Stretched tuning
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. 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). "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."[3]
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[edit]
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.). (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 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.
Intervals and inharmonicity[edit]
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.
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.
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.
Effects on tuning[edit]
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.