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> 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.

I don't know what the true answer is, but I'd consider the overtones. Vibrating strings produce overtones whose frequencies are multiples of the fundamental. So the first overtone is twice the frequency; an octave higher. When you play that note an octave higher, you're again producing overtones that perfectly line up with the overtones that were produced by the lower octave (if we just ignore the effect of string stiffness). So it's like the same note, only missing the fundamental frequency and odd multiples of it.

This is also why the feedback from a heavily overdriven guitar amp (which emphasizes overtones) can morph the sound to the same note at a higher octave. It's simply a matter of letting the fundamental frequency grow quiet in comparison to the powerful overtones which are further amplified by feedback. I don't know if that can ever produce a different note; I don't think it could. That would require actual change in frequencies.

Different notes, by comparison, produce overtones that only sometimes (or never) line up with the overtones of another note. I'm not sure but I think that also explains why some notes together sound dissonant while other combinations make good chords. If you're trying to transcribe a piece of music by analyzing its spectrogram (which I do because my ear isn't very good), you'll find that the notes that are usually hard to tell apart are the ones that harmonize well and have overtones that line up.

That’s the usual explanation I read, but I don’t fully buy it. Yes, two notes an octave apart have lots of overtones in common. But that’s true of two notes a perfect fifth apart as well. Or even better, two notes an octave and a fifth apart.

The overtones of two notes a perfect fifth apart do not line up, as already the perfect fifth itself is not an overtone of the base tone. This is indeed the case for the octave+fifth, but there you've got already clearly a second order, as it's exactly 3 times the frequency of the base while the octave is 2 times the base frequency.

For this exact reason, in music theory, the perfect fifth is seen as a note that amplifies/emboldens the root note of a chord and does not add a particular harmonic color.

In jazz it's often left out to open up chords more (i.e. not muddy the frequency distribution with too many adjacent notes unless you explicitly want the powerful sound). In pop the nondescript nature is used to create a strong sense of grounding in the scale; the root note of the scale and its fifth fit over every possible diatonic root note (play c-g in your right hand with root notes c,f,a,g in the left one after the other, it will sound familiar). In rock they literally call the perfect fifth a 'power chord' and they play melodies with it. It fattens up the melody note without adding any harmonic identity that would create tension with the rest of the harmonic content of the song.

For sure. It still doesn't explain why the octave is the specific point where the cycle of pitch perception repeats. The perfect fifth is very consonant, for sure, but there is no "perfect fifth equivalence" in any musical traditional as far as I know.

Note that there is no harmonic that is a perfect fifth above the fundamental; the first harmonic is the octave. So if cycles of perception have to be based on harmonics (multiples of frequencies), the next plausible one after octaves (2f) would be based on the perfect 12 (3f), rather than the 5th (3f/2). The 4f based cycle is really just octaves again, except we're skipping every other octave. It gets too distant after that. Maybe the 3f progression does contain a cycle of pitch perception; I will try giving that a listen.

I tried this! Eerily, I'm able to convince my ear/brain that this 12th interval is an octave-like relationship; that the two notes have a sameness (that I don't perceive in the case of the perfect fifth that we normally consider enharmonic with the 12th, giving it the same letter name).

Come to think of it, a lot of western harmony is based on a two-octave "gamut". Like for instance alterations to chords such as dominants take place close to this "3f octave" (dodecade?) 12th interval, like the like 11th, 13th. The usual explanation is that if these alterations are in the higher octave, it prevents certain dissonances. But from the "3f octave" view, we can just regard them as different notes; that the 13th is not simply the 6th, only one octave higher, but rather an "augmented dodecave" interval.

People have experimented with alternative "octaves". For example the Bohlen-Pierce scale (https://en.wikipedia.org/wiki/Bohlen–Pierce_scale) has an "octave" with a 3:1 ratio (and thirteen notes to the octave).

Vocal cords have even more pronounced overtones than strings. To a first approximation, they produce a series of pulses. And we are exposed to these harmonic-rich sounds from the womb.

It would be an interesting (and perhaps cruel) experiment to expose babies to synthesized sounds that have a harmonic structure with different mathematics. For example, sounds with lots of sqrt(2) frequency ratios.

The ear itself produces some harmonic distortion, at moderate levels. I don't know if that's important enough to play a role in our pitch perception though.

I disagree. A sine wave has no overtones but we can still easily perceive the octave notes.

Also overtones don't have to be even multiples of the fundamental. Many organs have stops that are not multiples of the fundamental. 2-2⁄3′, 1-3⁄5' are common flute stops.

> I disagree. A sine wave has no overtones but we can still easily perceive the octave notes.

Sure we can. Does the explanation given above require that we shouldn't? Of course, laying sines an octave apart is a good first step to building a simple additive synth that doesn't sound as dull as a pure sine. In that case, are you adding a note or an overtone? If anything, to me this only makes the given explanation more credible. The relationship between common naturally occurring harmonic overtones and the octave.

> Also overtones don't have to be even multiples of the fundamental. Many organs have stops that are not multiples of the fundamental. 2-2⁄3′, 1-3⁄5' are common flute stops.

Sure, there are dissonant sounding instruments as well as instruments that can sound more like a chord when a single note is played. Of course, we can synthesize anything. On the other hand, many instruments are prone to producing overtones that increasingly deviate from the harmonic series. And many tricks are employed to try and reduce that effect, because it doesn't sound good.

> Okay, not quite. You actually also need to know that the frequency ratios of all pairs of keys n keys apart should be the same, for all n.

We could have made a different second assumption instead, and remembered that not only is the note an octave higher twice the frequency, but also, the note a perfect fifth higher is one-and-a-half times the frequency. Propagating these axioms to tune all the notes would lead to Pythagorean tuning, which has some nicer-sounding intervals but makes playing in certain keys sound bad, since you can't make all n-key intervals exactly the same size.

> 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.