Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments and older, non-digital electric pianos (such as the Fender Rhodes piano and Wurlitzer electric piano) to accommodate the natural inharmonicity of their vibrating elements. In stretched tuning, two notes an octave apart, whose fundamental frequencies theoretically have an exact 2:1 ratio, are tuned slightly farther apart. This page is about musical systems of tuning, for the musical process of tuning see tuning. ... A musical instrument is a device that has been constructed or modified with the purpose of making music. ... An electric piano (e-piano) is an electric musical instrument of the 1960s or 1970s, which was built to replace a (heavy) piano on stage. ... A Rhodes piano is a musical instrument. ... Wurlitzer is the common name for band organs or orchestrions, vintage band organs and jukeboxes produced by the Rudolph Wurlitzer Company. ... In music, inharmonic refers to the degree to which the frequencies of the overtones of a fundamental differ from whole number multiples of the fundamentals frequency. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ...

Fundamentals and harmonics

In most musical instruments, the tone-generating component (a string or resonant column of air) vibrates at multiple 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.). (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. A string is a vibrating element used on many musical instruments, such as the violin, guitar, harp, and piano. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... The fundamental tone often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ... Pitch may refer to: Pitch is the property of a sound or musical tone measured by its perceived frequency Pitch, or tone of voice, refers to variation of tone in tonal language, and in languages with melodic accent Pitch, a throw of a baseball by a pitcher Pitch, part of... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ... An overtone is a sinusoidal component of a waveform, of greater frequency than its fundamental frequency. ... The wavelength is the distance between repeating units of a wave pattern. ... Pitched musical instruments are usually based on a harmonic oscillator such as a string or a column of air. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation. ... In music, timbre is determined by its spectrum, which is a specific mix of keynote, overtones, noise, tune behaviour, and envelope, as well as the temporal change of the spectrum and the amplitude. ...

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. 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 electric pianos, the motion of the vibrating element is sensed by an electromagnetic pickup and amplified electronically. This article is about the modern musical instrument. ... Harpsichord in Flemish style; for more info, click the image. ... Clavichord A clavichord is a small, very quiet, European keyboard musical instrument. ... A string is a vibrating element used on many musical instruments, such as the violin, guitar, harp, and piano. ... An electric piano (e-piano) is an electric musical instrument of the 1960s or 1970s, which was built to replace a (heavy) piano on stage. ... A Rhodes piano is a musical instrument. ... Wurlitzer is the common name for band organs or orchestrions, vintage band organs and jukeboxes produced by the Rudolph Wurlitzer Company. ... The fundamental tone often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ... Pitch may refer to: Pitch is the property of a sound or musical tone measured by its perceived frequency Pitch, or tone of voice, refers to variation of tone in tonal language, and in languages with melodic accent Pitch, a throw of a baseball by a pitcher Pitch, part of... An electric piano (e-piano) is an electric musical instrument of the 1960s or 1970s, which was built to replace a (heavy) piano on stage. ... A pickup device acts as a detector and captures mechanical vibrations (usually from suitably equipped stringed instruments such as the electric guitar, electric bass guitar and violin) and converts them to an electronic signal which can be amplified and recorded. ...

Intervals and inharmonicity

In tuning, the relationship between two notes (known musically as an interval) is determined by evaluating their common harmonics. 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. 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. This page is about musical systems of tuning, for the musical process of tuning see tuning. ... In music theory, an interval is the distance in pitch between two notes, the lower and higher members of the interval. ... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... The fundamental tone often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ...

On instruments strung with metal wire, however, neither of these assumptions is valid, and inharmonicity is the reason.

Inharmonicity refers to the difference between the theoretical and actual frequencies of the harmonics or overtones of a vibrating tine or string. 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. But on metal strings, tines, and reeds, the measured frequencies of those harmonics are slightly higher, and proportionately more so in the higher than in the lower harmonics. A digital emulation of these instruments must recreate this inharmonicity if it is to sound convincing. Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ... An overtone is a sinusoidal component of a waveform, of greater frequency than its fundamental frequency. ... A string is a vibrating element used on many musical instruments, such as the violin, guitar, harp, and piano. ... A string is a vibrating element used on many musical instruments, such as the violin, guitar, harp, and piano. ... A digital piano is a modern electronic musical instrument that is intended to function as a standard piano, often adding other features. ...

Note: Inharmonicity is not related to issues of temperament in musical tuning. Temperament is an issue that must be resolved on any instrument that plays fixed-pitch notes or that can play more than one note at a time (intervals or chords). Inharmonicity is a physical characteristic of each individual string, and it is the same regardless of temperament. This page is about musical systems of tuning, for the musical process of tuning see tuning. ... In music theory, an interval is the distance in pitch between two notes, the lower and higher members of the interval. ... In music and music theory, a chord (from the middle English cord, short for accord) is three or more different notes or pitches sounding simultaneously, or nearly simultaneously, over a period of time. ... A string is a vibrating element used on many musical instruments, such as the violin, guitar, harp, and piano. ...

Vibration of wire strings

When a stretched wire string is excited into motion by plucking or striking, a complex wave travels outward to the ends of the string. As it travels outward, this initial impulse forces the wire out of its resting position all along its length. After the impulse has passed, each part of the wire immediately begins to return toward (and overshoot) its resting position, which means vibration has been induced. Meanwhile, the initial impulse is reflected at both ends of the string and travels back toward the center. On the way, it interacts with the various vibrations it induced on the initial pass, and these interactions reduce or cancel some components of the impulse wave and reinforce others. When the reflected impulses encounter each other, their interaction again cancels some components and reinforces others.[1] A string is a vibrating element used on many musical instruments, such as the violin, guitar, harp, and piano. ...

Within a few transits of the string, all these cancellations and reinforcements sort the vibration into an orderly set of waves that vibrate over 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length of the string. These are the harmonics, and it is clear that the length of the vibrating string segment is less, and therefore the amplitude of its vibration is less, for higher harmonics than for lower. In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation. ...

Theoretically, vibration over half the string's length will be twice as fast, and vibration over one-third of the string will be three times as fast, as the fundamental vibration over the whole string's length. In the theoretical string, however, the only force acting to return a part of the string to its rest position is the tension between its ends, and the only force carrying that part of the string past its rest position is its momentum.

If you try bending a short piece of piano wire or guitar string slightly with your fingers, you can feel the wire's resistance to bending. In a vibrating string, that resistance adds to the effect of string tension in returning a given part of the string toward its rest position, and opposes the string's momentum as it moves past its rest position. The result is a frequency of vibration higher than the theoretical frequency. And because the wire's resistance to bending increases as its length decreases, its effect is greater in higher harmonics than in lower.

Tines and reeds

Tines and reeds differ from strings in that they are held at one end and free to vibrate at the other. The frequencies of their fundamental and harmonic vibrations are subject to the same inharmonicity as strings. However, because the comparative thickness of the bars that terminate the tines in an electric piano, the larger (and stronger) vibrations tend to "see" termination points slightly deeper in the bar than do smaller, weaker vibrations. This enhances inharmonicity in tines.

Effects on tuning

Inharmonicity "stretches" harmonics beyond their theoretical frequencies, and higher harmonics are stretched proportionally more than lower. Thus, in our example of an octave, exactly matching the lowest common harmonic causes a slight amount of stretch, matching the next higher common harmonic causes a greater amount of stretch, and so on. If the interval is a double octave, exactly matching the upper note to the fourth harmonic of the lower complicates the tuning of that upper note with the one an octave below it. In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... In music theory, an interval is the distance in pitch between two notes, the lower and higher members of the interval. ...

Solving such dilemmae is at the heart of precise tuning by ear, and all solutions involve some stretching of the higher notes upward and the lower notes downward from their theoretical frequencies. In the shortest pianos, where strings are short and wire stiffness is proportionately high, the stretch required for overall sonority is extreme; concert grand pianos require far less stretch. Online sources[2] suggest that the total amount of "stretch" over the full range of a concert grand piano may be on the order of +/- 35 cents. This page is about musical systems of tuning, for the musical process of tuning see tuning. ... The cent is a unit in a logarithmic scale of relative pitch or intervals. ...

References and further information

1. Five lectures on the acoustics of the piano
2. Inharmonicity in piano tuning

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