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 3.4×1013 n²d² / ν0²l4, where n is the mode number, d is the diameter of the wire in cm, l is the vibrating length in cm, and ν0 is the fundamental frequency. A value of Q/ρ = 25.5×1010 cm²/sec² was assumed for the steel wire, where Q is Young's modulus and ρ 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.
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 out1 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 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.
According to the usual simple theory, the fundamental frequency of a thin
flexible string vibrating transversely
between rigid supports is
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 inharmonicity² and to express it in cents
(hundredths of an equally tempered semitone). Thus the inharmonicity
δ is given by
It may be noted in passing that according to the definition used previously²,4 the inharmonicity was expressed relative to an integral multiple of the frequency of the first mode, rather than relative to nν0. 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 same5 as for a
flexible string, with nodes at x = 0 and x = l; namely,
_________________________________________________________________ Mode --> 1 2 3 4 5 6 7 8 _________________________________________________________________ String: A-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 _________________________________________________________________
With the help of the approximation in Eq. 4 it follows that
The coefficient of inharmonicity is a characteristic of the stiff string. A comparison of the relationships usually quoted9 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
from Eq. 1 it follows that
If one takes for steel wire, Q/ρ =
expresses d and l in centimeters and
in cycles per second,
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 rule10 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.
|Maker||No.||Style||Date||String length in cm.||Dia. in mm.||K
|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|
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 δ3 and δ1 are the inharmonicities of the third and first modes of vibration, respectively, then the relative inharmonicity is given by
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 A4 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 A5 + 3 cents, the third mode E6 + 8 cents, etc. The notes A4, A5, E6, A6, C#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 A4 and E6 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 δ3-1 = 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
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 A7 string is 4.6 cents/n². Then the second mode is more inharmonic than the first by δ2-1 = 4.6(2² - 1²) = 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
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 C4 (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/n², and thus the scatter is actually reasonable. Increased scatter between D#6 and B6 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/n².
Figure 2 is incomplete in two respects. There are plain wire strings for still another octave up to C8, but to determine the inharmonicity for this octave would require frequency measurements another octave beyond B7. Furthermore, the plain wire strings extend down to F2 in this particular piano, but for the sake of uniformity with later figures, the grid for Fig. 2 is started at C3. Excepting the measured coefficient for F2 which is 0.2 cent/n², both calculated and observed coefficients are less than 0.1 cent/n² 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 20×1011 dynes/cm² 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/cm³ which is implicit in tables of linear density published by the American Steel and Wire Company, resulted in Q/ρ = 25.5×1010 cm²/sec². 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/n² at C3 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
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#5 and A5 accompanies a sudden shift in the scaling of lengths occasioned by the presence of a bar in the iron frame.
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 C3, A3, A4, A5, and A6 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 D5 and D#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
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 (A0) 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/n². 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
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 F3, F4, and F5 strings. Those from the Haddorff came from measurements on only the F3 and F5 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 A4 was greater than 0.7 cent/n² 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 C4 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.
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 B2 and B♭2 strings are somewhat greater than for the strings nearby, although the value of 0.18 cents/n² for B♭2 does not seem really large. The coefficients for several wound strings below this point are less than 0.1 cent/n². Thus the inharmonicity of the B♭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♭2 and A2. 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.