The piano was founded on earlier technological innovations in [url=https://en.wikipedia.org/wiki/Keyboard_instrument]keyboard instruments[/url]. [url=https://en.wikipedia.org/wiki/Pipe_organ]Pipe organs[/url] have been used since antiquity, and as such, the development of pipe organs enabled instrument builders to learn about creating keyboard mechanisms for sounding pitches. The first [url=https://en.wikipedia.org/wiki/String_instrument]string instruments[/url] with struck strings were the [url=https://en.wikipedia.org/wiki/Hammered_dulcimer]hammered dulcimers[/url],[sup][url=https://en.wikipedia.org/wiki/Piano#cite_note-6][6][/url][/sup] which were used since the [url=https://en.wikipedia.org/wiki/Middle_Ages]Middle Ages[/url] in Europe. During the Middle Ages, there were several attempts at creating stringed [url=https://en.wikipedia.org/wiki/Keyboard_instrument]keyboard instruments[/url] with struck strings.[sup][url=https://en.wikipedia.org/wiki/Piano#cite_note-7][7][/url][/sup] By the 17th century, the mechanisms of keyboard instruments such as the [url=https://en.wikipedia.org/wiki/Clavichord]clavichord[/url] and the harpsichord were well developed. In a clavichord, the strings are struck by tangents, while in a harpsichord, they are mechanically plucked by quills when the performer depresses the key. Centuries of work on the mechanism of the harpsichord in particular had shown instrument builders the most effective ways to construct the case, soundboard, bridge, and mechanical action for a keyboard intended to sound strings (Wikipedia).

The invention of the piano is credited to [url=https://en.wikipedia.org/wiki/Bartolomeo_Cristofori]Bartolomeo Cristofori[/url] (1655–1731) of [url=https://en.wikipedia.org/wiki/Padua]Padua[/url], Italy, who was an expert harpsichord maker, and was well acquainted with the body of knowledge on stringed keyboard instruments; this knowledge of keyboard mechanisms and actions helped him to develop the first pianos[br][br]Cristofori named the instrument [i]un cimbalo di cipresso di piano e forte[/i] ("a keyboard of [url=https://en.wikipedia.org/wiki/Cypress]cypress[/url] with soft and loud"), abbreviated over time as [i]pianoforte[/i], [i]fortepiano[/i], and later, simply, piano.[sup][url=https://en.wikipedia.org/wiki/Piano#cite_note-11][11][br][/url][/sup][br]Cristofori's great success was designing a stringed keyboard instrument in which the notes are struck by a hammer. The hammer must strike the string, but not remain in contact with it, because this would [url=https://en.wikipedia.org/wiki/Damping_(music)]damp[/url] the sound and stop the string from vibrating and making sound. This means that after striking the string, the hammer must fall from (or rebound from) the strings. Moreover, the hammer must return to its rest position without bouncing violently, and it must return to a position in which it is ready to play almost immediately after its key is depressed so the player can repeat the same note rapidly.[br][br]Cristofori's early instruments were made with thin strings, and were much quieter than the modern piano, but they were much louder and with more [url=https://en.wikipedia.org/wiki/Sustain]sustain[/url] in comparison to the clavichord—the only previous keyboard instrument capable of dynamic nuance responding to the player's touch, or the velocity with which the keys are pressed. [br][br]While the clavichord allows expressive control of volume and sustain, it is relatively quiet. The harpsichord produces a sufficiently loud sound, especially when a coupler joins each key to both manuals of a two-manual harpsichord, but it offers no dynamic or expressive control over each note. The piano offers the best of both instruments, combining the ability to play loudly and perform sharp accents.

Manipulate the following applet in order to modify the amplitude, i.e. the loudness of the sound.

The applet shows sine functions which can be played as sound waves over your speakers. Both amplitude (perceived as loudness) and frequency (perceived as pitch) can be manipulated during playback.

Every sine function has an [b]amplitude[/b] and a [b]period[/b]. Amplitude is the distance between the center line of the function and the top [i]or[/i] bottom of the function, and the period is the distance between two peaks of the graph, or the distance it takes for the entire graph to repeat.

You can find both the amplitude and period of a sine graph without going through the entire process of graphing just by looking at the equation. The generalized equation for a sine graph is as follows:[br][br][math]\text{y[br]=[br]A[br]⋅[br]sin[br]([br]B[br]([br]x[br]+[br]C[br])[br])[br]+[br]D}[/math][br][br]Using this equation:[br]Amplitude =A[br]Period =[math]\frac{\text{\text{2[br]π}[br]}}{B}[/math][br]Horizontal shift to the left =C[br]Vertical shift =D

Remember that we are looking at our functions in terms of radians instead of degrees. Period is equal to 2πB because there are 2π radians in a full rotation.[br]We are only looking at amplitude and period for now, so we can simplify the equation to:[br][br][math]\text{y[br]=[br]A[br]⋅[br]sin[br]([br]B[br]x[br])}[/math]

The example graphed in the picture above is [math]\text{y[br]=[br]sin[br]([br]x[br])}[/math]. Just by looking at the equation we can see that both A and B equal 1. This is shown in the graph as the amplitude is 1 and the period is 2π.[br]Let's try a practice problem:[br]

Listen to the sine wave for various frequencie. Change the volume A.

Wikipedia. (2021). Piano. [url=https://en.wikipedia.org/wiki/Piano]https://en.wikipedia.org/wiki/Piano[/url][br]Heller, T. (2021). [url=https://www.expii.com/t/finding-amplitude-and-period-of-sine-functions-5228#:~:text=Amplitude%20is%20the%20distance%20between,the%20entire%20graph%20to%20repeat.&text=Using%20this%20equation%3A%20Amplitude%20%3DAPeriod,the%20left%20%3DCVertical%20shift%20%3DD]Finding Amplitude and Period of Sine Functions[/url]. Xpii.[br]Frequency and amplitud of sound. [url=https://www.geogebra.org/u/sengkwang]Tan Seng Kwang[/url], [url=https://www.geogebra.org/u/lacostinois]Pierre Lacoste[br][/url]Amplitude, Period, & Phase Shift. [url=https://www.geogebra.org/u/stuart.fortier]Stuart Fortier[/url]