Right, I made an edit which accounts for that while you wrote your comment. It's an important detail.

That's interesting, so they can get closer to "just intonation" then?

I assume it all breaks down if they need to play alongside a keyboard (or fretted) instrument.

Well yes, the pitch of any violin note except an open string is set by where the finger is placed.

However, being perfectly in tune is also a big red herring kind of thing. People, especially people who like seeing math in music, get obsessed with chasing ideas of perfection in music and music is art... it isn't supposed to be perfect. To have sounds at perfect intervals or sounds perfectly in tune is after a certain point just an annoying detail compared to literally every other aspect of a piece of music.

A lot of advanced digital synthesizers will carefully detune oscillators from each other so they aren't "perfectly in tune" in order to get thicker sounds.

> A lot of advanced digital synthesizers will carefully detune oscillators from each other so they aren't "perfectly in tune" in order to get thicker sounds

As noted in other comments, this also applies to singing and arbitrary pitch instruments, possibly at a subconscious level, and it has the opposite "mathematical" implication than you seem to think: any fixed tuning is a serious constraint that makes some chords sound wrong, and only being able to tune individual notes perfectly allows the introduction of aesthetically pleasing imperfections.

including multiple methods for the user to detune oscillators is quite common on modern synthesizers, advanced or otherwise. it’s almost never a fixed amount of detuning.

one of those methods is called a “chorus” effect. this is extremely common across effect platforms and is not limited to synthesizers / keyboard-type instruments.

How does all of that sound to people with absolute pitch?

Varies a lot depending on the person. "Absolute" pitch isn't really absolute, in the vast majority of cases. It's a degree of an ability to retain a given pitch and then produce it later without prompting or context.

Keep in mind also that a lot of musicians with "perfect" pitch have to deal with performing situations where the main pitch is not the standard A=440 Hz. For instance in the Baroque repertoire which I perform often, the most common pitch is around A=415, which is around a half step lower, but there are other tunings that pros have to deal with which are both above and below A=440 (European orchestras often tune higher, music before the Baroque is often at A=390, music from the classical period is often around A=430, etc.).

Violins and family(typically) tune their instruments with 3/2 just fifths. You get the A (440) from the oboe and tune the rest of your strings with perfect just fifths. That means sometimes the cellos' C strings will be noticably too low in some circumstances so you'll see them finger an "open C" just above the nut to make it sound right.

I actually tune my C string slightly sharp for this reason!

It's 2% of a semitone off by my understanding. I thought I had pretty good ears but I really doubt I could pick that. Open strings do often stick out in general on string instruments though, for a combination of reasons, lack of vibrato and ability to micro-adjust tuning presumably being the main ones (but even the tone is different, I assume based on the difference between having one end fixed by a soft fleshy substance vs the wooden nut).

the last reason you gave is exactly why open strings sound different. check out zero-fret guitars.

Except you very very often don't get an A=440, since a lot of orchestras don't tune to that pitch and early-music orchestras are a full half-tone below that, etc.

Huh, is that why an open C on a viola always buzzed wrong?

String players have no choice but to learn equal temperament as the vast majority of the time they're playing alongside other musicians, and it's what modern ears (since the late 18th century) expect to hear. It'd be a rare violinist these days that could actually accurately play something in any sort of intonation based entirely on just intervals. Note that almost any sort of vibrato is likely to "smother" the pitch difference between equal and just temperaments anyway - e.g. an equal temperament fifth is 2 cents off a natural fifth, but vibrato can cover a 50 to 70 cent range (opera singers often go over 100, which I find unpleasant to listen to personally - it's basically a trill!)

> String players have no choice but to learn equal temperament as the vast majority of the time they're playing alongside other musicians, and it's what modern ears (since the late 18th century) expect to hear. It'd be a rare violinist these days that could actually accurately play something in any sort of intonation based entirely on just intervals.

That's not true at all. A lot of string players learn to play in orchestras or chamber style, which means they're only playing with other stringed instruments, and they absolutely are taught dynamic tuning by ear, which uses just intervals.

I did say "based entirely on just intervals". But as a composer I most certainly wouldn't want string players choosing their temperament based on whether there happened to be other instruments in the ensemble capable of the same. And it sounds off for music that doesn't largely sit in a single key signature anyway, which is arguably most music composed since Beethoven. Though I did just read a classic example of where just intervals are often used is the opening of Das Rheingold, that sits on an E flat (not D#!) major chord for several minutes.

> But as a composer I most certainly wouldn't want string players choosing their temperament based on whether there happened to be other instruments in the ensemble capable of the same.

This is a weird way of looking at it. String players aren't sitting there consciously thinking of their tuning as they play - they're doing it by ear in real-time. The tuning they use will be the one that best harmonizes with the other notes being played at that moment.

> And it sounds off for music that doesn't largely sit in a single key signature anyway,

That's actually where the ability to adapt tuning dynamically is the most powerful - it allows you to be in tune relative to other pitches being played in that moment, not just in tune relative to some absolute benchmark that nobody is going to be able to hear anyway (because almost nobody has perfect absolute pitch).

Sure, I imagine it's not dissimilar to how we sing as choristers. But I've played on keyboards tuned to exact just temperament in a particular key and it starts to sound very weird very quickly the moment you veer off the reference key signature.

> But I've played on keyboards tuned to exact just temperament in a particular key

Well, that's your problem. You're using a keyboard, which doesn't permit you to harmonize dynamically the way an unfretted string instrument does.

Even within a particular key, the pitch that sounds the best for a particular note will depend on which other notes within that key you're attempting to harmonize with. A keyboard can't do that.

Btw, this is from the wikipedia article on Equal Temperament, and I'd say it aligns with my general understanding/ expectation:

"Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.[4]"

No and it's possible that as a pianist my ears are more attuned to prefer equal temperament than those of a string player. But I admit when singing a capella there are occasions particular chords just seem to sit better than when having to match a piano accompaniment, and to some extent that's likely to be the ability to use "purer" intervals.

Exactly. The only fixed-pitch instruments in an orchestra are the piano and the percussion section.

It was routine even in my high school brass section to pitch down the major 3rds.

Having briefly learned a few wind instruments (flute and horn primarily) I'm aware pitch adjustment is possible but the keys/valves are designed around equal temperament - for anything other than slower sustained passages (or potentially repeated notes) constantly trying to approximate just intervals doesn't seem sustainable. And again, absolutely not what I would want or except to hear as a composer.

skilled instrumentalists are quite capable of consistently reproducing intervals in a given tuning system. particularly thirds in just intonation. it’s not an approximation. it’s one of the reasons we spend so much time learning ear training in conservatory.

I argue all just about all intonation is some sort of approximation, unless you're playing an electronic instrument that doesn't allow pitch adjustments! And it does surprise me how little my ears seem to notice despite having zero tolerance for people singing even slightly off-key.

relative to mathematical perfection, of course it’s all an approximation when a human instrumentalist is involved. that’s the nature of our physical reality.

the most important element here is how it sounds to our ears. not how closely it tracks to an equation.

I'm only an amateur, but I doubt there are string players that "learn" equal temperament. I have no idea how I would find 440 * (2^(1/12) ^ n) Hz, for any n not a multiple of 12, in the way that I can find 440 * (4/3) Hz, or 440 * (3/2) Hz, etc. When playing with equaled tempered instruments like piano, you just listen for clashes and adjust dynamically, which is only going to happen in slower, sustained passages.

And you're right, we don't play "based entirely on just intervals." What we do is constantly adjust our intonation depending on whether we need it to be "just" with respect to something else (like other notes in a chord), or whether we are free to use a more "melodic" intonation. See https://www.youtube.com/watch?v=QaYOwIIvgHg for a good demonstration -- note that he talks in formal terms like "play x in the Pythagorean system," but I think you can largely see this as a rationalization of what players do naturally).

Finally, the presence of vibrato doesn't really obviate intonation concerns, sadly. There's a lot of theoretical debate about how the pitch of a vibrated note is perceived (is it the highest pitch in the range that determines whether the note sounds in tune? etc.), but in practice you can easily verify that adding vibrato to an out-of-tune scale will not make it sound any more in tune, nor will adding it to a shift mask a slightly-missed shift (if only!).

I chose the word "smother" deliberately, though maybe "blur" would be better. There's quite a bit of debate as to how the pitch of a note with vibrato is perceived. It definitely isn't right in the middle which might be the naive hypothesis.

Fretted instruments, especially electric guitars, are usually not strictly equal temperament and are made to have just intonation in at least some combinations of notes because equal temperament sounds bad with distortion.

There exist equal temperament guitars, but they're usually custom built:

https://guitargearfinder.com/faq/true-temperament-frets/

In any case most people don't mind such small differences, especially that guitars aren't terribly precise to begin with - a player can easily get 10 cents of a semitone on each individual string when playing a power chord with distortion, bringing the whole thing to just intonation.

Edit: I was wrong below.

---

Hi! I am not a musician. Did you mean that true temperament guitars are the ones with squiggly frets, instead?

My understanding was that true temperament [0] is not the same as equal temperament [1]. I also believe that both pianos and guitars are typically tuned to equal temperament [2], but you may well be right about guitars.

Maybe somebody can shed some more light on this. Thanks!

[0] https://www.truetemperament.com

[1] https://en.wikipedia.org/wiki/Equal_temperament

[2] https://youtube.com/watch?v=-penQWPHJzI

Guitars are indeed supposed to be 'ideally' equal temperament. But they're not.

Even if you take out the dynamics of vibrating strings, the idea of 'frets' is to 'pre-divide' the string into its intervals for you. For example, the 12th fret is the halfway point on the string.

But look at the bridge of any guitar. Clearly, the saddles are not all an equal distance from the nut, so the 12th fret can't be actually halfway down all of them.

For this reason, a guitar is fretted in a way that is actually more of an approximation of equal temperament than actual equal temperament. It's rarely far enough out to be bothered about.

'True temperament' is a bit of a misnomer. There's no such thing as 'true' temperament. 'Temperament' by definition means a 'tempering' of the 'true' interval (the pure/just intonation).

SOME sort of temperament is required on a fretted instrument precisely because of the question that this article addresses: on a guitar, you can only pick one 'pitch' for a fret, even though the 'correct' frequency for a D# may well be different than the 'correct' frequency for an Eb depending on the key in which they appear.

So calling it 'true temperament' is a bit naughty. All it means is that it's trying to iron out some of the approximations which are inherent in the instrument design to get it closer to 12-TET.

True temperament appears to be a marketing term for a fret system providing equal temperament.

A "spherical cow" model of a guitar would be equal temperament, but that ignores the messy reality of how strings behave - chiefly they need to be some distance above the fretboard and pressing them naturally bends the string ever so slightly.

so slightly that it can be on the range of 0-5 cents, provided the instrument is sufficiently constructed and the player is sufficiently skilled.

this is why a guitar using equal temperment can play consistently in-tune with itself as well as with other instruments tuned in the same system. it’s not about perfection according to some abstract mathematical model.

Kinda is, considering that the frequencies for each note are a standard and used by all equal temperament instruments.

Guitars are almost never perfectly in tune according to that standard, but like I said - people generally don't mind.

yes, I am aware that conventional guitars have fundamental issues with intonation in equal temperament systems.

this has not prevented it from being a versatile instrument that is quite capable of being played “enough” in tune with ensembles of other instruments, such that the vast majority of people hear zero problems.

how is what we hear in music less relevant than whether or not a given instrument is not perfectly in tune, mathematically speaking, if that variation in tuning is imperceptible to human hearing?

We actually tend to approach Pythagorean tuning, according to the Catgut Acoustical Society.

...because if you are playing with a piano, and you play those intervals "right", they will be out of tune.

Is a D-sharp/E-flat played on a piano or guitar closer to D-sharp or E-flat on the violin?

It's in the middle between the two!

It would be interesting to have an electronic keyboard that watches what you are playing and decides when you press the D-sharp/E-flat key, which note it should play.

Some old style organs that are not "well-tempered" have split keys for some notes, so that you can choose D# vs Eb (for example), depending what else is going on.

I'm sure I've see on here something that does not just that, but also remembers what it just did so when you play your next notes it doesn't jump to a different tuning.

There are also digital keyboards that let you bend pitch after you hit a note by shifting the pressure similar to a violin.

Could also be exactly the same as "the two", as violinists would also often just play those two at the traditional "piano" pitch, when playing alonside a piano and other such instruments.

Sorry but as a guitarist this just sounds like "violinists miss the half-step notes on purpose"

Which is okay. I like to bend notes too, but just call it what it is.

Depends on the temperament you are shooting for, at least that's my understanding after reading the article.

Would there be a specific notation for the merry-middle in-between note (D# and E)?

You mean D "three quarter" sharp? The name is a bit illogical because it's really "a sharp and a half", or "sharpened three quarters of a tone". The usual representation looks like a sharp with three vertical bars, and there's a unicode symbol for it (tried to cut and paste but no luck). Microtonality is really annoying on a piano though.

As it happens I've been trying to work out what exact intervals are used for the two-chord leitmotif heard in "The Sandman" series, I'm not sure if they're regular microtones or just some sort of eerie detuning (surprisingly I can't find any discussion of it online either).

I meant to ask regarding D-sharp and E-flat, but a typo led me to learn about microtonality. Bravo and thank you!

the sandman (*) intervals aren’t coming from microtonal tuning… it’s dynamically modulated detuning in equal temperament, just as you say. it’s an extremely common type of modulation, especially if there are synthesizers involved.

* what an incredible show!