Stretched octave

A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 (2.3:1 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.

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. See: stretched tuning.

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  play on clarinets  of the Bohlen–Pierce scale (3:1).

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