Stretched tuning
Stretched tuning is a detail of
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.[4] For example, the piano features both stretched harmonics and, to accommodate those, stretched fundamentals.
Fundamentals and harmonics
In most musical instruments, the tone-generating component (a
In the acoustic
Intervals and inharmonicity
In
On instruments strung with metal wire, however, neither of these assumptions is valid, and inharmonicity is the reason.
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.
Vibration of wire strings
When a stretched wire
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. 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.
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 of 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 alters harmonics beyond their theoretical frequencies. As the overtone series progresses, each partial becomes proportionally sharper. 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 two octaves plus a fifth (the favored means of cross-checking the stretch of the upper treble of the piano), exactly matching the upper note to the sixth harmonic of the lowest requires great sophistication of octave stretch to make the lower individual octaves, its double and triple octaves, and their other intervallic relationships to sound pure and balanced.
Solving such dilemmas 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 shorter pianos, the wire stiffness in the tenor and bass registers is proportionately high, and causes greater inharmonicity and hence greater stretch, negatively affecting timbre and creating serious compromises to what is considered acceptable tuning. On large grand pianos, and in particular concert grand pianos, this effect is greatly reduced. Online sources[2] 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.
See also
References
- ISBN 9780191591679. "In a properly tuned instrument the notes will be progressively sharper in the treble compared with the frequencies calculated for a particular tempered scale (Schuck and Young 1943). Likewise in the bass the notes will becomes progressively flatter. This effect is known as octave stretching."
- ISBN 9780323142755. "The tuning of pianos is usually stretched, that is, the high tones are higher and the lower notes are lower than would correspond to the tempered scale. This can be ascribed partly to the inharmonicity of piano strings (Schuck and Young, 1943)..."
- ISBN 9781563962837.
- ^ Hartmann (1997), p.276.
Further information
- Five lectures on the acoustics of the piano
- Inharmonicity in piano tuning at the Wayback Machine (archived 3 March 2016)