The Journal
of the
Acoustical Society of America
Volume 24 Number 3
MAY 1952

Inharmonicity of Plain Wire Piano Strings

Robert W. Young*

*This work was done privately without any official connection with the U. S. Navy Department.
U. S. Navy Electronics Laboratory, San Diego 52, California

(Received February 18, 1952)

The inharmonicity of plain wire strings in situ has been measured in six pianos of various styles and makes. By inharmonicity is meant the departure in frequency from the harmonic modes of vibration expected of an ideal flexible string. It is shown from the theory of stiff strings that the basic inharmonicity in cents (hundredths of a semitone) is given by 3.4*10^13*n^2*d^2/(nu-sub-0^2*l^4), where n is the mode number, d is the diameter of the wire in cm, l is the vibrating length in cm, and nu-sub-0 is the fundamental frequency. A value of Q/rho = 25.5*10^10 (cm/sec)^2 was assumed for the steel wire, where Q is Young's modulus and rho is the density. The observations are entirely compatible with the relationship given. In general terms the inharmonicity of the plain steel strings is about the same in all the pianos tested. being about 1.2 cents for the second mode of vibration of the middle C string. Above this point every eight semitones it is doubled. Below middle C the inharmonicity is consistently less in large pianos than in small ones.

INTRODUCTION

The simple theory for an ideal string depends upon the assumption of a thin, flexible string vibrating transversely between rigid supports. It has been pointed out*1 *1 W. E. Kock, J. Acoust. Soc. Am. 8, 227-233 (1937). that these assumptions are not entirely valid for actual piano strings, but there has been little quantitative evidence as to the extent of the failure of the simple theory for piano strings in situ. The ideal string has modes of vibration whose frequencies form a harmonic series; any departure from the series (i.e., any inharmonicity) is therefore a measure of the failure of the simple theory. Moreover, as pointed out in an earlier paper,*2 *2 O. H. Schuck and R. W. Young, J. Acoust. Soc. Am. 15, 1-11 (1943). this inharmonicity influences the tuning of the piano and also its musical quality.

In the present paper certain theoretical considerations are organized to facilitate study of the underlying physics of the problem. Detailed information is given herein about the inharmonicity of the plain steel wire strings in six pianos. The theory is shown to provide a very satisfactory explanation of the observed inharmonicity. In some respects the problem of wound strings may be looked on as a starting point for the more complex problem of wound strings.


THEORY OF INHARMONICITY

At least a century ago a formula was derived to predict how the stiffness of a piano string can cause it to vibrate at frequencies somewhat greater than those of the ideal string. It was Donkin's opinion,*3 *3 W. F. Donkin, Acoustics (Clarendon Press, Oxford, England, 1884), second edition, p. 187. however, that "the deviation of the upper tones from the harmonic scale is probably too small to be made sensible to the ear."

According to the usual simple theory, the fundamental frequency of a thin flexible string vibrating transversely between rigid supports is
Eq. 1
nu-sub-0 = c/(2*l),

where
Eq. 2
c = (T/[rho*S])^(1/2)

is the speed of wave propagation, l = free length of string, T = tension, rho = density, and S = cross section. The frequency of the nth normal mode of such a string is simply n*nu-sub-0.

The frequency vn of any given mode of vibration of an actual string departs more or less from that of the corresponding mode of an ideal string. It is convenient to call this departure the inharmonicity*2 and to express it in cents (hundredths of an equally tempered semitone). Thus the inharmonicity delta is given by
Eq. 3
delta = 1200 log2 vn/(n*nu-sub-0) &eqapprox; 1731*[vn/(n*nu-sub-0 - 1)]

or
Eq. 4
vn/(n*nu-sub-0) = 2^(delta/1200) = e^(delta/1731) &eqapprox; 1 + delta/1731.

The fractional error in delta resulting from the approximations is less than one percent if delta is less than 35 cents.

It may be noted in passing that according to the definition used previously*2,*4 *4 Franklin Miller, Jr., J. Acoust. Soc. Am. 21, 318 (1949). the inharmonicity was expressed relative to an integral multiple of the frequency of the first mode, rather than relative to n*nu-sub-0. It now seems preferable to employ Eq. 3, in that the ideal string is taken as the standard of comparison throughout.

It is well recognized that a stiff string acts as a dispersing medium in which the speed of a transverse wave increases with frequency. Suppose for the moment that this is the only effect of the stiffness on the vibration of a piano string. Then the form of the standing wave is (by assumption) the same*5 *5 P. M. Morse, Vibration and Sound (McGraw-Hill Book Company, Inc., New York, 1948), second edition, p. 84. as for a flexible string, with nodes at x = 0 and x = l; namely,
Eq. 5
y = A*sin(pi*n*x/l)*cos(2*pi*v*t - phi).

If this function be substituted in the usual (simplified) differential equation for the vibration of a stiff string under tension,*6 *6 Reference 5, p. 166. it follows at once that
Eq. 6
vn = n*nu-sub-0*[1 + n^2*pi^2*Q*kappa^2*S/(l^2*T)]^(1/2).

Here Q is Young's modulus, and kappa is the radius of gyration about the neutral axis of the cross section. For a wire of diameter d, kappa = d/4. Table I. Short-cut method of recording observations and finding relative inharmonicity. Stroboconn readings on line marked string. All entries in cents. See Text.

Mode &rtarrow; 1 2 3 4 5 6 7 8
String: A-sub-4 1 3 8 11 2 27 2 43
Difference 0 2 7 10 1 26 1 42
Correction 0 0 -2 0 14 -2 31 0
Relative Inharmonicity 0 2 5 10 15 24 32 42
This Eq. 6 is intrinsically the one used by Schaefer*7 *7 Otto Schaefer, Ann. Physik 62, 156-164 (1920). which, upon approximation by binomial theorem, yields also the form of correction given by Donkin and said by him*8 *8 G. E. Allan, Phil. Mag. 4, 1324-1337 (1927). to agree with experiment for wires stretched over bridges. The form does not, however, agree very well with that found satisfactory by Allan*8 for a monochord.

With the help of the approximation in Eq. 4 it follows that
Eq. 7
delta = (1731/2)*(pi^2*Q*kappa^2*S*n^2)/(l^2*T) = b*n^2.

The factor b defined by this equation is hereafter referred to as the basic coefficient of inharmonicity. Its unit is cents per square of mode number.

The coefficient of inharmonicity is a characteristic of the stiff string. A comparison of the relationships usually quoted*9 *9 R. S. Shankland and J. W. Coltman, J. Acoust. Soc. Am. 10, 163 (1939). for the clamped and unclamped string indicates that this basic inharmonicity, which changes with mode number, is the same in both cases.

Ordinarily it is easier to determine the frequency than the tension, so with the approximative nu-sub-0 from Eq. 1 it follows that
Eq. 8
b = [1731/(2*64)]*[(pi^2*d^2)/(nu-sub-0^2*l^4)]*(Q/rho).

It is significant that the inharmonicity varies inversely as the fourth power of the length&emdash;a marked dependence.

If one takes for steel wire, Q/rho = 25.5*10^10 (cm/sec)^2 and expresses d and l in centimeters and nu-sub-0 in cycles per second,
Eq. 9
b = 3.4*10^13*d^2/(nu-sub-0^2*l^4).

If d and l are in inches, then the constant 3.4*10^13 should be replaced by 5.3*10^12.

EXPERIMENTAL METHOD

Frequency measurements were made with a Stroboconn (chromatic stroboscope). The readings appear directly as cents deviation from some frequency of the standard equally tempered scale. In the frequency range of present interest the error of measurement was ordinarily less than one cent.

A filter such as the General Radio Type 760A was also used. The frequencies of the approximate harmonic series to which the filter must be set can be found with reasonable ease by a musical slide rule*10 *10 L. E. Waddington, J. Acoust. Soc. Am. 19, 878-885 (1947). which contains the frequencies of the equally tempered scale. The slide can be reversed so that the "mode of vibration" numbers can be matched with a bit of Scotch tape placed properly for each string to be tested; the frequencies then appear at the end of the slide.

Table II. Statistics on pianos under consideration.
String length in cm Dia. in mm K q
Maker Style No. Date A-sub-0 A-sub-4 A-sub-6 A-sub-4 A-sub-6 cents/n^2 s/m
Steinway D 273180 G 1931 201.9 40.5 11.7 0.99 0.89 0.61 0.038
Steinway A 145879 G 1911 141.5 39.3 10.6 0.97 0.86 0.67 0.041
Steinway 40 324766 U 1948 103.1 38.0 11.5 0.97 0.84 0.76 0.036
Mason and Hamlin B 56696 G 1949 122.0 41.2 10.8 0.99 0.86 0.57 0.047
Kranich and Bach 41215 G 1903 122.5 41.4 11.0 0.99 0.91 0.58 0.042
George Steck 151288 U 1947 98.7 39.2 10.7 0.99 0.84 0.73 0.039
Starck G 0.66 0.029
Haddorff U 1941 112.0 39.2 11.0 0.99 0.77 0.029

In this paper the name of the maker, the style, and serial number of each piano are reported. There is no intent of introducing any commercial bias. The information is simply offered as a weak substitute for a complete physical description of each piano. A future investigator may thereby be able to use the present data to better advantage.

It is convenient to work with the relative inharmonicity, i.e., the amount by which any given mode of vibration is more inharmonic than, say. the first mode. In symbols, if delta-sub-3 and delta-sub-1 are the inharmonicities of the third and first modes of vibration, respectively, then the relative inharmonicity is given by
Eq. 10
delta-sub-3-1 = delta-sub-3 - delta-sub-1.

The relative inharmonicity was obtained by a short cut which requires very simple operations but which gives the same result as the method previously described.*2 It depends upon keeping track of notes of the harmonic series. The method is illustrated in Table I for observations on the A-sub-4 string of a Steinway concert grand piano. (By the particular subscript notation here employed this is the A above middle C.) The first mode of vibration happened to be 1 cent sharp in comparison with the standard A of 440 cycles per second. The second mode was measured A-sub-5 + 3 cents, the third mode E-sub-6 + 8 cents, etc. The notes A-sub-4, A-sub-5, E-sub-6, A-sub-6, C-sub-7#, etc. are the familiar ones of the harmonic series. Upon subtracting the reading for the first mode from the others, the differences obtained are shown on the next line of the table. These are the amounts, in cents, by which each mode of vibration is sharper than its equally tempered representation. Now the equally tempered interval between A-sub-4 and E-sub-6 is 1900 cents (i.e., between the first and third modes), but a true 3/1 frequency ratio corresponds to an interval of 1902 cents; a correction of -2 cents must therefore be added to the 7 cents listed in the "difference" line of Table II. Thus the relative inharmonicity is delta-sub-3-1 = 5 cents. The method becomes particularly simple if only the first four modes of vibration are involved because steps two and three can be combined easily.

A check on a random sample of the data showed that the relative inharmonicity of any one string does ordinarily vary with the square of the node number as required by Eq. 7. Any constant is subtracted out in the calculation of the relative inharmonicity, so empirically
Eq. 11
delta = constant + b*n^2.

Thus the variable part of the inharmonicities of different modes of vibration can be described by a single coefficient of inharmonicity b.

If one wishes at any time to reverse the process to find the relative inharmonicity from the coefficient, he merely multiplies the coefficient of inharmonicity by the difference between the squares of the respective mode numbers. Suppose, for example, that the coefficient for a certain A-sub-7 string is 4.6 cents/n^2. Then the second mode is more inharmonic than the first by delta-sub-2-1 = 4.6*(2^2 - 1^2) = 13.8 cents.

If one sets up the usual conditions (the "least squares" method) for finding the coefficient which will minimize the sum of the squares of deviations of observed values from those predicted by the assumed relationship of Eq. 11, it follows that the "best" value of b is
Eq. 12
b = -0.027delta-sub-2-1 + 0.012delta-sub-3-1 + 0.066delta-sub-4-1,

for observations on the first four modes of vibration.

THREE STEINWAY PIANOS

For later study in connection with tuning, inharmonicity measurements were made on the plain wire strings of three sizes of pianos by a single maker. More specifically, observations were made only on the middle string of each group of three; also the inharmonicity could not be determined for the top octave, because the upper limit of the Stroboconn is B-sub-7. (The limit can be extended by a frequency divider.)

Relative inharmonicities for the plain wire strings in a Steinway concert grand piano are plotted in Fig. 1. Note that the inharmonicity is very small below C-sub-4 (middle C) and that it increases rapidly above this point. The smooth curves are drawn simply to aid the eye. Remember that the measurements were recorded only to the nearest cent.

When the information displayed in Fig. 1 is treated in the manner indicated by Eq. 12, one obtains the values for the coefficient of inharmonicity which are plotted in Fig. 2. Now 194 observations are simplified to a single set of points (the open circles) on an almost straight line.

The change to a logarithmic scale for the ordinate has the advantage of transforming the curve to a straight line. (Incidentally, the semitone steps for the abcissa also create a logarithmic frequency scale.) There is a slight nuisance, however, resulting from the fact that the scatter appears to be sizable at the lower left of the figure. Even though each point here is derived from four observations, the estimated standard error in the coefficient is 0.1 cent/n^2, and thus the scatter is actually reasonable. Increased scatter between D-sub-6# and B-sub-6 is to be expected since each point here is derived from observations on only the first two modes of vibration, and the standard error in the coefficient is probably 0.5 cent/n^2.

Figure 2 is incomplete in two respects. There are plain wire strings for still another octave up to C-sub-8, but to determine the inharmonicity for this octave would require frequency measurements another octave beyond B-sub-7. Furthermore, the plain wire strings extend down to F-sub-2 in this particular piano, but for the sake of uniformity with later figures, the grid for Fig. 2 is started at C-sub-3. Excepting the measured coefficient for F-sub-2 which is 0.2 cent/n^2, both calculated and observed coefficients are less than 0.1 cent/n^2 for all strings not represented in the graph.

The solid line in Fig. 2 represents the basic coefficient of inharmonicity as computed from Eq. 9 from the dimensions and frequency of each string. The individually computed coefficients do not always increase regularly from string to string; when the wire diameter is changed, it turns out that the coefficients are about the same for the adjacent strings of differing diameters. Such irregularities have been smoothed out in drawing the solid line. Nevertheless, at no place does the solid line deviate from a computed coefficient by more than two percent.

The agreement between theory and experiment is very good ‐ in fact much better than could reasonably be anticipated. Published values of Young's modulus and density vary several percent. A modulus of 20*10^11 dynes/cm^2 was measured on two samples of piano wire (diameters 0.64 and 0.99 mm, respectively) which happened to be at hand. This value, in combination with the density of 7.83 g/cm^3 which is implicit in tables of linear density published by the American Steel and Wire Company, resulted in Q/rho = 25.5*10^10 (cm/sec)^2. There is no easy way of knowing whether the wire in the pianos has exactly these same characteristics.

The significance of the agreement between measured and computed values is the indication that all the inharmonicity is basic; there is practically nothing left to be attributed to other causes. The order of magnitude may be remembered by noting that the coefficient of inharmonicity is 0.1 cent/n^2 at C-sub-3 and that it doubles every 8 semitones.

Consider next inharmonicity data on a somewhat smaller grand piano, as depicted in Fig. 3. The trend is very like that apparent in Fig. 2, but the values of the coefficients are slightly greater. The solid curve again passes within two percent of the coefficients computed for each string by Eq. 9. There is a bar in the iron frame at the place marked by a break in the solid curve.

In this piano the free string lengths in cm follow fairly well the rule
Eq. 13
l = 39.4*(1.94)^(-S/12) = 39.4*2^(-0.96*S/12) = 39.4*10^(-0.024*S),

except below D-sub-3 where they are somewhat shorter. S is the number of semitones above A-sub-4. The diameters in cm from E-sub-3 up are given within two percent by
Eq. 14
delta = 0.096*(1.005)^(-S/12) = 0.096*2^(-0.08*S/12) = 0.096*10^(-0.002*S).

Of course,
Eq. 15
nu = 440*2^(S/12) = 440*10^(0.025*S).

When these relations are substituted in Eq. 9, it turns out that the basic coefficient of inharmonicity for this piano is
Eq. 16
b = 0.67*2^(1.64*S/12) = 0.67*10^(0.041*S).

This is exactly the equation for the straight (solid and dashed) line of Fig. 3.

The inharmonicity in an upright piano is shown in Fig. 4. Again the solid curves depict the coefficient of inharmonicity computed from Eq. 9. The gap left between G#-sub-5 and A-sub-5 accompanies a sudden shift in the scaling of lengths occasioned by the presence of a bar in the iron frame.

THREE PIANOS OF VARIOUS MAKES

In the previous section measurements on consecutive strings of three pianos demonstrated that the existing inharmonicity is well predicted by the equation for basic inharmonicity. The three pianos were all of the same make. In the present section pianos of three different makes are studied but on only four strings per octave.

The coefficient of inharmonicity for a Mason and Hamlin medium grand piano is shown in Fig. 5. This graph appears somewhat simpler than the previous ones because it contains fewer points. Also the calculated (solid) curve was based only on dimensions and frequencies of the C-sub-3, A-sub-3, A-sub-4, A-sub-5, and A-sub-6 strings.

For the Kranich and Bach grand piano represented by Fig. 6 the coefficient of inharmonicity was calculated for at least four strings in each octave. The gap left in the calculated curve between D-sub-5 and D#-sub-5 corresponds to the position of a bar in the iron frame. The general trend is fairly well represented by the straight line, part of which is dashed.

The calculated curve in Fig. 7 for the George Steck upright piano is based only on dimensions and frequencies of the G-sub3, A-sub-3, A-sub-4, A-sub-5, and A-sub-6 strings. In accordance with the practice followed earlier, the smooth curve is drawn within two percent of the calculated values. Here the coefficients of inharmonicity were derived from measurements of only the first, second, and fourth modes of vibration, but the fit between observed and computed values is still very good.

COLLIGATION

It is a time-consuming job to collect data on the inharmonicity of piano strings in situ. It seems worthwhile, therefore, to tie together all information available to see what general principle may be formulated thereby.

Table II gives the names of the pianos discussed above, their style designations, serial numbers, and other collateral information. G or U indicates whether grand or upright. The year in which the piano was made is included if perchance any evidence should be forthcoming as to any effect ascribable to a change in design over the years. The length of the longest string (A-sub-0) is offered as a rough index of the "size" of the piano.

Superficial comparison of the preceding graphs reveals that (aside from minor interruptions) the coefficient of inharmonicity in all pianos increases regularly as strings become shorter and that the inharmonicity is roughly the same in all pianos from middle C upward. At this point the coefficient of inharmonicity is of the order of 0.3 cent/n^2. This is equivalent to an inharmonicity of 1.2 cents for the second mode of vibration of the middle C string. There is a region where, at least for a couple of octaves, the graph (with the logarithmic scales employed) is approximately a straight line. This means that the coefficient of inharmonicity can be represented by
Eq. 17
b = K*10^(q*S),

where K is the coefficient for the A-sub-4 string and S is the number of semitones above this point. Values of K and q are listed in Table II. Reference to the earlier figures will remind one that this simplified description is of varying validity in different pianos. It does, however, afford a method of computing quickly an approximate value of the coefficient for any one piano.

Table II includes some information derived from data reported earlier.*2 The Starck piano was previously described as a medium grand and the Haddorff Vertichord as a 40-inch console piano. The parameters listed for the Starck piano were here interpolated from inharmonicity measurements on only the F-sub-3, F-sub-4, and F-sub-5 strings. Those from the Haddorff came from measurements on only the F-sub-3 and F-sub-5 strings. The values of K for these two pianos are very like those for the comparable grands and uprights respectively listed higher in Table II, but the values of q are so small as to be suspect.

It is evident from Table II that the coefficient of inharmonicity for A-sub-4 was greater than 0.7 cent/n^2 for the three uprights, whereas it was less than this value for the five grands. A glance at Figs. 2 to 7 also shows that the inharmonicity below C-sub-4 is usually less for the larger pianos than for the smaller ones.

Either Eq. 9 contains an erroneous choice for the ratio of Young's modulus to density or practically all of the inharmonicity is indeed basic, i.e., it is a consequence only of the stiffness of the wire. The equation has not been checked, however, for the strings in the topmost octave. The eight pianos listed in Table II should not be regarded as necessarily a good sample of all the pianos that have been made in the last half century, but the consistency of results gives one considerable confidence in estimating by Eq. 9 the coefficient of inharmonicity of plain steel wire strings in all pianos of conventional design.

The evidence here assembled indicates that (a) the inharmonicity changes from string to string roughly in the manner prescribed by Eq. (17), (b) the inharmonicity is slightly less in larger pianos than in smaller ones, and (c) the inharmonicity is almost all basic and can be calculated simply from string dimensions and frequency.

DISCUSSION

It is beyond the scope of the present paper to treat thoroughly the musical implications of piano string inharmonicity. The influence of inharmonicity on the tuning of pianos studied above will be reported in in a companion paper. Some slight speculation as to the relation to musical quality may be in order here.

It was pointed out above that, while the differences in inharmonicity of plain wire strings are not great among different pianos, yet in the much used octaves near middle C the inharmonicity is measurably less in the larger pianos. Is this one of the reasons that grand pianos have been traditionally preferred for their musical excellence?

O. H. Schuck has commented on the fact that the natural evolution of piano design has been such that the inharmonicity changes smoothly from string to string without any sudden increase or decrease. It is plausible that the smoothly changing inharmonicity is the characteristic which the designer has attained unconsciously in his effort to create a good musical effect. It could well be that "half-size" piano wire was later introduced into the regularly numbered series of sizes to afford not only better gradation of tension but also better gradation of musical quality associated with inharmonicity.

The strings of the Steinway Style A piano happen to afford the means of a simple experiment. Figure 3 indicates that the coefficients for the B-sub-2 and B-flat-sub-2 strings are somewhat greater than for the strings nearby, although the value of 0.18 cents/n^2 for B-flat-sub-2 does not seem really large. The coefficients for several wound strings below this point are less than 0.1 cent/n^2. Thus the inharmonicity of the B-flat-sub-2 string is perhaps twice as great as for neighboring strings, albeit not very great. When the notes are played for about an octave in this neighborhood, a listener can identify with reasonably ease the place where the musical quality sudden changes between B-flat-sub-2 and A-sub-2. There may be an argument as to what constitutes the "best" musical quality, but it seems clear that inharmonicity influences judgment of quality.

One is thus stimulated to propose the construction of a piano in which the inharmonicity is kept to a low value throughout. It is probable that no one has ever had the opportunity to listen to such a piano. The size of a piano, of course, limits the length of bass strings, but it would seem that the treble strings could be lengthened without serious difficulty. Their inharmonicity could be halved, for example, by an increase in string length of less than 20 percent.

An attractive hypothesis for a psychological experiment is: Those pianos are "best" in which the inharmonicity is least and in which it changes smoothly. The Steinway concert grand piano (see Fig. 2) would satisfy this criterion.

ACKNOWLEDGMENTS

Fifteen years ago Professor O. L. Railsback aroused interest of which this paper is evidence; when he started to use the first chromatic stroboscope to test piano tuning, he very soon found evidence of string inharmonicity. The majority of the measurements reported here were made in the showrooms of the Thearly Music Company and the Southern California Music Company of San Diego. Officials of these companies were most gracious in repeatedly putting pianos at the disposal of the writer. Various details of piano construction were supplied by Theodore D. Steinway; his continuing interest in this problem is gratefully acknowledged.