The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.
You want (a) because it you gives you nice algebraic properties (the music structure is invariant under frequency shifts). You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency: f, 2f, 3f ... nf. Small-integer ratios are naturally occurring and very recognizable.
Modern tuning (C-f "12-TET" in the article) almost, approximately satisfies (a) and (b) simultaneously. "12" means there's twelve tones between f and 2f; the ratio between adjacent tones is defined to be 2^{1/12}. This tuning can't contain both f and 3f (so it fails (b)), but it *can* contain f and 2^{19/12}f ~ 2.9966f, which is actually close enough to 3f to be indistinguishable. (Almost works!) But as you build ratios out of larger integers, it audibly falls apart. The closest you can get to (5/3)f is 2^{9/12}f = ~1.6818f, which is already 10% of the way to the next note. And it rapidly gets worse.
This is why two on-paper-identical notes can end up audibly different, depending on what key you're starting with (and hence how they are approached). There's tension internal to music theory itself.
Now that we can have electronic instruments that "tune" themselves, could we compute song-optimal tunings that preserve the intervals used most in that song? Does this have a name?
As a guitarist we often swap guitars or retune to make certain songs easier to play, or to be able to get a certain tamber put of the note. But I never considered it as a way to address temperament.
It's interesting to think how much of music theory emerges out of reconciliation with available instruments, as opposed to reconciliation with the ear.
Since you're a guitarist, there's also this Swedish guitar, which purports to solve the tuning problem (which I tend to think is not a problem but an essential part of the instrument's sound) https://youtu.be/-penQWPHJzI
Wow, this guitar sounds so ultra-clean! Depending on song this could be pretty nice.
But the normal, "imperfect" guitar does not sound bad. I would also say, this "imperfection" gives a guitar its typical sound in the first place, so it's not a "problem".
Both guitars in that video are great, but indeed quite different.
My favorite style of music (Psytrance) almost requires digital "perfection".
It's even not really possible to create a "properly sounding" Psytrance bass-line¹, not even a most basic variant, without doing some math (or using tools that will do that math for you). Frequencies, pitch, tempo, and phase need to match constantly and absolutely perfect, or it won't sound properly. Any "humanization" on any preset would kill the sound instantly!
For that reason creating Psytrance is a very "mechanical" task that only machines can perform with the required precision. (And not every machine is good enough for that actually. You need for example oscillators with very high precision or you will experience unwanted artifacts, especially on higher frequencies, that could destroy the sound).
Something that could create "perfectly matching" chords that don't include any dissonance would be really useful to get the (most of the time) desired "ultra-clean" Psytrance sounds. The usual alternative is to filter out all dissonance. But that's a lot of work, or in "bigger" chords or soundscapes outright impossible (even when you slice the sound in the frequency spectrum with all kind of tricks; filters also produce artifacts… And trying to get rid of those artifacts, like phase imperfections, changes the sound again in often undesired ways. A "perfect" tuning form the get go would maybe help with such things).
___
¹ Here two of the better tutorials for Psytrance bass-lines:
I don't think all that electronic "magic" will go away anytime. It's more or less a prerequisite to create some kinds of sounds, which you just can't "manufacture".
It's not only about pure electronic music.
Almost all modern productions in all kinds of genres are depended on the usage of computers (and other machines). All that tech makes modern sounding peaces possible in the first place.
Even seemingly "maximal analog" music (like for example classic orchestra) sounds on a record the way it sounds because there was a lot of digital processing to remove all kinds of imperfections form the recorded raw audio, and of course a lot of of other post-processing.
Also I think it's clear what the majority prefers: People are asking actually for better produced pieces, otherwise the market wouldn't have moved that way.
Back in the day you would need to record and produce in one of that big, very expensive, well know studios somewhere around the world if you wanted to land a hit. The tech in those studios is worth millions to this day. Nobody would have invested in such a thing if people wouldn't have liked the results coming out there. (Today it's thankfully much cheaper to have a good recording; actually some small HW setup and a powerful PC with SW for only a few thousand bucks is more or less enough to sound absolutely professional, if you only know what you're doing of course).
On the other hand your actually right in some sense:
In a lot of genres "absolute perfection" isn't a goal. We use computers to remove "random (human / physical) imperfection" but add in on the other side some "humanization" into the sounds by digital means.
I think there is at the moment even a kind of trend to add more realistic, powerful, and interesting "humanization" possibilities to digital instruments. ("Humanization" or "randomization" knobs aren't anything new, but those functions get extended lately I think; for example you can use now AI instead of simply adding some randomness to some parameters).
So yes, we want to keep "natural" sounds to some degree. But by now we're using computers to artificially make the things that we're producing digitally sound more "natural". :-)
I think the core point here is control. To have something sound really great you need to be able to control every aspect of the sound, even the most tiny details. "Imperfections" are OK, or even actually desired, but only as long as they're added deliberately and remain controllable.
A license has to be purchased, and I think only Steinberg's Cubase and Apple's Logic Pro offer it as a feature. Since Steinberg is owned by Yamaha I suppose they might be allowed to use it in a hardware synthesizer, but as far as I know they do not.
Edit: this table says that Access have hardware synths with Hermode tuning: http://www.microtonal-synthesis.com/micro_af.html . Elsewhere I see Waldorf listed as having offered Hermode in some models.
> Now that we can have electronic instruments that "tune" themselves, could we compute song-optimal tunings that preserve the intervals used most in that song?
We've had self-tuning instruments for thousands of years. Vocal harmony has almost always been perfectly tuned for its key. Likewise orchestral strings are fretless and can produce perfect intervals. Equitemperment was an innovation in the 17th century because it approximated the perfect intervals very well ("sounded good") but also permitted the ability to simultaneously represent scales based on every note in the circle of fifths ("sounded interesting"). But the "real" chords were always (well, since the late middle ages) understood to be integer ratios.
This is a cool insight! Can choruses be shown to dynamically adopt "optimal" tunings for a particular song? I.e. the singers settle onto frequencies that make the song's intervals sound best?
To be clear, I'm trying to explore the idea that individual songs have optimal tunings because they only use certain intervals. So, something more fine grained even then singing for a particular key.
You can absolutely sing a perfect chord. That's most of the idea behind styles like barbershop, for example. But things start to fall apart when chords transition between each other. The first and third notes of the central chords in a key will line up on top of each other, but the middle notes of the chords and triads based on other notes don't. So just like an equitempered scale sounds a tiny bit off, harmony gets wonky too if you try to do interesting things.
So the compromise we've all settled on is that we play music in the equitempered scale, and only adjust a little bit here and there to exploit perfect tunings in limited, style-dependent ways.
Which is to say: perfect chords are interesting flavor, but at the end of the day kinda boring in isolation; "real" music needs more rules.
Awesome appreciate you explaining this. Hadn't considered the idea that transitions vs simultaneous notes "compete" on what the optimal note frequencies are. And very cool to understand that people are dealing with this pragmatically all the time.
Singers will do this via intuition - you don't think of, say, a perfect fifth as 2^(7/12) = 1.4983x over the root. You think of it as a particular pair of sounds that resonates well, much like when you picture "red" in your mind you're not thinking of exact HSV or Pantone values. At most, you'll think of a perfect fifth as exactly halfway between the octaves (1.5x over the root). As the sibling comment points out, this isn't the singers choosing a particular temperament for the entire song; it's them constantly tuning individual chords and intervals to each other and to their previous notes as the song goes on. The same note on paper can be several slightly-different frequencies in different parts of the song, and most singers won't even be able to tell you that they're doing that.
(This is also the same mechanism at work when an entire choir singing an unaccompanied piece goes flat without realizing it. Someone will not quite make an ascending interval, and everyone else will adjust to cover it.)
Does this apply also to sax? I've listened to some (mainly old) recordings where the sax seems clearly out of tune; sometimes it's subtle but there are recordings in which it's so off that one wonders if it's done on purpose (1) and personally I really dislike it. Back in the day there weren't digital effects or they were so primitive that applying pitch correction on the fly while maintaining sound quality and spectral integrity was out of question, still tape recorders allowed to finely set their speed, so tuning the song to a sax being recorded would have been trivial.
As far as I know it applies to both brass and woodwinds, though the degree of difficulty involved probably varies between types of instruments and also (at least on the woodwinds I'm familiar with) note to note.
A sax is really easy to control (or lose control) of the pitch. And in fact, many saxophonists will just shift various ranges around sharp or flat to suit their style (cough Phil Woods cough).
So basically, a woodwind like a sax will tune a few notes with a piano or whatever, but it's really up to the player to keep playing in tune. I would not even bother trying to autotune or use post-processing; it'll just sound weird.
This is also how you can get a room full of student musicians "tuned" but it still sounds like a disaster.
retuning via tape velocity modulation would be easiest if the instrument in question was consistently out of tune with the rest of the band - like if the sax was always 15 cents flat relative to the harmonic structure.
usually that’s not the case though. typically it’s individual notes. much harder to precisely and accurately modulate tape velocity (especially by hand).
Yep. And - where musically appropriate - if you know which note of the chord you're playing, you can tweak the pitch towards the Pythagorean tuning and get the harmony to "ring" as the harmonics of each note reinforce each other.
This sort of hybrid tuning is common in barbershop quartet singing as well.
Also, the tubes for the individual valves have their own tuning slides. A trumpet will typically have a little thumb-operated lever for one of those slides, to help with some of the notes. I saw a video of a tuba solo, and the tubist was working the tuning slides almost as much as the valves.
Thumb-operated levers on trumpets are uncommon (though, IMHO ergonomically superior). More common are a ring in which you place your left ring finger. The ring is directly attached to a slide on the third valve, so you can flatten notes by extending your left ring finger.
Those removable slides are usually just to get the spit out of the loops. I haven't seen the lever you mention - are you sure you are not just looking at the usual [water key](https://www.youtube.com/watch?v=vMbb8-WK_VM)?
With valved brass instruments you are trying to approximate a logarithmic relationship with a linear sum of components. Trumpets have a high resonant Q, so not using the valve slides is going to produce out of tune notes. I played horn once upon a time. Horns have low resonant Q, so you just “lip it in”.
Is "maximized consonance" what causes those extreme sharp sounding ring tones? (After listening to this peace my ears are still ringing; 2 min. after the fact).
Also the tonal glides sound like an old broken record player. (This creates a sensation of "wobbling speed", which sounds just wrong).
Hmm, my ears are still ringing, even while writing this; that was not a pleasant experience to be honest…
I guess I need some ear-bleach. Psytrance to the rescue! Let's see, maybe, hmm, Talpa¹, or maybe better that old Atma set²?
This is really cool. It sounds weird for about the first 45-90 seconds but then my mind adjusts and it sounds really pleasant. Would make a good context/theme for a video game soundtrack.
To me the individual notes sound fine and usually normal-ish (except for the really extended ones), but I have difficulty hearing the overall tune? Or, it sounds like there are parts of a tune with other parts on top which I don’t hear how they fit?
I think a clearer demonstration might be to have a side by side comparison of a fairly simple tune in 12TET vs in this dynamical tuning.
Agreed! I’m not hearing a very definable or memorable melody or harmony. The synth sounds chosen are kind of grating, which doesn’t help. I’d love to hear something more coherent in this sort of tuning to get a better understanding of it.
It really is kool. However, I have the feeling you can transport only a very limited range of emotions with it as we are accustomed to certain harmonics I guess.
Still, it's kinda like alien music and it's certainly creative.
But there seem to be differences. Some demos have those tonal glides (that I don't like) but some don't (and sound just great).
Could someone explain in a "TL;DR" what's going on here?
But I see, that wiki I just found seems to be full of info. But it will take time to read all that… Would prefer to have some VSTs to just play around with. Any tips?
And why spell it "color" like the Romans did when you can blithely attempt to imitate the French aristocracy by injecting arbitrary "u"s into random words, thus giving you license to complain about CSS keywords for the rest of recorded history? :P
The Norman conquest of England brought with it pork, beef, mutton and plenty of other adaptations of Old French words. The nobility ingratiated themselves by adopting the new vocabulary, and doing so (true for most of history, I imagine) stood out as a social status signal. The way of speaking filtered down to the lower classes over time.
Whenever I hear the phrase "that guy" in a guitar/music thread I can never not hear Guthrie Govan cracking jokes (also funny, in context to your comment considering it's pronounced "guh-van" despite the spelling)
The pronunciation is highly variable and the spelling has historically also been variable. When French words are imported to English, sometimes people try to retain the French pronunciation and other times they anglicize it. This word seems to have been handled both ways.
Another thing that happens is that both English and French change their pronunciation over time. After English imports a word, the French pronunciation may change making the English word look odd or not even look connected. Not sure that this happened to “timbre” but it did happen to words like “chief” and “chef”. Both were imported from French but at different times. “Chief” when French used the hard ‘ch’ sound and “chef” when French had switched to the soft ‘sh’ sound.
Maybe, but it needs to be the whole band not just one instrument. What notes the bass us hitting changes how the guitar needs to sound and vice versa. If you have a large orchestra it's gets hard, and even worse if someone hits a wrong note.
Okay but in 12-TET, there are only 12 notes. D# and E♭ are the same note, because there is only one note between D and E. On paper and in practice, the note between D and E is the same whether you write it as D# or as E♭. A piano doesn't know how the note is written in the sheet music.
EDIT:to be clear, I'm not disagreeing with most of what you're saying. 12-TET can't represent the desirable perfect fractions, and in a system which can (such as a just intonation system), the starting point does matter. And maybe a vocalist or a violinist would play D# and E♭ subtly differently, I don't know. My main point is just that in a whole lot of contexts, such as when playing a piano, there is no difference between the notes. Your comment made it look like there's always a difference between theory and practice which makes D# and E♭ different in practice, when that's often not the case. We do use 12-TET in practice.
A piano doesn’t know the difference and can’t differentiate them (on the fly), but a violinist certainly can and does. Most instruments have real time manual control over intonation and skilled musicians will bend pitch to best meet the current key and context.
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.
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.
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).
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.
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.
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:
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.
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!
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.
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?
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.
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.
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).
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.
As a life long string player I can tell you that there is no difference between E flat and D sharp. String players usually play with other instruments that are not so tuneable. Good intonation means playing in tune with the other players, not playing according to mathematics. If you don't have good intonation then you hide it with vibrato. Flats and sharps don't enter into consideration.
The one exception is harmonics which are based upon integer ratios rather than 12-TET.
I can't edit my comment above but I want to clarify: I don't mean to say that string players always play in 12-TET.
If you're playing a C in C major and I'm playing a G, it may sound best if my G is close to a perfect fifth from your C in just intonation. This is why string sections often sound so sickly sweet, like A Capella.
On the other hand, if you are playing a C and a G on the piano, and I'm also playing a G, then it will sound best if I play the same G as you in 12-TET. If I were to play the "correct" G against your "wrong" G, it would sound out of tune.
Context is everything.
As you may notice, G doesn't have a sharp or a flat in C major! If string players relied upon accidentals to tell them how to tune a particular note, they would be out of luck seven twelfths of the time.
That process of adjustment: called intonation, happens after resolving which pitch class I want to play. It's not something that an arranger can control through the use of enharmonic spelling, but it doesn't stop them from trying!
I think to understand the difference between the two notes is context. Like the word ‘read’. The same word is pronounced different according to context. ‘I read the book’ vs ‘Did you read the book’.
When you read music you expect a e flat not a d sharp and vise versa
For the random piano, you are right, there is no difference. For a paino being used to play a very specific piece, the tuning might be slightly different depending if the intended song is using D# or E♭, depending upon the key of the song. Though in such a case the piano might be tuned using a different standard that better fits the song.
One more extreme example is two pianos tuned to 12-TET, but one is half off. They are made to be played together by two closely in sync pianists for a few more complex songs that need 24 steps between octaves.
Overall I do find the system confusing enough to wonder if a better one won't one day catch on. And it might already have, I know some musicians who can't read sheet music but play by chords. It seems more limited in the level of detail you can specify, but works plenty well for the songs they want to play.
I worked as a harpsichord tuner during college, and this kind of tuning was generally only used when only string instruments were involved. Once a single instrument with holes, valves, or frets is involved, you have to use equal temperament. Almost nobody does specialty tuning.
Excellent explanation! It's not certain though that (a) is as desirable as we make it out to be. We accept that transposition is transparent but it could not be. Keys used to have a meaning attached to them and weren't interchangeable. The direction we have chosen made us lose that and it's a little bit sad IMHO.
In Go, and Chess, there are a number of "rules": you should never do this (move the same chess piece twice in the opening), you should do that, ... And then AlphaGo appeared and dismissed all this and did just the right thing for the particular game being played. Know the rules, but if you are an expert you can break the rules.
I wonder if AI will do the same thing in music, it will use the "perfect" tuning suitable for a particular piece of music and dismiss this idea of a universal tuning scale.
You can't realistically have a different guitar or a different saxophone for each and every piece you want to play, and those frets and holes can't be freely moved around. It gets that much worse when you consider "installation" instruments like carillons or pipe organs.
AI just literally, fundamentally can't "dismiss the idea of a universal tuning scale", because whatever per-piece optimisations it can come up with still need to be realised by physical instruments at some point. The idea of a good-enough compromise solution that allows you to play a wide variety of pieces on a single instrument is just too damn important.
There is more to music than just physical instruments.
In popular commercial music you do literally have a different instrument (synth setup) for each song.
But even if we talk about guitars and saxophones, I was speaking about AIs which directly output a sound file, not a music sheet. So they can synthesize a fake saxophone which is tuned in a weird non-physically possible way, as if each note was played by a different physical saxophone that the musician switches to.
You specifically brought up Alpha Go dismissing the conventional wisdom on how Go should be played. Many of the things we thought we knew about the game turned out to be wrong and the game as a whole was turned on its head.
None of that applies to music. Nobody who studies this stuff seriously is under any sort of illusion that 12-TET is the "right" way to play music. I know a fair few professional musicians, and I've "talked shop" with as many of them as I could, and the deficiencies of 12-TET recurringly come up. There is nothing here to "dismiss".
Don't get me wrong: The idea of computationally-optimised tuning sounds really interesting, and the discussion of what we should be optimising for would itself be fascinating to follow. It's just that people are already doing that sort of thing manually today, so there's no big "oh no we're doing it wrong" dismissal of the status quo waiting at the end.
But how would we know that? People thought music was figured out and then atonal music was invented/discovered/re-discovered (whatever you prefer).
We are somewhat speaking about different things. You talk about people playing instruments, and you are sort of right, all possibilities were explored.
I'm talking about audio files with songs, many of which are currently being produced with software using a specific tuning (typically 12-TET). But in this world the tuning is just an artifact of the production process, it's not fundamental like in your world.
The current picture producing AIs don't start with a blank digital canvas and drag digital brushes over it, they synthesize the image in a holistic way and in this world the "brush" can be unique at each position.
More precisely, I'm thinking that music producing AIs could make music where the first 5 seconds of the lead instrument uses 12-TET and then switches to another, the backing bass track uses a different tuning, the vocal sings to yet another one yet it all comes together beautifully. And the tunings used could morph during the song duration. In a way this means that there is no tuning at all.
Again, the point isn't that there's nothing left to learn. There's plenty to learn, and plenty to explore, and the whole field of applying computational methods of all sorts to music is a treasure trove waiting to be explored.
What I'm saying is that the situation with Go was completely different. The Go community was utterly convinced that the state of the art was within a couple stones of the hand of god, and Alpha Go thoroughly disabused them of that notion. The status quo was completely shattered, and the community's understanding of the game as a whole was completely upended. It's entirely fair to describe the situation as "and then AlphaGo appeared and dismissed all this"
The situation in music is very different. Ethnomusicology has been a thing since the mid-20th century, and musicology in general has swung away from prescriptivism and more towards descriptivism. There can be no earth-shattering revelations here, not because our current understanding of music is unassailable, but simply because there is no earth to shatter to begin with. AI-drive computational music might produce some innovative work around how we understand pitch and tunings, but that work won't dismiss our current understanding of those things, it'll sit alongside it.
Ok, this is fairly long winded, but the point is that I take issue with the "dismiss" part of it all, I guess.
I think the key difference is that playing go is about winning (at least, presumably that's what the AI is optimized for). Music is not.
(I also agree with others in this thread that the popular commitment to equal temperament is exaggerated -- it's not all that uncommon to hear good musicians of various styles playing/singing/synthesizing "out of tune" music for various effects).
> You can't realistically have a different guitar or a different saxophone for each and every piece you want to play
Looks quizzically at 44.1kHz-u16 audio sink.
Pretty sure I can, actually; my computer's speakers certainly do, barring a rare handful of groups of songs that were recorded at the same time and place.
People broke conventional rules with success in both Go and chess before AlphaGo and AlphaZero.
In a similar way, people have been using particular tunings for their songs for a long long long time. The idea of a universal tuning scale is relatively new. No need for AI to point us away from it, we already did that ourselves.
Yes and it was common for the player to tune (and presumably retune) fixed-intonation instruments like the clavichord... Bach famously had his own tunings that "sounded good". You still see this today with e.g. hurdy gurdy players who are constantly tweaking their tangents by ear.
>where he explored some (to me) weird tunings, starting with one where A = 432Hz
That's just changing a convention, not a tuning in the sense talked elsewhere in this thread (how we divide notes), but "what our starting frequency is".
A=432 and A=440 is just as arbitrary. They just had to pick something so they would all match.
The main difference is that 432 is associated with a set of new age, healing, "universe", etc. BS claims in certain "spiritual" circles...
Some late baroque-period harpsichords had a selectable A: you could chose ~430, ~410, or ~390. The adjustment came from sliding the keyboards to the left or right based on which A you wanted. Supposedly A = 390 or even lower was used by the French in the renaissance, so you wanted your harpsichord to be able to accurately play historical music.
Tuning your orchestra high was sort of the 19th century equivalent of the modern loudness war. The problem is that orchestras tuning to ever higher pitches meant that singers had to sing higher to match, and it was putting serious strain on their voices, which can easily lead to injuries.
Having some sort of agreement setting a standard was just something of an "enough is enough" sort of moment. It just amuses me to no end that this was achieved by writing it into the Treaty of Versailles, of all things. We're settling a freaking world war, so let's make sure we settle the issue of orchestra pitch as part of the treaty.
Though it is true that a lot of older string instruments weren't designed to take the tension of modern strings at modern pitch, and some of them really open up at a slightly lower pitch. I'm building lyres, and many lyre people are from that "A432 resonates with the universe" crowd, so I've been using it — and I can't deny that there seems to be a sweet spot in sound for a lot of instruments at that pitch. I honestly have wondered if there's some physiological reason so many people prefer it.
Aside from the universe, there's a very practical point related to this, which is that the instrument has other resonances besides those of the strings. E.g. I have read that the frames of harpsichords are tuned to particular resonances, which is part of what gives different keys different qualities.
Well he does go into Pythagorean tuning later in the video[1], both a proper one and one which was made to "look nice", so bit more to it than that from what I understood.
Or I might be wrong, I know nothing about music[2].
In the context of the difference between D# and Eb, 19-TET is very interesting to play around with. It adds an extra black key between every pair of white keys, and most songs intended for 12-TET still work fine, as long as you play sharps and flats as written. If you play a D# instead of an Eb, you suddenly get a very different sounding interval.
Also as long as sharps and flats are written in a very pedantic manner. For example a diminished C chord only sounds "right" if it's notated as C-E♭-G♭-B♭♭ rather than C-E♭-G♭-A.
On top of this, harmony may or may not work the same in 19-TET and 12-TET. With the same example of diminished chords, the diminished chord does not divide the octave in four equal parts in 19-TET. Adim and Cdim are enharmonic in 12-TET, but Adim in 19-TET is A-C-E♭-G♭♭; that is, only C and E♭ are the same.
In 19-TET, B♭♭ is an A#, which is two "steps" below B.
BTW I was wrong in that Adim is A-C-E♭-G♭. Still not enharmonic to Cdim, and probably diminished chords sound weird in general because G♭-A is four 19-TET steps, which is a 19-TET step smaller than a minor third. It probably sounds halfway between a semidiminished chord and a regular diminished chord.
Also want to plug my own project, Pivotuner: https://www.dmitrivolkov.com/projects/pivotuner/ . I believe it gives more flexibility and control to the performer than the others on that list. It's not publicly released yet (hopefully soon), but (anyone) feel free to email me if you're curious to try it out!
Though, this is more of a "toolset" to do custom tunings and apply them at various times in a DAW, than something actually implementing what the parent asked for.
In other words, it's something someone might use to implement what they asked for - but also lots of other things besides, and it's not meant specifically for that purpose.
(Former oboist) You can absolutely adjust the tuning of a note with embouchure, and in a group context will do so all the time to make chords tune better.
I hope this doesn’t come across as rude, but have you ever held a trumpet? Even if you’ve never played one, you can see they have adjustable tuning slides (a main one at the first bend, then a smaller one off each of the three valves). Maybe you’re thinking of a bugle? But any decent bugle player can bend notes up or down at least a little, probably to compensate for weather/temperature/etc.
I have held a trumpet, and an oboe, and every other instrument I cited here. Bending notes a little bit (which you can do on almost every instrument with varying amounts of effort) is not equivalent to playing in an unequal temperament.
My point was more about the tuning slides on a trumpet, and that it doesn’t have a “fixed tuning.” It’s almost like a trombone in a way: you can play any tone you want within a certain range by adjusting the slides.
(I do apologize if I came across poorly—I couldn’t think of another way to ask the question.)
wind instruments don't have fixed tuning. intonation allows you to bend notes enough to get the tuning you want. for a dramatic example of this, look at the clarinet solo at the beginning of rhapsody in blue.
The glissando at the opening of rhapsody in blue is not a counterexample to fixed tuning. It is a specific technique availed by having open holes under the fingers: by sliding the fingers slowly off the holes, and partially covering the holes, you can get a glissando effect. This same technique is used to create semitones.
Both of these are very difficult to do precisely, and come at a significant cost in the agility of the player. They are more equivalent to pitch bending on a guitar than adjusting tuning systems on a violin, which has almost no impact. Instruments with valves and hole covers, like bassoons, make techniques like this extremely difficult if not impossible.
However, the holes in the instrument are drilled at specific places along the length of the instrument corresponding to specific notes. This is what gives the instrument its tuning. Hole positions are calculated and drilled very precisely to make sure that the instrument is in tune. It is not accurate to say that these instruments do not have fixed tuning. The tuning is literally drilled into the body of the instrument.
Dude I've played clarinet for literally a decade (and a few years of saxophone). Anyone who's even a moderately talented amateur can bend notes enough to bend your note out of equal temperament. Sure you don't do this for anything fast, but if you have a longer chord this is a very common technique.
> The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.
I don't know much about this, but isn't (b) impossible even if you satisfy (a)? There is no sequence of numbers that contains any arbitrary integer ratio because there are infinitely many possible ratios but only finitely many ratios you can make out of a sequence of numbers.
(Obviously some ratios like 2:1 and 3:1 are more important than, say, 52697:16427. 12-TET chooses to permit 2^n:1 at the cost of all other ratios, which seems like a good tradeoff to me.)
(a) makes it much more restrictive though: you can't even have {f, 2f, 3f} simultaneously. (If 2f = a^m f and 3f = a^n f, then 2^n = a^{mn} = 3^m, which has no nonzero solutions. Equal temperament contains *no* integer ratios at all, other than whole-number multiples).
> You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency
I'd say it's more likely because intermodulation distortion between frequencies with low-complexity fractions tends to be low-frequency.
The explanation of enharmonic equivalent (though the author didn't use this term) is right, but I do take problem with this sentence:
> but what musical difference does it make? In the present day, the answer is, none whatsoever.
This is not quite right. On all fretless instruments, including most bowed strings and the human voice, enharmonic equivalent notes still have different pitches. The subtle differences in intonation is incredibly important and noticeable on the violin, for example.
You definitely risk a clash between a well-tempered instrument (like a piano) and a violin, given the piano is just one big compromise whereas the violin can hit the theoretically correct note. Either the violin (typically subconsciously) tweaks the tuning of a given note to match the piano, or the note is too short to notice (given the main way to notice the difference is by spotting pulses, or “beats”, which is the phase difference between the two notes - which could be measured in seconds if it’s <1Hz difference).
Khachaturian loved playing with enharmonics - the violin concerto has runs where you get D# and Eb (or similar) in different parts of the same run - or worse, two different Bb’s, as the run implicitly moves through different keys as it goes. This is then made particularly fun in the lead-up to the cadenza, where the violin duels with the clarinet, and to sound correct, you have to explicitly coordinate on which key the various phrases are actually playing in (given it effectively switches implicit key faster than the explicit key signature). From memory, you end up with the clarinet deliberately playing very different enharmonics to the violin, giving it an incredibly otherworldly feeling.
edit: to clarify, you literally have to say: “so play this Bb as the Bb in a G-minor scale, and this Bb as the Bb in a Ab-major scale” or similar - as they have different frequencies. Or more accurately “play this subphrase as if it’s in G-minor, and this phrase as if it’s in Ab-major”. Despite the Clarinet having fixed stops, you still “lip” the notes up and down to get the right frequency.
Yes and no, a violinist has to use their taste and experience to match the intonation of the fretted instrument in some cases (for example, when playing the exact same note) and not other cases (for example, perhaps a piano plays a C and a G and violin plays an E, the violinist will likely want to play a lower E than the piano would to get the exact ratios described in the article.)
Physical instruments can't be perfectly in tune even within themselves. Perfectly in tune is the preserve of electronic instruments. Even so I have a software synth that can adjust its own tuning within a chord to provide a more pleasing sound.
> Physical instruments can't be perfectly in tune even within themselves.
No, but they can sidestep the need to be perfectly in tune within themselves by allowing the player to produce notes unbounded by discrete steps or subdivisions thereof, e.g. fretless string instruments and trombones.
>by allowing the player to produce notes unbounded by discrete steps or subdivisions thereof, e.g. fretless string instruments and trombones
Then you have the problem that the player will himself be off, unwillingly, most of the time. Often more than the offsets of 12-tet to the "ideal" note.
> Then you have the problem that the player will himself be off
Maybe at first, but it takes all of two milliseconds to recognize "oh wow, I'm sharp/flat, I should move my hand a smidge further/closer". With more practice comes better muscle memory and better accuracy. Even high schoolers in marching bands can learn how to play a trombone in tune; a professional musician should have no trouble with that at all.
15 cents is huge. It's 15% of the way to the next note. Even 5 cents sounds noticeably out of tune. Trained musicians can easily tune to less than 2 cents without using tricks like beating to get even more accurate tuning.
Guitarists may be out of tune, but chances are more likely you're hearing a poorly intonated guitar. You can tune the open strings perfectly, but if your string scale-length deviates from what your fretboard expects, you'll have notes that progressively get more out of tune the further down the neck you play. You can't correct this with tuning, you need to adjust the tensions in your bridge saddles, and most amateur guitarists are afraid to do this.
Also you mention live settings, it depends on how big the group is I guess, but at smaller venues and smaller bands the stage monitoring is often so bad you can't hear your own guitar.
Guitars being out of tune happens because of the frets, not some lack thereof; most people tune each string with a tuner and/or by comparing to other strings, which means that some notes will be in tune and others will be very out of tune. This is one of many possible motivations (arguably the primary one besides ease of fingering) for tunings other than the "standard" EADGBE tuning: to change which subset of possible chords are in tune.
This is arguably a non-issue for fretless guitars, but those take more skill to play (and I'd imagine getting some chords perfectly in tune would entail nigh-impossible hand contortions even without there being frets involved).
Having to bend up to a given note each time you need to hit it will be slower and less precise.
Generally the bend is done after the fretted note is struck as well. I guess it would be possible to always pre-bend to a given alternate note if you wanted a constant tone, but it definitely seems like working against the grain of the tool versus just using a fretless instrument.
Eddie Van Halen was known to do this, though he was admittedly a freak of nature. He tuned the B string a few cents flat so that barre chords played up the neck would sound more in tune. If he needed to play, for example, a D chord in first position, he'd bend the D fretted on the B string slightly sharp.
There’s a video of a Van Halen concert where the synth track for Jump was played back at the wrong bitrate. Eddie worked furiously to find it on the guitar but couldn’t. It’s pretty wild to watch.
There are some players of guitars with a scalloped fretboard who do so to experiment with tempered tunings. It's definitely uncommon, but it's not unheard of.
Early electronic instruments, particularly before the 1980s had tuning all over the place, and you would have to wait 30 minutes to get anything resembling stability, even then nothing was guaranteed.
Think it would be possible to mockup some really interesting tunings/temperaments in BitWigs grid or Max4Live.
A tonewheel organ could be amplified acoustically, e.g. by physically touching an appropriately sized resonant chamber to the wheel when you press a key. The exact size of the resonator does not matter because it's mode locked to the wheel, which turns at a speed determined only by the gear train. Tonewheel organ gears traditionally do not have perfectly accurate tuning, but there's no reason they couldn't be built to match any tuning system within the limits of human hearing (at greater cost and complexity).
Which "modern rock musicians" do that? It surely is not widespread in rock.
Rock musicians change guitars (in a live or even studio situation) mostly to get a different sound or a different tuning (like going from "standard" to an open tuning). Not for microtuning adjustments, or because they have moved the frets to match an individual piece.
Some prog musicians might do it, but it surely is not a "rock" custom.
In arabic music, on the other hand, or renaissance music, and other genres, it is, and instruments there often have movable frets.
Huh…can I get a source on this that delves into it more? I played the cello in an orchestra, solos, and chamber music for about 10 years growing up and I’ve literally never heard anyone mention I or anyone else should’ve been putting my finger in a slightly different place for C# and Db. I suspect this is for all intents and purposes not true in the 21st century.
You will adjust without even realizing. You'll change your pitch to sound right in relation to everyone around you. This probably means that in practice your finger is slightly up or down depending.
It is also a reason why when playing an A, many prefer moving the hand to 4th position on the D string, rather than using the A-string with no finger. Partly because you can make a better tone (add vibrato if wanted), but you can also intonate.
> You will adjust without even realizing. […] This probably means
This might be true once in a while on very slow chords or the final resolving chord of a piece, maybe, but this sounds like assumption to me based on it being theoretically possible, and not evidence that it actually happens often. From experience, it sure seems like years upon years of equal tone muscle memory, from having to play with other instruments, is far more likely to dominate finger placement. Not to mention everyone being used to equal tone - having equal tone sensibility as to what sounds right. Sounding right in relation to everyone around you is still valid in 12-TET. Enharmonic micro intonations are almost certainly not happening during fast sequences, and because of that, the argument that it’s subconscious and imperceptible seems implausible - professional musicians absolutely would notice a change in finger placement depending on context, because of key changes, because of abrupt fast-slow resolutions, because of chords and arpeggios and situations where open strings are called for, etc. etc..
You didn't miss out on much. My cello teacher mentioned it to me early on, in passing, but it's basically useless trivia until you actually have enough control for it to make a difference. From observing my kids go through music study, I'd say it emerges as something to actually think about at the college level.
Instead, I switched to the double bass, joined the jazz band, and majored in physics. ;-)
I think "sweetening without realizing" may be a thing. You've assimilated the sound of classical (or whatever) music through listening. You can hear how you want the note sound in your head, and your finger goes there.
I’m responding to this part of the parent’s post: On all fretless instruments, including most bowed strings and the human voice, enharmonic equivalent notes still have different pitches. The subtle differences in intonation is incredibly important and noticeable on the violin, for example.
Yes exactly, and one would intonate it slightly differently by ear depending on what role the note has in the current harmony is the idea.
Presume a base note of A is being played and the violin plays a C# functioning as the major third of an A major chord. The ear would want to play the C# justly intonated to the root note A, or maybe a compromise somewhere between equal temperament and just intonation.
This is indeed a very weird thing to say by the author. Your "not quite right" is too kind to the author. As you wrote, good fretless instruments and singers will absolutely play/sing those notes differently, although maybe unconsciously.
In a given orchestra (for example), what instruments have fixed tuning? A piano yes (although lots of orchestral music doesn’t have piano.) The harp? I struggle to think of others.
Wind instruments essentially have continuous tuning because the player can adjust the pitch with their lips and vocal shape. Orchestral string instruments are all fretless (and thus continuously pitched.) Singers, same thing. Even fretted instruments are often played with a lot of vibrato that masks any true pitch problems.
I think the inability to play with a perfect pitch is more the exception than the rule (at least in “classical” music), it’s just that piano is such a popular instrument in the modern era that this becomes a problem.
Not that many: the piano, the harp, the glockenspiel, etc.
But the thing is, most modern music, is not with a classical orchestra, but can still have violin (and in some genres, like bluegrass, irish, country, etc. it often does).
>This is not quite right. On all fretless instruments, including most bowed strings and the human voice, enharmonic equivalent notes still have different pitches.
Not because we want it or it is some ideal situation, though.
Just because 12-tet can't get a single note to be in the exact right ratio. If it could, we'd play D# and Eb exactly the same.
Besides, if the fretless instrument is not soloing, it might still play it as the 12-tet single note compromise, to match what others play at the same time.
Usually the difference is a slight roll of the finger. But you can hear it. You might not know you can hear the difference in pitch, but you can hear one performer being cleaner than another and intonation accuracy is a huge factor.
the higher you go in frequency, the physical distance between each interval on the fretboard becomes smaller. so if a +/- 5 cent adjustment is 0.5 mm at the first “fret” after the nut, it will be something like be 0.1 mm when you are at the 7th “fret” location (i.e where the interval of perfect fifth, relative to the open string, is played).
My thought is, if you peel back the first layer of music theory, you discover a chaotic, lawless world. The main thing I've noticed is that this is extremely unnerving to engineers, who want to learn it as a precise hierarchical structure. Regular people are more focused on the fact that somebody is somehow making it all sound good, and want to learn how to do that.
On the other hand, most musicians are completely ambivalent to it, or even thrive in the chaos. Yet the "rules" are useful because they provide a common ground for forming ensembles, or connecting composer and performer. We've watched musicians go down the rabbit hole of nonstandard scales, innovative notation systems, etc., only to discover that nobody can play their material.
I'm a double bassist. I'm happy just to be able to coordinate my ears, brain, and hands, well enough to play the same note the same way twice if I want to. Claiming that I have conscious control over temperament would be laughable. I've got too much other stuff to think about: The notes on the page, the non-notated passages (many jazz bass parts are expected to be improvised), tempo and rhythm, connecting with the rest of the band and the audience, who's coming in the front door, and so forth. This stuff is all happening in real time.
> My thought is, if you peel back the first layer of music theory, you discover a chaotic, lawless world.
That's because some people think the theory comes first, and the music is based on it. But music is just art, like any other art. The rules are soft and broken and hardly gospel. And music theory is an attempt to have some way to communicate about music using ordinary language. It isn't math, it isn't science, it's just some basic terminology and observations, none of which have much to do with the actual artistic act of making music.
When you think of songs where 'bending' a note is used, or intentionally hitting a note a little bit flat or sharp for a desired aesthetic effect (or both), this all makes a lot more sense.
Music theory gives us a way to measure & more accurately describe what we were already doing.
Absolutely. Music theory is descriptive, not prescriptive. It just so happens that some things that sounded pleasant to people in the past still sound pleasant to modern ears, so you sometimes get into a bit of “tail wagging the dog” when people use those descriptive academic terms and concepts when creating music today, e.g. saying “I’m gonna write a 16-bar AABA tune that’s based on a I-vi-ii-V progression and modulates to the mediant in the B section”, and therefore think these are “rules” to abide. One of the more unfortunate misconceptions when it comes to the study of music theory.
Huh, that’s interesting. I bounced off learning music theory because it seemed to be all about putting everything into little boxes, and music doesn’t really work that way. What are some of the more interesting elements that you get to after the first layers?
As they say, music theory is descriptive not prescriptive. However...
A really rough analogy is a programming language. The rules of the language don't tell you what kind of program to write, but choosing a language gives you a huge jump start on creating interesting and useful programs. Likewise knowing algorithms and good patterns.
I think that very few people are interested in studying music theory as an end unto itself. Like, I have a friend who is a retired theory professor, and did his PhD in theory. (He also performs music, but treats it as a hobby). For everybody else, the purpose of learning theory is to make you a better musician. So you can take it as far as needed to make that happen within its applicability to the kind of music you're interested in.
And there are different approaches, such as "jazz theory," that doesn't spend a lot of time with (for instance) the forms of larger musical compositions, or Bach.
So, what aspect of your musicianship are you trying to improve? I can cite one example. I play mostly jazz. I'm not great at theory myself. Everybody I know who can compose good jazz, or create written arrangements for larger ensembles, studied theory in college. I'm stuck with playing their music, which I love, but am not capable of creating my own. The theory probably helps in terms of letting you go from a composition that "almost" works but has awkward bits, and make it really sparkle.
The distinctions between D-sharp and E-flat only make sense in the context of a key.
For instance, you construct a major key in E-flat and not D-sharp for the practical need to represent the scale nicely on the staff - so each tone in the scale should have a unique place in the staff.
You can construct the E-flat major scale with just three flat tones (Eb F G Ab Bb C D), whereas you would need four sharp and two double-sharp(!) tones if you started with D# (D# E# F## G# A# B# C##). And having to use F## and C## to refer to G and D tones is just ugly.
(I had made a mistake in the earlier version of this comment.)
I don’t think of it as ugly. It’s just what happens sometimes. Like if you start in G# minor and then use the leading tone. It’s way better to see F-double sharp than to see two different Gs fighting each other on the page. And it’s even worse to have to decipher those awful chromatic systems that are all painful to read.
Technically speaking that’s a Ab minor. New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s. Adding #’s are used for deriving new major scales. At least that’s how I understand it.
You can see this on the Wikipedia article on various minor[1] and major scales[2].
G# minor is used all the time as the relative minor of B major (5 sharps in the key signature). Ab minor is the relative minor of Cb minor (7 flats) and thus is almost never used except perhaps in passing for a modulation.
> New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s. Adding #’s are used for deriving new major scales.
No, a major scale can have both flats and sharps and the same for minor scales. In fact major scales often start on a flat while minor scales often start on a sharp. Major scales use Db Eb F# Gb Ab Bb as the roots of the scales (rarely C# and Cb), plus the white keys; while minor uses C# D# Eb F# G# Bb (rarely Ab and A#), plus the white keys.
> New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s.
I think you're confusing two different ways of constructing the minor scales.
One way is to start with the A minor scale (which has no sharps or flats) and to go around the circle of fifths[1] adding sharps or flats. Whether you add sharps or flats depends on whether you're going clockwise or counter-clockwise: for example, D minor[2] (one step from A minor going counter-clockwise) has one flat, and E minor[3] (one step from A minor going clockwise) has one sharp.
Another way to construct a minor scale is to start with its parallel major[4] and add a flat to the 3rd, 6th, and 7th. But note that the result can still have sharps (like in the E example above, where E major has 4 sharps).
In any case, G# minor is definitely a key that is used[5].
Technically speaking, if they said it's G# it's G#.
G# minor is a much better use of the key signature system than Ab: 5 sharps versus 7 flats. In practical terms, that's a proxy for it being more common.
Your vision that minor scales are constructed from A minor is valid; thinking it's by adding flats exclusively is misguided.
I'm not going to go out on a limb and defend the fact that sharps-based minor scales could be more common than flat-based, as that's likely not the case. A much easier argument against your logic is that flats-based major scales are used all the time.
Any given key signature can be either major or minor, be it made of sharps or of flats. It can be seen as altering C major or A minor indeed, but the alteration is allowed to go either way.
> awful chromatic systems that are all painful to read
The worst ones are the ones that petulantly stick to some theoretically-correct framework and produce a mishmash of accidentals that are canceled on the next note. If it's ascending, add sharps (or cancel the flat) on the second note. Let the key signature do the work instead of making me read all that to discover it's a simple chromatic run.
Of course people use double-sharp keys. And like you said, it is usually done in cases where it is the simpler notation to describe what is happening musically.
Simplicity is beautiful and construction of the major key in E-flat is decidedly simpler than in D-sharp.
Sorry, I had made a mistake. Wanting to create a major key starting with D# would end up looking like D# E# F## G# A# B# C##. The ugliness is even more stark.
- two double sharp keys
- F## to refer to G, C## to refer to D, B# to refer to C
If we keep the constraint that each letter has to be used exactly once when naming the notes of a major scale, but drop the constraint that the tonic has to be named using the same letter as the scale name (e.g., we can write G major starting at F##) then that pattern of sharps and flats generalizes nicely.
Number the 12 tones of 12-TET starting with C=0, C#/Db=1, ..., B=11. Then if you write a major scale starting at note N, the sum of all the accidentals counting sharps as +1 and flats as -1 will be equal to 7N mod 12.
For example G is note 7. G major then should have an accidental sum of 7 x 7 = 1 mod 12. We get that writing it G A B C D E F#. But it could also be written with a sum of 13, as F## G## A## B# C## D## E##, or with a sum of -11 as Abb Bbb Cb Dbb Ebb Fb Gb.
Note that because 7 x 7 = 1 mod 12, if we have to answer the question what scale N would have an accidental sum of K mod 12, we can solve 7N = K mod 12 by multiplying both sides by 7, giving N = 7K mod 12.
E.g., what major scale as 3 flats? 7 x -3 = -21 = 3 mod 12, which gives us the major scale starting at Eb.
Personally I find this approach a lot easier than memorizing the circle of 5ths to find key signatures given the key or to find the key given the signature.
A couple of questions naturally arise at this point. Why 7N? Why mod 12. The 12 part is easy to guess--it is because we are picking our major scale out of an underlying 12 tone scale. The major scale has 7 notes out of those underlying 12 notes, so a reasonable guess is that is where the 7 comes from.
But if you think about starting with C major (all white keys) and going up half a step, because the white keys are 0 2 4 5 7 9 11 12 (I've included the octave at 12 to make things clearer), and two of those (4 and 11) are white keys that do not have a black key immediately to the right, it might seem that how many accidentals get added or removed each time you go up in key half a step is going to vary a lot. Going from C to C#, every position goes black except those two. Those two will go black when you go C# to D, and all the ones on black will go to white.
The way the white and black keys are distributed gives you some different regions of the keyboard, each of which has a distinct pattern of adding and removing accidentals as you step through, and the overall pattern of accidentals is a result of those different patterns interacting. So maybe the 7 depends on those regions, and would be different if you had a 7 tone major scale chosen from 12 underlying tones but did not have the same pattern of white/black that we have.
I spent a while trying to show that the patterns would interact in such a way as to make 7N mod 12 work, but utterly failed.
To check that out we can try imagining alien music. Maybe some aliens who also use a 12-TET underlying scale and also have a 7 tone major scale have picked 0 2 3 4 7 9 10 as their major scale. Quite a different pattern. However, it turns out that 7N mod 12 works for that too. It also works even with alien music whose major scale is 0 1 2 3 4 5 6. You can have to use a crazy number of sharps or flats in that system!
What the pattern of white/black keys affects is which notes get accidentals when, not the total number of accidentals. By having the white and black keys spread out about as evenly as you can for a 7 white/5 black system we can write every key using the "right" starting note without needing any note to have more than one sharp or flat. Less even distributions of the black keys make it so you need multiple sharps and flats on some notes, but don't change the total number of accidentals mod 12.
Once you realize it really doesn't have anything to do with the pattern of black/white but only on the number of white keys, it is then not too hard to prove that it does indeed only depend on the number.
This can be further generalized. If aliens used a 5 note major scale, then the accidental sum of key N would be 5N mod 12. Since 5x5 = 1 mod 12, they could also go the other way and find the key from the accidental count K via 5K mod 12.
In general for a M note major scale from a T tone underlying scale, transposing that scale to note N uses NM mod T accidentals.
how would you use this approach in practical application?
i haven’t met many working musicians who had much difficulty learning the relationships between different keys, how they connect to the circle of fifths (fourths), and key signatures.
i get that it can seem overwhelming and non-intuitive, but it’s really not that complicated once you spend time playing and practicing music that illuminates these relationships (like playing ii-V-I progressions in every key, going around the circle of fifths). very little memorization involved; moreso muscle memory and an accumulation of applied theory in context.
most of the musicians i know are jazz players, where being able to play in any key is a critical aspect of mastering the genre. all the classical musicians i know are professional orchestral musicians, and they don’t seem to have any difficulty either.
>The next level of explanation is to say: “Yes, I recognize that D-sharp and E-flat sound the same, but they function differently, and the spelling communicates this functional difference.” This explanation always bothered me, because if the “function” is limited to the page and isn’t audible, then is it even a real thing?
A feature of notated music (which is what most of us mean when we say "[western] classical music") is that there can be things notated and not heard. Similarly, there are also different notations which correspond to the same sound. Notation is ambiguous, and this can be a source of both frustration (for the students) and invention (for the composers). Charles Rosen opens his book The Romantic Generation with a fascinating discussion about music which is seen and not heard which deals with this philosophical issue.
Of course this practice goes back much further. Composers have been playing with notation for a long time and it reached a peak of sophistication in the 15th century, as Emily Zazulia demonstrated in her PhD thesis and book[1]. This quality is obviously absent in musical cultures which do not rely on notation. I imagine to the outsider it appears as if the notation itself has taken on a life of its own to the detriment of the sounding music. Of course there is a certain elitism involved as well since explaining subtleties in notation is also a sure way of ostentatiously demonstrating one's erudition, which may explain why these kind of discussions are perennially popular here ;).
I would argue the difference is heard in music that broadly follows tonal harmony. Sure, the note it self sounds the same, but the difference is context. That context is there regardless of whether it is heard or seen.
Edit: just to add some detail: you can definitely hear if something sounds Lydian. If it sounds Lydian, you know that's a sharp 4, not a flat 5. Put it in C e.g., then it's an F# and not a Gb, and you can hear that.
The author frequently uses the term “same note” when the more accurate term in this context would be “same pitch.” Your comment and the next one up both clarify why having differently-named notes that use the same pitch matters.
My favorite illustrstion of this is “Call Me A Hole”, a mashup where the vocal track of NIN’s “Head Like A Hole” is played atop the music track of Carly Rae Jepsen’s “Call Me Maybe.” A vocal performance that was originally seething with rage is transformed into a disco pop anthem, and the main reason it works is because “Call Me Maybe” was written in the relative major key to “Head Like A Hole.” The same vocal pitches—the whole melodic structure—functions entirely differently, with hilariously effective results.
The mashup is an in-the-large example of the musical context you mention. D-sharp and E-flat is the same principle, just at a much more fine-grained scope.
Yes, exactly. A note is rarely found by itself, and looking at the context surrounding the note usually clears up the function of a note pretty quickly. Now that I think about it, the notation actually reduces ambiguity in this case since it specifies the function of notes which have the same sound.
Instruments without frets don't have this problem. I played violin for many years. When you play a double stop (two strings at the same time), since there are no frets, you can play true 3rds, 6ths etc. The harmony is so "pure" that it causes a third harmonic to ring (which is how you know you're doing it right). My violin teacher always insisted that e-flat and d-sharp are not the same. When you're playing in different keys you have to put your finger in a slightly different place.
Right. I had this too, but because I never got the explanation this post provides, I had to live with "because it's a different key" - but could never quite understand why it made me out of tune with an accompanying pianist. Now I know... this is awesome!
The people replying to this thread, the person who wrote this blog are so far away from me it is hard to say we inhabit the same field called 'music'.
I play a lot of improvised lead lines. I know my pentatonic scale shapes on the fretboard, but I also play lots of notes not in those shapes...because I like the way they sound. I also play a lot of 1/4 tone bends (notes between the piano keys) which don't even fit in the traditional system, but sound good. I say this as it is an interesting case of 'more than one way to skin a cat'
>I say this as it is an interesting case of 'more than one way to skin a cat'
You're basically skinning the cat the same way, you just don't know the terms of the steps involved or the theory (the "why") behind them, and can't generalize it to ways to skin all kinds of other animals, and even do taxidermy on them - things that the author does.
You however might have picked some special tricks of cat-skinning, and self-taughtingly built your own small conventions, that the author might not know, but which still follow music theory - which, in musicology, is way broader than "common practice" music theory -, (and the author could also delve into them and explain their function theoritically if he played the same genre and bothered to check them out).
Blues guitar is kind of the odd one out in terms of the field of music, its one of the few styles where you can get by quite well without note reading or theory. Everything changes if you have a horn section in your blues band though. Music notation exists to facilitate people with different instruments playing together.
As someone who learned the same way you did, on tab and modern pentatonic blues riffs and improv, but over time learned (still learning) more theory and reading music, I’d recommend learning more reading & theory because it really seriously expands on what you can do with blues guitar. A lot of early blues and pretty much all jazz don’t stay in the pentatonic scale rut, they move around and mix other scales. It’s really helpful to know which diatonic scales you can seamlessly blend with pentatonic, and the reverse: when you can blend pentatonic into a modal song structure, just for two simple examples.
BTW 1/4 tone bends are definitely in the traditional system, they are common even, and in fact are quite directly related to what this article is talking about. The “blue note” in blues is a well known microtone example, but microtonal music in general has theory and notation hundreds of years old, there’s a lot of stuff taking these ideas to new levels. Wikipedia’s article is just the tip of the iceberg, microtonal music history is bigger and broader than this suggests: https://en.wikipedia.org/wiki/Microtonal_music
Music is kind of great like that. Much of music theory is over my head but I have spent time to understand a good bit too and I’ve gained a lot of respect for it. Everything you’re doing as a blues guitarist can be explained well by a music theorist who really know their stuff.
I read an interesting article by a music theorist breaking down the song “Smells Like Teen Spirit” and remarking on the genius and uniqueness of the chord progression and how it violates a lot of what theory says would “sound good” and hence why it’s genius. It can be presumed Kurt Cobain was not too interested in music theory and if he would have been he may have never even considered the progression and other interesting aspects of that song.
I've said it before, I'll say it again. It's more productive to think of "music theory" as a way for musicians and composers to talk about what they just did, or what they are just going to do, than as a way to generate those things.
Giant Steps "violates" jazz music theory on the surface but sure enough Coltrane knew his basic chord progressions. If you look close enough Giant Steps builds on traditional ii-V-I progressions and applies (also well known) tritone substitutions to achieve quick key changes.
the progression you are referring to is I-IV-bIII-bVI, where are all these chords are “power chords” i.e. dyads comprised of a root and fifth.
it is an awesome progression but violates absolutely nothing in music theory.
we can find examples of similar progressions across jazz and classical music, most of which was composed by folks who have mastered western tonal harmony.
Same here, I know how to read music and tabs but gave up on it and now just play by ear, it’s way more satisfying. Western music theory is so baroque. The book Brainjo basically killed my interest in it.
A lot of those in-between notes hark back to times before the scales were rationalised into the "well tempered" system we mostly use today. Often they are harmonics like the 5th harmonic which lives between the minor and major 3rd on the scale.
Obviously Ethan knows this and just isn't going into it because this is more a history lesson than a theory lesson, but the same applies to white keys. So B♯ and F♭ are perfectly valid notes. C♯ major for example contains B♯, despite that there's no black key between B and C.
For that matter, F𝄪 (F double sharp) and A𝄫 (A double flat) are both legitimate alternatives/equivalents to G in some situations, by extrapolating the sequences. (And if you extrapolate far enough, any note has multiple possible alternatives—for example, you could get a G♮ that’s kinda more G♯♭ or G♭♯ than just straight G, to use super fuzzy terminology.)
There is some basic information that is very wrong in this article. For example:
"My track is tuned in a system called five-limit just intonation via the magic of MTS-ESP. It’s the basis for all the tuning systems used in Western Europe between about 1500 and 1900."
No - at no point in the last 500 years was 5-limit just intonation ever the predominant tuning system used anywhere in Western Europe. The real predominant tuning system was called "meantone temperament," to which this article sadly devotes only about 3 words - and those words are only about 12 tone meantone keyboard layouts, not about the bigger, abstract idea of meantone temperament in general as it was understood and taught by practitioners of the day.
There is a very important difference between meantone and just intonation. The goal of meantone was to have four tempered perfect fifths (approximately a 3/2 frequency ratio) add together to approximate the fifth harmonic (or a 5/1 frequency ratio). Thus, the major third from the circle of fifths would approximate a 5/4 ratio with the tonic, and the major chord would approximate a very crunchy sounding 4:5:6 ratio. In order to do this, fifths are all flattened slightly to make the tradeoff - flattening the fifths by 1/4 of a "syntonic comma" was typical, or "quarter-comma meantone". Even though keyboard instruments evolved in a more well-tempered direction, meantone was the way that teachers of the common practice era (such as Leopold Mozart) still taught and thought about this stuff.
Meantone sounds noticeably different from just intonation, where the major third from the circle of fifths is a syntonic comma sharp of a 5/4 ratio (about 22 cents). In just intonation, if you want your major chords to be 4:5:6, you need to bring in this other, different, independent 5/4 major third that is not on the circle of fifths. As a result, certain chord progressions that are common in Western music will tend to exhibit strange sounding "comma drifts" if you play them in just intonation. Adam Neely has a good video on "Benedetti's Puzzle" about this for those who are interested.
Of course, there is nothing wrong with just intonation, and comma shifts can sound interesting if you want to deliberately use them in some kind of modern microtonal setting, but it simply isn't the tuning historically used in common practice Western music.
Anyway, though, if you go through the article with a marker and replace all instances of "just intonation" with "meantone," the general idea is mostly correct.
Hi, I'm the author of the blog post. I said that 5-limit is the basis for systems like meantone, which is true. Meantone systems take 5-limit as their starting point and then modify it. I deliberately skated over the specifics of how meantone works on purpose, because I have too much experience watching my students' eyes glaze over when I talk about this kind of thing. I'm trying to strike a balance between giving correct information and not turning people away.
I can't comment regarding how you think is best to teach your students. This is now a popular blog post that has gone viral on HackerNews to a much wider audience of well-educated people, so you should expect people will clarify these things on here. I'm talking mostly about stuff like this:
> Five hundred years ago, however, it would have made a very big difference. Before the advent of temperament systems, D-sharp and E-flat were two different notes. They weren’t just written differently; they sounded different. You can compare the historical versions of these notes yourself in this track I made... My track is tuned in a system called five-limit just intonation
^ These are not the historical versions of those notes. 5-limit just intonation was not in widespread use in Western Europe 500 years ago. 500 years is not before the advent of temperament systems. And so on. Teach this to your students however you think is best, but people on here may be interested to know that.
I know that pure five-limit just wasn't ever a prevalent tuning system, but all the temperament systems are based on it, so it seems reasonable to present it as a kind of baseline, the "ground truth" from which the meantone and well temperaments are departing to a greater and lesser degree. As to the dates, I have read a bunch of histories of tuning, and the main conclusion they present is that there was no uniform (or even predominant) tuning standard until 12-TET, so simplification is necessary. I am definitely open to the idea that 1500 isn't close enough to the "real" widespread advent of temperament to use as an approximation and that I should instead be saying 1400 or whenever.
What's weirder is that if you're using just intonation in, say, the key of C major, there's two different D's. One is 9/8, and the other is 10/9. The note we call D in equal temperament is about half-way between the two.
If you want to play these chords in C major: Cmaj, Fmaj, Gmaj, Dmin, Emin, and Amin, you'll need to use both of those D's: 9/8 for the 5th of Gmaj, and 10/9 for the root of Dmin. If you try to play Dmin with the 9/8 instead, it sounds absolutely awful.
In other words, if you want to play those six chords that are regarded as belonging to C major, you'll need two different D's. Which means the C major scale should really have 8 notes instead of seven. But we don't have a symbol to distinguish between 9/8 and 10/9 in standard notation, they're both just plain D.
Some 12-EDO music makes use of the ambiguity between these two notes (or any two notes with the same relation to each other) to string together chord progressions that don't actually make sense mathematically. If you used those progressions in just intonation, you'd find you don't return to the chord you started on, you actually shifted up or down by a small interval of 81/80. It's sort of the musical equivalent of some formula that only works if you assume that pi is equal to exactly three.
Perhaps more succinctly, you always write any two consecutive notes of a diatonic scale on two different lines of the staff. It would be bad if your in-key scale looked like it had two notes on the same line and then a jump of a third. Note this is true even when you have accidentals! It’s a way to keep the intention or semantics of notes more clear, and more easily readable, regardless of the pitch interval. Like how the article talks about the distinction between an augmented 2nd and a diminished 3d, the notation is designed to help clarify that distinction.
Right, but if your tune mostly uses conventional major and minor scales (which most do!) you mostly won’t need accidentals.
Also, you’ll be able to transpose to any other key just by shifting the letter names up or down and changing the key signature. That’s a really interesting and useful property.
Also also, the notes with accidentals won’t change when you transpose (although the accidentals themselves will need rewriting).
Transposing music would be hellish without this system!
D-sharp and E-flat are two different notes used to describe the same physical vibration for the same reason "father" and "son" are two different words that could describe the same person. It's just a way to communicate contextual relation.
Because for most instruments, it is. A violin player won't move up for a full half tone for a D sharp, so there'll remain a small pitch difference between them.
It's only for the small subset of keyed instruments like pianos that pitch is quantized into 12 subtones. But even there, organs use a different pitch to key mapping than keyboards.
For a very interesting rabbit hole, search for "Wolfsquinte", which is a chord that sounds nice on keyboard but horrible on organ.
I am not aware of any manuscript having different tuning of organs vs other keyboards in the Renaissance or Baroque eras... can you cite one?
The Wikipedia article for Wolfsquinte makes it clear that it has nothing to do with keyboards or organs: it's a feature of your choice of tuning. Perhaps you're used to organs and keyboards with different tuning choices?
I don't think that's quite GP's claim. They're not necessarily saying that organs and keyboards of the same heritage had different tuning (though it's well documented that instruments had different tuning according to region even through the 1700s and 1800s), rather that historic organs which retain their original temperament sound very different to modern keyboards.
Here's an article supporting this. It refers to different rates of beating in different tuning regimes, comparing equal temperament to 'cornet-ton' type things.
Because it's common for organs to be hundreds of years old, and it's common for people to want historic organs to sound as close to how they did when they were made as possible because it's uncommon for them to play in non-vocal ensemble, this leads to a relatively common situation where an organ played today may well be tuned very differently to a piano played today. Depending on the organ.
Here is a second article on organs by one manufacturer tuned in 'meantone' https://www.bach-cantatas.com/Topics/Meantone.htm. It's also the case that harpsichords were commonly tuned in meantone https://www.harpsichord.org.uk/wp-content/uploads/2015/04/te... which might actually support the claim that historic non-organ keyboards sounded different from organ keyboards (specifically, if Bach didn't like the mean-tone organs he played and it was common to tune harpsichords in meantone, that would seem to provide some evidence for both temperaments existing and sounding different on the different types of instrument in the same historic period).
I don't think GP was particularly making the point that it was innately impossible to tune an organ and a piano the same, just that it's common for them to be tuned differently (especially in European churches). Same with harpsichord and piano (where a harpsichord is not tuned to concert A).
Either way, hope you enjoyed the citations - I found them interesting - the one about Bach writing in specific keys so as to match the instruments he's working against reflects a different kind of craftsmanship and concern than one would see from most composers in the 21st centruy!
I tune organs professionally, and in the US most instruments are tuned to equal temperament. For the performance of pieces originally composed on unequal-tempered instruments, though, something is lost on equal-tempered organs: the movement through harmonic progressions on unequal temperaments creates a dramatic tension between consonance and dissonance, with dissonance increasing the farther you get from the more "in-tune" keys and decreasing as the progression returns to them. Similarly, pieces composed in keys that are some distance away from the "purest" key, gain their own distinctive colors. If you're used to equal temperament and then hear a big major chord in the temperament's home key on an organ with a historic temperament, the impact is really quite something as the thirds and fifths are much closer to the natural overtones of the unisons and the whole chord draws together into a gloriously-coherent tonality.
Pipe organs often contain stops called mutations (whose frequencies are non-integer multiples of unison-rank frequencies), and others called mixtures (where there are multiple such pipes per note, generally rather small and high-pitched). These are both intended to reinforce natural harmonics, and as such are tuned pure -- even on equal-tempered instruments! The exception is highly-"unified" instruments where one rank has been wired to play at both unison and mutation pitches (to save money and/or space); this sorta works for quints (fifths), but is pretty bad for tierces (thirds), and don't even try it with a septième (seventh).
While electronic tuners are often used to set an initial temperament on a reference rank (it can also be set by listening to the contrasting rates of the differential waves between fourths and fifths), we generally tune other ranks to the reference rank, listening to the differential waves created by the two ranks to discern in/out-of-tuneness. For mixtures and mutations, the trick is to be able to recognize differential beating with partials of the reference rank that are higher than the fundamental; and for very high pitches, listening for sub-harmonics comes into play (frequencies can align in a way that creates the illusion that they are harmonics of a fundamental that's not actually being played, and our brains fill in the fundamental; this phenomenon is sometimes used to create the illusion of extremely low "resultant" Pedal-division ranks sounding an octave lower than the root of the fifth that the pipes are actually playing, and the use of an independent pure-tuned quint rank produces the most convincing result).
>Why are D-sharp and E-flat considered to be two different notes?
Officially, it's only on paper.
It kind of makes the key signatures come out more sensible because you don't want to have a signature where there are both sharps & flats in one key.
>electronic tuners are often used to set an initial temperament on a reference rank
>tune other ranks to the reference rank, listening to the differential waves created by the two ranks to discern in/out-of-tuneness.
The equivalent on guitar is to use the tuner for reference on the high E string only, then tune the low E to match perfectly by ear. You're going to be hearing a lot of these two, and they better be able to make you happy to begin with.
Then tune the middle 4 strings according to what the hands will be doing in relation to the reference strings, as well as who you will be playing with and how they are tuned.
Without an electronic tuner a single tuning fork is enough for this, and it's actually better than having a set of 6 forks at the nominal even tempered frequencies.
E=329.6 is the fork you want so you don't have to fret the high string to match an A=440 fork.
This discussion definitely sets a record for "people writing the most words to explain to me things I already know". Hopefully some spectators got something out of it.
Fun story, I once volunteered to play a piece at 440 and a piece at 415 in the same concert, not realizing that it would take a long time for the instrument (a viola da gamba) to "settle" after that drastic of a change.
The organs made by the Silbermann family are tuned with non-equal key spacing. And those are among the ones Bach played on. So if you play the same notes on a digital organ, or on a keyboard, the harmonies won't work as intended.
Native Instruments also offers to switch the tuning mode for their virtual/digital instruments, BTW, so that you can compensate for that in software if needed.
There's a ton of good information here, but it seems to assume a guitar or piano, where there's only one key or fretted space for each note.
For a fretless stringed instrument, they are indeed different notes, and the same note within a single piece can sound different depending on whether the line is moving up or down.
If that sounds heretical: I got this from the Alexander String Quartet, in the Q&A session after their performance. They have a measurement of microtones (I think they're called "clicks" but I forget), and all four of them have to agree on how many clicks up or down from the center of the note they're using.
I find that a little odd when vibrato can be as wide as 70 cents (70% of a semitone) further up the fingerboard. It makes sense for certain chords in highly tonal music though.
I asked about the movie A Late Quartet (a great movie, btw, with the immortal Phillip Seymour Hoffman), and they said, "in the movie, they say 'our vibratos aren't lining up' and that's something I actually would say in a rehearsal."
Some bands with more "approachable" sounds (vs the xenharmonic playlist, which gets spicy) known for microtonal work:
Psychedelic rock:
- King Gizzard and the Lizard Wizard (namely albums Flying Microtonal Banana, KG, LW)
- Altin Gün
- Gaye su Akyol
Classic blues rock (Led Zeppelin and the like, but also OG blues like Robert Johnson, that's another rabbit hole) also has a surprising amount of off-12Tet notes, because of the blues scale
In the electronic realm, Aphex Twin does some interesting stuff with microtones
- Jacob Collier - musical prodigy, mostly a capella/vocal arranging, but is a genre polymath, does some incredible stuff with just harmony
If anyone has anything to add, please do! I can't get enough of this stuff
> Before the advent of temperament systems, D-sharp and E-flat were two different notes.
D# and Eb were defined as elements of a temperament system, namely meantone temperament.
> My track is tuned in a system called five-limit just intonation ... It's the basis for all the tuning systems used in Western Europe between about 1500 and 1900.
The basis for tuning systems in Western Europe between 1400 and 1800 was meantone temperament. 5-limit JI has never been the basis of a common practice music culture anywhere or at any time in world history.
> We can do this because Western people consider octaves to be equivalent.
All people experience octave equivalence.
> Ultimately, split black keys did not catch on.
Split keys were fairly common on keyboard instruments for about a century.
> Bach wrote The Well-Tempered Clavier to show off how one well temperament system (no one knows which one) sounds okay in every major and minor key.
Bach was a proponent of well temperaments in general, not any specific one.
Meantone is a modification of three-limit just intonation to try to better approximate the thirds from five-limit just intonation. I know that it is historically inaccurate to say that meantone follows from five-limit, but in order to understand the point of meantone's existence, you have to know about five-limit as the standard that meantone is aiming for. I should probably rephrase my post to say that meantone follows logically from five-limit, not that it follows historically.
All humans (and some monkeys) can detect octaves, but that does not mean that all humans consider octaves to be equivalent. That's a culturally specific convention. It's a very widespread convention thanks to Europe's wide cultural reach, but it's very much not a universal one.
Split keys may have been common for a while, but I think we can agree that in the long term, they did not catch on.
>Bach wrote The Well-Tempered Clavier to show off how one well temperament system (no one knows which one) sounds okay in every major and minor key. The keys closer to C sound sweeter and more euphonious, while the more distant keys sound darker and edgier.
Fairly recent research has shown that Bach may have been very explicit in specifying a temperament system. A series of what appear to be decorative swirls at the top of the title page of the WTC has been conjectured to actually be instructions for tuning to the temperament system he favoured.
I wrote a blog post summarizing this research. The idea is that the swirls specify turns of the tuning pegs to modify meantone temperament. It's more or less pure speculation, but it does produce a very nice-sounding tuning, almost equal temperament but not uniform across the keys. https://www.ethanhein.com/wp/2020/what-does-the-well-tempere...
It would be remiss of me not to mention the archicembalo[0], which was a keyed instrument that allowed a musician to experiment with this distinction to a degree.
As a former jazz musician, I always find the classical perspective on theory interesting. This article touches on Pythagorean tuning techniques, which if you ever find yourself in front of a good a cappella choir, they’ll be tuning to the true temperament tuning scheme described here.
A fun comparison to make in the jazz world is enharmonic usage for the purpose of readability. Jazz chords are very dense and short lived compared to the very clean and predictable counterpoint found in classical music, so “correctness” doesn’t really matter as much. Most charts are sight read, so even though the band is sounding some flavor of a B chord, if you’re playing the 3rd, there’s a chance there may be a written E flat instead of a D sharp simply because E flat is a more commonly written note for horn players.
Pythagorean tuning is somewhat different from what’s described.
In Pythagorean tuning, your E would be 81/64 above C, or equal to four fifths minus two octaves. This is slightly higher than E in the article, and the difference (81/80) is called the syntonic comma.
Different tuning systems were invented in order to resolve this discrepancy, and without these advances, jazz would be radically different. One of the things about jazz is that you see distant movements that only really make sense as enharmonics—like how Coltrane’s “Countdown” uses the familiar ii-V-I, but modulates in major thirds, which only makes sense when you allow the final modulation te be the same as the first—something that only works enharmonically.
I'm the guy who wrote the blog post, and I want to offer some clarity. Usually when people say "Pythagorean" tuning they are talking about three-limit just intonation, not five-limit. In three-limit, the major third is indeed tuned to 81/64, which sounds pretty terrible. That might explain why medieval people thought thirds were a dissonant interval. Five-limit just intonation tunes major thirds to 5/4.
Cool article, by the way! When I sang in choir in undergrad, we were exposed to these subjects. Since I was buried in jazz music, your blog was a cool primer on tuning choices.
For sure, the broad similarity I’m trying to touch on is the focus on mathematical resonance and context of a key. Equal temperament removed a lot of that context, but definitely opened the door for further harmonic experimentation. Giant Steps is also a good example of what you’re talking about too.
I'm struggling to fully understand this tbh, but was recently exposed to these subtle differences when writing a mini organ synth.
When implementing the draw bars (dictating the harmonics comprising each key) I realised the true harmonics of a note and neighbouring notes calculated in the 2^n/12 way are sometimes the same and sometimes slightly off... organs just kinda ignore this fact and use the closest neighbouring notes for the draw bars anyway so that they don't need a million different oscillators, so technically the draw bars are just chords on the keyboard using the same oscillators and not harmonics (well some of them happen to be exactly the same as harmonics, others not).
The author asks an interesting question which I didn’t see the answer to in his reply: what DO you call this note in the A minor blues scale?
He says you’re only supposed to use each letter once, so am I supposed to call this an E flat and then refer to the following note as an F flat?
I struggle a bit with considering the ‘blue note’ a variation of the fifth - it feels more to me like a variation of the fourth, so is it in fact a D sharp? But then the fourth would be C double-sharp, and the (flat) third would be B sharp… right?
With TTET, we really need to drop note names just use scale degrees (aka ‘Nashville Number System’). This would remove a ton of confusion when texting music theory.
I started asking my father, a concert pianist, composer, and teacher - this question when I was 4. He explained. In my head I said, “Bullshit.” I continued to press him on it periodically until I was a teenager. I still shook my head and thought, “What is wrong with these people.” Now, I can read sheet music just fine, but I still feel like … never mind. I’ve done a ton of composing without once taking any notice of what key any of it was in. And it all sounds fucking great. I prefer to do as much as possible “by ear”. I’m unbelievably stubborn.
tl;dr - Somebody really should have picked up on the autism when I was 4, and MIDI rolls don’t give a shit about keys and accidentals.
MIDI rolls don't, by themselves, make any noise or even inherently define any note frequencies at all.
The frequency of the sound produced by a given synthesizer when it receives any particular MIDI note number is up to the synthesizer. This is part of the point of the MIDI tuning system. The synthesizer and/or the tuning system may very much care about keys and accidentals.
I went to the Wikipedia link in the article to https://en.wikipedia.org/wiki/Five-limit_tuning and saw that the first example there shows two very slightly different frequencies, which is a good way of setting the stage for reading the rest of the article.
Guitarist might find three videos recently posted to classical guitarist and lutist Brandon Acker's YouTube channel interesting.
Lutes and early guitars (before around 1800) did not have metal frets. Instead they used pieces of string tied around the neck.
They did this because the strings were very expensive, with a set of strings for a lute often costing more than the rest of the lute, and strings were not as robust as more modern strings. With metal frets strings would wear out faster. You could easily end up spending more per year on strings than you had initially spent on your instrument. By making the frets of the same material as the strings they didn't need to change strings as often.
Acker and luthier M.E. Brune took a classical guitar Brune was building but had not yet put frets on and played around with putting on tied gut frets and gut strings. In the first video [1] they just go over the history of tied on frets, and do some comparisons with metal frets.
With tied on frets it is relatively easy to try tunings other than 12-TET. You can change the position of a fret, or you can add extra frets. You can also add partial frets. Renaissance lutists would often glue on small pieces of string behind or in front of a fret. The fret would give them some particular note from a sharp/flat pair, and the little mini fret, called a tastino, would give them the other note from that pair.
The second video [2] explores the tuning possibilities of tied on frets and tastinos. Acker plays a bunch of things in 12-TET and in other tunings more suitable for the particular piece, and also gives some examples of how bad other tunings can sound when you are playing a piece in a key that doesn't fit the key your instrument was tuned for.
The third video [3] is just playing around after the tied on fret experiment is over but the guitar has not yet had its metal frets installed. Acker tries to play it without frets. That turns out the be quite a mixed bag.
Given that integer ratios tend to sound better, are there songs edited to use notes that maximize the number of integer ratios rather than the standard tuning?
It would seem an easy hack to make people like your new pop track better. But then, I'm no musician, so maybe I'm oversimplifying.
You are correct. For instance, traditionally, in barbershop vocal music, singers are trained to deliberately deviate from 12 equal, towards an ever-shifting kind of just intonation, in order to maximize the extent to which the voices blend. Auto tune, on the other hand, just tunes things straight to 12 equal. Melodyne fares a little bit better in that it lets you tune to custom microtonal scales, or fudge things a little bit here and there, etc. Interestingly, Logic Pro X has a "Hermode Tuning algorithm" that will basically do the dynamic adjustment toward just intonation for you, but it only works for MIDI instruments and not auto tune as far as I know.
Autotune software has different parameters available to it. These include pitch correction speed, how close a singer has to be to the note in order to start/stop pitch-correction, which pitches to correct for.
The T-Pain effect, which is the autotune sound you’re probably thinking of, cranks most of those parameters all the way up in order to get to that robot voice: “instantly lock the vocals to one of these set pitches, if the singer goes lower than X, immediately switch to the next lower pitch in the set.” More subtle usage makes for a performance that is more in tune overall but keeps much more of the vocalist’s expression and pitch variation intact. Its goal usually is to not be noticed.
I don’t think I understand your question (edit: about it being a bug), so I won’t attempt to answer it directly, but maybe the above info is helpful in thinking about it.
You wrote, "your choices were these: you could limit your harpsichord playing to certain keys, or you could have a bewilderingly complex instrument that you would spend half your life tuning."
May I respectfully suggest that an alternative option exist now, which did not exist then?
First, please note that the tunings that you are describing are all extended meantone tunings (more or less), derived from stacking perfect fifths (P5's) that have been more-or-less tempered away from their just ratio (702 cents). If you widen the P5 to 720 cents, you get 5-tone equal temperament (5-tet). If you narrow the P5 to 686 cents, you get 7-tet. In between, you get the historical meantone tunings and today's 12-tet (P5=700). They are all defined by the same three elements:
1. Their period of repetition, the perfect octave (P8) which maps the 2nd partial to the P8.
2. Their generator, the tempered P5, which maps the 3rd partial to the P5.
3. Their comma sequence, which starts with the syntonic comma, which maps the 5th partial to the M3.(If the first comma in the sequence were the schisma, then the 5th partial would be mapped to the d4, defining the schismatic temperament.)
This combination of a P8 period, P5 generator, and comma sequence that starts with the syntonic comma, is called the "syntonic temperament" in "Dynamic Tonality" (see Wikipedia for both quoted terms).
An "isomorphic keyboard" is a keyboard that is generated by the same period, generator, and comma sequence as the temperament with which it is said to be isomorphic.
Isomorphic keyboards had not yet been discovered before the West standardized on 12-tet. Had they been known then, then the course of musical history might have been quite different.
With brings me back to your quote. A keyboard that is isomorphic with the syntonic temperament is brain-dead simple. It has the same fingering in every octave of every key of every tuning of the syntonic temperament. 12-tet, 1/4 meantone, 32-tet, 53-tet, infinity-tet -- same fingering. One can even bias the mapping of pitches to notes to support irregular tunings such as well-temperaments.
Therefore, IF ISOMORPHIC KEYBOARDS HAD EXISTED during the period that you describe, then they would have had a shot at becoming the de facto standard keyboard instrument subsequently.
Today, isomorphic keyboards DO exist. Anyone who is serious about exploring the frontiers of tonality should check them out (and Dynamic Tonality, which requires an isomorphic keyboard).
tl;dr: The notation system predates the modern, commonly-used 12-Tone Equal Temperament, for which there are (at least two) ways to describe any note within the (12-Tone) octave, either by sharps (D sharp) or by flats (E flat). In 12-TET, there are exactly twelve notes in the octave, and sharps and flats can be said to "overlap".
In earlier temperament systems, these notes may have been distinct (or in some cases unavailable), as the relationships between notes were based on non-equal, if not more mathematically perfect, ratios.
my TL;DR, as a western classically-trained amateur musician largely unfamiliar with Indian music: Harmonium (a small reed organ) is widely used in Indian classical music but its use is controversial because it doesn't allow for the fine adjustments in pitch (roughly analogous to the D#/Eb distinction discussed here) that is seen as central to most/all Indian styles.
Suppose we start with 12-TET and ask what simple integer ratios each note is close to. To do that we need some notion of what it means for a simple integer ratio to be a good approximation to some arbitrary given number.
Consider trying to approximate some number x with an integer ratio n/m. For a given m all we can guarantee is that we can find some m so to |x-n/m| <= 1/2m. One way to define good approximation is if for a given m, we can get a lot closer than 1/2m to x then that is close.
For example if we want to approximate pi with m = 6, 7, or 8, the closest we can get is 19/6, 22/7, and 25/8. The absolute errors are about 1/40, 1/790, and 1/60, respectively. They are all doing better than 1/2m, but 6 and 7 are only about 3.5 times better 1/2m, but 7 is 56 times better than 1/2m. So we say that 22/7 is a good approximation to Pi. That doesn't mean it is particular close--just that it is way closer than other approximations with similar sized denominators.
For a given number x there is a way to find such good approximations. You figure out the continued fraction for x. For Pi that is 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/(1 + ..., then even though that goes on forever you type ")))))" so that the unbalanced parens don't drive you crazy, and take the sequence you get by taking finite sections from the left side of that continued fraction. So for Pi we get 3, 1 + 1/7, 3 + 1/(7 + 1/15), ..., which when simplified give 3, 22/7, 333/106, 355/113, 103993/33102, .... Note that 22/7 is in there, which is the good approximation from early.
All of those numbers from taking the left parts of the continued fraction, which are called convergents of the continued fraction, are good approximation in the sense above: they are way closer to Pi than anything else with similar denominators.
What we can do then to find good integer ratios that are close to the notes of 12-TET is for each 12-TET note, take its frequency divided by C's frequency, compute the continued fraction of that, and compute the first few convergents. Here are the results. I've omitted convergents with a denominator of 1 or with a denominator > 500.
The difference is obvious even those who don't distinguish pitches much (can you hear the difference in the video linked in the article - https://www.youtube.com/watch?v=7GhAuZH6phs&t=21s ) because the 'wrong' one played in a scale or a chord sounds really, really horrible; that's why we have all the work done on 'tempered' tuning to reduce the gap so that everything is just slightly off.
The relationship between pitch and frequency is not simple. The physical frequency for a pitch can be derived in several ways.
One example is the Pythagorean system where the interval of fifth is set as a frequency ratio of 3/2. This system yields clearly distinct frequencies for D# and Eb.
In 12 tone equal temperament, a semitone is set as a frequency ratio of 2^(1/12). In this system you get the same frequency.
You can also derive frequencies from simple fractions of the scale root. In this instance you would generally obtain D# and Eb from unrelated roots.
It would be possible for most humans to notice something but feel indifferent, or for a sizeable minority to really hate something. So just abstractly I'd say that yes you do need more than those two questions.
Kind of like saying that “cell” and “sell” are different words. Obviously they’re different words, even though you can’t hear the difference. Just like it’s obvious that Ab and G# are different notes, even though they may sound the same.
Ask an English speaker to interpret a text about a sails man who sales around the world and cells sell phones, at have price for any guessed who sends him a facts to his office in grease.
It’s harder to read when you use the wrong words, just like how a score is harder to read if you use the wrong notes.
Short answer: they aren't. This is the musical equivalent of the fact that English has "guarantee" and "warranty" and they mean the same thing.
Long answer: they aren't. They are enharmonic equivalents in the vast majority of music that uses any of the conventional Western systems of tuning (as the author sort of goes out of their way to demonstrate in the article), and if you use or invent a different system then what you call the notes is kind of up to you since it's your system.
Not on a piano, but for all the other instruments with variable pitch (e.g. fretless strings, voice, trombones) they are. The enharmonic is useful information and we'll know whether to place that note just a little under or above its usual pitch to make the chord more in tune. We don't need to invent a new system to do this, we do it every day within the 12-tet system we already have.
Most of the time, even if you're playing a variable pitch instrument you're going to be tuning to fixed pitch because you'll have at least one fixed pitch instrument (eg a piano) and if you don't you'll just sound out of tune.
In cases (eg a consort group or string quartet or something) where you're all variable pitch, you'll be tuning to each other and to the scale/key as appropriate to the music and whatever sounds good. You may well sweeten the thirds or widen the fifths a bit etc but that doesn't apply to this question here because you're really not going to see the enharmonic equivalents in the same piece the absolute vast majority of the time for stylistic reasons and if you ever did you would just be tuning to each other to make the vertical incidences sound good rather than thinking consciously of tuning a d-sharp one way and an e-flat another way.
Source: Have a degree and postgrad in music, used to be a professional double bass player[1], spouse has a degree and postgrad in music and teaches at 2 conservatoires in London as well as performing professionally, mostly early music in small consort groups where this sort of tuning thing comes up a lot.
[1] So yeah you can make the standard joke about what do double bass players know about tuning.
I guess it really depends on your instrument, taste, style and the group you are playing with. While studying Cello I actually had a lot of lessons with string quartet where we were analysing the score (harmony) for intonation and it happens quite often in modulations that enharmonic equivalents were used to distinguish whether a chord belongs to the old or the new harmony. And sometimes we really needed to make a difference between an e flat and a d sharp to match an open string or to get a desired tension.
For me the enharmonic equivalent is usually just a totally different harmony, so that is what I tune to. As a result they are quite different notes. I try to do that consciously - also while playing with fixed pitch instruments when possible (like the grandparent comment explained).
Even if playing with a fixed-pitch instrument, it only really sounds out of tune if they're playing the same notes. Which in the styles I play isn't an issue.
So I guess how often this happens in practice varies between styles and eras of music, which would make sense to me. I haven't ever done early music and know nothing about it (other than trombones used to be designed terribly and we now know how to make better ones ;)
Source: Also have a postgrad in music, probably from one of the conservatories your spouse teaches at, and still play trombone professionally.
They are enharmonic equivalents in the vast majority of music that uses any of the conventional Western systems of tuning (as the author sort of goes out of their way to demonstrate in the article)
I don’t think that’s correct -- they are identical in 12-TET, but all the other tuning systems either treat them as different notes, or attempt to compromise between the alternatives in a way that favours certain keys over others.
Maybe this is isn’t critical info for a lot of people, but it is important foundational knowledge if you’re a music student, or just interested in music theory.
Actually most of the time if you're playing in a different tuning (eg quarter-comma meantone or pythagorean or whatever) where they would be different, you're playing a type of music where you exclusively would play one note or the other, so the fact that they are theoretically different doesn't arise.
This is not true in general. It is only true for instruments that use the well tempered tuning, e.g. a piano. But for example, the violin and cello do not.
Today is a great day for you, because you get to learn something new! Specifically, that if you derive pitches of notes from harmonics, D sharp and E flat are slightly different pitches! There's actually a great article about exactly this you might want to read, and it's handily linked above.
Today is a great day for you, because you get to feel good about yourself by being a first-order pedant and making basic assumptions about what I know and don't know and whether or not I've read the article.
I actually did read the article and even prior to that do know about deriving pitches from harmonics. What I posted was still correct in spite of the downvotes.
The problem is you are incorrect. The deeper your knowledge of music theory, and the more experience you have with a capella choir music or certain instruments where they can be played differently, The more apparent this will become.
Trying to sing a D# in a B major chord the same way you would a Eb in a C minor won't be a great experience.
Most of the adjustments will happen automatically if you listen to your fellow singers and have experience. But they do happen.
Addition: when you choose a tonality in which you write a piece of music, it may define its standard set of flats and sharps, to simplify building chords using uniform rules. Because of this, it is convenient to name the same note using different names, relative to its neighbors.
Expansion: in non-tempered, natural tuning, such as often used when playing a violin, there are differences between some sharps and flats built from different notes, because natural harmonic intervals, based on frequency ratios like 3:2, do not split the frequency range in a completely log-linear way. This is why, say, G# and Ab may be not the same for purposes of pure natural harmony [1].
Equal temperation was invented to overcome this. J.S.Bach wrote a great showcase for it, Woll-Tepmeriertee Klavier, which involves harmonies and chord progressions that are hard or impossible to achieve with natural tuning without producing weird dissonances.
It's unknown whether Bach wrote the WTK for equal temperament (the modern standard) or for a well temperament (something that tempers all keys enough to be usable but does not make all keys sound identical).
50 years ago, unless you were studying at the highest levels of theoretical/historical music research, you'd likely have been taught wrong.
The “Bach standardized the world on 12TET” trope is old and enjoyable enough to make a good story that unspecialized music teachers have parroted along for generations.
We've got better access to information now. I've corrected music teachers on this specific topic in the past. Some gratefully accept. Some pull out the “but the teacher here is me card”, so I'm sure a few more generations are going to be needed.
You want (a) because it you gives you nice algebraic properties (the music structure is invariant under frequency shifts). You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency: f, 2f, 3f ... nf. Small-integer ratios are naturally occurring and very recognizable.
Modern tuning (C-f "12-TET" in the article) almost, approximately satisfies (a) and (b) simultaneously. "12" means there's twelve tones between f and 2f; the ratio between adjacent tones is defined to be 2^{1/12}. This tuning can't contain both f and 3f (so it fails (b)), but it *can* contain f and 2^{19/12}f ~ 2.9966f, which is actually close enough to 3f to be indistinguishable. (Almost works!) But as you build ratios out of larger integers, it audibly falls apart. The closest you can get to (5/3)f is 2^{9/12}f = ~1.6818f, which is already 10% of the way to the next note. And it rapidly gets worse.
This is why two on-paper-identical notes can end up audibly different, depending on what key you're starting with (and hence how they are approached). There's tension internal to music theory itself.