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.
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.
I have software I use when I tune my Bosendorfer 290 that calculates the strech. Of course, the final tweaks are done by ear.
However if you want more notes than that to be their best you're going to have to compromise and work at it a bit.
Now if you want the instrument to sound its absolute best on its own solo, a slightly different place for some strings.
And then depending on other musicians you are playing with and the way their tuning has achieved perfection (or not), some further tweaking can make a big difference.
And that's after accepting that the "knobs can only be in one place".
For students to get really good at the tuning process can require a few extra years of everyday practice more than it does to learn to play a few pieces.
Part of the limitation is the way only a few minutes of tuning are spent for every hour of practice, if that.
I'm a relatively new adult beginner on the violin, and one of the fascinating (and extremely difficult) things about un-fretted string instruments is the player has the freedom to shift the tuning around to fit the context. On the violin, we normally play melodies and scales using Pythagorean tuning (which is actually a misnomer as Pythagoras didn't invent it, the ancient Mesopotamians did), which is based on the circle of fifths and leads to wider whole steps and narrower half steps than equal temperment tuning. But then for double stops (i.e. chords), and especially when playing in a string quartet, just intonation, which is based on the harmonic series, is used so the notes sound concordant. This page describes all the different tuning systems a violinist may use, also including 12 TET when trying to match a piano: https://www.violinmasterclass.com/posts/152.
This video shows how challenging it can be when trying to adjust intonation when playing in a string quartet: https://youtu.be/Q7yMAAGeAS4 . Interestingly, the very beginning of that video talks about what TFA discussed that when you tune all your strings as perfect fifths your major thirds will be out of tune.
I'll also put in a plug for light note, an online music theory training tool that was mentioned on HN a decade ago: https://news.ycombinator.com/item?id=12792063 . I'm not related to the owner in any way, I just bought access a few years ago and think it was the first time I really understood Western music theory. The problem with music theory is that the notation is pretty fucked up because it includes all this historical baggage, and lots of music theory courses start with what we've got today and work backwards, while I think it's a lot easier to start with first principles about frequency ratios and go from there.
Other notes (pun intended!): The violin is great for learning music theory because you can actually see on the string how much you're subdividing it - go one third of the way, that's a perfect fifth, go halfway, that's an octave, etc. Harmonics (where you lightly touch a string) are also used all the time in violin repertoire. Finally, the article mentions Harry Patch, but you should also check out Ben Johnston, a composer who worked with Patch and was famous for using just intonation. Here is is Amazing Grace string quartet, and you can really hear the difference using just intonation: https://youtu.be/VJ8Bg9m5l50
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
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.
Another way to think of it is that they have to hit every pitch without assistance from the instrument anyway, so they learn to make every note sound “good” rather than hitting a mathematically defined frequency.
Can't we have a system that is optimized for the notes that are actually played in a song rather than the hypothetical set? And what if the optimization is done per small group of notes rather than over an entire song?
https://strandbergguitars.com/en-WW/magazine/true-temperamen...
They solve exactly for this issue, and sound amazing in use. The downside is that you are somewhat locked into a given tuning.
Alternatively you can take the approach of guitars with movable frets so you can adjust them per tuning.
https://youtu.be/EZC69A8TsJ8?si=7hUIb7FEKb45eV_L
These are generally used for microtonal playing but can also effectively be true temperament as well.
Guitars with gut frets used to have adjustable positions, which allowed for some mitigation via changing fret positions too
But, unless you mainly play stacked fourths, why would you make it a requirement? You can, for instance, tune instead to get pure fretted fifths between adjacent strings, and fretted octaves between strings one removed.
The real reason you can't get your guitar in tune is one which makes none of the above matter. Most guitars don't have good intonation. Most acoustic guitars don't have movable saddles to set intonation at the bridge. Electric ones do. For accurate tuning, you need not only compensation at the bridge, but also at the nut.
https://guitarnutcompensation.com/
On my main axe, I installed a small screw next to the nut, right under the G string. Just doing the G string makes a huge difference!
Here is a test: play an open D power chord (open D, A on G string, D on B string) it is very clean. Now release the A to play a 1-4-8 G power chord (open D, open G, D).
On my compensated guitar, both of them are crisply in tune. Without nut intonation, one of the two will have ugly beats. If you tune one, the other goes wonky.
When I first heard how good it is after putting in the compensating screw, I was astonished and at the same time filled with the regret of not having done it decades earlier.
Why the G? The unwound G string on electrics is the most susceptible to bad intonation at the nut, because it undergoes the greatest pitch change when it is fretted. Guitarists like to bend that one for the same reason. Fretting it at the first or second frets makes it go markedly sharp; for that reason we need to shorten the distance between the nut and the first fret to get that sharpened interval back down to a semitone.
This is less of a problem on guitars with a wound G, which has a lot more tension in it to compensate for its weight, and doesn't pitch-bend nearly as easily.
Advanced banjo players will sometimes use harmonics for a ‘bell’ effect. Here’s a short video from Alison Brown, a great player.
Music doesn't live in an abstract realm of perfections, it is an expression however formed. The fact that we can measure it is one thing. But the music or instruments do not need conform to discrete measurements to satisfy.
I know engineers hate this, but ask any musician. It's like arguing that a sitar and its scales aren't right. Absurd.
The most guitars today are still made in the style of the 1950s Gibsons and Fenders, including the neck and tuner layout. Most guitar buyers focus on the aesthetic and not the quality. I switched to a headless guitar where the tuners are at the bridge and it has a fanned fretboard giving the strings more natural tensions, the thing stays in tune and is intonated at the frets extremely well.
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.
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.
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.
Define off tune? 12 TET? Just intonation? Bohlen-Pierce (56 TET) ?
The "in tune" notes are as much a function of culture as physics.
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.
> 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.
(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.)
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.
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 :(
I agree with this in spirit, but there are practical ramifications of getting the frequency domain wrong. The human brain is very particular in this space. Even for completely untrained listeners. It's nothing like the human visual system. You're working on timescales measured in microseconds with auditory signals. Even where the instruments are physically positioned on stage is significant. Getting their pitch slightly wrong can be catastrophic.
The 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.
Just because we live with the trade-off doesn’t make it correct in any other sense.
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.”
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.
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!
When the string's action is higher above the frets, the tension increases more when fretted than open, to a greater degree than low action.
So the saddle for that string needs to be positioned such that the plucked portion of the string is slightly longer than it would need to be if the tension were the same as the open string.
James Taylor compensates by tuning everything down a few cents, between -12 at the low E and -3 at the high E, with a little break in the pattern with -4 cents at the G to deal with its weirdness. Good electronic tuners have "sweetened" presets which do something similar.
Disclosure: String player.
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.
Additionally, some songs even change keys, which makes “per-song” not enough of a constraint.
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.
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.
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...
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.
Proof is left as an exercise to the student. ;-)
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).
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.
You can listen to variations here: https://youtu.be/kRui9apjWAY?t=622
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. . .
That is not really true. You usually have a couple of clock sources on a MCU, but the clock gets propagated down the clock tree and the source, and most of the times, the PWM has the same source clock as the CPU. Indeed, I think if you're before the PLL the clock is more accurate as in you get less jitter but the overall drift is the same. You might have distinct clock sources but you need a specific hw and a specific configuration.
This 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%.
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/
0: https://www.guyguitars.com/truetemperament/eng/tt_techdetail...
Been thinking of going a bit lighter recently, and also getting a classical.
Short answer: because math. Longer answer: because prime numbers don’t divide into each other evenly.
To understand what follows, you need to know some facts about the physics of vibrating strings:
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. The weird and interesting thing about harmonics is that each one produces a different pitch. So when you play a note, you’re actually hearing many different pitches at once.
It’s not difficult to isolate the harmonics of a vibrating string and hear their individual pitches. Harmonics are very useful for tuning your guitar – here’s a handy guide for doing so. They are also the basis of the whole Western tuning system generally.
As a string vibrates, its longer subsections produce lower and louder harmonics, while its shorter subsections produce higher and quieter harmonics. Click the image below to hear the first six harmonics of a string:
Remember that in a real-world string, you are hearing all these harmonics blended together. However, you can isolate the harmonics of a guitar string by lightly touching it in certain places to deaden some of the vibrations.
Imagine that you have a guitar string tuned to play a note called “middle C,” which has a frequency of 1 Hz. (In reality, middle C has a frequency of 261.626 Hz, so if you want to think in terms of actual frequencies, just multiply all the numbers in the following paragraphs by 261.626.)
The first harmonic is the string vibrating along its entire length, otherwise known as the fundamental frequency. When we say that your C string is vibrating at 1 Hz, that really means that its fundamental has a frequency of 1 Hz. The other harmonics all have other frequencies, and we’ll get to those, but the fundamental is usually the loudest harmonic, and it’s usually the only one you’re aware of hearing.
The second harmonic is the one you get from the string vibrating in halves. Each half of the string vibrates at twice the frequency of the whole string. The 2:1 relationship between the pitches of the first and second harmonics is called an octave. (I know that the word suggests the number eight, not the number two. Don’t worry about it.) The pitch that’s an octave above middle C has a frequency of 2 Hz, and it is also called C. Both of these notes have the same letter name because in Western convention, notes an octave apart from each other are considered to be “the same note“. The important concept here is that you can move up an octave from any pitch by doubling its frequency. You can also move down an octave from any pitch by halving its frequency.
The third harmonic is the one you get from the string vibrating in thirds. Its frequency is three times the fundamental frequency. Since your C string’s fundamental is 1 Hz, the third harmonic has a frequency of 3 Hz, and it produces a note called G. The interval between C and G is called a perfect fifth, for reasons having nothing to do with harmonics. I know it’s confusing.
The fourth harmonic is the one you get from the string vibrating in quarters, at 4 Hz. This note is an octave higher than the second harmonic, and so is also called C. (The eighth harmonic will also play C, as will the sixteenth, and the thirty-second, and all the powers of two up to infinity.)
The fifth harmonic is the one you get from the string vibrating in fifths. Its frequency is 5 Hz, and it produces a note called E. The interval between C and E is called a major third, which is another name that has nothing to do with harmonics.
There are many more harmonics (infinitely many more, in theory) but these first five are the most audible ones.
The ancient Greeks figured out that if you have a set of strings, it sounds really good if you tune them following the pitch ratios from the natural harmonic series. In such tuning systems, you pick a starting frequency, and then multiply or divide it by ratios of whole numbers to generate more frequencies, the same way you figure out the frequencies of a single string’s harmonics. The best-sounding note combinations (to Western people) are the ones derived from the first few harmonics. In other words, you get the nicest harmony (for Western people) when you multiply and divide your frequencies by ratios of the smallest prime numbers: 2, 3, and 5.
So, let’s do it. Let’s make a tuning system based on the harmonics of your C string. First, we should find the C, G and E notes whose frequencies are as close to each other as possible.
When you play 1 Hz, 5/4 Hz and 3/2 Hz at the same time, you get a lovely sound called a C major triad.
So far, so good. Let’s find some more notes!
We can extend our tuning system by thinking of G as our base note, and looking at its harmonics. When we do, we get two new notes. The third harmonic of G is D at 9 Hz. (Thanks to octave equivalency, we can also make Ds at 9/2 Hz, and 9/4 Hz, and 9/8 Hz, and 18 Hz, and 36 Hz, and so on.) The fifth harmonic of G is B at 15 Hz. (There are also Bs at 15/2 Hz, and 30 Hz, and so on.)
The notes C and G feel closely related to each other because of their shared harmonic relationship. The chords you get from their respective overtone series also feel related. If you alternate between C major and G major chords, it just about always sounds good.
Now let’s extend our tuning system further by treating D as our base note. The harmonics of D give us two more new notes: the third harmonic is A at 27 Hz (and 27/2 Hz and 27/4 Hz and 27/8 Hz), and the fifth harmonic is F-sharp at 45 Hz (and 45/2 Hz and 45/4 Hz and 45/8 Hz).
G major chords and D major chords have the same relationship as C major and G major chords, and they sound equally good when you alternate them. Also, C major, G major and D major chords all sound good as a group, in any order and any combination. Western people just really like the sound of shared harmonics. Last thing: notice that you can combine the harmonics of C, G and D to form a G major scale.
Now let’s make some more notes by treating A as our base and looking at its harmonics. The third harmonic of A is E at 81 Hz (and 81/2 Hz and 81/4 Hz etc).
But wait. We already had an E, at 5 Hz. If we put these two E’s in the same octave, then one of them is at 80/64 Hz, and the other is at 81/64 Hz. That may not seem like much of a difference, but even untrained listeners will be able to hear that they are out of tune with each other. Furthermore, if we use the E derived from C, then it will be out of tune with A. However, if we use the E derived from A, then it will be out of tune with C. This is going to be a problem.
Let’s forget about that conflict for a second. Instead, we’ll try a different method of expanding our tuning system, by going in the opposite direction from C. Let’s think about a note that contains C in its harmonic series. That would be F at 1/3 Hz. The third harmonic of F is C at 1 Hz, as expected. The fifth harmonic of F is A at 5/3 Hz.
Uh oh. This new A conflicts with the one we already had at 27 Hz. That is not good. But let’s bracket that and keep expanding.
We can push further left by finding the note whose overtone series contains F. That would be B-flat at 1/9 Hz. Its third harmonic is F at 1/3 Hz, and its fifth harmonic is D at 5/9 Hz. And now we have a new problem: this D clashes with our existing D at 9 Hz.
Can you see the pattern here? Anytime you want to use intervals based on third harmonics, you’re multiplying and dividing by 3, but anytime you want to use intervals based on fifth harmonics, you’re multiplying and dividing by 5. (Notice that the conflicting notes always conflict by the same amount, too, a ratio of 81/80.) Starting from C, it’s possible to produce any note if you multiply or divide your frequencies by 3 enough times, but those notes won’t be in tune with the notes you’d get multiplying or dividing your frequencies by 5, because 3 and 5 don’t mutually divide evenly. This is not just an abstract mathematical issue. It’s the reason that it’s impossible to have a guitar be in tune with itself.
Imagine that the guitar’s low E string has a frequency of 1 Hz. (It’s really 82.4069 Hz; feel free to multiply everything in this next section by that number if you want actual frequencies.) Ideally, you want your high E string to be tuned two octaves above the low one, at 4 Hz. Let’s see if you can get there by tuning the strings pairwise.
This is not good. We wanted the high E to be at 4 Hz, which is the same as 324/81 Hz. We’re 4/81 Hz flat! That difference is big enough to make your guitar tuning sound like warm garbage.
Let’s try a different strategy. I said you should tune the B string a major third above G. However, you could just as easily retune the B string so it’s a fifth plus an octave above the low E string. You do this by multiplying 1 Hz by 3/2, and then doubling it, which puts your B at 3 Hz. Now the B string sounds perfectly in tune with the low E string at 1 Hz, and with the high E string at 4 Hz. Unfortunately, the B string is now out of tune with the G string at 64/27 Hz.
So maybe you should just retune the G string a major third below your new B, at 12/5 Hz. That makes the G and B strings sound great together. Unfortunately, now the G string is out of tune with the D string at 16/9 Hz.
You could retune the D string to be a fourth below G… but now the D string will be out of tune with the A string. If you retune the A string based on your new D, then it will be out of tune against the low E string. And if you retune the low E string based on your new A, then it will be out of tune with the high E string.

The bottom line: there is no way to tune the guitar so that every string is in tune with every other string.
The mathematical awkwardness of harmonics-based tuning systems has caused Western musicians a lot of pain over the past thousand years. Depending on your starting pitch, some intervals can be perfectly in tune, but others can’t be. And the more harmonically complex you want your music to be, the worse the tuning issues become.
In the 16th century, Chinese and Dutch musicians independently came up with an alternative system to harmonics-based tuning, called 12-tone equal temperament, or 12-TET. It’s the system that the entire Western world uses today. The idea behind 12-TET is to have everything be pretty much in tune, which you accomplish by having everything be a little bit out of tune. Is this a worthwhile compromise? Let’s do the math and find out.
In 12-TET, you divide up the octave into twelve equally-sized semitones (the interval between two adjacent piano keys or guitar frets). To go up a semitone from any note, you multiply its frequency by the 12th root of 2 (about 1.05946). To go down a semitone from any note, you divide its frequency by the 12th root of 2. If you go up by an octave (twelve semitones), you’re multiplying your frequency by the 12th root of 2 twelve times, which works out to 2. That’s a perfect octave, hooray! Unfortunately, you can’t exactly create the other harmonics-based intervals by adding up 12-TET semitones; you can only approximate them.
Remember that the pure fifth you get from harmonics is a frequency ratio of 3/2. In 12-TET, however, you make a fifth by adding up seven semitones. This means that you multiply your frequency by the 12th root of two seven times, which comes to about 1.498. That’s close to 3/2, but it’s not exact. As a result, fifths in 12-TET sound a little flat compared to what your ear is expecting from natural harmonics.
Major thirds are worse in 12-TET. Recall that the major third you get from the overtone series is a frequency ratio of 5/4. In 12-TET, you make a major third by adding four semitones, which means that you multiply your frequency by the 12th root of 2 four times. That comes to 1.25992, which is noticeably higher than 5/4. Thirds in 12-TET are quite sharp compared to what your ears are expecting from natural harmonics.
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. You don’t have to worry about whether your notes are derived from the third harmonic of some note or the fifth harmonic of some other note; they all just work together, kind of. If you use a digital guitar tuner, you are tuning your strings to the 12-TET versions of E, A, D, G and B. None of them will be perfectly in tune with each other, but they will all be wrong by an acceptable amount. Also, songs in the key of E won’t sound any better or worse than songs in the key of F or E-flat.
Not everyone in history thought that 12-TET was an acceptable compromise. Johann Sebastian Bach thought we should use other tuning systems that made better-sounding thirds and fifths in some keys in exchange for worse-sounding thirds and fifths in others. In Bach’s preferred tuning, each key had its own distinctive blend of smoothness and harshness. However, Bach did not get his way. We as a civilization have collectively decided that we want all our keys to be interchangeable. There are good reasons to want this! In 12-TET, all intervals and chords are built from standardized, Lego-like parts. You don’t have to keep track of a complicated web of different-sized intervals in every key. If you move a song from C to C-sharp or D or anywhere else, you can be confident that it will still sound “the same.”
Some musicians don’t want to accommodate to 12-TET, insisting instead that we should continue to use pure intervals derived from harmonics the way God and Pythagoras intended. Harmonics-based tuning systems are collectively known as just intonation systems. This is a poetically apt term, because it implies fairness. By contrast, the implicit message of 12-TET is that life isn’t fair. Just intonation systems give you some lovely pure intervals, but you can’t change keys unless you retune all your instruments. In other world cultures, this is not necessarily a problem. Hindustani classical music uses just intonation over an omnipresent drone, so everything is always in the same “key.”
Meanwhile, a few Western oddballs and nerds have explored just intonation systems that use bigger prime numbers than 2, 3 and 5 to generate finer pure intervals. Harry Partch used the primes up to eleven to make a tuning system that divides up the octave into 43 pure parts rather than 12 impure ones. You can try the Partch 43-tone scale using the Wilsonic app or Audiokit Synth One. It’s extremely strange! But, I guess, it’s strange in a pure way. I have made some music of my own with exotic just intonation tunings.
Just intonation may also play a role in the blues. There is a theory that the blues originates from the natural overtone series of I and IV. If this is true, then the characteristic chords and scales of the blues are really 12-TET approximations of the original just intonation blues scale. It’s conventional to say that blues musicians and singers bend notes to make them go out of tune, but it may be that they are actually bending the 12-TET pitches to get them in tune instead.
Anyway, outside of the blues and the avant-garde, most Western musicians just live with everything being a little out of tune. If you’re a guitarist, you know that no matter how you tune your guitar, it won’t stay in tune for long anyway, so how much does any of this even matter? There’s a joke among guitarists: we spend half our lives tuning, and the other half wishing we were in tune. There are lots of reasons why tuning is hard: you might be hampered by having a poorly made guitar, or by having a guitar that’s not set up correctly, or by using old worn-out strings, or by changes in temperature or humidity, or just by a lack of patience or time. At least you can be secure in the knowledge that some of your tuning struggles are due to the basic unfairness of the universe, and not just the limitations of your ears or your equipment.
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