Interval of equivalence

The equave (/ˈiːkwɪv/ EE-kwiv or /ˈiːkwəv/ EE-kwəv), also called interval of equivalence, equivalence interval, formal octaveor pseudo-octave, is the interval such that pitches separated by it are considered psychoacoustically or formally equivalent and are elements of the same pitch class.

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Pseudo-octave

If a periodic scale has an equave, the equave is typically the same as the period or a multiple thereof.

Etymology

The term equave was coined by Inthar. It is a portmanteau of equivalence and octave.

Examples

• In octave-repeating scales, the equave is typically 2/1.
• In Bohlen–Pierce, the equave may be taken as 3/1.
• In edfs, the equave may be taken as 3/2 or less commonly 9/4.

Mathematical interpretation

If intervals and notes an equave apart are considered to be wholly equivalent to one another, and are collapsed down to a single representative interval (as is usually the case when constructing lattices), this is mathematically identical to tempering out the equave, as it is an interval separating notes that are treated as the same thing. This gives us a tool to formalize the notion of equivalence in the language of regular temperament theory – for example, octave-equivalent meantone is a rank-1 temperament that tempers out 81/80, but also "tempers out" 2/1 (although the kinds of "tempering" are treated completely differently musically, both define an equivalence class of intervals)

See also

• Period
• Stretched tuning
• Equave limit