2pi....full circle....every negative even interger........
In order to understand what an octave and a harmonic is, you need to study the phenomenon called the overtone series or the harmonic series.
A single note vibrating produces other notes above itself that we call harmonics. An octave is one of these harmonic notes produced by the original note vibration. If the frequency of your fundamental note is 110Hz, then an octave higher is twice that (220Hz) and another octave higher than 220Hz is 440Hz and so on. Both of these higher octaves are produced by the fundamental note. These are even numbered harmonics (2, 4, 8 etc). The odd numbered harmonics make notes that are more distinctly different than the fundamental. If your fundamental is A you would get the following:
A A E A C# E G A
1 2 3 4 5 6 7 8
These notes would all be harmonics of A and the octave is just one of those harmonics.
These harmonics are much quieter than the fundamental. We can make these notes louder by playing them on other strings in a multi string instrument or with another instrument. This series is how we know what notes sound consonant or dissonant compared to another note.
An octave is just an important special case of a Harmonic. A harmonic is a frequency that is a whole number multiple (2, 3, 4 etc) of some fundamental frequency, and an octave is double the fundamental.
Harmonics are important for two reasons: (i) they tend to arise naturally whenever you have a long thin "linear" resonator like a violin string or an organ pipe and (ii) any sound that has a periodic waveform, even if it's not from a linear resonator (e.g, a person's singing voice) can be decomposed into harmonics by the technique of Fourier series, and our ears are especially sensitive to harmonic combinations.
Octaves are important musically because the 2:1 ratio is so natural to the ear that the octave sounds like "the same note just higher".
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