Stretched and compressed tuning
Tunings do not necessarily need equaves to be tuned to their exact ratios, and in some cases, octaves are best stretched or compressed.
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. Perhaps the most notable instance of this is in the case of acoustic pianos. Since a piano's overtones tend slightly sharp from their ideal natural harmonics and do not exactly line up with the harmonic series, 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, or to achieve other artistic goals.
Examples include (but are not limited to):
- 23edo and octave stretching
- 27edo and octave compression
- Musical cells
- Pelog
- Phoenix tuning
- Zeta peak index tunings
- Optimal octave stretch (zeta stretch)
- 5- to 10-tone scales in 47zpi
- Stretched harmonic series