Stretched and compressed tuning

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Stretched tuning

In stretched tuning, two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence).

In compressed tuning, also known as narrowed tuning, two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or narrowed equivalence).

In 12edo

Stretched tuning is used even outside of a xenharmonic context. Most acoustic pianos have overtones which do not exactly line up with the harmonic series, so stretched octaves are usually used to compensate.

In xenharmonic music

Within a xenharmonic context, stretched or compressed tuning may be used to reduce the harmonic entropy of a scale without sacrificing its melodic shape.

Examples include:

• 23edo and octave stretching
• 5- to 8-tone scales in zeta stretched 15edo
• Musical cells
• The Riemann zeta function and tuning#Optimal octave stretch