> If you watch slow-motion video of a guitar string vibrating, you’ll see a complex, evolving blend of squiggles. These squiggles are the mathematical sum of all of the string’s different harmonics.
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.
Hold on. Your first video is indeed a rolling shutter artifact. But your second video never shows enough of the string to see the harmonics. When you (for example) pluck with a finger on the 12th fret, you absolutely do have a real physical squiggle vibrating in the string, with one node and two antinodes. With a 7th fret harmonic, there are 3 antinodes, with a 5th fret harmonic there are four. There are squiggles, and you can see them with real slowmo.
While you're of course righ, in a certain way, the squiggles _are_ a function of the frequnencies that the chords are vibrating at. What you see is the interaction of the two frequencies, your the interaction depends on both frequencies.
Not sure if I’m misunderstanding your claim. A string does vibrate as the sum of the string’s harmonics. That’s how pinch harmonics work, and they wouldn’t work if that wasn’t the case.
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 can't see harmonics happening to the string.
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.
Interestingly, with an oscilloscope you can see the harmonics in all their gory detail :)
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.
> If you watch a video like [0], the squiggles aren't real, they're an artifact of a rolling shutter camera.
...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? :-)
Every pixel of every frame was really captured by the camera from the source, but it’s being played back to you very differently than how the source actually looked.
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.
How could you tell the note by looking at a string? Unless you’re talking by about marking timestamps and measuring the time between peaks. A 42 gauge string tuned to E or D or any other note are going to look basically the same.
> 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
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.
Actually is not a guitar problem, but all 12-TET tuned instruments have this, it is just a side effect of harmonic math. In the guitar case it is not only the tuning that counts, also the material the string are made and the diameter of the strings count to the final frequency, and we are using parallel frets so applying the same distance to different strings. There are guitars with not parallel frets that try to compensate for the diameter variation. But that’s all math and understanding, cause when you tune your guitar and just play you are in another world were "thought is the killer of flow"; so just play and enjoy the sound. :D
There are two type of “not parallel” frets and neither have anything to do with the diameter of the strings.
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.
Yes, the diameter is compensated at the same time as the tension.
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.
Another thing that’s not been mentioned here: there is a relationship between volume and pitch. In short, you strike a string hard and it goes a bit sharp. The issue is that the tonal math makes a linearization of the string physics, but the highly activated string is effectively a little tighter than the idealized version.
Humans are also not perfect at fretting with the exact same pressure every time, or without inducing some bend in the strings. This is really noticeable with the G string which always sounds out of tune while playing, because our tuning system gives it a half-step-down intonation as a trade-off to make it easier to form chords.
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.
Peterson guitar tuners can do custom tunings, and have the James Taylor tuning built in as a preset. (On Peterson tuners, it's called the 'acoustic' preset, but is actually the JT tuning.)
I was born with something not quite like perfect pitch, but when something is even slightly off tune it caused physical discomfort for me.
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.
Do say more! What was the problem with the clock, more exactly? I believe you, I've had issues caused by clock skew and CAN bus for example, when you have a small error that is amplified on beach bit enough time, errors add up and you eventually get out of synch.
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!
One simplistic way is to successively add a small constant to a large integer, and generate the waveform from the most significant bits. A "cent," which is 1/100 of a semitone, is a factor of about 580 parts per million, so you can work out the precision needed for the constant. On a microcontroller, you can control the timing with a PWM, which runs independently of the processor and its timing foibles.
> On a microcontroller, you can control the timing with a PWM, which runs independently of the processor and its timing foibles.
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.
Is it enough to have an audible effect? We’re not talking cesium clock levels of stability here. Now my curiosity is piqued, I have to figure out a way to measure this.
I've been doing audio software for 25-30 years. I have no idea what sort of synthesis you'd be doing where the processor clock played any roll at all. Waveform synthesis is normally done in buffers (8 to 8192 samples), and the "clocking" to convert the sample stream into an analog waveform is done by the audio interface/DAC, not the CPU. If you were basically implementing a DAC, then yes, the clock would matter a lot ... is/was that the issue?
You've not done it long enough to have worked with machine language programs that used instruction timing to click a speaker.
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.
> 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
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%.
My first guitar teacher told me that someday I'd start to notice that you can't get all strings perfectly in tune. At that point, he said, you'll know you're getting somewhere on the guitar.
With an ordinary fretted guitar, you can sort of perfectly tune it to what you play but not perfectly tune it in a global sense.
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).
Classical guitarists are used to pushing nylon strings into consonance by compressing the string either towards the nut or the bridge. Not so easy with steel, where players will just preemptively retune to whatever chords are most prominent in the song.
I play with generally lighter strings. 8.5-40 mighty slinky fender scale. I noticed when I switched my fingers pay much more attention to pressure, and being in tune with microbends.
Been thinking of going a bit lighter recently, and also getting a classical.
Get obsessed over the perfect tuning. Blame the imperfections on the quality of the guitar. Don't play until you get a better guitar. Repeat until you give up. Then actually start playing the damn thing.
Yes exactly. Although I didn't buy a new guitar, I bought 5 tuners and at least a dozen apps. It finally clicked when I got one that was "real time" enough to see how the tuning shifts from high to low.
Doesn't help that most tuners are (still!) crap, 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 and it's probably the right approach for most, I just don't like it.
This is why string instrument players sometimes prefer to play a note not on the empty string (let's say play a A on the A-string on a cello), but instead on a lower string (e.g. first finger, fourth position on the lower D string) to accord for these imperfections. As a string instrumemt player, you pretty much only have to worry about these notes on empty strings, every other note you can "wiggle into place".
Instruments which have a non-discrete set of pitches (as well as voices) will tend towards the more harmonious (so to speak) tuning when playing in harmony. You’ll notice this in choirs, for example, where singing a capella, the chords will follow nice integral ratios of frequencies. Fretless string instruments and the trombone are obvious cases of instruments which can do micro-tuning, but it’s worth noting that brass instruments have finger loops on some of the valve loops to allow adjustment of pitch. Micro-tuning of the pitch can also be managed in wind instruments through adjustment of the embouchure so while woodwinds seem like they would be only capable of discrete pitches, there is some ability to adjust the pitch during performance.
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.
Not all instruments are limited to a fixed set of pitches. A good classical string player micro-adjusts each individual note to adapt to its harmonic context. For example, making all the thirds and fifths sound good even when the key changes, or adjusting a leading tone up or down very slightly to make it even more leading.
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.
Yes! If you broaden your scope beyond “Western diatonic” you get even more opportunity. “Why can’t you tune your Turkish microtonal guitar” would also be an interesting follow-up.
A good analogy for equal temperament might be the Gregorian calendar, as a year does not evenly divide perfectly into 365 days. So in order to compensate for that we adjust the calendar by a leap year every so often to make the calendar more accurate in the longer term. That's kinda similar to how every note is a little off in equal temperament so that at the larger scale of being able to play all intervals works out.
> If thirds and fifths are so out of tune in 12-TET, why do we use it? The advantage is that all the thirds and fifths in all the keys are out of tune by the same amount. None of them sound perfect, but none of them sound terrible, either.
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?
The higher the variety of notes (out of the overall 12 sounds in an octave) in the song, the less this becomes possible.
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.
Yes, people try this. Check out dynamic tonality. It doesn't necessarily need a system. Experienced guitar players often find themselves unconsciously making little microtonal adjustments through bends and other techniques when playing leads. I found myself doing this just because it sounded better to me. I didn't even notice there was a consistent pattern until I eventually learned the math. For example I'd always want to bend minor thirds slightly sharp and bend the neck to slightly detune major thirds.
> 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 note rather than over an entire song?
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.
In addition to singers, adaptive tuning is something which happens naturally for fretless stringed instruments (violin, etc), brass instruments with slides (most prominently the slide trombone but in fact many (most?) others), woodwind instruments where the pitch can be bent like saxophone, and so on.
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.
Played trombone many years ago, but never well enough to ever adjust that finely (at least not consciously?). The tuning slide on the third valve on a trumpet usually has a finger fork/loop so that it can be tuned in realtime. I believe the first valve on higher end trumpets similarly has a thumb fork for the same reason.
I played trombone in high school, never very well, but I definitely adjusted like this. Actually, although it was a slide trombone, I'm talking about adjusting automatically with embouchure. Someone would play the reference note, I'd match (in 1st position) but bend my pitch to match. The band teacher once complimented me on the adjustment. Which was stupid, because (1) I wasn't doing it intentionally, and (2) the adjustment only lasted during tuning; as soon as we started playing, I was right back out of tune. I never did learn to suppress the adjustment so I could actually fix the tuning.
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.)
Actually Bach's Well Tempered Clavier IS a book written in a single set of tuning system that actually lost/forgotten. We still have discussions about how it's constructed. For more information google "Well Tempered Clavier interpretation"
Neither lost nor forgotten! It's the basis for the "Thidell Formula 1" temperament [0], which is what produces those squiggly frets on expensive guitars. It also works well for multiple keys (at the expense of others), making it a compromise for a range of music rather than a single song.
It doesn’t work per-song. Songs have multiple chords, some even with alterations. If you tune an E so that it is perfectly a major third above C, then that E won’t be a perfect fifth above an A note. The Am chord has the notes A, C and E, so Am has notes that all belong to C major.
Additionally, some songs even change keys, which makes “per-song” not enough of a constraint.
Singers drift because they use relative pitch, because most musicians dont have perfect pitch.
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
You can with instruments without fixed pitches, like human voice and string instruments, in fact choirs and string quartets do play this way, adjusting each note.
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.
Kyle Gann's Arithmetic of Listening goes deeply into this. Given an infinite number of ways of dividing the range from f to 2f, some other equal-division temperaments (31 or 53, for example) get closer than 12TET to maintaining low-integer ratios across key centers, but each additional pitch adds complexity. I'd recommend that book in particular. https://www.kylegann.com/Gannbooks.html
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!
Article reads like a well akchuallly to is your guitar in tune.
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.
For some reason it’s taken me decades of playing guitar to become good enough at tuning and also sensitive enough to really feel the fact that I can’t tune the guitar. Recently I finally grokked the simple reason that 12 TET cannot be perfect, and it doesn’t take a long article to see it. I was kind-of aware of the major third problem, but I naively thought fifths were still perfect.
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.
True temperament solves for a _different_ issue than what OP's post talks about. From the strandberg true temperament page:
> 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.
A true temperament isn’t just about compensating for the string mass differences but also for their differing intonation points. A true temperament won’t get to the same level as a movable fret system but it does also compensate to a certain degree for the differing intonation points across strings at different tunings (what it refers to as speaking length which captures both point to point length and mass related deformation length). They’re different but inherently associated issues.
This feels like a really puritanical take on things. Fanned frets and multiscale absolutely help with the playability of an instrument. It’s physics, there’s nothing mystical or gimmicky about it.
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.)
The other problem I always notice on top of all this is that when you pluck a string, it adds tension to it temporarily, so the pitch when you first play it is a little higher than the pitch as it settles down. The louder you play it, the more the effect.
There is no way to tune your guitar so that all the successive open fourths (and the one major third) are pure, without the high E being quite off pitch relative to the low one.
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.
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.
It doesn't matter how good the intonation at the bridge or nut. There's the mathematical fact that we cannot get pure thirds and even fifths in modern equal temperament system. If you have a good ear you'll feel the subtle difference between notes, and they can never get exactly where you want. It's something you have to live with in modern music.
The thing is, unless you're playing with other instruments, no one is forcing you to tune to equal temperament. E.g., it's very common to tune a violin's A string to an A440 reference, then tune the other strings to 3:2 perfect fifths by ear. It just gets more complicated for fretted instruments like the guitar.
If you tune the strings in a guitar to a perfect 4th, which is the case except the 3rd between strings b-e, the lower e will significantly differ from the high e open string. There's no way to get around it.
Wanting to play in any key and not be locked into a key automatically pushes musicians toward equal temperament, even when playing solo, and even on a violin. Saying no one’s forcing you is technically true but sounds pretty naive, and (forgive the pun) tone deaf to me; there’s no realistic alternative for modern music. Some people do choose to play with other tuning systems on occasion, but there’s a reason why 12 TET is so popular and widespread.
Wanting to change keys freely only pushes fixed-pitch musical instruments toward equal temperament. Since many important instruments are like that, and virtually all instruments that are capable of accurate intonation not relying on ear are like that.
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.
> It doesn't matter how good the intonation at the bridge or nut.
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.
Actually, it will solve most of it for guitarists, because the tuning problems that most guitarists blame on equal temperament are actually coming from their bad intonation. The fourths and fifths in equal temperament are not nearly as bad as they imagine. The equal temperament fifth is only 1.955 cents sharper than the pure fifth (3/2 ratio). Just under 2 cents.
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.
> You can, for instance, tune instead to get pure fretted fifths between adjacent strings, and fretted octaves between strings one removed.
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.
Frets are already on equal temperament on the vast majority of mainstream guitars; you're not getting away from equal temperament unless all you play is open strings, like a koto/harp.
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.
That’s not very helpful. You can only do that one string, and it breaks the perfect octave starting on the detuned string. As soon as you also try to detune the next string up, you break the octave you started from. This is a house of cards idea that falls apart immediately.
Have you actually tried this? What songs work? To me this sounds totally impractical and useless, like it’s a logical technicality for the purposes of this discussion and not something a real musician would ever do. You’re making tuning a pain, breaking the E-E octaves (all barre chords), breaking octaves with 2 strings between, breaking the next-string octaves, breaking a lot of scales and jazz chords, and to top it off it would still only work for certain keys and not others. I’ll pass.
1. Yes; I've been tuning along those lines for nearly four decades. What songs work: anything with power chords that benefit from sounding sharp, free of flutter.
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.
> Not everyone in history thought that 12-TET was an acceptable compromise. Johann Sebastian Bach thought we should use other tuning systems
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.
I always thought it's an optimization problem: the headstock is pulled by the strings towards the guitar's body, and whenever you get one string in tune, the change in (the distribution of) force changes the tuning of all other strings as well. So ideally they should be moved into an acceptable configuration collectively. People with six hands should be able to do it.
If you've been storing it with strings loose and are bringing it into tune you definitely gradually tighten all the strings rather than just doing it in sequence, for that reason
I thought this was going to be about stiffness of the strings and how the modes even on a single string are not in tune compared to the mathematical model, which assumes infinitely flexible strings.
Even if you tuned two string to ensure that two specific notes on them vibrated at a perfect interval, there are non-multiplicative overtones modulated by resonance with the rest of the instrument. Those intervals are ideals for minimizing dissonance. Practically, the dissonance of 12TET intervals falls below the noise floor of all the other acoustic distortions that give instruments character.
fun fact: some bands, like red hot chilli peppers, will tune the G string slightly flat such that major thirds become just, for some of their riffs. Listen to "scar tissue" for example
I wondered why this article was so high on the front page but now I realize it’s simply because everyone else wanted to “um actually” it. I guess that makes sense.
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:
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.
Piano tunings are also "stretched" so that the harmonics are more in tune. This is especially needed on verticals and short "baby" grands. See https://en.wikipedia.org/wiki/Stretched_tuning
I have software I use when I tune my Bosendorfer 290 that calculates the stretch. Of course, the final tweaks are done by ear.
Well, there's only 6 knobs and if you want to be "in tune with the world" those six knobs can only be in one place.
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.
Woah, so cool when a topic I was going into in depth gets to HN.
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
Absurd. A guitar within tolerance is in tune. It's a fundamental feature of the instrument. Not a flaw.
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.
> Music doesn't live in an abstract realm of perfections
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.
Many musicians can readily confirm that the difference between temperaments can be felt and heard by trained ears. A guitar tuned to equal temperament has major thirds that warble audibly. It feels different when you use just intonation, which isn’t generally possible on a guitar.
For the reasons the article explains. You can use “just intonation” a bit on a guitar, but it will only work for certain chords in certain positions. BTW note that just intonation is different from string intonation - I wasn’t talking about making sure the 12th fret is the same note as the 12th fret harmonic on a single string, I was talking about the tuning system called “just intonation” that defines what certain intervals are, and allows for perfect thirds and perfect fifths in some keys. But it won’t work everywhere on a guitar. It’s not possible to get (for example) perfect fifths on all string combinations in all positions, but it is possible to tune the guitar so you have a perfect fifth when crossing 1 string while in 5th position.
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.
This article is just an introduction to the math behind 12-TET, why it exists, the tradeoffs, etc.
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.
Even those are not "true-temperament" instruments like a piano, they adapt a "well-temperament" tuning system like Bach's keyboard tuning [0]. The end result is closer to true-temperament in some keys, and farther in others.
Oh, and that applies to standard tuning only. YMMV with alternative tunings, especially the open tunings.
I don't think this is generally true for the guitar. There are even songs that have notes intentionally out of tune (e.g. Scar Tissue by the Red Hot Chili Peppers).
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.
Outside of people like van halen also pretty much no one is exploring the entire neck on a single song. So the issue of the guitar not being perfectly intonated is irrelevant since they are using just a piece of its range.
A lot of simple songs are just open "cowboy" chords for sure. But those are played on the first frets while the guitar is typically intonated at the 12th fret and tuned with open strings. I would expect those first frets to be fairly "out" vs. the open strings.
I am a jazz guitarist and am sympathetic to this comment: the way I tune my guitar these days is hitting an E tuning fork, playing a particular E7 chord, and deciding if it sounds good:
e —0–
B —0–
G —7–
D —6–
A —7–
E —0–
Learned it from Jimmy Bruno. I despise digital tuners. However it is worth noting: a properly-tuned guitar will never be able to play a “barbershop seventh,” which hits the natural harmonic dominant 7th and is so flat compared to TET that it’s really almost a 6th. The chord itself sounds more bittersweet and less “funky” than a TET dominant 7th. OTOH the TET chord is an essential part of modern blues-influenced music: being “out of tune” makes the chord sharp and strong, almost like a blue cheese being “moldy.” So I’m not beaten up about the limitations, it’s just worth keeping in mind: no instrument can beat a group of human voices.
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 ;)
Fixed it: “Why can’t you tune your poorly made guitar?”
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.
Even your fancy guitar is not exempt from harmonics math. TFA has nothing to do with the quality of a guitar and everything to do with 12-Tone Equal Temperament.
You’re assuming that the goal for a guitar player is to have perfectly optimal instrument when in reality many players want an instrument that feels and sounds like the artists that inspire them. Aesthetics is part of that but if they enjoy the sound of the instrument then who’s to say that another one is “better”?
Tuning guitar strings by hand is trivial. Try to tune a piano. There have up to 3 strings per key, and quite a lot of keys. Cannot be perfect, but should sound "warm" enough.
> If you watch slow-motion video of a guitar string vibrating, you’ll see a complex, evolving blend of squiggles. These squiggles are the mathematical sum of all of the string’s different harmonics.
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
Hold on. Your first video is indeed a rolling shutter artifact. But your second video never shows enough of the string to see the harmonics. When you (for example) pluck with a finger on the 12th fret, you absolutely do have a real physical squiggle vibrating in the string, with one node and two antinodes. With a 7th fret harmonic, there are 3 antinodes, with a 5th fret harmonic there are four. There are squiggles, and you can see them with real slowmo.
Yes, I think that with all these videos what you actually see is aliasing...
While you're of course righ, in a certain way, the squiggles _are_ a function of the frequnencies that the chords are vibrating at. What you see is the interaction of the two frequencies, your the interaction depends on both frequencies.
Not sure if I’m misunderstanding your claim. A string does vibrate as the sum of the string’s harmonics. That’s how pinch harmonics work, and they wouldn’t work if that wasn’t the case.
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 can't see harmonics happening to the string.
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.
If you stretch a rope or chain several meters long, you can simultaneously see the fundamental and first harmonic standing wave.
Interestingly, with an oscilloscope you can see the harmonics in all their gory detail :)
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.
> If you watch a video like [0], the squiggles aren't real, they're an artifact of a rolling shutter camera.
...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? :-)
Every pixel of every frame was really captured by the camera from the source, but it’s being played back to you very differently than how the source actually looked.
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.
How could you tell the note by looking at a string? Unless you’re talking by about marking timestamps and measuring the time between peaks. A 42 gauge string tuned to E or D or any other note are going to look basically the same.
GP is talking about seeing the string oscillations alias against the 30 Hz camera frame rate.
I've never tried it.
> 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
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.
Actually is not a guitar problem, but all 12-TET tuned instruments have this, it is just a side effect of harmonic math. In the guitar case it is not only the tuning that counts, also the material the string are made and the diameter of the strings count to the final frequency, and we are using parallel frets so applying the same distance to different strings. There are guitars with not parallel frets that try to compensate for the diameter variation. But that’s all math and understanding, cause when you tune your guitar and just play you are in another world were "thought is the killer of flow"; so just play and enjoy the sound. :D
There are two type of “not parallel” frets and neither have anything to do with the diameter of the strings.
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.
Yes, the diameter is compensated at the same time as the tension.
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.
Another thing that’s not been mentioned here: there is a relationship between volume and pitch. In short, you strike a string hard and it goes a bit sharp. The issue is that the tonal math makes a linearization of the string physics, but the highly activated string is effectively a little tighter than the idealized version.
Humans are also not perfect at fretting with the exact same pressure every time, or without inducing some bend in the strings. This is really noticeable with the G string which always sounds out of tune while playing, because our tuning system gives it a half-step-down intonation as a trade-off to make it easier to form chords.
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.
Peterson guitar tuners can do custom tunings, and have the James Taylor tuning built in as a preset. (On Peterson tuners, it's called the 'acoustic' preset, but is actually the JT tuning.)
I was born with something not quite like perfect pitch, but when something is even slightly off tune it caused physical discomfort for me.
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.
Do say more! What was the problem with the clock, more exactly? I believe you, I've had issues caused by clock skew and CAN bus for example, when you have a small error that is amplified on beach bit enough time, errors add up and you eventually get out of synch.
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!
One simplistic way is to successively add a small constant to a large integer, and generate the waveform from the most significant bits. A "cent," which is 1/100 of a semitone, is a factor of about 580 parts per million, so you can work out the precision needed for the constant. On a microcontroller, you can control the timing with a PWM, which runs independently of the processor and its timing foibles.
Proof is left as an exercise to the student. ;-)
> On a microcontroller, you can control the timing with a PWM, which runs independently of the processor and its timing foibles.
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.
Is it enough to have an audible effect? We’re not talking cesium clock levels of stability here. Now my curiosity is piqued, I have to figure out a way to measure this.
> I was born with something not quite like perfect pitch, but when something is even slightly off tune it caused physical discomfort for me.
Define off tune? 12 TET? Just intonation? Bohlen-Pierce (56 TET) ?
The "in tune" notes are as much a function of culture as physics.
I've been doing audio software for 25-30 years. I have no idea what sort of synthesis you'd be doing where the processor clock played any roll at all. Waveform synthesis is normally done in buffers (8 to 8192 samples), and the "clocking" to convert the sample stream into an analog waveform is done by the audio interface/DAC, not the CPU. If you were basically implementing a DAC, then yes, the clock would matter a lot ... is/was that the issue?
You've not done it long enough to have worked with machine language programs that used instruction timing to click a speaker.
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.
> 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
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%.
yes it was this
My first guitar teacher told me that someday I'd start to notice that you can't get all strings perfectly in tune. At that point, he said, you'll know you're getting somewhere on the guitar.
With an ordinary fretted guitar, you can sort of perfectly tune it to what you play but not perfectly tune it in a global sense.
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).
Classical guitarists are used to pushing nylon strings into consonance by compressing the string either towards the nut or the bridge. Not so easy with steel, where players will just preemptively retune to whatever chords are most prominent in the song.
I play with generally lighter strings. 8.5-40 mighty slinky fender scale. I noticed when I switched my fingers pay much more attention to pressure, and being in tune with microbends.
Been thinking of going a bit lighter recently, and also getting a classical.
Get obsessed over the perfect tuning. Blame the imperfections on the quality of the guitar. Don't play until you get a better guitar. Repeat until you give up. Then actually start playing the damn thing.
Yes exactly. Although I didn't buy a new guitar, I bought 5 tuners and at least a dozen apps. It finally clicked when I got one that was "real time" enough to see how the tuning shifts from high to low.
Doesn't help that most tuners are (still!) crap, 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 and it's probably the right approach for most, I just don't like it.
This is why string instrument players sometimes prefer to play a note not on the empty string (let's say play a A on the A-string on a cello), but instead on a lower string (e.g. first finger, fourth position on the lower D string) to accord for these imperfections. As a string instrumemt player, you pretty much only have to worry about these notes on empty strings, every other note you can "wiggle into place".
Indeed, and another factor is that a fingered note has a different tone quality.
Disclosure: String player.
And the thicker strings sound a bit different as well.
And the fingering for a given melody may just lay across the strings better one way than another.
Instruments which have a non-discrete set of pitches (as well as voices) will tend towards the more harmonious (so to speak) tuning when playing in harmony. You’ll notice this in choirs, for example, where singing a capella, the chords will follow nice integral ratios of frequencies. Fretless string instruments and the trombone are obvious cases of instruments which can do micro-tuning, but it’s worth noting that brass instruments have finger loops on some of the valve loops to allow adjustment of pitch. Micro-tuning of the pitch can also be managed in wind instruments through adjustment of the embouchure so while woodwinds seem like they would be only capable of discrete pitches, there is some ability to adjust the pitch during performance.
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.
Not all instruments are limited to a fixed set of pitches. A good classical string player micro-adjusts each individual note to adapt to its harmonic context. For example, making all the thirds and fifths sound good even when the key changes, or adjusting a leading tone up or down very slightly to make it even more leading.
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.
This sensitivity to intonation is why all the great quartet composers were either deaf, Hungarian, or both.
fretless and continuous instruments are not confined to 12-TET
Yes! If you broaden your scope beyond “Western diatonic” you get even more opportunity. “Why can’t you tune your Turkish microtonal guitar” would also be an interesting follow-up.
A good analogy for equal temperament might be the Gregorian calendar, as a year does not evenly divide perfectly into 365 days. So in order to compensate for that we adjust the calendar by a leap year every so often to make the calendar more accurate in the longer term. That's kinda similar to how every note is a little off in equal temperament so that at the larger scale of being able to play all intervals works out.
> If thirds and fifths are so out of tune in 12-TET, why do we use it? The advantage is that all the thirds and fifths in all the keys are out of tune by the same amount. None of them sound perfect, but none of them sound terrible, either.
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?
The higher the variety of notes (out of the overall 12 sounds in an octave) in the song, the less this becomes possible.
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.
Yes, people try this. Check out dynamic tonality. It doesn't necessarily need a system. Experienced guitar players often find themselves unconsciously making little microtonal adjustments through bends and other techniques when playing leads. I found myself doing this just because it sounded better to me. I didn't even notice there was a consistent pattern until I eventually learned the math. For example I'd always want to bend minor thirds slightly sharp and bend the neck to slightly detune major thirds.
https://en.wikipedia.org/wiki/Dynamic_tonality
> 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 note rather than over an entire song?
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.
In addition to singers, adaptive tuning is something which happens naturally for fretless stringed instruments (violin, etc), brass instruments with slides (most prominently the slide trombone but in fact many (most?) others), woodwind instruments where the pitch can be bent like saxophone, and so on.
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/
Played trombone many years ago, but never well enough to ever adjust that finely (at least not consciously?). The tuning slide on the third valve on a trumpet usually has a finger fork/loop so that it can be tuned in realtime. I believe the first valve on higher end trumpets similarly has a thumb fork for the same reason.
I played trombone in high school, never very well, but I definitely adjusted like this. Actually, although it was a slide trombone, I'm talking about adjusting automatically with embouchure. Someone would play the reference note, I'd match (in 1st position) but bend my pitch to match. The band teacher once complimented me on the adjustment. Which was stupid, because (1) I wasn't doing it intentionally, and (2) the adjustment only lasted during tuning; as soon as we started playing, I was right back out of tune. I never did learn to suppress the adjustment so I could actually fix the tuning.
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.)
Actually Bach's Well Tempered Clavier IS a book written in a single set of tuning system that actually lost/forgotten. We still have discussions about how it's constructed. For more information google "Well Tempered Clavier interpretation"
You can listen to variations here: https://youtu.be/kRui9apjWAY?t=622
Neither lost nor forgotten! It's the basis for the "Thidell Formula 1" temperament [0], which is what produces those squiggly frets on expensive guitars. It also works well for multiple keys (at the expense of others), making it a compromise for a range of music rather than a single song.
0: https://www.guyguitars.com/truetemperament/eng/tt_techdetail...
It doesn’t work per-song. Songs have multiple chords, some even with alterations. If you tune an E so that it is perfectly a major third above C, then that E won’t be a perfect fifth above an A note. The Am chord has the notes A, C and E, so Am has notes that all belong to C major.
Additionally, some songs even change keys, which makes “per-song” not enough of a constraint.
That's how it works when you sing! But if you have an instrument you need to tune it would be annoying if you had to retune it between every song.
Or in the middle of a song -- lots of songs modulate between different keys.
Ok, does this explain why singers drift to a different key when there is no accompaniment?
Singers drift because they use relative pitch, because most musicians dont have perfect pitch.
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
Yes, I mean would it be an additional factor?
I highly recommend the book “How Equal Temperament Ruined Harmony (and Why You Should Care)” if you are interested in this subject.
You may be interested in:
https://en.wikipedia.org/wiki/Just_intonation
https://en.wikipedia.org/wiki/Comma_pump
Logic Pro has Hermode tuning, which does this per chord: https://support.apple.com/guide/logicpro/hermode-tuning-lgcp...
You can with instruments without fixed pitches, like human voice and string instruments, in fact choirs and string quartets do play this way, adjusting each note.
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.
Kyle Gann's Arithmetic of Listening goes deeply into this. Given an infinite number of ways of dividing the range from f to 2f, some other equal-division temperaments (31 or 53, for example) get closer than 12TET to maintaining low-integer ratios across key centers, but each additional pitch adds complexity. I'd recommend that book in particular. https://www.kylegann.com/Gannbooks.html
I think you can only be "perfectly" in tune for a single mode so a multi-modal song would become very difficult to play?
Not really, because notes do double duty.
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!
Article reads like a well akchuallly to is your guitar in tune.
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.
For some reason it’s taken me decades of playing guitar to become good enough at tuning and also sensitive enough to really feel the fact that I can’t tune the guitar. Recently I finally grokked the simple reason that 12 TET cannot be perfect, and it doesn’t take a long article to see it. I was kind-of aware of the major third problem, but I naively thought fifths were still perfect.
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.
The most surprising thing is that everyone here is an expert on true temperament. Who would have guessed.
I’m surprised they don’t show true temperament frets.
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
True temperament solves for a _different_ issue than what OP's post talks about. From the strandberg true temperament page:
> 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.
A true temperament isn’t just about compensating for the string mass differences but also for their differing intonation points. A true temperament won’t get to the same level as a movable fret system but it does also compensate to a certain degree for the differing intonation points across strings at different tunings (what it refers to as speaking length which captures both point to point length and mass related deformation length). They’re different but inherently associated issues.
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.
TT necks are an absolute waste of money. At least if you went with the Earvana nut you only wasted $5 and the time to replace the thing.
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...
This feels like a really puritanical take on things. Fanned frets and multiscale absolutely help with the playability of an instrument. It’s physics, there’s nothing mystical or gimmicky about it.
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.
Here's the YouTube link without that tracking code for those of us on mobile:
https://youtu.be/EZC69A8TsJ8
(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.)
I like the music of Aloboi who makes electronic music, often with just intonation
https://m.youtube.com/watch?v=WkYJb21hpyA
The other problem I always notice on top of all this is that when you pluck a string, it adds tension to it temporarily, so the pitch when you first play it is a little higher than the pitch as it settles down. The louder you play it, the more the effect.
In heavy metal, especially in modern down-tuned genres, this is often used as an artistic choice.
There is no way to tune your guitar so that all the successive open fourths (and the one major third) are pure, without the high E being quite off pitch relative to the low one.
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.
It doesn't matter how good the intonation at the bridge or nut. There's the mathematical fact that we cannot get pure thirds and even fifths in modern equal temperament system. If you have a good ear you'll feel the subtle difference between notes, and they can never get exactly where you want. It's something you have to live with in modern music.
The thing is, unless you're playing with other instruments, no one is forcing you to tune to equal temperament. E.g., it's very common to tune a violin's A string to an A440 reference, then tune the other strings to 3:2 perfect fifths by ear. It just gets more complicated for fretted instruments like the guitar.
If you tune the strings in a guitar to a perfect 4th, which is the case except the 3rd between strings b-e, the lower e will significantly differ from the high e open string. There's no way to get around it.
Wanting to play in any key and not be locked into a key automatically pushes musicians toward equal temperament, even when playing solo, and even on a violin. Saying no one’s forcing you is technically true but sounds pretty naive, and (forgive the pun) tone deaf to me; there’s no realistic alternative for modern music. Some people do choose to play with other tuning systems on occasion, but there’s a reason why 12 TET is so popular and widespread.
Wanting to change keys freely only pushes fixed-pitch musical instruments toward equal temperament. Since many important instruments are like that, and virtually all instruments that are capable of accurate intonation not relying on ear are like that.
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.
> It doesn't matter how good the intonation at the bridge or nut.
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.
Intonation is just the beginning. It won't solve the deeper issues with equal temperament.
Actually, it will solve most of it for guitarists, because the tuning problems that most guitarists blame on equal temperament are actually coming from their bad intonation. The fourths and fifths in equal temperament are not nearly as bad as they imagine. The equal temperament fifth is only 1.955 cents sharper than the pure fifth (3/2 ratio). Just under 2 cents.
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.
> You can, for instance, tune instead to get pure fretted fifths between adjacent strings, and fretted octaves between strings one removed.
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.
Frets are already on equal temperament on the vast majority of mainstream guitars; you're not getting away from equal temperament unless all you play is open strings, like a koto/harp.
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.
That’s not very helpful. You can only do that one string, and it breaks the perfect octave starting on the detuned string. As soon as you also try to detune the next string up, you break the octave you started from. This is a house of cards idea that falls apart immediately.
The detuned string in between likewise has a perfect octave relationship to strings that are two removed.
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.
Have you actually tried this? What songs work? To me this sounds totally impractical and useless, like it’s a logical technicality for the purposes of this discussion and not something a real musician would ever do. You’re making tuning a pain, breaking the E-E octaves (all barre chords), breaking octaves with 2 strings between, breaking the next-string octaves, breaking a lot of scales and jazz chords, and to top it off it would still only work for certain keys and not others. I’ll pass.
1. Yes; I've been tuning along those lines for nearly four decades. What songs work: anything with power chords that benefit from sounding sharp, free of flutter.
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.
> Not everyone in history thought that 12-TET was an acceptable compromise. Johann Sebastian Bach thought we should use other tuning systems
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.
I always thought it's an optimization problem: the headstock is pulled by the strings towards the guitar's body, and whenever you get one string in tune, the change in (the distribution of) force changes the tuning of all other strings as well. So ideally they should be moved into an acceptable configuration collectively. People with six hands should be able to do it.
If you've been storing it with strings loose and are bringing it into tune you definitely gradually tighten all the strings rather than just doing it in sequence, for that reason
Relevant MinutePhysics video which dives deep: https://www.youtube.com/watch?v=tCsl6ZcY9ag
I thought this was going to be about stiffness of the strings and how the modes even on a single string are not in tune compared to the mathematical model, which assumes infinitely flexible strings.
Even if you tuned two string to ensure that two specific notes on them vibrated at a perfect interval, there are non-multiplicative overtones modulated by resonance with the rest of the instrument. Those intervals are ideals for minimizing dissonance. Practically, the dissonance of 12TET intervals falls below the noise floor of all the other acoustic distortions that give instruments character.
https://web.archive.org/web/2019if_/http://www.ethanhein.com...
fun fact: some bands, like red hot chilli peppers, will tune the G string slightly flat such that major thirds become just, for some of their riffs. Listen to "scar tissue" for example
Close enough for rock'n roll, as it is called.
I wondered why this article was so high on the front page but now I realize it’s simply because everyone else wanted to “um actually” it. I guess that makes sense.
I read the title and felt personally attacked :(
Interesting.
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
Steve Hackett takes advantage of guitar harmonics in a piece inspired by Bach's prelude to the first suite for solo violincello:
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.
Great article. Also, awesome comments; thanks everyone.
Piano tunings are also "stretched" so that the harmonics are more in tune. This is especially needed on verticals and short "baby" grands. See https://en.wikipedia.org/wiki/Stretched_tuning
I have software I use when I tune my Bosendorfer 290 that calculates the stretch. Of course, the final tweaks are done by ear.
Isn’t there a similar issue with pianos?
Yes, and besides that no matter what you do you just can't get these virtuosos to tune their pianos on the spot ;)
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 :(
Well, there's only 6 knobs and if you want to be "in tune with the world" those six knobs can only be in one place.
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.
Woah, so cool when a topic I was going into in depth gets to HN.
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
Absurd. A guitar within tolerance is in tune. It's a fundamental feature of the instrument. Not a flaw.
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.
> Music doesn't live in an abstract realm of perfections
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.
Many musicians can readily confirm that the difference between temperaments can be felt and heard by trained ears. A guitar tuned to equal temperament has major thirds that warble audibly. It feels different when you use just intonation, which isn’t generally possible on a guitar.
Why can’t you use intonation? Isn’t that just confirming the note is the same on different strings? And also the goal of bridge adjustment?
For the reasons the article explains. You can use “just intonation” a bit on a guitar, but it will only work for certain chords in certain positions. BTW note that just intonation is different from string intonation - I wasn’t talking about making sure the 12th fret is the same note as the 12th fret harmonic on a single string, I was talking about the tuning system called “just intonation” that defines what certain intervals are, and allows for perfect thirds and perfect fifths in some keys. But it won’t work everywhere on a guitar. It’s not possible to get (for example) perfect fifths on all string combinations in all positions, but it is possible to tune the guitar so you have a perfect fifth when crossing 1 string while in 5th position.
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.
This article is just an introduction to the math behind 12-TET, why it exists, the tradeoffs, etc.
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.
Have you heard of even tempering, on piano?
Engineers hate it and so they invented the true temperament guitar. It’s like a regular guitar except the frets are a bit funky.
Even those are not "true-temperament" instruments like a piano, they adapt a "well-temperament" tuning system like Bach's keyboard tuning [0]. The end result is closer to true-temperament in some keys, and farther in others.
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...
Any musician with enough training will tell you which notes are out of tune on their well-tuned instrument, and how they correct for it as they play.
Just because we live with the trade-off doesn’t make it correct in any other sense.
I don't think this is generally true for the guitar. There are even songs that have notes intentionally out of tune (e.g. Scar Tissue by the Red Hot Chili Peppers).
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.
Frusciante is an amazing guitar player — check out his solo albums. He knew what he was doing putting those out of tune notes in.
Outside of people like van halen also pretty much no one is exploring the entire neck on a single song. So the issue of the guitar not being perfectly intonated is irrelevant since they are using just a piece of its range.
A lot of simple songs are just open "cowboy" chords for sure. But those are played on the first frets while the guitar is typically intonated at the 12th fret and tuned with open strings. I would expect those first frets to be fairly "out" vs. the open strings.
“Whole neck” is irrelevant. Things can go out of tune at any fret.
EVH famously tuned his B string slightly flat to make his D (on the 3rd fret) sound better. Look it up.
I am a jazz guitarist and am sympathetic to this comment: the way I tune my guitar these days is hitting an E tuning fork, playing a particular E7 chord, and deciding if it sounds good:
Learned it from Jimmy Bruno. I despise digital tuners. However it is worth noting: a properly-tuned guitar will never be able to play a “barbershop seventh,” which hits the natural harmonic dominant 7th and is so flat compared to TET that it’s really almost a 6th. The chord itself sounds more bittersweet and less “funky” than a TET dominant 7th. OTOH the TET chord is an essential part of modern blues-influenced music: being “out of tune” makes the chord sharp and strong, almost like a blue cheese being “moldy.” So I’m not beaten up about the limitations, it’s just worth keeping in mind: no instrument can beat a group of human voices.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.”
There's more than one way to be sympathetic :)
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. . .
Fixed it: “Why can’t you tune your poorly made guitar?”
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.
Even your fancy guitar is not exempt from harmonics math. TFA has nothing to do with the quality of a guitar and everything to do with 12-Tone Equal Temperament.
Weird to describe 99%+ of all guitars, including some of the best made guitars in the world, as "poorly made."
This article is relevant to a theoretic perfectly designed and built guitar.
You’re assuming that the goal for a guitar player is to have perfectly optimal instrument when in reality many players want an instrument that feels and sounds like the artists that inspire them. Aesthetics is part of that but if they enjoy the sound of the instrument then who’s to say that another one is “better”?
Tuning guitar strings by hand is trivial. Try to tune a piano. There have up to 3 strings per key, and quite a lot of keys. Cannot be perfect, but should sound "warm" enough.