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my question is about the importance of Octaves vs harmonics? What is the difference between these two concepts?

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    $\begingroup$ What do you mean by importance? Is there a certain application you are interested in or question you are trying to answer? $\endgroup$ – BioPhysicist Mar 4 '18 at 20:24
  • $\begingroup$ We can tell you that an octae is meant to have twice the fundamental frequency, so that it enhances maxima of the fundamental frequency when played in phase, that the ear recognises it as the same note... but as said, it's not clear what you're asking. $\endgroup$ – FGSUZ Mar 4 '18 at 20:30
  • $\begingroup$ Oh thanks this is what I want to know ( to enhance the maxima ) $\endgroup$ – Day Light Mar 4 '18 at 20:33
  • $\begingroup$ But I really cant understand something :(I know the difference between harmonic and octave and both enhance the maxima but why do both concepts exist $\endgroup$ – Day Light Mar 4 '18 at 20:36
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One octave is a factor of two in frequency. The importance of it is when you study sound generated by a string that it pinched at both ends, or a pipe that is open at both ends, the main frequencies it generates will be at its fundamental (lowest) frequency and one octave up (the first harmonic). Higher frequencies that are integer multiples of the fundamental will also be present, but to a lesser extent.

A musical instrument generates the frequencies it does because it has a resonant response to them. So, when produce any of the harmonic frequencies the instrument responds to by some other means, the instrument will vibrate in response. It is in that sense that the sounds are similar.

The other factor to consider is a phenomenon called "beat frequency". Here's a video demonstrating both resonant response and beat frequency. What's happening is that the effects of the wave in air literally add up. Sometimes, when both wave peaks and troughs line up, they produce a louder sound. Other times, when the peaks of one wave are lined up with the troughs of the other, they cancel each other out. Because their frequencies are different, they alternate between the two states, producing a "beating" change from loud to quiet. This happens at a rate fixed by the difference in the two tones, called the "beat frequency".

When the beat frequency is lower than the lower limit of human hearing, it sounds like a modulation in loudness of an audible tone, as in that demo. When it is within the realm of human hearing, the beat frequency will sound like a tone. If you use an audio program, like Audacity, to generate a 250 Hz tone and a 350 Hz one, and play them simultaneously, you'll hear pitch power than either, concentrated at about 100 Hz (the beat frequency). It won't sound like a pure tone, though, it'll have a rougher feel to it. That's not accidental. Human screams and other animal alarm calls are known to oscillate rapidly in loudness, so hearing a moderately high beat frequency triggers unpleasant feelings in us.

Why bring up beat frequencies at all? Well, frequencies that are an octave apart produce a beat frequency at the frequency of the lower frequency tone, making a waveform that doesn't have so much of the rapid volume oscillations. Since such small integer harmonics (or near so) are an unavoidable part of generating any sound, they don't raise alarm bells in our psyche.

Now, what I've said here should not be generalized too much, since we obviously enjoy hearing chords played during a musical performances that should, in this naive picture, produce dissonant beat frequencies.

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