> Of course, there are so many more details on every side of this topic. Why 12 divisions per octave (this certainly isn’t the case for all musical traditions)?Earlier systems had you tune instruments so that notes within an octave would be tuned to a few different simple harmonic relationships to a root key. This has the downside that playing a tune in a different key yields entirely different frequency relationships to the root. Twelve tone equal temperament is a sort of compromise in that it produces pitches that only roughly correspond to these simple relationships, which it trades for consistent relationships in any key. A few of these simple relationships are common across multiple musical traditions. In the Arab tone system there are musically significant intervals that don't have any nears in 12-TET. At some point it was modernized to an equally tempered scale, but with 24 tones per octave instead of 12.
If you listen to a string ensemble or a choir, you can however often hear that they tend towards the simple harmonic relationships rather than their corresponding equal-tempered frequencies.
> And, most importantly to me, since it’s the question I can find the least solid information on: why do we take for granted that one note with twice the frequency of another note sounds so similar that we call both notes by the same name? I have a pretty good understanding of other details of music theory, but I’ve never gotten a straight answer on why (and indeed to what extent) octave equivalence exists.
Maybe thinking of sounds in terms of sets of overtones is useful. If we define sound similarity (of largely harmonic and overtone-rich sounds) as the size of the intersection of their sets of harmonic overtones, the octave is the most similar you'll get to the root because half of the harmonic overtones of the root also appear in the octave. The next best interval in these terms is the perfect 12th, which has a third of all the overtones in the octave.
Of course this is oversimplifying and the overtones should probably be weighed according to timbre and the limits of hearing, but I think it gets the general idea across.
These integer harmonic intervals (octave and perfect 12th) are also unique in that played together with the root, they don't produce any undertones.
The relationships within the octave are more complex and don't have integer harmonic relationships to the root. They occur early in the harmonic series, but the root does not coincide with the base frequency of the series, so they produce undertones. For example, a perfect fifth (3:2) played together with the root will produce an undertone an octave below the root. If you sum two sine waves with this frequency relationship together, you'll see that the resulting waveform cycles at half the frequency of the root. For very small ratios like that of the minor second (16:15 in some tuning systems) the overtone series in which both appear starts at a 1/15 of the frequency of the root note, so the length of this cycle is much longer, thus appearing dissonant. If the interval is really small, though, or if you play a very low root note, this undertone ends up at an inaudible frequency and will be perceived rather as a slow timbral modulation than a dissonance.
Of course, the perfect fifth and the perfect 12th don't appear at all in 12-TET.
