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Evolution of the Piano
Figure 3 shows several other components of the piano. The string is the vibrating element that determines the frequen- cies that will be present in the final tone. Most notes involve more than one string, as we will explain shortly. In a modern piano, strings in the midrange and above are composed of steel with a diameter of about 1 mm. The strings in the bass range have a more complicated design that we will describe below. The strings run from a pin (called a hitchpin) at the back of the piano, pass over a bridge glued to the sound- board and over a "nut" or other structure, with the strings’s end wound around a tuning pin at the front of the piano (Figure 3). The force of the string is transmitted through the bridge to the soundboard, setting the soundboard into mo- tion and producing the sound of the instrument. The tuning pin allows the tension in the string to be adjusted to achieve the desired fundamental frequency for the string. In the next sections I describe how a number of the components in Fig- ure 3 were constructed in Cristofori's pianos and how they have changed as the instrument evolved into the modern piano.
How Many Notes Should a Piano Have?
The Cristofori piano in Figure 1 has a 4-octave range, with the fundamental frequencies of the notes varying from about 65 Hz to 1,048 Hz (assuming the now standard pitch of 440 Hz for the A above middle C). However, it did not take long for this range to expand. Much of the baroque key- board music of composers such as Bach and Scarlatti can fit in 4 octaves, but later composers wanted more. The range expanded to 5 octaves in the late 1700s (Mozart), then to 6 octaves by the very early 1800s (Beethoven), to 7 octaves by about 1840, and the 71⁄3 octave range we have today arrived around 1860. The piano's range thus nearly doubled in less than a century, but it has not changed further in the 150 years since then. Technologically, there is no fundamental limit to this range. We also know that human hearing extends over a broader range, which leads to the question: "Why don't we have more notes?"
The lowest note on a modern piano has a fundamental fre- quency of 27.5 Hz and for the highest note it is 4,186 Hz. (These are the ideal frequencies of these notes. For real pia- nos these frequencies deviate slightly from these ideal val- ues; Giordano, 2010.) Human hearing is certainly able to de- tect sounds well beyond these frequencies, both lower and higher. However, the way such tones are perceived is very interesting. Sounds with frequencies much below about 25 Hz are perceived as rapid clicks rather than as a typical mu-
14 | Acoustics Today | Spring 2016
sical tone (Plack and Oxenham, 2005). Such clicking sounds are probably of little use musically, so there is no value in extending the piano's range to lower pitches.
Human auditory perception at frequencies above about 5,000 Hz is limited in a different way. The relationship be- tween two or more tones is used often in music, producing musical intervals and chords that are pleasing or provide a particular musical effect. It turns out that humans are not able to perceive such tonal relations at frequencies above about 5,000 Hz (Plack and Oxenham, 2005). That is, while tones with fundamental frequencies of, say, 4,000 Hz and 8,000 Hz can both be heard, and we can tell that the pitch of one tone is higher than the pitch of the other, most peo- ple are not able to judge that they are an octave apart. Since tones above about 5,000 Hz cannot be used to form musical intervals, they are not much use to a composer. In this way, human perception has determined the upper limit for the notes of a piano.
Piano Shape and Design of the Strings
All three of the existing Cristofori pianos have been restored multiple times, although much is not certain about their original state. It is believed that they were originally strung with brass wire in the bass and iron wire in the midrange and treble, and that the string diameter increased somewhat in going from the treble to the bass. If we assume for simplicity that all the strings in the piano in Figure 1 have exactly the same diameter, density, and tension (which would be only a very rough assumption for the string diameters), then the length of the vibrating portion of the string should increase by precisely a factor of two as one moves an octave toward the bass. For the 4-octave piano in Figure 1 this means that the string length of the lowest bass string would be 16 times longer than the highest treble string. Since the strings must fit inside the case, this variation in length determines the shape of the piano, and with our slightly idealized assump- tions, we arrive at the familiar "wing" shape of a grand piano. Indeed, the Cristofori pianos do have this shape and do have string lengths that vary (approximately) by a factor of 2 per octave within a case that is about 2 m from front to back on the bass (left) end.
This simple scaling of the string length is fine for a 4-octave instrument but leads to problems for a modern grand piano. With the lowest note of a modern piano being more than an octave below the lowest note on a Cristofori piano, scal- ing the string lengths by a factor of 2 per octave would pro- duce a piano more than 5 m long. This is obviously too big