I find this hard to believe from a mathematical point of view - octaves are as far as you can get from arbitrary which is I guess what gives the headline it’s shock value.
There are multiple definitions of the word octave. Some have to do with frequencies and some have to do with musical scales. Furthermore, no common instrument emits a pure tone, there are always overtones and other frequency components (aka timbre). If the 'note' on the instrument is based on the fundamental frequency, there's no guarantee that all of the modes in the timbre are going to re-align at double the frequency of the fundamental.
So, basically, we're all talking past each other (again).
Having a harmonic at twice the frequency is not arbitrary. Lots of things that vibrate will have a fundamental frequency and harmonics at integer ratios. The name "octave" implying 8 is a bit more complicated, but if you just treat it as a name for a moment, the "octave" is the ratio 2:1. Another common ratio is 3:2, and again, if you ignore the meaning of the name, it's called the "fifth".
So why 8 notes per octave? Well, it's really 12 notes per octave. Maybe think of it as the white and black keys on a piano. 8 white keys gets you back to where you were, but you skipped some black keys along the way. So why 12 notes per octave? Well, that's because (3/2)^12 (1.5 to the 12th power) is almost a power of 2. So if you step up by fifths 12 times, you very nearly land 7 octaves up. Each of the notes you stepped on along the way becomes one of the 12.
Heh, there are also "fourths" at a ratio of 4:3. So music has a "fourth" plus a "fifth" equals an "octave"! It's kind of silly :-)
Anyways, that's the quick version. If you go further down this road, there are "wolf fifths", various tunings with subtle (but perceptible) differences, and you can even find 19 and 31 tone scales. It kind of goes on and on.
There are 12 semitones in an octave because western scales are based on the mathematical coincidence that 1.5^12 ~= 2^7
Because of consonance/dissonance, on most of our instruments, some of these 12 notes sound better in combination than others.
Imagine you ran a clustering analysis to group the ones that sound better together and ended up with two clusters, a cluster of 5 and a cluster of 7.
The cluster of 7 gives us the name "octave" as musicians double-count the first note when it is repeated at 2x the frequency at the top end of the scale.
There is no particular reason, that part is certainly cultural. There aren't even seven, they are twelve equal slices, called semitones. There are infinite slices an octave can be divided into, we just picked one kind of division completely arbitrarly ~2000 years ago because we needed to build organs that sound good together and those people then thought those divisions were enough to express all kinds of emotions. Of course that process took a long time to finalize to what we call now western music.
The ear doesn't care too much about which frequencies, but that they are the same ones. That said, an octave is the doubling or halving of those frequencies which we identify as them being the same note. So, I find it very hard to believe what the article is claiming, also because one reason we can listen to music in low quality speakers is that our brain fills in the missing fundamental[0] so the doubling/halving part seems to be integral to our perception. I don't see how such a psychoacoustic effect be trained or be culturally based.
But on the other hand, tuning into a particular division is very hard to get rid of, westerners can't hear scales that have more than 12 semitones, arabic, indian and eastern music in general is like that and to a westerner ear these sound mostly dissonant and you need to spend a lot of time listening to start appreciating the expression.
Mostly because primitive flutes and horns historically had those seven notes (plus octave) [1], or they sounded good when sung or played in sequence or together. It "just made sense" to stumble upon them, because of the harmonic series [2].
Some intervals were more obvious (octave, perfect fifth, major third). Others probably took hundreds or thousands of years to be discovered.
But keep in mind that different cultures used different scales. It wasn't always the same. Some had minor seventh, others major seventh. Mesopotamians used a sharp fourth. This is still visible in different cultures today (blues/rock uses a lot of flat seventh!).
As for "why seven?": Since "mixing" major/minor sevenths/thirds/etc is very dissonant and weird, people ended up having seven notes regardless.
I would say that "note choice" was more cultural, but the options were obviously influenced by the harmonic series. After a while people started seeing patterns and those historical scales converged into the major scale we know today.
Later in the 1500s some geniuses found a way to transpose the scales but still maintain the ratio between notes, but without having to retune the instrument. The trick was to divide the octave in 12 notes but only use seven at a time [3]. It wasn't "perfect" like just intonation [4], but it was in the ballpark. That became the new normal. Equal temperament is not perfectly in tune with the harmonic series, but people got used to it (to the point that just intonation sounds "off" to a lot of musicians).
The harmonic series (of which the first three intervals are octaves, fifths, fourths in pythagorean tuning) is not arbitrary.
How you construct a more useful set of notes from the harmonic series is arbitrary. The 1.5^12 ~= 2^7 coincidence I note above allows you to construct a scale using octaves and fifths. You can just as easily do it with other similar coincidences, use fifths and fourths instead, etc etc.
Check the smoothness/roughness functions in my blog link - the reason you think e.g. a 3rd sounds as 'right' as a 7th is likely purely cultural. Other cultures have other scales. One of the Indian ragas has over 100 notes.
(I'm aware OP claims it's ALL learned, including the octave, but even if true I don't think that means it's all cultural).
An octave is the interval between one musical pitch and another with double its frequency, which isn’t arbitrary; however the number of notes within an octave (and, I think, their exact tuning) is arbitrary.
If you play two sine wave tones of different frequencies at the same time you get beats[0] caused by the alternating constructive and destructive interference.
Helmholtz hypothesized[1] that the dissonance of a pair of sine wave tones was related to these beats. Slow beats sound like a pleasant vibrato effect. Extremely fast beats are not perceived as beats at all, with only two separate tones heard. Only moderately fast beats sound dissonant.
This was confirmed experimentally[2] by Plomp and Levelt.
Sethares generalized this relationship to arbitrary sounds[3], finding that an amplitude-weighted sum of the consonance of all pairs of partials ("partial" meaning one of the sine waves that forms part of the waveform, as can be found by Fourier transform) well approximated perceived consonance.
Most Western musicals instruments are harmonic or approximately harmonic[4]. They produce a waveform with partials of frequencies that are an integer multiple of the lowest frequency partial (called the "fundamental").
Increasing pitch by an octave doubles the frequency of all partials. An integer multiplied by two is still an integer, so if you play harmonic notes separated by octaves the partials will overlap. All pairs of partials will be either identical or far apart, so none form dissonant beats. This maximizes consonance.
But music with only octave intervals would be very boring, so the octave in standard Western music theory is divided into 12 equal parts. This is an excellent choice for harmonic instruments, because it closely approximates several small-integer ratios. The interval of a "fifth" (actually seven steps away in the octave, but music theory uses strange numbering to simplify playing the most common musical styles) is a frequency ratio of 3:2. This results in half the partials overlapping, and the other half still being positioned so they avoid dissonant beats, so the fifth is also highly consonant.
Wikipedia has a graph comparing equal divisions of the octave with small-integer ratios:
You can see that 12 divisions has many useful approximations. It represents a good balance between complexity and musical utility, so I don't think it's surprising that it became the standard.
But note that small-integer ratios are only consonant with harmonic timbres! If the partials are not integer multiples of the fundamental, as is often the case in tuned percussion, you need a different tuning system. Indonesian classical music[5], which makes heavy use of tuned percussion, is famous for this. You can use Sethares' model to generate tuning systems suitable for arbitrary timbres, e.g. https://sethares.engr.wisc.edu/mp3s/morphine_crystal.html
I'm a musician but I was completely tone-deaf before I started studying, and took me a while to recognise even octaves. In my head it's still the "SomeWHERE Over the Rainbow" interval ¯\_(ツ)_/¯.
Said that, when playing two octaves together vs other intervals it's easy for a layman to notice how in tune and how consonant they are, because of beating.
> Said that, when playing two octaves together vs other intervals it's easy for a layman to notice how in tune and how consonant they are, because of beating.
I think the point was that this beating depends on the timbre of the instruments (e.g., the fact that overtones are integer multiples of the fundamental). For many percussion instruments this is not the case. You can actually shape a collection of bells so that octaves sound very dissonant (lots of beating) while a slightly different interval sounds consonant.
