Why are D-sharp and E-flat considered to be two different notes?
ethanhein.com
ethanhein.com
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.
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.
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.
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.
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).
* what an incredible show!
Which is okay. I like to bend notes too, but just call it what it is.
I assume it all breaks down if they need to play alongside a keyboard (or fretted) instrument.
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.
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.).
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.
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.
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.
It was routine even in my high school brass section to pitch down the major 3rds.
the most important element here is how it sounds to our ears. not how closely it tracks to an equation.
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).
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.
"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]"
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!).
There exist equal temperament guitars, but they're usually custom built:
https://guitargearfinder.com/faq/true-temperament-frets/
In any case most people don't mind such small differences, especially that guitars aren't terribly precise to begin with - a player can easily get 10 cents of a semitone on each individual string when playing a power chord with distortion, bringing the whole thing to just intonation.
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Hi! I am not a musician. Did you mean that true temperament guitars are the ones with squiggly frets, instead?
My understanding was that true temperament [0] is not the same as equal temperament [1]. I also believe that both pianos and guitars are typically tuned to equal temperament [2], but you may well be right about guitars.
Maybe somebody can shed some more light on this. Thanks!
[0] https://www.truetemperament.com
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.
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.
Guitars are almost never perfectly in tune according to that standard, but like I said - people generally don't mind.
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?
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.
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.
That would something that only happens very rarely, like for just 1/10000th the pianos people will encounter in their lives...
The one exception is harmonics which are based upon integer ratios rather than 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 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.
And for earlier stages you can have human raters or similarity with past successful songs.
As someone who hasn't taken any musical theory or similar, it was quite interesting to hear.
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...
In baroque music, ~430 and 415 are also very common since they are thought to be the historical pitches of "A"
That's so low, it's more like playing the piece a three semitones lower than an alternate choice for A4.
If the piece was in A, it would be more like playing it in F# (while still using A=440).
(Of course if you did that, the "sweet-spots of 12TET and its off-notes would be different, than if you played with A=360)
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.
Or I might be wrong, I know nothing about music[2].
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.
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.
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.
(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).
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.
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.
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.
A computer generated piano performance could pretty easily pick versions of each note that are in harmony with the other notes played at that moment.
I wonder if that would be worth doing? Has it maybe already been done?
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.
(I do apologize if I came across poorly—I couldn’t think of another way to ask the question.)
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.
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!
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.
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.
(1) Example: "get up an get out" by Iggy Pop. https://www.youtube.com/watch?v=R1ld5jG3f-M
usually that’s not the case though. typically it’s individual notes. much harder to precisely and accurately modulate tape velocity (especially by hand).
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.
This sort of hybrid tuning is common in barbershop quartet singing as well.
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.
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.
(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.)
Still, it's kinda like alien music and it's certainly creative.
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.
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?
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https://en.xen.wiki/w/Adaptive_just_intonation
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?
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.
https://www.merriam-webster.com/dictionary/timbre https://en.wiktionary.org/wiki/Timbre
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
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.
Prediction: society demotes all of the auto-tuning and AI art to training status, and actual art produced by himans with pulses is preferred.
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:
https://www.youtube.com/watch?v=m40xkEkrEKo
https://www.youtube.com/watch?v=-4B1NcdNJjE
And if you're lazy see here for a VST plugin send form the gods:
B) this is a subjective point, and one accepts your POV as valid
C) your POV may prove a minority one over time. Or not.
B), C)
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.
How to tune synthesizers this way? What and where to buy?
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.
I'd say it's more likely because intermodulation distortion between frequencies with low-complexity fractions tends to be low-frequency.
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).
> 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.
Think it would be possible to mockup some really interesting tunings/temperaments in BitWigs grid or Max4Live.
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.
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.
In fretted instruments, bending is done for effect, not for adjusting each and every note.
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.
Any proof to this? 12 tet can vary by 15 cents from just intonation. Even an amateur musician can hear how out-of-tune a 15 cent difference is.
Judging from all kinds of out-of-tune players in live settings, and youtube videos (especially guitar, which I follow a lot) I kind of doubt that...
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.
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).
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.
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.
In modern rock music, some musicians will change guitars every song to have sweeter chords depending on the particular chords in that song.
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.
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.
See for example https://music.stackexchange.com/questions/113812/violin-tuni...
there's a lot of nuance. A lot of playing it by ear. :)
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.
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..
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.
Notice how the fifths are perfectly "still", no beating. It can almost sound out of tune if you're too used to tempered tuning.
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).
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.
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).
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.
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.
Music theory gives us a way to measure & more accurately describe what we were already doing.
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.
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 presume you mean having to call the F tone E#.
- two double sharp keys - F## to refer to G, C## to refer to D, B# to refer to C
Yikes.
Most people write “x” for double-sharp, instead of ##, in order to match how it looks on a score.
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.
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.
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].
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].
[1] https://en.wikipedia.org/wiki/Circle_of_fifths
[2] https://en.wikipedia.org/wiki/D_minor
[3] https://en.wikipedia.org/wiki/E_minor
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.
> 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.
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.
Easy to explain to a beginner.
C e g b db f a - Cmaj13b9
C db g b d f a - confusion (for beginners) as that would not have a third....'d' is the 2nd letter alphabetically.
Any key sig can be represented...
Ie 'E' has to be followed by a g 'of some kind', so even e# can be followed by a gb g or g# to construct a chord.
Easy to see on a guitar, especialky with multiple positions to sound the same note.
e g b d f a c e g b d f a c e....rotates forever, whatever the starting note.
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.
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.
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 ;).
[1] https://global.oup.com/academic/product/where-sight-meets-so...
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.
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.
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 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.
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.
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).
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
"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.
> 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.
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.
And why is that important? Answer: This lets you write the key signature once, and then not have to bother with accidentals in front of notes.
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!
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.
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?
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.
https://www.eunomios.org/contrib/francis2/francis2.pdf
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).
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!
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).
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.
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.
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.
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.
there are 100 cents between each 1/2 step in 12-tone equal temperament.
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."
- xeharmonic wiki: https://en.xen.wiki/w/Main_Page
- xenharmonic playlist: https://open.spotify.com/playlist/1OcPPq0ohnUarvDCERaxaR?si=...
- Tolgahan Çoğulu and his awesome microtonal guitar creations: https://youtu.be/iRsSjh5TTqI
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.
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.
Hell, doesn't this apply to the white keys too?
Not in 12-TET because the ratio between every semitone is the same.
In other temperaments the frequency of every note can be different in every key.
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.
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.
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).
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?
All of that just sounds mad.
It might be worth a listen for anyone who finds the topic interesting, but the answer is obviously already found in the top response too.
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.
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.
obligatory self-promotion, https://www.youtube.com/channel/UCUmdU7WpuhAv3imtVqkGpIA
https://www.youtube.com/watch?v=vapZZdog0NI
Here's a small sampler. I include one that's maybe a bit similar, the one called 'crsh'
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.
[1] https://www.youtube.com/watch?v=--y_vf-Kg-w
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.
I'm assuming, but I want to check with you all: does Autotune always "correct" to the exact center of the note? I assume the answer is probably Yes.
If so, that's a bug, is it not?
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.
The comment was asserting (or questioning) the T-Pain effect. I honestly didn't know if that was what everyone was using in Autotune or not.
(flippant, but also accurate! of course explaining why that matters is a whole different beast. Lots and lots of good books available though...)
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).
Respectfully,
Jim Plamondon
Music theory must be racist!!!
/sarc
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.
https://news.ycombinator.com/item?id=31476078
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.
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.
C#/Db: {17/16, 18/17, 89/84, 196/185}
D: { 9/8, 55/49}
D#/Eb: { 6/5, 19/16, 25/21, 44/37}
E: { 4/3, 5/4, 29/23 34/27, 63/50, 286/227, 349/277}
F: { 3/2, 4/3, 295/221}
F#/Gb: { 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408}
G: { 3/2, 442/295}
G#/Ab: { 3/2, 8/5, 19/12, 27/17, 100/63, 227/143, 781/492}
A: { 5/3, 37/22}
A#/Bb: { 7/4, 9/5, 16/9, 41/23, 57/32, 98/55}
B: {15/8, 17/9, 168/89, 185/98}Play the piano on your computer with Python:
https://jugad2.blogspot.com/2013/04/play-piano-on-your-compu...
Some of the comments gave some interesting music theory info.
Can most humans hear a difference?
Do we need more than the two questions above?
Can end-users of the program tell which one was used?
Do we need more than the two questions above?
If programmers thought like this HN would be empty!
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
Now I'm wondering if I was told wrong 50 years ago, or if this is new research?
https://en.m.wikipedia.org/wiki/The_Well-Tempered_Clavier#We...
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.