tenacity to the latter, possesses, on the other hand, the great
advantage of higher ductility, so that its elastic qualities are very
marked. It is thus evident that copper is a more suitable material for
the generation of musical sound than is iron, and the qualities which
we have just noted as pertaining to it are precisely those most useful
in the production of harmonic progressions of partial tones. It is
therefore clear that as between copper and iron all the advantages lie
with the former.
The thickest wire used for the uncovered strings is generally No.
24. In beginning the scaling of the bass strings, however, we choose
No. 17 or No. 18 for the notes nearest to the treble. The covering
is usually from No. 25 to No. 28 (standard, not music, wire gauge)
according to the size of the piano and the practicable string-length.
Of course, longer strings may be covered with lighter wire. The first
covered string is generally approximately one-sixth shorter than the
string immediately above it. This proportion, as suggested above,
may, however, be profitably disregarded, if it thereby be possible to
lengthen the bass strings. There are always two of these strings to
each tone and the thickness of covering wire must be progressively
increased as the scale descends. A descending increase of one number in
thickness of the covering wire for each pair of strings may properly be
allowed, unless the lengths are too closely alike, or vice-versa, in
which cases suitable modifications may be made. But assuming that the
descending lengths are arranged in arithmetical progression with a mean
of 3/4 of an inch, and supposing the highest covered string to be 45
inches long; then the suggested increase of thickness should under all
circumstances hold good. It may often be found, however, that the space
limitations of an instrument or other practical considerations make it
impossible to follow out these rules with exactitude. In any case, we
must remember that all such rules are themselves the fruit of empirical
observation and to such observations we must look, when it becomes
necessary to revise them in order to satisfy the requirements of some
particular situation.