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Comment by wonger_ | Hacker News Reader

I am very unfamiliar with the topic. Here's my takeaways from the article:

  - almost all music I've heard uses 12TET
  - JI is microtonal / an alternative tuning to 12TET
  - you're the author of the lovely ambient.garden, which uses JI
  - the audio widgets and visualizations were very helpful
  - JI has perfect harmonies, and 12TET approximates those enough to still sound good
  - JI opens up cool chord options
  - but there's extra pitfalls when making JI chords compared to 12TET 
  - still not sure what consonant, continuous, or misleading chords are. I only know what dissonance sounds like
  - still haven't looked at numbers; all I know is that there's ratios in there somewhere

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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 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

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.

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.

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.

One definition of consonant intervals are those that share many overtones/partials in common [1]. This reduces the phenomenon of "beating" which is readily perceived as unpleasant dissonance. (Though small degrees of "beating" can be pleasant.)

[1] https://en.wikipedia.org/wiki/Consonance_and_dissonance#Acou...

> still not sure what consonant, continuous, or misleading chords are. I only know what dissonance sounds like

Consonance is the opposite of dissonance; so while a dissonant chord sounds jarring and unpleasant with beating, a consonant chord sounds harmonious, pleasant, smooth.

I'm not sure about the other two either, I don't think they're commonly used terms.

Maybe continuous refers to "continuo" in Baroque music, where the bassline and chords are separate but work together harmonically - in some tunings, perhaps this is harder.

I think misleading refers to chords where their function harmonically is ambiguous until they resolve "home" at a later time - for example, not using the root note of a chord, instead using an inversion or modulation that feels like the chords could lead in several directions, creating a kind of uncertainty and tension in the piece that feels satisfying to resolve when the later chord indicates which of the paths through the harmony were the "correct" one.

this might help, primer on JI ET and pythagorean intervals for violinists in string quartet context. Good demos 30 seconds in

https://www.youtube.com/watch?v=QaYOwIIvgHg

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