A pseudo-octave, pseudooctave,[1] or paradoxical octave[2] in music is an interval whose frequency ratio is not 2:1 (2.3:1[1] or 1.9:1, for example), that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency. When used as a basis for an equal temperament, the pseudo-octave may also be called the Interval of Equivalence (IoE), the Repeat Ratio, and the nonoctave.
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The stretched octave, for example 2.01:1, sounds out of tune when played with true harmonic overtones, but in tune when played with tones whose overtones are stretched equivalently.
In piano tuning, stretched octaves are commonly encountered, where the inharmonicity caused by string thickness and tension makes it necessary to widen every interval very slightly.
The octaves of Balinese gamelans are never tuned 2:1, but instead are stretched or compressed in a consistent manner throughout the range of each individual gamelan, due to the physical characteristics of their instruments. Another example is the tritave of the Bohlen–Pierce scale.
Stretched octaves are caused by the physics of standing waves in a stretched wire. The pitch of each overtone produced by a piano string is determined by the ratio of the string's restoring force (expressed as a spring constant), divided by its mass per unit length.
In an ideal piano string, the only restoring force would be due to the tension in the string. In practice, piano strings are made from high-carbon steel, which is stiff. The stiffness adds an extra restoring force to each string; the amount of this extra force depends on the amount of bend being induced in the string. Higher normal modes bend the string more, inducing more stiffness-related force and sharpening the pitch of the resulting overtone.
Octave stretching is less apparent on large pianos which have longer strings and hence less curvature for a given displacement; that is one reason why orchestras go to the expense of using very long concert grand pianos rather than shorter, less expensive baby grand, upright, or spinet pianos. (The other reason is that long strings under high tension can store more acoustic energy than can short strings, giving larger instruments more volume and better sustain than similar, smaller instruments).