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
You poke a spot where a given harmonic doesn’t vibrate, and that takes energy away from the other harmonics that do need to vibrate at that spot.
If we’re just talking about visually being able to see them, I suppose that’s a different question. Maybe on an incredibly low pitched string, or with a strobe light playing at a synced frequency? But in terms of what the string is doing, it is vibrating as the sum of its harmonics.
But you absolutely can if you rest a finger on a node and pick it, producing primarily the harmonic. You can even see the lesser vibration at the nodes with your eyes.
Actually depending on microphone or instrument interface, you can see stuff that's beyond the range of hearing too.
Also, on a low-frequency long-string like an upright bass, if it is bowed at the halfway node, you still hear mainly the fundamental but the second harmonic is naturally emphasized more than usual, and you can also see half the string making its contribution as pictured, with the naked eye.
...is this correct? You can say this about any oscillating phenomenon - that doesn't mean it's not 'real'. The "squiggles" are an artifact of the frequency of the string and the scan rate of the rolling shutter. You'll also see artifacting from a global shutter camera, where the "squiggles" will be an artifact of the string frequency and the frame (rather than scan) rate.
Or do I misunderstand?
I've been playing guitar for 25 years, and it seems to me that I can look at the "squiggles" from a rolling shutter capture of a string and tell you which string it is (and possibly, if I'm having a particularly sharp day, whether it's E or drop-D). I've never tested myself this way - am I certain to fail? :-)
The most obvious example of this would be the wagon-wheel effect, where a spoked wheel can appear to rotate at a different speed and direction than its true rotation when captured by a camera under certain conditions.
I've never tried it.
How do you distinguish vibration from squiggles? To me these seem like the same concept, at the very least over time. The moment simply doesn't matter except to neurotic people without a solid understanding of harmonics and especially of sound.
Different guitarists use different diameter strings because the diameter determines the tension when you tune to pitch. Different people prefer different tension. Most shredders prefer light tension. Most jazz players prefer high tension.
The diameter is compensated at the bridge and in some guitars the nut. When you press a thin string to a fret, the center of the string is closer to the fret than when a thick string is pushed to the fret. Thicker strings compensate for this by using slightly longer length which you can adjust at the bridge.
One type of non parallel frets are called true temperament frets. They are sort of parallel but squiggly. This results in better intonation closer to that of a piano.
Another type of non parallel frets is multi scale or fanned frets. This allows the bass strings to have a longer scale length, which allows you to use relatively thinner strings for bass notes. This is important because when strings get thicker relative to their length, they start to behave more like cylinders with thickness rather than ideal springs, and sound rather nasty because harmonic overtones are out of tune with the fundamental.
When the string's action is higher above the frets, the tension increases more when fretted than open, to a greater degree than low action.
So the saddle for that string needs to be positioned such that the plucked portion of the string is slightly longer than it would need to be if the tension were the same as the open string.
James Taylor compensates by tuning everything down a few cents, between -12 at the low E and -3 at the high E, with a little break in the pattern with -4 cents at the G to deal with its weirdness. Good electronic tuners have "sweetened" presets which do something similar.
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.
Though yaman raga is very popular and has a regular third, while other ragas still have a third-ish note, but microtonically adjusted up/down from the major and minor variants.
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.
Define off tune? 12 TET? Just intonation? Bohlen-Pierce (56 TET) ?
The "in tune" notes are as much a function of culture as physics.
Huh? Pitch ratios are not a social construct, it's just arithmetic.
But in the case if sound, I would have expected the skew to be less of a problem. Also surprised how the orof instantly know. It took me a while to figure out. How did you fix it? Cool story!
Proof is left as an exercise to the student. ;-)
That is not really true. You usually have a couple of clock sources on a MCU, but the clock gets propagated down the clock tree and the source, and most of the times, the PWM has the same source clock as the CPU. Indeed, I think if you're before the PLL the clock is more accurate as in you get less jitter but the overall drift is the same. You might have distinct clock sources but you need a specific hw and a specific configuration.
This worked well in 1980's microcomputers which used an accurate, crystal oscillator clock. IC's like the MOS6502 or Intel 8086 don't have built-in clocking. The boards were large and costly enough to afford a clock; and often it was dual purposed. E.g. in Apple II machines, the master oscillator clock from which the NTSC colorburst clock was derived also supplied the CPU clock.
These processors had no caches, so instructions executed with predictable timing. Every data access or instruction fetch was a real cycle on the bus, taking the same time every time.
Code that arranged not to be interrupted could generate precise signals.
Some microcomputers used software loops to drive serial lines, lacking a UART chip for that. You could do that well enough to communicate up to around 1200 baud.
This sounds like they were most likely bit banging square waves into a speaker directly via a GPIO on a microcontroller (or maybe using a PWM output if they were fancy about it). In that case, the audio frequency will be derived directly from the microcontroller's clock speed, and the tolerance of an internal oscillator on a microcontroller can be as bad as 10%.
That’s an issue with tuning instruments in general, and why pianos are generally slightly out of tune as a compromise.
As you get used to a particular guitar and strings, as you train your ear, you can also learn to work around the imperfections by adjusting how you hold down the strings (even with a fretted guitar, you can slightly repitch a string by holding it differently).
Been thinking of going a bit lighter recently, and also getting a classical.
Doesn't help that most tuners are still dog slow, none of the beginners courses properly tell you how the guitar actually works, or what a "chord" really is. They're all just "play this and don't worry about it". To be fair it does get you going.
Disclosure: String player.
And the fingering for a given melody may just lay across the strings better one way than another.
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.
Most brass instruments have three valves. The first lowers the pitch by a tone. The second lowers the pitch by a semitone. The third lowers the pitch by a tone and a half. If you need to lower the pitch by two tones, then you press the second and third valves at the same time, and that works fine. However, if you need to lower the pitch by three tones, then you need to press all three valves at the same time. However, that adds the length of all the valve loops together to the total length of the instrument, whereas to lower the pitch by a fixed interval you need to multiply the length of the instrument by a certain amount, and so to truly lower the pitch by three tones you need to add a little bit more length beyond that supplied by pressing all three valves together. That's what the finger loop on the tubing for the third valve is for, so you can slide it out a bit for certain low notes.
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.
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?
If your song is really simple, e.g. only consists of the 3 notes that make up a major triad (root, third, fifth), then this is definitely possible and you can just use natural thirds and natural fifths.
But as you start adding more notes, more chords and perhaps change of keys etc, it starts to break down.
That's the reason why J. S. Bach wrote The Well-Tempered Clavier. It's a collection of 24 preludes and fugues, in each possible major and minor key.
The basic idea was that if every prelude and fugue sounded good on an instrument (organ, harpsichord etc.), than it meant that the instrument was "well-tempered".
Using natural tuning instead of 12-TET would have resulted in some pieces sounding very good and other sounding very bad.
(Though something that happens in just intonation is that you often find out you need more notes than you might have originally thought, because JI makes distinctions between notes that are treated as the same in 12-TET. For instance, you might have 10/9 or 9/8 as your major second, or your minor seventh might be 9/5, 16/9, 7/4, or 12/7 depending on context.)
I don't think any just intonation guitar has been mass produced, but you can definitely build one or modify an existing guitar if you have the right tools and are willing to do a bunch of math and learn how to install frets.
This page is about a JI keyboard I built a while back, but there's also a few pictures of a couple old Harmony guitars I adapted to JI: https://jsnow.bootlegether.net/jik/keyboard.html
Here's a so-so performance of myself playing a Bach piece on a newer and vastly improved version of that just intonation keyboard: https://www.youtube.com/watch?v=rqbWnDhip0A
In 12-EDO the song has 11 distinct pitch classes. (Bach used the tritone, but not the minor second.) In my straightforward JI interpretation, I use 15 pitch classes. (I would have used 16, but my keyboard simply doesn't have a key for that note.)
You can. It’s called adaptive tuning, or dynamic just intonation, and it happens naturally for singers with no accompanying instruments.
It’s impractical on a real instrument, but there’s a commercial synthesiser implementation called hermode tuning.
You’re trading one problem for another, though. No matter how you do this, you will either have occasional mis-tuning or else your notes will drift.
I used to play fretless bass in a garage hip hop troupe that played with heavily manipulated samples that were all over the place in terms of tuning instead of locked to A440, forcing adaptations like "this section is a minor chord a little above C#".
Adaptive tuning is hard to do on a guitar because the frets are fixed. String bending doesn't help much because the biggest issue is that major thirds are too wide in equal temperament and string bending the third makes pitch go up and exacerbates the problem.
You can do a teeny little bit using lateral pressure (along the string) to move something flat. It's very difficult to make adaptations in chords though. A studio musician trick is to retune the guitar slightly for certain sections, though this can screw with everybody else in the ensemble.
Attempts to experiment with temperament using squiggly frets make it clear how challenging this problem is: https://stringjoy.com/true-temperament-frets-explained/
But with the way I played, I'm not even sure how much it mattered. The best tool for enhancing my playing would've been a mute. (And it would have been most effective lodged in my windpipe.)
You can listen to variations here: https://youtu.be/kRui9apjWAY?t=622
0: https://www.guyguitars.com/truetemperament/eng/tt_techdetail...
Additionally, some songs even change keys, which makes “per-song” not enough of a constraint.
With relative pitch music sounds the same even if you deviate from the original equal temperament pitch of the key you started singing even changing the key.
For the same reason if there is a fixed instrument playing at the same time, like a piano accompaniment, it's sound would be used as a reference and the singers would not drift
But for instruments with fixed pitches, like guitar or pianos,12 equal temperament is the best compromise to be able to play in all keys.
You might play a G# note in the context of an E chord (where it's the third), and then you might play it in the context of a C# (where it's the fifth).
These are discernably different pitches, but the same "note", in the same key, in the same 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
> Let’s begin by describing the issue with standard equal tempered frets; standard fret spacing is calculated from one single piece of information about the guitar, the scale length. This principle ignores that the frequency of a vibrating string is calculated by three factors: the mass of the string, the tension applied and the speaking length. All three of these factors are affected to different degrees each time a string is pressed down on a fret. The only way to correctly compensate for all three of these parameters is to adjust each string-to-fret connection point independently, until each note plays the correct frequency. This issue, which is impossible to solve with standard tempered frets, is what True Temperament solves.
So the true temperament system is compensating for the fact that a thicker string behaves differently when fretted than a thinner string. It still provides a 12 TET system however.
What you are probably thinking of, is a _just intonation_ fretboard, which exists and looks very different: https://projectionsliberantes.ca/en/guitars-tuning-system/
You can see that rather than squiggles, different strings have frets in completely different places.
More information is here https://www.thatguitarlover.com/blog/what-is-true-temperamen...
But this is also why I mention both fret compensation systems in my original post.
I've known a lot of musicians that have used the necks but mainly only while sponsored and none of them prefer them. Big names in the guitar world.
You're better off spending that money on a better-constructed guitar. And lessons.
So many people mistakenly think that gimmicks will make them sound better. TT necks. Fanned frets/multiscale. The right effects chain...
Maybe YOU don’t want it, but it prevents strings from going flabby without needing much heavier gauges. Which does help with a wide range of playing styles and genres.
Unless you also believe that all guitars should have a single scale length or something, and a single neck profile and fingerboard radius. Otherwise if you concede that it comes down to feel+preference then there’s no argument to make against multiscale instruments.
(I wish Firefox on iOS had a "open clean link" option, but I'd wish Mozilla would fix other more important stuff first, like letting me search/open bookmarks from a private tab.)
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.
If an ensemble includes instruments that are equal temperament, then the non-fixed-pitched instrumentalists adjust their pitch to sound good with those.
An ensemble consisting only of instruments that can play any interval can change keys by pure intervals.
E.g. switching from the original major key to the relative dominant key can mean changing the root by a pure fifth. In equal temperament, this modulation is done by altering only a single note: sharpening the subdominant. All other notes are from the original scale. If we change key by a pure fifth, that is obviously not so; all notes are detuned off the original scale.
If we change through all the keys along the circle of fifths, perfectly purely, we arrive at the Pythagorean comma: the gap between the destination root and the original.
Another possibility is to progress the roots through the diatonic fifths of the original scale, rather than pure fifths. Like, we start with a pure, just intonated C major, and then change keys through G,D,A,E,B,F#,C#,Ab,Eb,Bb,F back to C using the notes of that pure C major scale, or sharps/flats relative to those. Then we don't run into the Pythagorean comma; but of course all the pure scales we end up using are detuned from C major, and in a different way from following pure fifths.
Yes, it does.
> There's the mathematical fact that we cannot get pure thirds and even fifths in modern equal temperament system.
Those are the pennies that don't matter, if your instrument has dollar problems.
If you don't have good intonation, then you can't even properly get the approximations provided by equal temperament.
With good intonation, compensated on both ends, you have a much better experience making tuning adjustments to get better compromises for the music you are playing.
Guitar intonation that is accurate to 2 cents is very good, I would say above average.
Another way to look at the pitch error in the ET perfect fifth is as a percentage of the pitch, which is about -0.169 %.
Suppose a 1200 Hz tone (quite a high note, somewhere between D6 and D#6) is played together with one that is 0.169 % flat. That flat one will have a frequency of 1198 Hz. The difference is 2 Hz, and so a 2 Hz beat will be heard: two volume swells per second.
Much lower down, at 120 Hz, that will be 0.2 Hz: two volume swells every ten seconds. Basically nothing. It makes no difference to guitar chords played in the first four fret box down by the nut.
The equal temperament error is worse for some other intervals; the ET major third is a percent sharp, or around 13.6 cents, which is a lot. It is pretty jarring, even in lower registers.
That's not what the submitted article is about; tuning in such a way as to fixing the tiny error in the fourths/fifths will not repair the major third.
No, you can’t. If you tune so that octaves with one string between are correct everywhere on the neck, that will force the tuning to be 12 tone equal temperament, and a fifth in 12 TET cannot be a perfect fifth.
If octaves are perfect with one string in between, the in between string can be slightly detuned from equal temperament to provide a clean fifth, free of beats. Then it also provides a clean fourth up to the octave. That's a useful thing that will make certain chords sound good.
The E, D and B strings are turned such that they yield clean octaves (and other equal-temperament intervals).
Then so are the A, G and E.
But these two groups are slightly detuned, so that the fifths are clean from the E to A string, D to G, and B to E.
2. The error between the equal temperament perfect fifth and the pure one (3/2) is just less than 2 cents. So the difference I'm talking about is at the same level of accuracy as that of pretty excellent guitar intonation. The corrections are not simply for equal temperament; they are not separable from the condition of the instrument and its intonation. The given instrument is what it is, and to get those 1-5-8 power chords to sound clean you do whatever you have to.
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.
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/watch?v=ubadQ1jcWOM
And the late Jaco Pastorius with the bass harmonics song that would have broken the Internet if we had had the internet when he released his first solo album:
https://www.youtube.com/watch?v=nsZ_1mPOuyk
Speaking as a person who owns basses... I like the sound of harmonics on a bass better. I think it's something to do with the longer strings giving more play to the overtones.
I have software I use when I tune my Bosendorfer 290 that calculates the stretch. Of course, the final tweaks are done by ear.
So you've got to tune your guitar to sound good with them and probably not just matching your open strings to their corresponding notes.
While your electronic tuner flashes an ugly warning or the strobe tuner won't stand still :(
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
If they did, why did they hold it back just to speak so contemptuously of a subject that is actually interesting and reasonably well explained?
Generosity is worth having by default, though. Filter people out when they burn it explicitly.
Whether it's testes or testy language, getting personal and insulting does not meet my personal standard for assuming good intent and being worthy of an open-minded attempt to create constructive dialogue.
But I applaud you for wanting to lift the standard of discourse!
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.
I agree with this in spirit, but there are practical ramifications of getting the frequency domain wrong. The human brain is very particular in this space. Even for completely untrained listeners. It's nothing like the human visual system. You're working on timescales measured in microseconds with auditory signals. Even where the instruments are physically positioned on stage is significant. Getting their pitch slightly wrong can be catastrophic.
The goal of regular guitar intonation and bridge adjustment is to get the guitar as close as possible to 12 tone equal temperament (TET), which is slightly ‘out of tune’ as the article describes. 12 TET is the best you can do if you want something equally close to perfect fifths (or thirds, etc.) in all positions in all keys across all string combinations; that’s what 12 TET is for, it’s designed to minimize the worse case, at the expense of losing the best case.
The only thing that is absurd here is your bizarre strawman that discussing equal temperament is somehow non-musical and that engineers can’t understand what music is because they want to measure things.
Oh, and that applies to standard tuning only. YMMV with alternative tunings, especially the open tunings.
0: https://www.guyguitars.com/truetemperament/eng/tt_techdetail...
Just because we live with the trade-off doesn’t make it correct in any other sense.
Agree with the OP that the characteristics of the guitar, including its "out of perfect tune", is what gives its music its unique characteristic. It's not a bug it's a feature. There might be some people with perfect pitch who get annoyed but for most people that's "colour" and the sound they expect and associate with their favorite music. If you played those songs on an "ideal" guitar they would not sound right.
EVH famously tuned his B string slightly flat to make his D (on the 3rd fret) sound better. Look it up.
e —0–
B —0–
G —7–
D —6–
A —7–
E —0–
In general your ears do not hear these little arithmetical games around mismatched harmonies. They hear things like “this chord sounds warm and a little sad, this one is bright and fun.”
With 12 of the strings on a sitar having equal (thin) diameter, but different lengths so they can be tuned to the 12 notes in the scale, these are also unplayed strings which contribute to the sound by resonating underneath the main course of strings which are the ones fretted and manually played on.
That's so endearing I guess that's why they call them sympathetic strings ;)
While my guitar gently weeps, etc. . .
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.