The theoretical frequency of the second harmonic is twice the fundamental frequency, and of the third harmonic is three times the fundamental frequency, and so on. Inharmonicity refers to the difference between the theoretical and actual frequencies of the harmonics or overtones of a vibrating tine or string. On instruments strung with metal wire, however, neither of these assumptions is valid, and inharmonicity is the reason. Theoretically, this means the fundamental frequency of the upper note is exactly twice that of the lower note, and we would assume that the second harmonic of the upper note will exactly match the fourth harmonic of the lower note. For example, we say two notes are an octave apart when the fundamental frequency of the upper note exactly matches the second harmonic of the lower note. In tuning, the relationship between two notes (known musically as an interval) is determined by evaluating their common harmonics. In electric pianos, the motion of the vibrating element is sensed by an electromagnetic pickup and amplified electronically. Each note on the keyboard has its own separate vibrating element whose tension and/or length and weight determines its fundamental frequency or pitch. In the acoustic piano, harpsichord, and clavichord, the vibrating element is a metal wire or string in many non-digital electric pianos, it is a tapered metal tine ( Rhodes piano) or reed ( Wurlitzer electric piano) with one end clamped and the other free to vibrate. (See harmonic series.) The fundamental note and its harmonics sound together, and the amplitude relationships among them strongly affect the perceived tone or timbre of the instrument. In most musical instruments, the tone-generating component (a string or resonant column of air) vibrates at many frequencies simultaneously: a fundamental frequency that is usually perceived as the pitch of the note, and harmonics or overtones that are multiples of the fundamental frequency and whose wavelengths therefore divide the tone-generating region into simple fractional segments (1/2, 1/3, 1/4, etc.). For example, the piano features both stretched harmonics and, to accommodate those, stretched fundamentals. Melodic stretch refers to tunings with fundamentals stretched relative to each other, while harmonic stretch refers to tunings with harmonics stretched relative to fundamentals which are not stretched. "For a stretched tuning the octave is greater than a factor of 2 for a compressed tuning the octave is smaller than a factor of 2." In stretched tuning, two notes an octave apart, whose fundamental frequencies theoretically have an exact 2:1 ratio, are tuned slightly farther apart (a stretched octave). Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos (such as the Fender Rhodes piano and Wurlitzer electric piano), and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements. If the widths of the keys of a piano keyboard were stretched as the intervals between the corresponding notes are in stretched tuning, it would look something like the above.
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