Stretched tuning

If the widths of the keys of a piano keyboard were stretched as the intervals between the corresponding notes are in stretched tuning,^[1]^[2] it would look something like the above.

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.

^ Campbell, Murray and Greated, Clive (1994). The Musician's Guide to Acoustics, p.257-58. 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."
^ Carterette, Edward (1978). Hearing, p.525. 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)..."
^ Hartmann, William M. (1997). Signals, Sound, and Sensation, p.275. ISBN 9781563962837.
^ Hartmann (1997), p.276.

[1] Campbell, Murray and Greated, Clive (1994). The Musician's Guide to Acoustics, p.257-58. 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."

[2] Carterette, Edward (1978). Hearing, p.525. 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)..."

[3] Hartmann, William M. (1997). Signals, Sound, and Sensation, p.275. ISBN 9781563962837.

[4] Hartmann (1997), p.276.

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