This is incorrect. If you watch a video like [0], the squiggles aren't real, they're an artifact of a rolling shutter camera. A real slowmo camera will correctly show the entire string vibrating[1].

The rest of the article is correct, but you can't see harmonics happening to the string.

[0] https://youtu.be/XOCGb5ZGEV8 [1] https://youtu.be/6sgI7S_G-XI

He keeps writing "for western people" but some parts of these are inherent in the human ear evolution and rather universal. All around the world we can find pentatonic music for example, even from ancient peoples, and this includes e.g. West African cultures, China, etc. And traditions that have microtonal inflections will still place the same emphasis on the octave, the fifth, major/minor third, etc the microtones add different flavors but it's not some widely different thing, which is why e.g. middle eastern or Indian songs e.g. can still be played on pianos, simplified (to the nearest approximation) but still retaining a lot, just losing their full flavor.

My cs department had a cool project class where you built what was basically a raspberry pi with a microcontroller by hand, and you had to use the dumb speaker and controller to make your own music firmware to produce notes. the challenge involved, was basically, the processor’s clock wasnt fine grained enough to produce perfect notes. I wanted to make a simon says toy but the notes were off. I approached my professor with my problem and he said I could cheat the processor clock in a clever way to get what i wanted and it was such a “oh wow computers are magic” to me, i got the notes i wanted. disappointingly the TA grader wasnt that impressed but that proff ended up offering me a job before I graduated.

A 12 TET chromatic is 2^(1/12), and a 12 TET fifth would be 2^(7/12). A perfect fifth is a 3:2 ratio. Those numbers are slightly different, and that’s enough to understand it. Another way of thinking about it is that if you were to complete the cycle of fifths purely by stacking fifths, you should end up on the note you started with but many octaves higher. But you should be able to see that starting on C1 and going by octaves will produce a number that is purely powers of 2, whereas stacking fifths will necessarily involve powers of both 2 and 3, so they can never be equal, I can stack fifths and never land on my original note’s octaves.

On a church gig in the 90s, I encountered an organ which was not tuned in equal temperament so that playing guitar with the organ always sounded out of tune (something I only discovered once Mass began since we had rehearsed with a piano) and I had to switch to bass to be able to play an accompaniment that sounded decent.

Another way to think of it is that they have to hit every pitch without assistance from the instrument anyway, so they learn to make every note sound “good” rather than hitting a mathematically defined frequency.

Can't we have a system that is optimized for the notes that are actually played in a song rather than the hypothetical set? And what if the optimization is done per small group of notes rather than over an entire song?

I probably haven't tuned my guitar to concert tuning for a long time.

I tried rocksmith and often tuned to that otherwise I just keep it in tune with itself and what approximately sounds right to me.

My fingers are too fat for any precision to matter too much. So long as it's in tune with itself intonation is vaguely right and the action is acceptable no one will notice my solo playing in the garage by myself is out of tune are the fifth harmonic.

https://strandbergguitars.com/en-WW/magazine/true-temperamen...

They solve exactly for this issue, and sound amazing in use. The downside is that you are somewhat locked into a given tuning.

Alternatively you can take the approach of guitars with movable frets so you can adjust them per tuning.

https://youtu.be/EZC69A8TsJ8?si=7hUIb7FEKb45eV_L

These are generally used for microtonal playing but can also effectively be true temperament as well.

Guitars with gut frets used to have adjustable positions, which allowed for some mitigation via changing fret positions too

But, unless you mainly play stacked fourths, why would you make it a requirement? You can, for instance, tune instead to get pure fretted fifths between adjacent strings, and fretted octaves between strings one removed.

The real reason you can't get your guitar in tune is one which makes none of the above matter. Most guitars don't have good intonation. Most acoustic guitars don't have movable saddles to set intonation at the bridge. Electric ones do. For accurate tuning, you need not only compensation at the bridge, but also at the nut.

https://guitarnutcompensation.com/

On my main axe, I installed a small screw next to the nut, right under the G string. Just doing the G string makes a huge difference!

Here is a test: play an open D power chord (open D, A on G string, D on B string) it is very clean. Now release the A to play a 1-4-8 G power chord (open D, open G, D).

On my compensated guitar, both of them are crisply in tune. Without nut intonation, one of the two will have ugly beats. If you tune one, the other goes wonky.

When I first heard how good it is after putting in the compensating screw, I was astonished and at the same time filled with the regret of not having done it decades earlier.

Why the G? The unwound G string on electrics is the most susceptible to bad intonation at the nut, because it undergoes the greatest pitch change when it is fretted. Guitarists like to bend that one for the same reason. Fretting it at the first or second frets makes it go markedly sharp; for that reason we need to shorten the distance between the nut and the first fret to get that sharpened interval back down to a semitone.

This is less of a problem on guitars with a wound G, which has a lot more tension in it to compensate for its weight, and doesn't pitch-bend nearly as easily.

Maybe this was implied by the "just tuning"?

https://m.youtube.com/watch?v=WkYJb21hpyA

This is presented as fact, but as I understand it there is no conclusive evidence for what Bach intended wrt temperament. There is a theory that the title page of the Well-Tempered Clavier encodes Bach’s preference in the calligraphic squiggles, but this is a recent theory and speculative. I don’t believe there are any direct statements by Bach as to his intention.

Our perception of these things (for most, not all) is incredibly fluid, much like our perception of time. Music that moves us tends to have the right "technical imperfections". Too much and it comes off as amateur, too little, and it comes off as sterile.

Even on a production-level, the right amount of harmonic distortion/non-linearity can be a huge benefit to how sounds are perceived. The amount of soft-saturation tooling in modern electronic/in-the-box music production is wild. Almost every modern plugin seems to include some kind of "warmth" control now.

Yet another example how perfect reproduction doesn't sound quite right.

Advanced banjo players will sometimes use harmonics for a ‘bell’ effect. Here’s a short video from Alison Brown, a great player.

https://www.youtube.com/shorts/NJDgpw2jIdc

I have software I use when I tune my Bosendorfer 290 that calculates the stretch. Of course, the final tweaks are done by ear.

However if you want more notes than that to be their best you're going to have to compromise and work at it a bit.

Now if you want the instrument to sound its absolute best on its own solo, a slightly different place for some strings.

And then depending on other musicians you are playing with and the way their tuning has achieved perfection (or not), some further tweaking can make a big difference.

And that's after accepting that the "knobs can only be in one place".

For students to get really good at the tuning process can require a few extra years of everyday practice more than it does to learn to play a few pieces.

Part of the limitation is the way only a few minutes of tuning are spent for every hour of practice, if that.

I'm a relatively new adult beginner on the violin, and one of the fascinating (and extremely difficult) things about un-fretted string instruments is the player has the freedom to shift the tuning around to fit the context. On the violin, we normally play melodies and scales using Pythagorean tuning (which is actually a misnomer as Pythagoras didn't invent it, the ancient Mesopotamians did), which is based on the circle of fifths and leads to wider whole steps and narrower half steps than equal temperment tuning. But then for double stops (i.e. chords), and especially when playing in a string quartet, just intonation, which is based on the harmonic series, is used so the notes sound concordant. This page describes all the different tuning systems a violinist may use, also including 12 TET when trying to match a piano: https://www.violinmasterclass.com/posts/152.

This video shows how challenging it can be when trying to adjust intonation when playing in a string quartet: https://youtu.be/Q7yMAAGeAS4 . Interestingly, the very beginning of that video talks about what TFA discussed that when you tune all your strings as perfect fifths your major thirds will be out of tune.

I'll also put in a plug for light note, an online music theory training tool that was mentioned on HN a decade ago: https://news.ycombinator.com/item?id=12792063 . I'm not related to the owner in any way, I just bought access a few years ago and think it was the first time I really understood Western music theory. The problem with music theory is that the notation is pretty fucked up because it includes all this historical baggage, and lots of music theory courses start with what we've got today and work backwards, while I think it's a lot easier to start with first principles about frequency ratios and go from there.

Other notes (pun intended!): The violin is great for learning music theory because you can actually see on the string how much you're subdividing it - go one third of the way, that's a perfect fifth, go halfway, that's an octave, etc. Harmonics (where you lightly touch a string) are also used all the time in violin repertoire. Finally, the article mentions Harry Patch, but you should also check out Ben Johnston, a composer who worked with Patch and was famous for using just intonation. Here is is Amazing Grace string quartet, and you can really hear the difference using just intonation: https://youtu.be/VJ8Bg9m5l50

Music doesn't live in an abstract realm of perfections, it is an expression however formed. The fact that we can measure it is one thing. But the music or instruments do not need conform to discrete measurements to satisfy.

I know engineers hate this, but ask any musician. It's like arguing that a sitar and its scales aren't right. Absurd.

The most guitars today are still made in the style of the 1950s Gibsons and Fenders, including the neck and tuner layout. Most guitar buyers focus on the aesthetic and not the quality. I switched to a headless guitar where the tuners are at the bridge and it has a fanned fretboard giving the strings more natural tensions, the thing stays in tune and is intonated at the frets extremely well.