Course:Phys341 2020/Piano

The piano is an instrument that can create different sounds through various mechanisms. A pianist can manipulate volume, how long they hold notes for, and what notes they combine in order to do so.This page outlines the basic physics involved in these techniques, starting with the basic mechanisms of how sound is produced in a piano and continuing on to explain the factors of volume, note duration, and note combination.

Fig. 1: The inside of an upright piano. The hammers are the bottom row of wood and felt pegs. The dampers are just above, resting on the strings. Behind the strings the soundboard can just be seen. Own work.

Basic Mechanisms

While from the outside it appears as though simply pressing a key creates sound, there is much more happening within the piano that leads to the production of sound.There are strings for each key linked to bridges which are attached to a soundboard. The number and type of strings depends on how high or low the pitch (frequency) is; the lowest being 27.5 Hz (A₀) and the highest at 4186 Hz(C₈). The lowest pitches need one thick, wire wound string while the highest pitches require three thinner steel wire strings. There is a hammer for each set of strings of a key; when a key is pressed, the hammer strikes the string(s). When the strings are hit by the hammer, they vibrate and displace air molecules, which creates a sound wave. The soundboard acts as a large wooden resonator, amplifying the sound waves the strings create. There are dampers for each set of strings, which are small felt pads attached to wooden blocks that stop strings from vibrating . In the natural resting position of a key, the damper lies on the strings and the hammer is lifted (see fig. 1). When a key is pressed, the damper lifts up and the hammer strikes; as long as the key is pressed down, the damper remains lifted. Once the key returns to resting position, the damper returns to the strings and stops the vibration, and hence the sound altogether, although some resonance may remain through the soundboard.

Factors of Musicality

Fig. 2: This diagram depicts the differences in how volume levels affect which partials are the strongest. Loud volume (ff) corresponds with the second partial being strongest. Medium volume (mf) shows the first and second partial being close in strength. For quiet volume (pp), the first partial is the strongest. These measurements were done by Nick Giordano.

There are techniques used by pianists to create different types of sounds, such as using more force when pressing down on a key to increase volume or sustaining a note to let it ring longer than it would if simply pressed and released. As well, the combination of notes used in a song greatly influences the listener's experience and interpretation of the sounds.

Volume

Changing note volume involves interactions of amplitude and resonance. Hitting a key harder makes the hammer strike the string with more force, creating a larger amplitude of the sound wavelength and more resonance overall. While this increases decibels, it can also change the tone color of the note (timbre). Playing a note more loudly affects which partial shows up the brightest on a spectrogram (a tool which visualizes frequency spectrums), indicating which one is most powerful. Figure 2 shows that the louder the note is, the stronger the second partial will be. This change in which partial is most powerful alters timbre despite still hearing the fundamental frequency most clearly . This is due to the difference in which partials are most present; if the fundamental frequency is most present, the tone sounds more dim, if the higher partials are most present, the tone sounds brighter. Even when the increase in volume is accounted for, there are still visible differences between the wave shapes and resulting tones from having more or less force.

Duration of Note

Fig. 3: Spectrogram of a "staccato" note versus one that is sustained. Note the difference in how long the frequencies appear on the spectrogram, clearly indicating the bottom one as the staccato note and the upper one as the legato (sustained frequencies appear brighter for much longer). Own work.

Increased duration of string vibration, and thus sound, is created either by physically holding a key down or using the sustain pedal, which when engaged keeps the dampers lifted off the strings, allowing them to vibrate freely. When a note is held down, a smooth continuous sound is created, and when done with multiple notes in a row is called "legato"-- meaning smooth and without breaks between notes. Holding down the sustain pedal lifts all dampers from the strings and allows sound waves to travel freely along the strings until the pedal is released and the dampers return to rest on the strings. Playing a note quickly causes the dampers to lift and fall back on the strings with great speed, causing the vibrations to start and stop quickly. This is called "staccato"-- sharp, or detached. Figure 3 shows the different spectra for a short and long note. Here is a video which shows the inside of a piano as a short note is played, followed by a long note using the sustain pedal.

Note how quickly the dampers return to the strings for the short note, cutting off the vibrations and ending the sound. Alternatively, for the long note the dampers remain lifted while the sustain pedal is engaged; this allows the strings to vibrate freely and the sound to continue, even after the pianist has removed pressure from the key.

Note Combination

Fig. 4: Simple diagram of the first few partials in a perfect 5th and tritone. Every third partial of C lines up with every second of G. None of F# or C's partials align here, but are very close, which creates beats. Own work.

At the root of piano theory lies the basic physics required to understand harmony and what makes different combinations of notes sound "good" or "bad." Each note has a specific frequency that determines the fundamental frequency and partial tones that lie above. When playing two notes that have small frequency ratios in relation to each other, their partials line up frequently and there is little dissonance, creating a very harmonious sound. For example, C₄ and G₄ have a ratio of 3/2. When these notes are played together, every third partial of C₄ lines up with every second partial of G₄ (see fig. 4). This interval is called a perfect fifth; here is an audio demonstration.

In contrast, when two notes have large frequency ratios, their partials will not line up frequently and may create beats. Beats describes when two frequencies that are very close to each other interfere and alternate between loud and soft volume-- a "beating" sound.. For example,C₄ and F#₄ have a ratio of √2 (approximately 45/32). When these notes are played together, they create a dissonant sound since so few partials align, but many are close to lining up, creating beats (see fig. 4). This note combination is called a tritone, and can be found in history as the "diabolus in musica"-- devil in music. The peculiar note interval was given this title due to its unpleasant and "devilish" sound . Here is an audio demonstration of a tritone (C₄ and F#₄) played on a piano.

Combined Elements

These three factors largely constitute the dynamics and style with which the piano is played and can translate to a number of other instruments. Combining them in different ways creates varying experiences for those listening. For example, "Berlin Song" by Ludovico Einaudi largely utilizes quiet tones and simple frequency ratios, such as thirds and fifths. Its swells in volume add passion before returning to a calm, quiet state. It is easy to listen to due to the consonance and smooth tone. This can be contrasted with Beethoven's 5th Symphony, as transcribed by Listz for the piano, which begins with thundering volume and minor chords. These minor chords have more dissonance than major ones and the loud volume conveys intensity and brooding passion. The song later moves to quiet volume and major chords; the difference is striking. Underlying these changing emotions and feelings evoked by the piano remains the physics of how sound is produced and works in difference ways.