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to be practical, a problem that was solved by redesigning the strings. The redesign involved two changes, one simple and one not so simple. The first change was to increase the string diameter as one moves from the treble into the bass. For the modern piano in Figure 2, the string diameter varies from about 0.8 mm in the extreme treble to 1.0 mm for the strings an octave below middle C. If the tension is held fixed, this al- lows the strings to be shortened by nearly 50%, which helps but does not solve the problem completely. One might think that simply increasing the diameter by even more could be possible, but such thick strings lead to a different problem.
The vibrational frequencies of an ideal flexible string form a perfectly harmonic relationship, and this relationship leads to a musical tone with components that are harmonic. Real piano strings have a small amount of stiffness that adds a restoring force for the string due to the bending stiffness, causing the vibrational frequencies to deviate from an ideal harmonic series. For this reason the vibrational modes of a stiff string are called "partials" instead of harmonics. It turns out that increasing the diameter of steel string to val- ues greater than about 1.2 mm would make the deviations from harmonicity so large as to result in unpleasing tones. The upshot is that piano designers needed a different way to increase the effective string diameter without causing a significant increase in the stiffness.
The solution to increasing the effective string diameter is to use what are called wound strings (Figure 4) in which a cop- per wire is wound around a central steel core. The diameter of the core is usually about 1 mm, while the diameter of the winding typically varies from about 0.5 mm to more than 1 mm as one moves farther into the bass region (and some pianos have bass strings with two sets of windings). The cop- per windings add mass to the string without an appreciable increase in stiffness, so much shorter strings can be used in the bass than would otherwise be possible. In this way the length of a modern grand piano is typically around 2 m (Fig- ure 2) and is rarely more than the 3 m of a concert grand piano. This also means that the case deviates from the wing shape and is "rounded off" on the bass end (also visible in Figure 2).
Stringing the Instrument
Essentially all modern pianos now have 88 notes. While this number is standardized, piano designers still have some flexibility in how many strings they use for each note. Cris- tofori's pianos all employed two strings per note. This choice was probably copied from Italian-style harpsichords of his
Figure 4. Expanded view of the lower left corner of Figure 2, showing the ends of the first few bass strings. These strings consist of copper wire wound around a central steel core. The agraffe plays the role of the nut in Figure 3. From Giordano (2010) with permission from Oxford University Press.
time. In Italian harpsichords each note had a pair of strings of essentially the same length that were plucked at slightly different distances from their ends. This caused the strings to produce tones with the same pitch, but with different tim- bres due to a different balance of the fundamental compo- nent and the harmonics. The effect is present for the piano too (even with just a single hammer for each note) but to a much lesser degree than for the harpsichord.
The fact that multiple strings are involved in a single note gives rise to other interesting properties of the piano. For a note involving two strings, there is a strong interaction be- tween the strings since they are tuned to have essentially the same frequency (i.e., they can be thought of as degenerate oscillators). The way in which the two piano strings interact is an interesting problem that was elucidated by Weinreich (1977). He showed how the coupling of two such strings through their connection to the bridge causes their vibra- tional frequencies to split, in the same way that the interac- tion of two nominally identical harmonic oscillators results in modes with slightly different frequencies. The slight split- ting of the mode frequencies gives a certain richness to the overall tone, in the same way that the sound of two violins simultaneously playing the same note has a very appealing richness. Another reason for using multiple strings for each note is to obtain a greater total volume of sound, an impor- tant consideration as concert halls have increased in size since Mozart's time. For these reasons, piano designers have followed Cristofori's lead and employ multiple strings per note for nearly all notes.
For the piano in Figure 2, the lowest 10 bass notes are all produced by a single wound string. The next 15 notes em- ploy two wound strings per note, and all of the higher notes
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