Stretched and compressed tuning
Equal divisions of the octave tunings do not necessarily need octaves to be exactly 2:1, and in some cases where
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, 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