They are made up of a series of overtones. By far the loudest of these overtones is the 2nd harmonic... which is double the base frequency aka one octave.
This is not some artificial construct but a fundamental part of how things vibrate.
Not true in general. There is no reason to expect the 2nd harmonic to be louder than the others. Some instruments produce mostly odd-numbered harmonics, such as the clarinet.
> This is not some artificial construct but a fundamental part of how things vibrate.
That's only true in a one-dimensional world. Instruments in the three-dimensional world are either mode-locked (not freely vibrating) or have some amount of inharmonicity. This is why the octaves on a piano are not tuned to 2:1 ratios, as I mentioned.
From the article you cited:
> This list of frequencies is for a theoretically ideal piano. On an actual piano the ratio between semitones is slightly larger...
For more details, see:
https://en.wikipedia.org/wiki/Stretched_tuning
Or,
https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_...
It seems the same to us because we're building on hundreds of years of music that tells us it is. If you're comfortable with the tonic moving, the octave is no more "the same" than any other note.
Most of what we take for granted in music is learned through exposure.
Just because a mathematical relationship exists doesn't mean that we perceive it.
Not all music is built around 2:1 octaves. People who grew up listening to music with 2:1 octaves can identify pitch classes, people who grew up in isolated tribes without exposure to that kind of music--hear C and C' simply as different notes. People with absolute pitch sometimes have the opposite problem--they can accurately recognize the pitch class of C or C', but are more likely to mistake C for C' or vice versa.
Real music often deviates from the 2:1 ratio anyway. Only certain instruments produce a near pure harmonic spectrum, most instruments have some inharmonicity, and these instruments are often played according to their spectrum, rather than the harmonic series.