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Off-key is musical content that is not at the expected frequency or pitch period, either with respect to some absolute reference frequency, or in a ratiometric sense (i.e. through removal of exactly one degree of freedom, such as the frequency of a keynote), or pitch intervals not well-defined in the ratio of small whole numbers.
The term may also refer to a person or situation being out of step with what is considered normal or appropriate.
Explanation of on-key
The opposite of off-key is on-key or in-key, which suggests that there is a well defined keynote, or reference pitch. This does not necessarily have to be an absolute pitch but rather one that is relative for at least the duration of a song. A song is usually in a certain key, which is usually the note that the song ends on, and is the base frequency around which it resolves to at the end.