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The term 'octave' is the name for an interval(space) between two notes.

The frequency of the note is doubled, or halved, depending on whether the interval goes up or down in frequency. The two notes are heard toghether as the same, and will also have the same name.

For example,

an octave above A(440 Hz) is A(880 Hz)

an octave under A(440 Hz) is A(220 Hz)

To answer the question, the similarity between the keynote and an octave above it, is that the ratio between the frequency of these notes is 1:2 and of course, their names.

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Q: What does the keynote and octave have in common?
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