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Example: If I play an octave on the piano starting on middle C.

To my limited understanding the simple ratio of 2:1(due to sound-waves matching) and the multiple overtones that match up in this octave create a pleasing sound (consonance).

1. I was wondering what of these two phenomenon has more importance or impact? Does the overtones just add more richness to the sound/consonance?

2. Does simple ratios and musical intervals come from the overtones or vice versa?

3. Is it just a coincidence that there existed "secret" harmonics when Pythagoras first started researching intervals and ratios in music?

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To my limited understanding the simple ratio of 2:1(due to sound-waves matching) and the multiple overtones that match up in this octave create a pleasing sound (consonance).

For the purposes of thinking about consonance, don't think about the fundamental and the overtones of a sound separately; think about the frequencies of all the partials (the fundamental and all the overtones). This answer explains how the relative frequencies (and strengths) of the full set of partials combine to give the perceived consonance.

I was wondering what of these two phenomenon has more importance or impact?

As above, there aren't two phenomena - just one.

Does simple ratios and musical intervals come from the overtones or vice versa?

Normally, most of the overtones of a note have a simple frequency relationship with its fundamental - so if the fundamentals of two notes have a simple frequency ratio, there will be many pairs of harmonics that also have simple ratios in their frequency relationships, leading to a consonant sound.

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I'm not sure that either is of great importance in the following sense. The consensus (at least in European and European-influenced music) about consonant and dissonant intervals hasn't changed much. Styles have changed but the assignment of octaves, fifths and fourths into "perfect consonances"; major and minor thirds and sixths into "imperfect consonances" and the rest into "dissonances" hasn't changed much. What has changed is that use of dissonances follows different rules from the past. The most obvious (because it's been extensively publicized for 400 years) is that one can play a dominant sevenths chord without preparing the seventh (although the seventh is often resolved.) In blues, these sevenths are neither prepared nor resolved.

These judgments seem to be the same in 12-tet, Pythagorean or just intonation.

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