Nicholas Giordano – firstname.lastname@example.org
Department of Physics
West Lafayette, IN 47907
Popular version of paper 3aMU6
Presented Wednesday Morning, May 25, 2011
161st ASA Meeting, Seattle, Wash.
The piano was invented in the first decades of the 18th century by the Italian Bartolomeo Cristofori, and wide adoption of the instrument began in the later part of that century. During the period from 1750 to the late 1800s, the piano evolved considerably, as composers such as Mozart, Beethoven, Liszt, and others demanded more and more from the instrument. These demands included the desire for more sound—that is, greater volume—to fill the new, larger concert halls of the time. The desire for more sound required designers to place the strings under greater tension, so as to exert greater forces on and produce greater motion of the soundboard. This in turn led to many other changes in the instrument, including the addition of an iron plate to strengthen the case so it could withstand the increased force of the strings. By the late 1800s the piano reached its modern form; for example, the Steinway model D concert grand piano that is widely used today was designed in 1884. In this paper, I describe how improvements in music wire allowed the necessary increases in string tension, and especially how improvements in the tensile strength of music wire affected the evolution of the instrument. I illustrate this evolution by an analysis of four instruments: two relatively early pianos designed in 1815 and 1820, which use iron strings; an 1850 piano designed using some early steel music wire; and a modern piano designed in 1912 that uses steel wire with a tensile strength more than three times greater than that of the iron wire used in the earliest pianos. I also speculate about what might be possible in the future with the advent of new string materials.
The challenge faced by the piano designer can be expressed as follows. The designer wants to increase the string tension (to get a more powerful sound), but one is limited by the tensile strength of the music wire. For a given material, the only way to increase the usable tension is to increase the string diameter. This increases the bending stiffness of the string (since a thick string is harder to bend than a thin one), but if the stiffness is too great, the tone quality is adversely affected. The designer must therefore make a compromise involving tension versus stiffness to achieve more sound while maintaining an acceptable tone quality.
The physics of how string stiffness affects the quality of a piano tone is based on the properties of a vibrating string. For an “ideal” piano string, that is, a piano string with only a very small (or zero) stiffness, the vibrational frequencies approximately follow the mathematical pattern
fn = nf1 n = 1, 2, 3, …
where f1 is the fundamental frequency of the string. The frequencies corresponding to n = 2, 3, … are called harmonics. For a piano string with a fundamental frequency of 440 Hz (the note A above middle C), the harmonics are at 880 Hz, 1320 Hz, etc., and the corresponding piano tone would contain components at these frequencies. The strengths and frequencies of these components give a tone its characteristic timbre. Most importantly, musical tones that consist of harmonic components are generally judged to be more pleasing than tones that do not.
This simple pattern of harmonic frequencies does not quite apply to real piano strings. All real strings have some stiffness, which depends on the string diameter and other material properties. String stiffness causes an upward shift in the vibrational frequencies, which then follow the pattern
fn = nf1 (1 + a n2)
This new pattern means that the higher frequencies (larger n) are shifted more and more above the ideal harmonic values, an effect called anharmonicity. For this reason, these vibrations are not referred to as harmonics but as “partials.” The partial frequencies are important because piano tones become harsh and unpleasant if the deviations from harmonicity (which can be measured by the value of the anharmonicity parameter a) are too large.
I have analyzed how four piano designers working in the period from 1815 to 1912 dealt with the compromise between the desire for more sound (greater string tension) and tone quality (keeping the anharmonicity to an acceptably small value). One of my results is shown in the figure below, which plots the amount of anharmonicity (the value of a) for notes across the piano keyboard. The results show that the anharmonicity increases for the higher notes, a fact that is well known. However, a new result here is that these very different pianos had essentially the same anharmonicity. This suggests that different designers, and different piano listeners, have had the same musical “taste” in piano tones for more than a century.
While the pianos studied here had similar values of anharmonicity, they did so with very different strings. The two earliest pianos studied (from 1815 and 1820) used iron wire while the later pianos (from 1850 and 1912) used much stronger steel wire. The technology for making steel piano wire was developed in the mid-1800s, and it enabled the later pianos to use much greater tensions than allowed by the iron wire used in early pianos. However, the technology of steel wire making improved considerably from 1850 to 1900, achieving improvements in tensile strength of about 30% over that period. Surprisingly, the 1912 piano did not appear to take full advantage of the stronger wire that was presumably available. It is possible that some other properties of the wire were a limiting factor, but for now our analysis suggests that the designer of the 1912 piano was a bit conservative. For some, it may be heretical to suggest that the Steinway Company could have done better, but that may have been the case.
The pianos of today are all designed for the steel piano wire that was available more than a century ago. Is it time for further evolution of the instrument? With regards to steel music wire, there has been no significant improvement in tensile strength since the late 1800s. So, for a piano that uses steel music wire, the limitations faced in 2011 are the same as those faced in 1900. However, new materials, such as carbon fiber strings, may be worth exploring. While a piano with carbon fiber strings has not yet (to the best of my knowledge) been made, a theoretical analysis suggests that their combination of stiffness and tensile strength could make them preferable to steel strings. This analysis assumes that stiffness and anharmonicity are the dominant design factors, but other factors may also be important. Probably the only way to know for sure will be to try it.
Caption. Anharmonicity factor a (defined above) for the four pianos analyzed in this study, with the makers and the years the pianos were designed or built listed in the inset. The note A0 is the lowest note on a modern piano and note A6 is in the treble, more than two octaves above middle C. The results show that the designers of these instruments achieved very similar levels of anharmonicity, even though they used very different strings (iron wire in the two early pianos, steel in the others), string diameters, and string tensions.