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Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital (such as the Fender Rhodes piano and Wurlitzer electric piano), and some sample-based synthesizers based on these instruments, to accommodate the natural of their vibrating elements. In stretched tuning, two notes an octave apart, whose fundamental frequency theoretically have an exact 2:1 ratio, are tuned slightly farther apart (a stretched octave). If the frequency ratios of octaves are greater than a factor of 2, the tuning is stretched; if smaller than a factor of 2, it is compressed."Hartmann, William M. (1997). Signals, Sound, and Sensation, p.275. .
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.Hartmann (1997), p.276. For example, the piano features both stretched harmonics and, to accommodate those, stretched fundamentals.
In the acoustic piano, harpsichord, and clavichord, the vibrating element is a metal wire or string; in many non-digital , 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 , the motion of the vibrating element is sensed by an electromagnetic pickup and amplified electronically.
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 frequency of the or 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 piano of these instruments must recreate this inharmonicity if it is to sound convincing.
The theory of temperaments in musical tuning do not normally take into account inharmonicity, which varies from instrument to instrument (and from string to string), but in practice the amount of inharmonicity present in a particular instrument will effect a modification to the theoretical temperament which is being applied to it.
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 . As a rule, the amplitude of its vibration is less for higher harmonics than for lower, meaning that higher harmonics are softer—though the details of this differ from instrument to instrument. The exact combination of different harmonics and their amplitudes is a primary factor affecting the timbre or tone quality of a particular musical tone.
In an ideal plain string, vibration over half the string's length will be twice as fast as its fundamental vibration, vibration over a third will be three times as fast, and so on. In this kind of string, the only force acting to return any part of it to its resting position is the tension between the string's ends. Strings for low and mid-range tones, however, typically consist of a core that is wound with another, thinner piece of wire. This makes them naturally resistant to being bent, adding to the effect of string tension in returning a given part of the string toward its resting position; the result is a comparatively higher frequency of vibration of wound strings. Since rigidity is constant, its effect is greater for shorter wavelengths, i.e. in higher harmonics.
Solving such dilemmas is at the heart of precise musical tuning by ear, and all solutions involve some stretching of the higher notes upward and the lower notes downward from their theoretical frequencies. In shorter strings (such as on spinet pianos), the wire stiffness in the tenor and bass registers is proportionately high; this leads to a timbre which is generally poorer, due to the higher inharmonicity and octave stretch, creating significant compromises to what is considered acceptable tuning. On longer strings, such as on concert grand or even moderately sized grand pianos, this effect is greatly reduced. Online sources suggest that the total amount of "stretch" over the full range of a piano may be on the order of ±35 cents: this also appears in the empirical Railsback curve.
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