Editing Pseudo-octave (section)

==Stretched octave==
{{main|Stretched tuning}}

The '''stretched octave''', for example {{nobr| 2.01 : 1 ,}} rather than {{nobr| 2 : 1 }} (an 8.6&nbsp;[[musical cent|cent]] pitch difference), sounds ''out of tune'' when played with ideal [[harmonic]] [[overtones]], but ''in tune'' when played with lower notes whose overtones are themselves naturally stretched by an equivalent amount.

In [[piano tuning]], stretched octaves are commonly encountered in instruments where string thickness and high string tension causes some strings to approach their [[elastic limit]], which makes the string respond to stretching and bending with a pull to restore its original shape and position a little [[non-linear|out of proportion]] to how far it was bent or stretched. That [[non-linearity]] causes small differences between the string's [[partial (music)|real overtone]] frequencies and the mathematically ideal [[simple harmonic oscillator]]'s integer multiple [[harmonic]]s. The so-named ''"piano-tuners' octave"'' used to compensate for the non-harmonic partials is well approximated by the [[Piano_acoustics#The_Railsback_curve|''Railsback curve'']] (which see).

The effect of strings' small [[elasticity (physics)|inelastic response]] is that rather than the simple [[harmonics]] expected for its overtone series, which would all be integer multiples of the [[fundamental frequency]], the [[timbre]] of the note that the string actually produces has slightly [[inharmonicity|inharmonic overtones]]. In detailed discussions of pitch and tuning the actual overtones in the sounded note are called [[partial (music)|''partial tones'' or ''partials'']], in order to avoid confusing them with the more familiar, mathematically simple integer harmonics; both are often relevant in the same sentence. Partials measured in the sounds produced by real musical instruments almost always have a slightly higher pitch than the corresponding idealized harmonic, with the discrepancy being less important for high-pitched instruments (above {{nobr|{{gaps|5|000}} [[Hertz (unit)|Hz]])}} whose high-level overtones fall above the range of [[human hearing]].

The practical consequence of the discrepancy between the sharpened pitches in a [[bass (sound)|bass note's]] overtone series that [[treble (sound)|treble notes]] must match, makes it necessary to widen every interval very slightly. Generally, it's more than sufficient to sharpen only whole octaves slightly, rather than separately modifying all intervals that reach individual pitches in the upper octaves (''see'' [[stretched tuning]]).

The octaves of [[Bali]]nese [[gamelan]]s 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.{{citation needed|date=August 2018}} Another example is the tritave {{Audio|Tritave on C clarinet.mid|play on clarinets}} of the [[Bohlen–Pierce scale]] (3:1).

Octave stretching is less apparent on large pianos which have longer strings and hence less curvature for a given [[Displacement (vector)|displacement]]; that is one reason why orchestras go to the expense of using very long [[grand piano|concert grand piano]]s rather than shorter, less expensive [[baby grand piano|baby grand]], [[upright piano|upright]], or [[spinet|spinet piano]]s.{{citation needed|date=March 2020}}  (Another reason is that long strings under high tension can store more acoustic [[energy]] than can short strings, making larger instruments [[loudness|louder]] (hence making a single piano better able to be perceived over the volume of an entire orchestra) and giving them longer sustain than similar, smaller instruments.{{citation needed|date=March 2020}})