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 /
*ν*_{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 *ν*_{0} is the
fundamental frequency. A value of *Q*/*ρ* =
25.5×10^{10} 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 out^{1} 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

Eq. 1

where

Eq. 2

is the speed of wave propagation,

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

Eq. 3

or

Eq. 4

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

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
*nν*_{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} as for a
flexible string, with nodes at x = 0 and x = *l*; namely,

Eq. 5

If this function be substituted in the usual (simplified) differential equation for the vibration of a stiff string under tension,

Eq. 6

Here Q is Young's modulus, and

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 —> | 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 |

_________________________________________________________________ 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

Eq. 7

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}
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
*ν*_{0}
from Eq. 1 it follows that

Eq. 8

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

If one takes for steel wire, Q/*ρ* =
25.5×10^{10}
cm²/sec² and
expresses *d* and *l* in centimeters and
*ν*_{0}
in cycles per second,

Eq. 9

If

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}
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 cents/n² |
q s/m |
||||

A_{0} |
A_{4} |
A_{6} |
A_{4} |
A_{6} |
|||||||

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 theEq. 10

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_{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_{5}
+ 3 cents, the third mode
E_{6}
+ 8 cents, etc. The notes
A_{4},
A_{5},
E_{6},
A_{6},
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
A_{4}
and
E_{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
δ_{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.

Eq. 11

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_{7} string is
4.6 cents/n². Then the second mode is more inharmonic than the
first by δ_{2-1} = 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

Eq. 12

for observations on the first four modes of vibration.

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_{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/n²,
and thus the scatter is actually reasonable. Increased scatter between
D#_{6} and B_{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/n².

Figure 2 is incomplete in two respects. There are plain wire strings
for still another octave up to C_{8}, but
to determine the inharmonicity for this octave would require frequency
measurements another octave beyond B_{7}.
Furthermore, the plain wire strings extend down to F_{2} in this particular piano, but for the sake
of uniformity with later figures, the grid for Fig. 2 is started at
C_{3}. Excepting the measured coefficient
for F_{2} 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 20×10^{11} 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* = 25.5×10^{10}
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 C_{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

except below D

Eq. 14

Of course,

Eq. 15

When these relations are substituted in Eq. 9, it turns out that the basic coefficient of inharmonicity for this piano is

Eq. 16

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#_{5} and
A_{5}
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
C_{3},
A_{3},
A_{4},
A_{5},
and A_{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_{5}
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
G_{3},
A_{4},
A_{5}, and
A_{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.

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_{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/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

Eq. 17

where K is the coefficient for the A

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_{3},
F_{4}, and
F_{5} strings. Those
from the Haddorff came from measurements on only the
F_{3} and
F_{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_{4}
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
C_{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.

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_{2}
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 A_{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.