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The key to the article, and to all of Western music, is about a set of weird mathematical coincidences involving the twelfth root of two. In particular, that tw

The key to the article, and to all of Western music, is about a set of weird mathematical coincidences involving the twelfth root of two. In particular, that twelfth-root-of-two to the fourth power is almost exactly 4/3, and twelfth-root-of-two to the fifth power is is almost exactly 3/2. And, of course, twelfth-root-of-two to the 12th power is exactly 1. For some reason, not entirely well understood, when you play frequencies that are nice multiples together, they make pleasant patterns ("consonance"). They interfere and reinforce at regular intervals, and it sounds nice. When you don't get nice ratios, they make irregular interference patterns, and that sounds unpleasant ("dissonance"). So... take a scale (one frequency to twice-that-frequency), divide it into 12 equal parts (geometrically), and you get of those nice patterns. Each successive note is the frequency of the previous note times the-twelfth-root-of-two. That's the "twelve tone equal temperament". It makes it easy to write pretty music just by following some mathematical rules. There's a Wikipedia article on the-twelfth-root-of-two: https://en.wikipedia.org/wiki/Twelfth_root_of_two https://en.wikipedia.org/wiki/Twelfth_root_of_two EXCEPT... that those coincidences aren't perfect. They're very close, but tiny differences matter. Some intervals that should work, don't. Different scales take on different feelings, even though in theory they're just the same thing nudged up a few hertz. Anyway... that set of coincidences is the basis of all of the music you're familiar with. Other musical cultures also take advantage of it. But there's no fundamental reason why those particular coincidences should be the only interesting music, and so people look around (often, to yet more world musical cultures) for alternatives.

cjbprime 2y ago
I think you're maybe missing the main motivation, which is that "correctly tuned" overtones are in nature, in physics, in all sounds -- they aren't a social construct, unlike everything else about music. I think the argument from JI advocates is that our body's perception knows very well what a justly-tuned chord sounds like, etc.
- TheOtherHobbes 2y ago
  They aren't though. No acoustic instrument has perfectly linear overtones. And even something like a flute, which probably comes closest, has an overtone system that varies with force and input excitation. The idea of the harmonic series is a useful approximation, not a physical truth. Fourier applies to completely static waveforms, and says nothing about how waveforms change over time. If you try use Fourier for dynamic waveforms - which is most of them - you run into all kinds of problems and tradeoffs. So the concept of whole number consonant ratios is an attempt to impose a mathematical ideal of perfection where it very much doesn't exist. Clearly music is related to the harmonic series, but it's related around it, not trapped inside it. The dissonances and deviations create colour, tension, and movement. This doesn't mean xenharmonic experiments aren't worth doing, and we should all just use 12TET and not think about anything else. But it does mean these are experiments away from real acoustics into computed perfection, not back towards a perfect ratio heaven that music was exiled from by 12TET.

cjbprime 2y ago

I think you're maybe missing the main motivation, which is that "correctly tuned" overtones are in nature, in physics, in all sounds -- they aren't a social construct, unlike everything else about music. I think the argument from JI advocates is that our body's perception knows very well what a justly-tuned chord sounds like, etc.

TheOtherHobbes 2y ago
They aren't though. No acoustic instrument has perfectly linear overtones. And even something like a flute, which probably comes closest, has an overtone system that varies with force and input excitation. The idea of the harmonic series is a useful approximation, not a physical truth. Fourier applies to completely static waveforms, and says nothing about how waveforms change over time. If you try use Fourier for dynamic waveforms - which is most of them - you run into all kinds of problems and tradeoffs. So the concept of whole number consonant ratios is an attempt to impose a mathematical ideal of perfection where it very much doesn't exist. Clearly music is related to the harmonic series, but it's related around it, not trapped inside it. The dissonances and deviations create colour, tension, and movement. This doesn't mean xenharmonic experiments aren't worth doing, and we should all just use 12TET and not think about anything else. But it does mean these are experiments away from real acoustics into computed perfection, not back towards a perfect ratio heaven that music was exiled from by 12TET.