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).
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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.
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.