Some additional thoughts/corrections: 1) The rhythm is actually present throughout variation 5, I just picked bars 6 and 7 because they were the first ones with no ties and I didn't want to add that complication. 2) Yes, I know that in both PEMDAS and BODMAS the intention is for multiplication and division to exist on the same level, because they're the same operation. But that doesn't change the fact that, as evidenced by these annoying twitter debates, many people _learned_ that there was a correct order to do them in. Notation doesn't have right and wrong, it just has clear and unclear. 3) It's not technically true that we have _no_ idea what Brahms's performances sounded like. We don't have any recordings, but there are people who were alive when he would have played these and then went on to become pianists themselves in an era where recording technology _did_ exist, so we could theoretically infer from the recordings those artists made what they had originally heard Brahms play. I don't know of any specific individuals who definitely heard him play this particular piece and then recorded their own version, but early recordings of it also seem to stick with the triplet reading so I would assume that was in line with what people who'd heard it prior to the advent of recording technology would be expecting to hear as well. 4) Technology isn't the only reason modern rhythmic culture is so different from Romantic-era Europe. There's also, of course, the influence of other traditions, most notably African and African American dance music, which much more heavily prioritize strong, steady rhythms, to allow for more complex rhythmic developments. 5) I should note that, while many of Hook's examples can be explained using a similar interpretation (Basically, the second voice is meant to highlight an important melodic sub-line, not inform rhythm) there are some where it doesn't really work. For instance, in his paper (found at www.mtosmt.org/issues/mto.11.17.4/mto.11.17.4.hook.html ) example 10 definitely has two distinct voices with their own melodic shapes that happen to intersect at a point that is metrically undefinable. In those cases I'd just revert back to the rubato thing and say you're supposed to play it somewhere in between and not worry about it beyond that.
Oh. It's so weird to think that there was an overlap between Brahms' life and recording technology. Like, that just feels strange even though when you actually look at the dates that makes sense.
I may remember it wrong, but it appeared to me the same problem of notation happens in a far more famous piece, the triplets in the orchestral accompaniment in the chorale "Jesus bleibet meine Freude", BWV 147 by J.S. Bach.
I think pianists might be more familiar with this kind of notation being used as shorthand for playing instructions as notation like this is fairly common in older complex works. For example a tremolo is commonly used as shorthand for a very fast repeated figure - where they notate the figure for up to two bars then just switch to tremolo for the rest of the piece. Similar to repeat notations for guitars and drums, I think it might just be unfamiliar to non pianists.
@@crisdekker8223 You're right, but it's not the same problem. In that piece, one part plays triplets (or sometimes it's in 9/8) and the other plays a dotted rhythm. I think this is the same practice that @The ModicaLiszt refers to in their comment (ua-cam.com/video/xXIG8DiMmw8/v-deo.html&lc=Ugy-i20SohlWYVDZHzN4AaABAg), and might be related to the typesetting conventions of the time (I wonder if there's a manuscript that is written differently).
@@catomajorcensor I was curious myself. I went to IMSLP and saw Bach's original manuscript and he actually does juxtapose three triplets with a dotted quarter + eighth, so mathematically the last triplet and the eighth don't line up. Must be just shorthand. And you're right, I should've said similar in stead of the same problem.
Frankly, until you pointed out the missing triplet notation from the bottom line I could not even understand what was wrong about the rhythm. The number of examples where triplets are implied in notation of this period is so staggeringly high that I didn't even think twice about interpreting this as a constant stream of triplets.
so, just assume the triplet notation isnt written but implied in the lower line? edit: thats how i first thought it was, and after im quite certain thats still how it is.
@@gagaronpew4382 Yes. The only thing that is _somewhat_ confusing is why Brahms chose to connect the double bar line (notating two 16th notes) instead of notating an 8th note and a 16th note. Notation was nowhere near as universal back then (let's remind ourselves that many composers a century before Brahms didn't even notate dynamics) so I have no reason to doubt that this is a constant stream of 16th note triplets, as it's the most obvious rhythm that makes sense within the style of the period.
Next time, the general one instead of the stationary one: :p And then become more ambitious with Dirac's and then the Klein-Gordon one, in whatever order they like :p
16th century composer: I guess that note is sharpened, but there's no need to write the sharp in because every single musician alive already knows it's a sharp. 21st century musicologist: oh crap.
Not a Brahms scholar, and definitely under studied, but as someone who has learnt 100s of piano pieces both as a soloist and accompanist, and a composer FOR piano, as well, I can say with as much certainty as possible that this form of notation is used to highlight important melodic and phrasing information. I am not really sure if this was ever a thing that composers did in their own manuscripts, or something that has become a more recent editorial addition, but I will always prefer complicated decorated pieces to have melodic phrasings like this. Being able to see the lines through the piece (especially in a case where you don't have a lot of time to do it) is so important for legibility. It can be the difference between making a piece challenging to perform OR challenging to read.
Would you interpret the highlighted melody as to play it with a slight accent, or maybe slightly ghost the non-melody notes? Or perhaps slightly stretch those melody notes so they have emphasis? Or would you think "Oh that's the melody I've seen a hundred times before, just with an embellishment. I don't need to invest as much brain power reading it."?
@@randysterbentz5599 Logically, the latter - stylistically, some part of the former would enter in to it, as well. That really depends on the piece and performer though. Realistically, you're going to want to provide some kind of emphasis to the the melodic line over the decoration, and writing it out in this way makes that distinction a lot more obvious. Though as to how exactly you'd do that, it's really up to the performer, their interpretation, and the general style of the piece. Having not played this one myself, I can't speak to it, though glancing through the score, I would personal lean slightly in to the main melodic line, while trying to play the decorative notes more lighter.
i was literally about to write something similar to this. especially being a classically trained pianist. this is "common knowledge" but now that i sit down and see it under these lenses it kinda makes you wonder.
Yeah, I play violin and definitely intuited this pretty immediately. You've got triplets and markings to demonstrate which notes to emphasize/which notes are similar to ones before. no point in re-writing all the notation for a triplet when you already know they're triplets.
6:48 "Not stylistically appropriate" *12tone proceeds to draw a realistic pachyderm* And my absolutely uneducated interpretation: the first and third note in each triplet are accented and the middle note is closer to a grace note. I know I'm wrong, but I like the sound of it :)
The 2nd approach is the stylistically appropriate one and most certainly the correct one. The triplet rhythm is the underlying, correct rhythm and the added 16th note downward stems are to convey voicing/melodic lines as you pointed out (not to be played rhythmically like straight 16th notes). This type of notation is fairly common in romantic/classical literature, which is why pretty much every professional recording plays it that way. You basically hit the nail on the head at the end. A seasoned classical pianist wouldn't even think twice about how to perform this.
Exactly. This video is over-complicating things, seeing things that aren't there. This is a very common notational standard. It's not "impossible", it's just a limitation of written notations, and it's 100% obvious what is intended, it's not even weird. There ARE times in Brahms where he wants 2 against 3 in passages somewhat similar to this, but they don't share noteheads -- it's very obvious that they're spaced differently.
Can we take a moment to truly appreciate how amazing this guy's annotation skills are? I think you could build anywhere from 50 to 99% of a new Emoji dictionary just using his little elephants.
@@dying_allthetime The whole "this is an intentionally poorly written equation and it should have parentheses if it wants to be clear" thing flew over your head?
@@dying_allthetime not only did you not get the point with clearer notation, the parentheses added "changes" the answer to 16. Also, please don't use diagonal fraction lines like that. If you want something to be in the denominator just put it UNDER the line, not somewhere to the right of it.
Two thoughts: 1st, could this notation have been used to indicate stress/emphasis? Maybe Brahms wanted a "swing" feel with emphasis on the first and third part of the triplet and the middle one like a ghost note? One way to check that hypothesis would be to check whether he uses things like >, . and _ in his other scores? 2nd, what does the original look like? If that is a variation, maybe the source was two-voiced and for those who were familiar with the piece, he wanted to indicate "here, i joined those two voices into one" or something?
Exactly. Implied triplets of swing were my first thought. The notes with shared stems are the swung melody and the middle notes of the triplets act as a kind of pedal. The straight 16th beams just indicate the melodic line.
That was pretty fascinating to watch ngl As an arranger with no formal musical education, if I want things to be phrased a certain way on piano I usually write a slur over sections of music that I want to link together and if certain tones need to be emphasized over others in a stream of notes that happen in quick succession i'll just mark those notes with an accent. I don't really know if that's the proper way to do it, but it makes sense to me from an intuitive level
04:25. As a mathematician I like your statement "this is bad mathematical notation". Indeed, many people learned one of the precedence rules about multiplication and division, and think the rule they have learned is a dogm, and think all others are wrong. They forget that a notation is only a convention and nothing fundamental in the mathematics itself. And indeed, until you specify clearly which precedence rule you are using, the notation is ambiguous and to be avoided.
1:22 I don't think that's how this beaming works; I've seen this kind of beaming before and all it's basically doing is beaming over the triplet rest. This beaming style is often used to specifically demarcate a contiguous line that should stand out in a denser texture, I.E. the down-stemmed notes are all part of one idea, the up-stemmed notes are just the full rhythm with the middle beat in each triplet group being a filler note not part of the main melody. Not sure how you came to your conclusion or interpretation at all.
Exactly. To make Dr Hook happy, Brahms would have had to have either written in tiny 16th note triplet rests for the bottom line, or given up on highlighting the descending voice that the pianist is supposed to be bringing out. The first option would be unnecessarily cluttered and create more confusion (note that apparently every pianist who has ever played this, except Dr Julian Hook, has understood how to play this as written without any trouble); the second option would fail to communicate the intention to the player (which 12tone explains very nicely). (I should point out that I have not read Dr Hook's paper so maybe he makes clear that he knows perfectly well what it means but intended to highlight the very point 12tone makes at the end - musical notation isn't meant to be like a programming language, it is human readable, not machine readable.) I wonder whether this was even Brahms's doing or his publisher. Sadly, the manuscript isn't on IMSLP. The first edition only actually includes the triplet symbols the first time this appears - plainly, the triplet feel is implicit.
PEMDAS with respect to the multiplication and division in my school in the US was not taught "multiplication first." It was taught to read left to weight and perform multiplication and division before any addition or subtraction. The M and D were treated as a single step read from left to right without necessarily doing one or the other first. Same with addition and subtraction. One single step. 4 steps total: P: parentheses E: exponents MD: multiplication and division AS: addition and subtraction 6*7/8 = multiplication first because it comes first 6/7*8 = division first because it comes first
It’s purely a notational thing…… Brahms indicates the bottom voice to be brought out in the triplets. Very standard Romantic practice. Chopin for example uses dotted rhythms in a melody to mean triplet crochet + quaver if there are triplets in the accompaniment.
@@triple_croche Well it's not too much considering most people don't know how to read music. So he's explaining why there is a misunderstanding in the first place.
@@astra1288 He does have normal triplets. The bottom noteheads indicate that those notes should be emphasised or at the fore of the texture because they create a melody among the decorative triplets.
I think about this issue in terms of how I write music. Unlike a musical staff, I rely on the piano roll in my DAW. It places each note and it's duration on a grid with less ambiguity. Whether recorded, edited, or entered directly by hand, it places each note and it's duration where you can see it exactly. It doesn't completely remove ambiguity, because the synth patch may have a slow attack, for example, and you hear the note appear later than the grid of the piano roll would suggest. Or it may have an effect or slow release that makes it sound longer than the end point would suggest. This is clearly in the modern mindset you mention, but it's much easier to read and control than trying to do it with musical notation. When I first started with computer music, the programs I had access to expected you to enter notes on a musical staff.
@@jacksonbarker7594 yes but, getting a synth patch to respond exactly as you want doesn't necessarily conform to the written notation if you want it to last a little longer or shorter than the notation would indicate. Especially if you've got, say a delay that needs it to release earlier so it doesn't retrigger one too many times
I honestly love the idea of written music being pretty open to interpretation or, in this case, conveying structural context. I'm sure you can add more nuance or clarity to compositions this way than through a literal reading. Cool video!
As always this was a great video that had an awesome approach to the different possibilities. I would like to offer one more possibility that is definitely wrong but also would be hilarious: Brahms wanted you to play all the dowmstems with your thumb as if the piano was a guitar.
TLDR: Pianists do a thing called voicing (making melody lines stand out in relative loudness) and I think that’s what these extra stems are about. As someone who plays classical piano at a semi-advanced amateur level, there’s something that comes to me immediately that doesn’t seem to be addressed at all in the video (or at least not well): voicing (not to be confused with chord voicings). This is when the pianist makes the notes of the melody louder than those of the accompaniment (and/or the accompaniment softer), and sometimes even plays a secondary melody louder than the accompaniment but softer than the primary melody. The practice of voicing is expected to the point that a score doesn’t usually have any indication. But if a certain melody line is intended to be voiced but not obvious composers may choose to highlight its presence in a variety of ways (also depending on musical context). In the example in the video, I would use the triplet rhythm in the left hand, voicing only the notes indicated by the additional stems as a melody, and I would view the offset from the r.h. rhythms as an effect of sorts, rather than as a failure to mirror it precisely (‘cause lets be honest: romantic and later composers were often quite liberal in their explorations of earlier idioms), one consequence of which is helping the l.h. melody stand out against the r.h. Meanwhile the middle triplet notes would be part of the broader triplet rhythm but not treated as part of the second melody line, i.e. as accompaniment. On further thought, this *may* also affect phrasing, which for pianists is a dynamic device often realized by making a phrase grow louder, and then the last note of the phrase softer, but also, and more relevantly to this thought, often making longer notes softer than shorter ones within the broader arc.
As for example 10 from the paper referenced by 12Tone, I might prioritize the triplet rhythmically but voice the note as part of the top line. I’m less confident in my assessment/interpretation of this rhythm, though.
This reminds me of parts of Petrushka. I played it under a conductor who studied under the successor to the guy that premiered it. There are sections notated as a measure-long septuplet in 3/4 - these were actually meant to be two eighth notes, a triplet, and two more eighth notes, but the composer couldn't translate the line he came up with to that notation when he wrote it, so he decided to just write that there are seven notes.
6:13 is personally my favorite interpretation. I think it sounds unusual but I actually like the sound of it. Also just because the right hand is doing something cool doesn't mean that the left hand needs to have a line that doesn't sing on its own too and needs to be not distracting because you arbitrary suggest that the focus "should" be on the right hand. Yeah it's hard to play but it can be played.
To add a complication to this, in the romantic era it was not uncommon for triplet-notated right hand to be accompanied by a straight-notated 16th left hand with the understanding that the left hand should be played aligned with the triplets (as a sort of swing) and not as a polyrhythm. This is an example of that practice except that the notes are shared and are played in one hand.
It was also common to treat dotted rhythms as equivalent to triplet swing. (An example that springs to mind is the coda of the slow movement of Beethoven's pathetique sonata.)
Fantasiestücke, Op.73 (Schumann, Robert) was my own introduction to this rhythm, and my own confusion, but it does sound awesome when performed. In it the clarinet doesn't use triplets, the piano uses 8th notes and 8th note triplets as in the Brahms. But at certain cadences Schumann uses hemiola and visually offsets the 8th notes, the 8ths and triplets play out smoothly and not rhythmically jarring as your interpretation examines.
I feel like you would be perfect teaching us the content of music theory, you clearly have a deep understanding and make it fun. I would love to hear what you think of this because I would pay serious money to learn through a course you’ve made
I know it was just for an analogy to the musical example, but THANK YOU FOR THE PEMDAS/BODMAS COMMENTARY ABOUT THOSE RIDICULOUS ONLINE ARTHMETIC ARGUMENTS! It is such a pet peeve of mine, particularly because of how poorly math in general, and specifically order or operations, is taught in school. We learn the mnemonic and not why the order was chosen this way in the first place.
All math is arbitrary. The order is chosen to be taught that way because that is the way standardized way the order is read and all that matters with math is logical consistency. Which is, incidentally why I'm mad at them for perpetuating a misconception that there is more than one answer to that problem. There is not. Math is standardized. People aren't taught different orders of operations. They're taught different mnemonics for the same order. Some people just didn't learn or misremember. They are right, you can add parentheses to the problem for added clarity, but the problem isn't poorly written. You plug it into any calculator on earth capable of taking that string and you'll get the same answer. The *ONE* issue I'll say exists with those troll problems is if they are handwritten, which they usually aren't, and a / is used instead of a ÷, which they usually aren't, it can be misread as a ⁄ and the problem interpreted as a vulgar fraction which would change the answer. ⁄ and / can look alike. But the order of operation fights are just a bunch of people being wrong.
@@strifera That's demonstrably incorrect, there are plenty of calculators and other programs, and even programming languages, that use different orders of operations. I regularly work in a programming language that's only OOO rule is to go left to right unless something is parenthesized. There's nothing specifically wrong with that. I wouldn't recommend the language I used for lots of reasons, but I also think you should be clear when notating your math to be clear what you want and don't just rely on OOO if you can help it. You also have other notational systems like NPN and RPN, which move beyond infix operators altogether and thus remove the ambiguity. Notational conventions are used to make things easier, but they are not the math itself. There isn't some universal truth about having to multiply before adding, its a useful convention. The point is the OOO we learn in elementary school was designed with specific use cases in mind, and they do not include trying to trick people into doing 3 + 4 x 7 in the wrong order. A good example is for polynomials, so when you write something like 3x^2 + 2x - 7, you don't have to parenthesize everything to indicate the exponents should be applied first, then the multiplication, then the addition.
@@spinaltap526 I'm not incorrect. You're equivocating (ironically). Multiple languages obviously won't be standardized across one another. That's what makes them *separate languages*. Any calculator that can parse strings and OOO calcs (so you aren't actually just doing a sequence of operations, entirely different math), will give the same answer. That's like saying pan doesn't mean skillet or sound like fan if you're speaking in Japanese or sound it out letter by letter. You're changing the context of the discussion. In standard math, there is only one OOO.
@@doublespoonco Centuries? PEMDAS, BIDMAS, BODMAS, and BEDMAS are the same thing. Multiplication and division are grouped together, as are addition and subtraction.
It all depends also on the publisher. It would be interesting to see the manuscript. A lot of publisher used to change things in pieces just because they were assuming that it would be easier to read for amateur pianist. Chopin had that problem more than once. So maybe Bramhs notated something very precise but his publisher decided that it wasn't clear enough and changed it.
the notation of music is not always mathematical, it's sometimes psychological. And Brahms wasn't the first to use it. You can find similar cases in Chopin's and Schumann's pieces
You're correct that the stems sharing the same note head shows that the two voices show be played at the same time. In my opinion as someone who studied classical piano, you're interpreting the notation too strictly. I'm quite positive that it is meant to be a triplet rhythm with the first and third note of each triplet emphasized. Brahms is trying to communicate that the notes with two stems are a melody and the notes with one stem are more harmonic than melodic (or at least secondary to the notes with two stems). This is also supported by the often stepwise melodic motion of the notes with two stems. The notes with one stem sometimes form dissonant melodic intervals from the notes with two stems, for example there is a leap of an augmented 4th (C# to G) in the second triplet. The performances I've heard support this conclusion as well.
Brahms is the master of precise ambiguity. Check out the piu lento section of Op. 10, #4 for more of this type of notation, sometimes in one hand, other times alternating between two.
@@Leadvest You're right, my bad. Onion is one of my favorite side characters and the fact that he's troubled and not just "a troublemaker" is the kind of thing that brings the show from good to great, and I should take that lesson to heart. Thanks ❤
Also 12tone: plays the triplet version at 7:00, which he admits is also the correct version at around 11:11, and based on half-remembered memories of Brahm pieces sounds correct.
I am not an expert by any means but I like the stuttering approach by attacking the triplet and the sixteenth notes at different times. It's like a baroque version of a blues rhythm.
For that math equation, 8/2(1+3). The correct way to solve it is 8/2(4), add the 1+3 because parentheses always go first, then you do 8/2 because it always goes left to right either the multiplication and division/addition and subtraction. So that would be 4(4) and that equals 16
So, if I'm understanding correctly, the confusion lies in a phenomenon similar to how modern readers get bogged down in Shakespeare: It's not that he was purposely being obscure, or that he was bad at writing, it's that he was writing in a language that is fundamentally different from the language we use today and/or using conventions (iambic pentameter) that were well understood at the time but have fallen out of use. That makes all kind of sense to me.
This is the most possible accuracy to expect from any musical notation. I think the correct interpretation lies on the fact that Brahms did this deliberately. Also, there's a rhythmic tradition about triplet-duplet relation suggesting to assimilate both whenever it fits. The contradiction is actually not about measurement or metrics, every classical musician could make sense of it. Real struggle is to choose whether or not to assimilate the rhythm (both in left and right hands) in either direction. My theory is that Brahms is taking advantage of this dilemma in notation to create a fragile rubato in the melody. He was living in a time in which polyrhythms like this existed but still was exotic.
I’m cozily soothed by the notion that the interpretation is meriting not only intuition but context l, and that you’re doodling that context into existence in some way. Not that I’ll ever play Brahms… but it’s comforting nonetheless
You could use Relativity to play this tune. Using Einstein's train and station thought experiment, the lower voice is played in a train moving close to c, while the upper voice is played in the station. You will have to compensate the doppler effect by transposing the lower voice a dozen octaves down (so the musician in the train wouldn't even listen to what he is playing) and adjusting the tuning of the instrument to compensate for the doppler effect.
Oh my god. As soon as you said "we assumed that Brahms intended this to be read precisely" I actually yelled lmfao. I'm learning (or claiming to learn) a Debussy piece at the moment and I really should have seen that coming given the era that Brahms came from.
Yeah but hear me out... What if we're supposed to play just one attack but in the middle point between where a second sixteenth note and a third sixteenth note triplet would be ? I feel that doing it like this while emphasing the top voice, if played correctly and with the right expression, could sound dope
Talk about a mountain out of a molehill. Plenty of Romantic music in triplets is written without actually indicating the "3" for triplets where it was assumed to be obvious. Beethoven's Moonlight Sonata is a perfect example. It runs almost exclusively in triplets without a single 3 in sight in most early editions. It is not omitted out of laziness or imprecision, but because it is obvious and not needed, and omitting it keeps the score clean and easy to read. Same thing here. The 3's are already present in the top voice. The lower "voice" it simply added to draw attention to pedal note in the middle of each triple and confirm that it is the other notes that should be most prominent.
There's a whole paper about this non-issue? This was never about rhythm and it's very obvious. You could have solved it right away by mentioning this variation is an inverted canon and Brahms is merely pointing out the theme you're supposed to 'sing' in your left hand.
Asking again for tally hall's ruler of everything since it recently gained popularity despite being released in the early 2010s, as well as having some interesting musical choices
You were good when YT recommended your videos to me a while back but of course (? … !) you get better doing it… and this video is a masterpiece! I love it. The topic, the framing, the in-jokes, on both the visual and the audio level are so well crafted and the harmony when they come together is delicious! Thank you!
Just a quick note: he wasn't speaking a different musical language, that would be like someone speaking dumbek rhythms or Indian classical music's swara (ICM equivalent of solfege), he's speaking a different musical dialect. Similar to how Shakespeare's literature takes some decoding to be understood by modern speakers, even though the writing is the same language.
I remember hearing a (more than likely apocryphal) story of how Brahms held a soprano soloist out of a window while telling her she would sing his piece as written. The man had opinions on phrasing and ornamentation. (And some of the best women's choir music, imho)
So to draw an analogy, this is like a martial arts instructor saying target an inch into your target" Not really something you actually do, but a useful mental image.
I actually write like that sometimes. For example right hand plays accompanying constant 1/8 notes but some of them are the main melody and should be played louder. So i added accents beneath but also additional stems pointing in the other direction. And eventhough the melodie played dotted 1/4 notes i didnt actually include the dot or wrote any rests for the melodie, because that would just make it worse to read. So i interpreted the 1/16 notes exactly like that. They just indicate which notes are more important.
So, as someone who writes music and studies notation A LOT, it's fairly common, especially in solo pieces, when a line has more than a melody, but also contains the melody, to outline it with EITHER accents, or second stems for the melody rhythm. The section from Var I you showed, is likely meant as "play as if two were playing these three notes" and giving emphasis into the next line, not just a way to connect the two lines. That's my interpretation anyways.
This actually made me research when the first double keyboard piano was produced...and as far as I can tell, it was some time in either the late 1930s or early 1940s...because I was trying to think of a way for a straight literal and precise reading to occur live without two performers. Not that it sounds particularly good this way (I transcribed it into FLStudio with cloned piano channels...I cannot play anything this complex. I produce and compose...I can barely play), but still had to try it out that way.
If you learned PEMDAS correctly you learned that multiplication AND division are performed in order from left to right, so everyone still gets 16, because math is math here or there or anywhere.
Besides the Rhythm, for pemdas and bodmas, it’s actually supposed to go from left to right, meaning is division is first of the left you do division and vice versa for multiplication, meaning no matter if you used pemdas or bodmas, you should still get the same answer if you follow it correctly.
Thank you for not giving in to the pseudomath BS on that problem. I get so frustrated with people who push that as "see, you're wrong about math!" nonsense, it's comforting to see someone acknowledge that it's wrong (and, yes, by ISO standards that have been around longer than the internet, it is wrong notation).
Notation. It's like a book. "To be or not to be, that is the question " That's what always Hamlet will say, but the intonation and stress makes these apparently identical phrase every time sounds different, sometimes you feel like one performer nailed it and made you smile with approval
I think you made the point, that the last triplet belongs to the next note, so (representing the first, second and third triplets in each set of triplets as 'da, bo, da') a cross rhythm is created linking these last and first notes (da-da), which form the principle notes of the line; common in Brahms. The middle (and low) note of triplet is 'dropped' in and forms the secondary (bo) notes of the line, da-da bo da-da bo da-da bo
I’m pretty sure the lines indicate the accenting of the music rather than the actual notes. The triplets are the base line and the sixteenths are what should be accented. I know there’s a section in Rach’s third which has similar notation, but is written with extra lines to indicate the melody in the mess of notes
The straight 16th note voice is written to show the pianist how the main melody is embedded in the triplets. It's basically Brahm's way of telling the pianist "hey, I don't know if you noticed, but the first and third notes of each triplet are actually the same notes used in the 16th note theme, so when you play it, bring them out a bit so the listener can hear that the theme in there." For what it's worth, the classical composers tended to play very loose with triplet notation. A lot of the time, they wouldn't even mark triplets explicitly at all, and you just had to figure it out by looking at how many notes were barred together and how they lined up vertically on the page. And yes, in some cases, it could be really ambiguous.
10:26 PEABODY!! Man, I haven't thought about that dog for a very long time. Oh, and by the way - I enjoy your many doodles. You have a definite knack for it.
It's called swing rhythm. Jazz musicians play this rhythm all the time. Notes are notated as equal length but played alternately long and short. It's pretty common in baroque music as well.
Jazz as well as rock and r&b traditionally didn’t use notation but relied on the musicians hearing each other and learning by ear, and that’s still true in many cases, so there are a lot of timings that would be hard to “accurately” capture on paper.
4:03 Multiplication and division are on the same level. Where the confusion comes from is the bracket. People assume multiplying the bracket is the first thing you do, when it itsnt. 8/2(3+1) = 8/2(4) = 8 ÷ 2 × 4. When laid out like this, its easy to percieve easier. Since division and multiplication are treated the same, we do whicheverbis earlier first. 8 ÷ 2 = 4. 4 × 4 = 16. 8/2(3+1) = 16 I might have forgotten the actual equation but my point still stands.
I learned the math rules as PEMDAS, as I am from America, with the caviot that it is multiplication and divide basied on left to right, and adition and subtraction from left to right. So I can say that I am an exception to Americans learn it as Multiplication always first then division.
your youtube channel is absolutely amazing! working on my own music ytb channel rn and ur one of the inspirations when it comes to scripting videos with scholarly sources, amazing!
In 7th grade (I live in the US) I learned that PEMDAS should really be written as this: PEMA D S What this means is that Multiplication and Division are in the same step and the same for addition and subtraction. so the answer to the problem would be 16. 8/2(4) would mean that the next step would be both multiplication and division, in this case division would come first because it is to the left of the 4. if it were (4)8/2 then it would be 32/2 (which is also 16, I realized as I wrote it out)
There's same thing in Chopin's Nocturne C minor 48/1 (3rd part), but there no note heads are shared. Just third one of a triplet is written in the same place as the second one of the doublet. And usually they are struck on the sane time, although you might as well create a 2:3 polyrhythm there. I used to tend to the latter (albeit more technically difficult) version, but I suppose Chopin intended the first, more popular one - just like you concluded about Brahms here
I want to feed it into a synthesizer and see what comes out. Maybe its like 1812 Overture and the piece just can't be played as intended until there's a technological advance.
File under prescriptive & descriptive notation, subset Western late 19th century solo literature. For comparison, see also Bartok transcription of Eastern European traditional dance & 20th century Jazz notation.
The “problem” feels like the same problem just intonation runs into, while normally rhythms are played by separate instruments so we don’t need a unified “tuning” for polyrhythms, here we have one key for both attacks so we need to address the rhythmic comma.
Having written music with passages like this, my intention is simply to highlight the more melodic notes within a complicated passage, and that the overall triplet rhythm is the one meant to be played. If there was no second voice, one might just identify all the notes within complicated passage as melodic, leading to a much more clanky-sounding performance than intended. My guess about why Brahms didn't change the second voice into triplets is simply for ease of reading and writing the passage, where instead of having to write down one 16th-note and and one 8th-note (and potentially a triplet sign) every time, you can just write six 16th-notes where you want them to be played, bending the rhythmic rules in the process but still getting the point across - at least, that's what I have done. I have also seen other pieces with similar passages, but written with the 16th-8th-triplet double voice (as opposed to the way Brahms wrote it) - however, it seems like it's become such common practice to do it the "incorrect way" that it ends up being that both ways of writing it are valid. However, if one really wishes to be meticulous about finding the truth about this rhythm, perhaps the best course of action would be to find other music written at that time with similarly-written passages, and find the earliest recording of it, and at best find one played by the composer. There are real 19th-century recordings of pianists, which are incredible to hear, and the composers Rachmaninoff and Debussy made recordings of themselves playing their own music, as well as music by other composers, so that would probably be the best option to finding out how such passages were actually meant to be played. That is, of course, if the composer just just verbally write at the bottom of the score how it's meant to be played. This issue reminds me of the performance practices of classical-era grace notes. Music with grace notes from more recent eras tend to have them played slightly before the beat of the "main note". But in classical-era music (especially in music by Mozart), grace notes are often played as if they're extra "main" notes within the melody, so if a grace note appears before a quarter-note, the grace- and quarter-notes would be played as if they were both 8th-notes on-beat.
Some additional thoughts/corrections:
1) The rhythm is actually present throughout variation 5, I just picked bars 6 and 7 because they were the first ones with no ties and I didn't want to add that complication.
2) Yes, I know that in both PEMDAS and BODMAS the intention is for multiplication and division to exist on the same level, because they're the same operation. But that doesn't change the fact that, as evidenced by these annoying twitter debates, many people _learned_ that there was a correct order to do them in. Notation doesn't have right and wrong, it just has clear and unclear.
3) It's not technically true that we have _no_ idea what Brahms's performances sounded like. We don't have any recordings, but there are people who were alive when he would have played these and then went on to become pianists themselves in an era where recording technology _did_ exist, so we could theoretically infer from the recordings those artists made what they had originally heard Brahms play. I don't know of any specific individuals who definitely heard him play this particular piece and then recorded their own version, but early recordings of it also seem to stick with the triplet reading so I would assume that was in line with what people who'd heard it prior to the advent of recording technology would be expecting to hear as well.
4) Technology isn't the only reason modern rhythmic culture is so different from Romantic-era Europe. There's also, of course, the influence of other traditions, most notably African and African American dance music, which much more heavily prioritize strong, steady rhythms, to allow for more complex rhythmic developments.
5) I should note that, while many of Hook's examples can be explained using a similar interpretation (Basically, the second voice is meant to highlight an important melodic sub-line, not inform rhythm) there are some where it doesn't really work. For instance, in his paper (found at www.mtosmt.org/issues/mto.11.17.4/mto.11.17.4.hook.html ) example 10 definitely has two distinct voices with their own melodic shapes that happen to intersect at a point that is metrically undefinable. In those cases I'd just revert back to the rubato thing and say you're supposed to play it somewhere in between and not worry about it beyond that.
Oh. It's so weird to think that there was an overlap between Brahms' life and recording technology. Like, that just feels strange even though when you actually look at the dates that makes sense.
I may remember it wrong, but it appeared to me the same problem of notation happens in a far more famous piece, the triplets in the orchestral accompaniment in the chorale "Jesus bleibet meine Freude", BWV 147 by J.S. Bach.
I think pianists might be more familiar with this kind of notation being used as shorthand for playing instructions as notation like this is fairly common in older complex works. For example a tremolo is commonly used as shorthand for a very fast repeated figure - where they notate the figure for up to two bars then just switch to tremolo for the rest of the piece. Similar to repeat notations for guitars and drums, I think it might just be unfamiliar to non pianists.
@@crisdekker8223 You're right, but it's not the same problem. In that piece, one part plays triplets (or sometimes it's in 9/8) and the other plays a dotted rhythm. I think this is the same practice that @The ModicaLiszt refers to in their comment (ua-cam.com/video/xXIG8DiMmw8/v-deo.html&lc=Ugy-i20SohlWYVDZHzN4AaABAg), and might be related to the typesetting conventions of the time (I wonder if there's a manuscript that is written differently).
@@catomajorcensor I was curious myself. I went to IMSLP and saw Bach's original manuscript and he actually does juxtapose three triplets with a dotted quarter + eighth, so mathematically the last triplet and the eighth don't line up. Must be just shorthand. And you're right, I should've said similar in stead of the same problem.
Frankly, until you pointed out the missing triplet notation from the bottom line I could not even understand what was wrong about the rhythm. The number of examples where triplets are implied in notation of this period is so staggeringly high that I didn't even think twice about interpreting this as a constant stream of triplets.
Yeah, that’s exactly what i thought as well, I can think of so many other pieces that are technically also impossible
Of course, the triplets on the lower part would mean more unnecessary stuff to read.
so, just assume the triplet notation isnt written but implied in the lower line?
edit: thats how i first thought it was, and after im quite certain thats still how it is.
@@gagaronpew4382 Yes. The only thing that is _somewhat_ confusing is why Brahms chose to connect the double bar line (notating two 16th notes) instead of notating an 8th note and a 16th note. Notation was nowhere near as universal back then (let's remind ourselves that many composers a century before Brahms didn't even notate dynamics) so I have no reason to doubt that this is a constant stream of 16th note triplets, as it's the most obvious rhythm that makes sense within the style of the period.
And you are correct to do so. It's that simple.
3:35 something about writing an equation like this backwards is deeply unsettling
He had to tap into the quantum realm.
pretty sure that is what he intended
@@victorfunnyman intended or not, very fitting!
3:36 Awarding you a virtual medal for nonchalantly slipping Schrodinger's equation into a music video.
If the same note is played on the piano at the same time, it's dead. It's definitely dead 💀🤣
@@AudunWangen But until someone tries to play it, it could both be alive and dead.
Next time, the general one instead of the stationary one: :p
And then become more ambitious with Dirac's and then the Klein-Gordon one, in whatever order they like :p
@@AudunWangen never said there had to be one piano?
this piece has too many notes to solve for the rhythm exactly
16th century composer: I guess that note is sharpened, but there's no need to write the sharp in because every single musician alive already knows it's a sharp.
21st century musicologist: oh crap.
Not a Brahms scholar, and definitely under studied, but as someone who has learnt 100s of piano pieces both as a soloist and accompanist, and a composer FOR piano, as well, I can say with as much certainty as possible that this form of notation is used to highlight important melodic and phrasing information. I am not really sure if this was ever a thing that composers did in their own manuscripts, or something that has become a more recent editorial addition, but I will always prefer complicated decorated pieces to have melodic phrasings like this. Being able to see the lines through the piece (especially in a case where you don't have a lot of time to do it) is so important for legibility. It can be the difference between making a piece challenging to perform OR challenging to read.
Would you interpret the highlighted melody as to play it with a slight accent, or maybe slightly ghost the non-melody notes? Or perhaps slightly stretch those melody notes so they have emphasis?
Or would you think "Oh that's the melody I've seen a hundred times before, just with an embellishment. I don't need to invest as much brain power reading it."?
@@randysterbentz5599 Logically, the latter - stylistically, some part of the former would enter in to it, as well. That really depends on the piece and performer though. Realistically, you're going to want to provide some kind of emphasis to the the melodic line over the decoration, and writing it out in this way makes that distinction a lot more obvious.
Though as to how exactly you'd do that, it's really up to the performer, their interpretation, and the general style of the piece. Having not played this one myself, I can't speak to it, though glancing through the score, I would personal lean slightly in to the main melodic line, while trying to play the decorative notes more lighter.
I agree
i was literally about to write something similar to this. especially being a classically trained pianist. this is "common knowledge" but now that i sit down and see it under these lenses it kinda makes you wonder.
Yeah, I play violin and definitely intuited this pretty immediately. You've got triplets and markings to demonstrate which notes to emphasize/which notes are similar to ones before. no point in re-writing all the notation for a triplet when you already know they're triplets.
6:48 "Not stylistically appropriate" *12tone proceeds to draw a realistic pachyderm*
And my absolutely uneducated interpretation: the first and third note in each triplet are accented and the middle note is closer to a grace note. I know I'm wrong, but I like the sound of it :)
And proceeds to draw a perfect Peabody, too
Agreed, in fact you could think of the first and third form one principle line and the middle 'grace-like' notes form a sub-line.
6:45 I absolutely love the illustration for “not stylistically appropriate”. Perfect pictorial analogy.
Of course, that means it *was* stylistically appropriate…
@@aspidoscelis Well, it was thematically appropriate and narratively appropriate, but, as intended, it was stylistically jarring.
The 2nd approach is the stylistically appropriate one and most certainly the correct one. The triplet rhythm is the underlying, correct rhythm and the added 16th note downward stems are to convey voicing/melodic lines as you pointed out (not to be played rhythmically like straight 16th notes). This type of notation is fairly common in romantic/classical literature, which is why pretty much every professional recording plays it that way.
You basically hit the nail on the head at the end. A seasoned classical pianist wouldn't even think twice about how to perform this.
Exactly. This video is over-complicating things, seeing things that aren't there. This is a very common notational standard. It's not "impossible", it's just a limitation of written notations, and it's 100% obvious what is intended, it's not even weird.
There ARE times in Brahms where he wants 2 against 3 in passages somewhat similar to this, but they don't share noteheads -- it's very obvious that they're spaced differently.
Can we take a moment to truly appreciate how amazing this guy's annotation skills are? I think you could build anywhere from 50 to 99% of a new Emoji dictionary just using his little elephants.
And in a few hundred years the history will repeat itself and some future musicians will be trying to interpret these writings.
Oh, I thought they were piggies😁
Wow a music channel finally actually explained those BS math posts better than any single person in the comments of those posts ever has.
nope
No he didn't. He added a parentheses that was unneeded and further clouded this problem. The answer is 1.
@@dying_allthetime
The whole "this is an intentionally poorly written equation and it should have parentheses if it wants to be clear" thing flew over your head?
@@dying_allthetime not only did you not get the point with clearer notation, the parentheses added "changes" the answer to 16. Also, please don't use diagonal fraction lines like that. If you want something to be in the denominator just put it UNDER the line, not somewhere to the right of it.
@@JrgenHelland00 Funny, almost as if it was Intentionally poorly written. ;-)
I love that you put a Golden Strawberry from Celeste when you said "Physically Demanding". Great stuff!
Uhh, if you wanna do more music theory mysteries I am totally here for it... just throwing that out there.
I wish to second this
@@oscargill423 Excellent, can we get someone to perfect fifth this too? I don't know if 12tone can resist a RequestSus2
@@ImSquiggs can I be a third though?, could be cool too, right? a bit...?
But he has to narrate it as if it's a true crime story. "UNSOLVED MYSTERIES: MUSIC THEORY EDITION."
Idk what’s even happening so I’m just going to play a minor sixth in second inversion
How I wish that all the great masters would have had the ability to record so we could here how they played it.
6:47 OH MY GOD I love the golden strawberry from Celeste as an illustration of "physically challenging"
Two thoughts:
1st, could this notation have been used to indicate stress/emphasis? Maybe Brahms wanted a "swing" feel with emphasis on the first and third part of the triplet and the middle one like a ghost note? One way to check that hypothesis would be to check whether he uses things like >, . and _ in his other scores?
2nd, what does the original look like? If that is a variation, maybe the source was two-voiced and for those who were familiar with the piece, he wanted to indicate "here, i joined those two voices into one" or something?
Exactly. Implied triplets of swing were my first thought. The notes with shared stems are the swung melody and the middle notes of the triplets act as a kind of pedal. The straight 16th beams just indicate the melodic line.
That was pretty fascinating to watch ngl
As an arranger with no formal musical education, if I want things to be phrased a certain way on piano I usually write a slur over sections of music that I want to link together and if certain tones need to be emphasized over others in a stream of notes that happen in quick succession i'll just mark those notes with an accent. I don't really know if that's the proper way to do it, but it makes sense to me from an intuitive level
I bet this guy saves all the drawings he made in his videos and when he is old and retired he can read them in nostalgic awe :D
When he's old, he can sell them on Ebay for his retirement funds.
04:25. As a mathematician I like your statement "this is bad mathematical notation". Indeed, many people learned one of the precedence rules about multiplication and division, and think the rule they have learned is a dogm, and think all others are wrong. They forget that a notation is only a convention and nothing fundamental in the mathematics itself. And indeed, until you specify clearly which precedence rule you are using, the notation is ambiguous and to be avoided.
1:22 I don't think that's how this beaming works; I've seen this kind of beaming before and all it's basically doing is beaming over the triplet rest. This beaming style is often used to specifically demarcate a contiguous line that should stand out in a denser texture, I.E. the down-stemmed notes are all part of one idea, the up-stemmed notes are just the full rhythm with the middle beat in each triplet group being a filler note not part of the main melody. Not sure how you came to your conclusion or interpretation at all.
Yeah I see this as two ninelets
Exactly. To make Dr Hook happy, Brahms would have had to have either written in tiny 16th note triplet rests for the bottom line, or given up on highlighting the descending voice that the pianist is supposed to be bringing out. The first option would be unnecessarily cluttered and create more confusion (note that apparently every pianist who has ever played this, except Dr Julian Hook, has understood how to play this as written without any trouble); the second option would fail to communicate the intention to the player (which 12tone explains very nicely).
(I should point out that I have not read Dr Hook's paper so maybe he makes clear that he knows perfectly well what it means but intended to highlight the very point 12tone makes at the end - musical notation isn't meant to be like a programming language, it is human readable, not machine readable.)
I wonder whether this was even Brahms's doing or his publisher. Sadly, the manuscript isn't on IMSLP. The first edition only actually includes the triplet symbols the first time this appears - plainly, the triplet feel is implicit.
PEMDAS with respect to the multiplication and division in my school in the US was not taught "multiplication first." It was taught to read left to weight and perform multiplication and division before any addition or subtraction. The M and D were treated as a single step read from left to right without necessarily doing one or the other first. Same with addition and subtraction. One single step.
4 steps total:
P: parentheses
E: exponents
MD: multiplication and division
AS: addition and subtraction
6*7/8 = multiplication first because it comes first
6/7*8 = division first because it comes first
It’s purely a notational thing…… Brahms indicates the bottom voice to be brought out in the triplets. Very standard Romantic practice. Chopin for example uses dotted rhythms in a melody to mean triplet crochet + quaver if there are triplets in the accompaniment.
16 minutes on what is indeed just notational, is really too much
There's some video on his clickbait
@@triple_croche Well it's not too much considering most people don't know how to read music. So he's explaining why there is a misunderstanding in the first place.
im confused by what you mean. why would he notate it this way instead of normal triplets?
@@astra1288 He does have normal triplets. The bottom noteheads indicate that those notes should be emphasised or at the fore of the texture because they create a melody among the decorative triplets.
I think about this issue in terms of how I write music. Unlike a musical staff, I rely on the piano roll in my DAW. It places each note and it's duration on a grid with less ambiguity. Whether recorded, edited, or entered directly by hand, it places each note and it's duration where you can see it exactly. It doesn't completely remove ambiguity, because the synth patch may have a slow attack, for example, and you hear the note appear later than the grid of the piano roll would suggest. Or it may have an effect or slow release that makes it sound longer than the end point would suggest. This is clearly in the modern mindset you mention, but it's much easier to read and control than trying to do it with musical notation. When I first started with computer music, the programs I had access to expected you to enter notes on a musical staff.
There’s no ambiguity with notation if you wrote it yourself with a modern midi program
@@jacksonbarker7594 yes but, getting a synth patch to respond exactly as you want doesn't necessarily conform to the written notation if you want it to last a little longer or shorter than the notation would indicate. Especially if you've got, say a delay that needs it to release earlier so it doesn't retrigger one too many times
I honestly love the idea of written music being pretty open to interpretation or, in this case, conveying structural context. I'm sure you can add more nuance or clarity to compositions this way than through a literal reading. Cool video!
As always this was a great video that had an awesome approach to the different possibilities. I would like to offer one more possibility that is definitely wrong but also would be hilarious: Brahms wanted you to play all the dowmstems with your thumb as if the piano was a guitar.
Uncomfortably high budget for sharpies 🤣
This is why you should patreonize 12tone 😁
TLDR: Pianists do a thing called voicing (making melody lines stand out in relative loudness) and I think that’s what these extra stems are about.
As someone who plays classical piano at a semi-advanced amateur level, there’s something that comes to me immediately that doesn’t seem to be addressed at all in the video (or at least not well): voicing (not to be confused with chord voicings). This is when the pianist makes the notes of the melody louder than those of the accompaniment (and/or the accompaniment softer), and sometimes even plays a secondary melody louder than the accompaniment but softer than the primary melody. The practice of voicing is expected to the point that a score doesn’t usually have any indication. But if a certain melody line is intended to be voiced but not obvious composers may choose to highlight its presence in a variety of ways (also depending on musical context).
In the example in the video, I would use the triplet rhythm in the left hand, voicing only the notes indicated by the additional stems as a melody, and I would view the offset from the r.h. rhythms as an effect of sorts, rather than as a failure to mirror it precisely (‘cause lets be honest: romantic and later composers were often quite liberal in their explorations of earlier idioms), one consequence of which is helping the l.h. melody stand out against the r.h.
Meanwhile the middle triplet notes would be part of the broader triplet rhythm but not treated as part of the second melody line, i.e. as accompaniment. On further thought, this *may* also affect phrasing, which for pianists is a dynamic device often realized by making a phrase grow louder, and then the last note of the phrase softer, but also, and more relevantly to this thought, often making longer notes softer than shorter ones within the broader arc.
As for example 10 from the paper referenced by 12Tone, I might prioritize the triplet rhythmically but voice the note as part of the top line. I’m less confident in my assessment/interpretation of this rhythm, though.
This reminds me of parts of Petrushka. I played it under a conductor who studied under the successor to the guy that premiered it. There are sections notated as a measure-long septuplet in 3/4 - these were actually meant to be two eighth notes, a triplet, and two more eighth notes, but the composer couldn't translate the line he came up with to that notation when he wrote it, so he decided to just write that there are seven notes.
6:13 is personally my favorite interpretation. I think it sounds unusual but I actually like the sound of it. Also just because the right hand is doing something cool doesn't mean that the left hand needs to have a line that doesn't sing on its own too and needs to be not distracting because you arbitrary suggest that the focus "should" be on the right hand. Yeah it's hard to play but it can be played.
To add a complication to this, in the romantic era it was not uncommon for triplet-notated right hand to be accompanied by a straight-notated 16th left hand with the understanding that the left hand should be played aligned with the triplets (as a sort of swing) and not as a polyrhythm. This is an example of that practice except that the notes are shared and are played in one hand.
It was also common to treat dotted rhythms as equivalent to triplet swing. (An example that springs to mind is the coda of the slow movement of Beethoven's pathetique sonata.)
Something is Wrong with the captions, at 12:06 they start from the beginning again.
Alphonse Elric is an excellent doodle for representing the concept of equivalence. Props dude!
Fantasiestücke, Op.73 (Schumann, Robert) was my own introduction to this rhythm, and my own confusion, but it does sound awesome when performed. In it the clarinet doesn't use triplets, the piano uses 8th notes and 8th note triplets as in the Brahms. But at certain cadences Schumann uses hemiola and visually offsets the 8th notes, the 8ths and triplets play out smoothly and not rhythmically jarring as your interpretation examines.
I feel like you would be perfect teaching us the content of music theory, you clearly have a deep understanding and make it fun. I would love to hear what you think of this because I would pay serious money to learn through a course you’ve made
I know it was just for an analogy to the musical example, but THANK YOU FOR THE PEMDAS/BODMAS COMMENTARY ABOUT THOSE RIDICULOUS ONLINE ARTHMETIC ARGUMENTS! It is such a pet peeve of mine, particularly because of how poorly math in general, and specifically order or operations, is taught in school. We learn the mnemonic and not why the order was chosen this way in the first place.
All math is arbitrary. The order is chosen to be taught that way because that is the way standardized way the order is read and all that matters with math is logical consistency.
Which is, incidentally why I'm mad at them for perpetuating a misconception that there is more than one answer to that problem. There is not. Math is standardized. People aren't taught different orders of operations. They're taught different mnemonics for the same order. Some people just didn't learn or misremember.
They are right, you can add parentheses to the problem for added clarity, but the problem isn't poorly written. You plug it into any calculator on earth capable of taking that string and you'll get the same answer.
The *ONE* issue I'll say exists with those troll problems is if they are handwritten, which they usually aren't, and a / is used instead of a ÷, which they usually aren't, it can be misread as a ⁄ and the problem interpreted as a vulgar fraction which would change the answer. ⁄ and / can look alike. But the order of operation fights are just a bunch of people being wrong.
@@strifera That's demonstrably incorrect, there are plenty of calculators and other programs, and even programming languages, that use different orders of operations. I regularly work in a programming language that's only OOO rule is to go left to right unless something is parenthesized. There's nothing specifically wrong with that. I wouldn't recommend the language I used for lots of reasons, but I also think you should be clear when notating your math to be clear what you want and don't just rely on OOO if you can help it.
You also have other notational systems like NPN and RPN, which move beyond infix operators altogether and thus remove the ambiguity. Notational conventions are used to make things easier, but they are not the math itself. There isn't some universal truth about having to multiply before adding, its a useful convention.
The point is the OOO we learn in elementary school was designed with specific use cases in mind, and they do not include trying to trick people into doing 3 + 4 x 7 in the wrong order. A good example is for polynomials, so when you write something like 3x^2 + 2x - 7, you don't have to parenthesize everything to indicate the exponents should be applied first, then the multiplication, then the addition.
@@strifera one order of operations? Since when
@@spinaltap526 I'm not incorrect. You're equivocating (ironically). Multiple languages obviously won't be standardized across one another. That's what makes them *separate languages*. Any calculator that can parse strings and OOO calcs (so you aren't actually just doing a sequence of operations, entirely different math), will give the same answer.
That's like saying pan doesn't mean skillet or sound like fan if you're speaking in Japanese or sound it out letter by letter. You're changing the context of the discussion. In standard math, there is only one OOO.
@@doublespoonco Centuries? PEMDAS, BIDMAS, BODMAS, and BEDMAS are the same thing. Multiplication and division are grouped together, as are addition and subtraction.
It all depends also on the publisher. It would be interesting to see the manuscript.
A lot of publisher used to change things in pieces just because they were assuming that it would be easier to read for amateur pianist.
Chopin had that problem more than once.
So maybe Bramhs notated something very precise but his publisher decided that it wasn't clear enough and changed it.
the notation of music is not always mathematical, it's sometimes psychological. And Brahms wasn't the first to use it. You can find similar cases in Chopin's and Schumann's pieces
You're correct that the stems sharing the same note head shows that the two voices show be played at the same time. In my opinion as someone who studied classical piano, you're interpreting the notation too strictly. I'm quite positive that it is meant to be a triplet rhythm with the first and third note of each triplet emphasized. Brahms is trying to communicate that the notes with two stems are a melody and the notes with one stem are more harmonic than melodic (or at least secondary to the notes with two stems). This is also supported by the often stepwise melodic motion of the notes with two stems. The notes with one stem sometimes form dissonant melodic intervals from the notes with two stems, for example there is a leap of an augmented 4th (C# to G) in the second triplet. The performances I've heard support this conclusion as well.
Brahms is the master of precise ambiguity. Check out the piu lento section of Op. 10, #4 for more of this type of notation, sometimes in one hand, other times alternating between two.
I would love it if 12tone talked about hyperpop. Not a lot of music theory talk in that realm and I’m really interested to hear what 12tone thinks
I absolutely love your choice of Onion for poster child of starting shit for no apparent reason.
Explicitly saying he's a troublemaker, without acknowledging his troubles and triumphs is problematic. Onion's characterization has layers.
@@Leadvest You're right, my bad. Onion is one of my favorite side characters and the fact that he's troubled and not just "a troublemaker" is the kind of thing that brings the show from good to great, and I should take that lesson to heart. Thanks ❤
12tone: “this is impossible”
Me: *patiently waits for him to play it anyway*
Also 12tone: plays the triplet version at 7:00, which he admits is also the correct version at around 11:11, and based on half-remembered memories of Brahm pieces sounds correct.
I am not an expert by any means but I like the stuttering approach by attacking the triplet and the sixteenth notes at different times. It's like a baroque version of a blues rhythm.
For that math equation, 8/2(1+3). The correct way to solve it is 8/2(4), add the 1+3 because parentheses always go first, then you do 8/2 because it always goes left to right either the multiplication and division/addition and subtraction. So that would be 4(4) and that equals 16
So, if I'm understanding correctly, the confusion lies in a phenomenon similar to how modern readers get bogged down in Shakespeare: It's not that he was purposely being obscure, or that he was bad at writing, it's that he was writing in a language that is fundamentally different from the language we use today and/or using conventions (iambic pentameter) that were well understood at the time but have fallen out of use. That makes all kind of sense to me.
Wow. You are an incredibly talented madman! Love it.
This is the most possible accuracy to expect from any musical notation. I think the correct interpretation lies on the fact that Brahms did this deliberately. Also, there's a rhythmic tradition about triplet-duplet relation suggesting to assimilate both whenever it fits. The contradiction is actually not about measurement or metrics, every classical musician could make sense of it. Real struggle is to choose whether or not to assimilate the rhythm (both in left and right hands) in either direction. My theory is that Brahms is taking advantage of this dilemma in notation to create a fragile rubato in the melody. He was living in a time in which polyrhythms like this existed but still was exotic.
I’m cozily soothed by the notion that the interpretation is meriting not only intuition but context l, and that you’re doodling that context into existence in some way. Not that I’ll ever play Brahms… but it’s comforting nonetheless
You could use Relativity to play this tune.
Using Einstein's train and station thought experiment, the lower voice is played in a train moving close to c, while the upper voice is played in the station.
You will have to compensate the doppler effect by transposing the lower voice a dozen octaves down (so the musician in the train wouldn't even listen to what he is playing) and adjusting the tuning of the instrument to compensate for the doppler effect.
In one way obscure as heck, but in another way highly interesting. Well done, mate! 🙂
Oh my god. As soon as you said "we assumed that Brahms intended this to be read precisely" I actually yelled lmfao. I'm learning (or claiming to learn) a Debussy piece at the moment and I really should have seen that coming given the era that Brahms came from.
Yeah but hear me out... What if we're supposed to play just one attack but in the middle point between where a second sixteenth note and a third sixteenth note triplet would be ? I feel that doing it like this while emphasing the top voice, if played correctly and with the right expression, could sound dope
That's what I first thought when I saw this. I think that's how it was meant and it would sound very cool.
Talk about a mountain out of a molehill. Plenty of Romantic music in triplets is written without actually indicating the "3" for triplets where it was assumed to be obvious. Beethoven's Moonlight Sonata is a perfect example. It runs almost exclusively in triplets without a single 3 in sight in most early editions. It is not omitted out of laziness or imprecision, but because it is obvious and not needed, and omitting it keeps the score clean and easy to read. Same thing here. The 3's are already present in the top voice. The lower "voice" it simply added to draw attention to pedal note in the middle of each triple and confirm that it is the other notes that should be most prominent.
There's a whole paper about this non-issue? This was never about rhythm and it's very obvious. You could have solved it right away by mentioning this variation is an inverted canon and Brahms is merely pointing out the theme you're supposed to 'sing' in your left hand.
Asking again for tally hall's ruler of everything since it recently gained popularity despite being released in the early 2010s, as well as having some interesting musical choices
3:56 actually, we learn it as PEMDAS (P, E, MD, AS) and do the grouped ones from left to right.
You were good when YT recommended your videos to me a while back but of course (? … !) you get better doing it… and this video is a masterpiece! I love it. The topic, the framing, the in-jokes, on both the visual and the audio level are so well crafted and the harmony when they come together is delicious! Thank you!
Just a quick note: he wasn't speaking a different musical language, that would be like someone speaking dumbek rhythms or Indian classical music's swara (ICM equivalent of solfege), he's speaking a different musical dialect. Similar to how Shakespeare's literature takes some decoding to be understood by modern speakers, even though the writing is the same language.
Shawn Crowder: CHALLENGE ACCEPTED
Best
I remember hearing a (more than likely apocryphal) story of how Brahms held a soprano soloist out of a window while telling her she would sing his piece as written. The man had opinions on phrasing and ornamentation. (And some of the best women's choir music, imho)
4:11 despite being from the states, I feel lucky to have been taught that they are equal and you reed the equation left to right in problems like this
haha the doodling made me focused on his words... distracting myself subconsciously isn't needed xd
So to draw an analogy, this is like a martial arts instructor saying target an inch into your target"
Not really something you actually do, but a useful mental image.
I'm about halfway, still thinking Buddy Rich or Louie Bellson could have figured something out.
I actually write like that sometimes. For example right hand plays accompanying constant 1/8 notes but some of them are the main melody and should be played louder. So i added accents beneath but also additional stems pointing in the other direction. And eventhough the melodie played dotted 1/4 notes i didnt actually include the dot or wrote any rests for the melodie, because that would just make it worse to read. So i interpreted the 1/16 notes exactly like that. They just indicate which notes are more important.
So, as someone who writes music and studies notation A LOT, it's fairly common, especially in solo pieces, when a line has more than a melody, but also contains the melody, to outline it with EITHER accents, or second stems for the melody rhythm. The section from Var I you showed, is likely meant as "play as if two were playing these three notes" and giving emphasis into the next line, not just a way to connect the two lines.
That's my interpretation anyways.
I like how his Mewtwo is simple and recognizable, but still looks rushed. Yet, his Alphonse is almost perfect.
I like your drawing ✍️ Please keep doing more of it 😊 thank you!
Shout-out to Dr. Hook - my post-tonal theory class with him used a textbook that laid out the palindromic rhythms of Messiaen... citing his paper.
This actually made me research when the first double keyboard piano was produced...and as far as I can tell, it was some time in either the late 1930s or early 1940s...because I was trying to think of a way for a straight literal and precise reading to occur live without two performers. Not that it sounds particularly good this way (I transcribed it into FLStudio with cloned piano channels...I cannot play anything this complex. I produce and compose...I can barely play), but still had to try it out that way.
I think he just want the sensation of an ostinato AND this melody.
Its relevant with the extract you discuss at 13:00 too.
6:49 "very physically challenging" with a golden berry lmaooo
If you learned PEMDAS correctly you learned that multiplication AND division are performed in order from left to right, so everyone still gets 16, because math is math here or there or anywhere.
This is standard notational practice and has been for centuries, but I guess content is gonna content.
Besides the Rhythm, for pemdas and bodmas, it’s actually supposed to go from left to right, meaning is division is first of the left you do division and vice versa for multiplication, meaning no matter if you used pemdas or bodmas, you should still get the same answer if you follow it correctly.
Thank you for not giving in to the pseudomath BS on that problem. I get so frustrated with people who push that as "see, you're wrong about math!" nonsense, it's comforting to see someone acknowledge that it's wrong (and, yes, by ISO standards that have been around longer than the internet, it is wrong notation).
Notation.
It's like a book.
"To be or not to be, that is the question "
That's what always Hamlet will say, but the intonation and stress makes these apparently identical phrase every time sounds different, sometimes you feel like one performer nailed it and made you smile with approval
I think you made the point, that the last triplet belongs to the next note, so (representing the first, second and third triplets in each set of triplets as 'da, bo, da') a cross rhythm is created linking these last and first notes (da-da), which form the principle notes of the line; common in Brahms. The middle (and low) note of triplet is 'dropped' in and forms the secondary (bo) notes of the line, da-da bo da-da bo da-da bo
I’m pretty sure the lines indicate the accenting of the music rather than the actual notes. The triplets are the base line and the sixteenths are what should be accented. I know there’s a section in Rach’s third which has similar notation, but is written with extra lines to indicate the melody in the mess of notes
The straight 16th note voice is written to show the pianist how the main melody is embedded in the triplets. It's basically Brahm's way of telling the pianist "hey, I don't know if you noticed, but the first and third notes of each triplet are actually the same notes used in the 16th note theme, so when you play it, bring them out a bit so the listener can hear that the theme in there."
For what it's worth, the classical composers tended to play very loose with triplet notation. A lot of the time, they wouldn't even mark triplets explicitly at all, and you just had to figure it out by looking at how many notes were barred together and how they lined up vertically on the page. And yes, in some cases, it could be really ambiguous.
I wonder if it could be a notation of two pianos playing at the same time with them coming together at that point
Somehow I simultaneously didn't understand a word of that, and understood it completely.
10:26 PEABODY!! Man, I haven't thought about that dog for a very long time.
Oh, and by the way - I enjoy your many doodles. You have a definite knack for it.
It's called swing rhythm. Jazz musicians play this rhythm all the time. Notes are notated as equal length but played alternately long and short. It's pretty common in baroque music as well.
Jazz as well as rock and r&b traditionally didn’t use notation but relied on the musicians hearing each other and learning by ear, and that’s still true in many cases, so there are a lot of timings that would be hard to “accurately” capture on paper.
6:10 I think the first version is the right one. It sounds awesome! :D
4:03 Multiplication and division are on the same level. Where the confusion comes from is the bracket. People assume multiplying the bracket is the first thing you do, when it itsnt. 8/2(3+1) = 8/2(4) = 8 ÷ 2 × 4. When laid out like this, its easy to percieve easier. Since division and multiplication are treated the same, we do whicheverbis earlier first. 8 ÷ 2 = 4.
4 × 4 = 16.
8/2(3+1) = 16
I might have forgotten the actual equation but my point still stands.
I learned the math rules as PEMDAS, as I am from America, with the caviot that it is multiplication and divide basied on left to right, and adition and subtraction from left to right. So I can say that I am an exception to Americans learn it as Multiplication always first then division.
your youtube channel is absolutely amazing! working on my own music ytb channel rn and ur one of the inspirations when it comes to scripting videos with scholarly sources, amazing!
Loved the golden strawberry and Onion drawings
Can I just say well done for quietly slipping the best analysis of the whole bodmas nonsense I've seen on this whole website into a video about Brahms
In 7th grade (I live in the US) I learned that PEMDAS should really be written as this:
PEMA
D S
What this means is that Multiplication and Division are in the same step and the same for addition and subtraction. so the answer to the problem would be 16. 8/2(4) would mean that the next step would be both multiplication and division, in this case division would come first because it is to the left of the 4. if it were (4)8/2 then it would be 32/2 (which is also 16, I realized as I wrote it out)
There's same thing in Chopin's Nocturne C minor 48/1 (3rd part), but there no note heads are shared. Just third one of a triplet is written in the same place as the second one of the doublet. And usually they are struck on the sane time, although you might as well create a 2:3 polyrhythm there. I used to tend to the latter (albeit more technically difficult) version, but I suppose Chopin intended the first, more popular one - just like you concluded about Brahms here
I was thinking about that particular example! Thank you haha
13:25 you can tell he just started drawing random shit at the end
I’ve been watching your videos for YEARS and only just realized you’re left handed. 😅
Great video as always, and interesting breakdown!!!
I want to feed it into a synthesizer and see what comes out. Maybe its like 1812 Overture and the piece just can't be played as intended until there's a technological advance.
Extremely informative, interesting and entertaining video!
File under prescriptive & descriptive notation, subset Western late 19th century solo literature. For comparison, see also Bartok transcription of Eastern European traditional dance & 20th century Jazz notation.
The “problem” feels like the same problem just intonation runs into, while normally rhythms are played by separate instruments so we don’t need a unified “tuning” for polyrhythms, here we have one key for both attacks so we need to address the rhythmic comma.
i don't know how these visuals are made but i am absolutely tripping at the backwards writing @5:10
Having written music with passages like this, my intention is simply to highlight the more melodic notes within a complicated passage, and that the overall triplet rhythm is the one meant to be played. If there was no second voice, one might just identify all the notes within complicated passage as melodic, leading to a much more clanky-sounding performance than intended. My guess about why Brahms didn't change the second voice into triplets is simply for ease of reading and writing the passage, where instead of having to write down one 16th-note and and one 8th-note (and potentially a triplet sign) every time, you can just write six 16th-notes where you want them to be played, bending the rhythmic rules in the process but still getting the point across - at least, that's what I have done. I have also seen other pieces with similar passages, but written with the 16th-8th-triplet double voice (as opposed to the way Brahms wrote it) - however, it seems like it's become such common practice to do it the "incorrect way" that it ends up being that both ways of writing it are valid.
However, if one really wishes to be meticulous about finding the truth about this rhythm, perhaps the best course of action would be to find other music written at that time with similarly-written passages, and find the earliest recording of it, and at best find one played by the composer. There are real 19th-century recordings of pianists, which are incredible to hear, and the composers Rachmaninoff and Debussy made recordings of themselves playing their own music, as well as music by other composers, so that would probably be the best option to finding out how such passages were actually meant to be played.
That is, of course, if the composer just just verbally write at the bottom of the score how it's meant to be played.
This issue reminds me of the performance practices of classical-era grace notes. Music with grace notes from more recent eras tend to have them played slightly before the beat of the "main note". But in classical-era music (especially in music by Mozart), grace notes are often played as if they're extra "main" notes within the melody, so if a grace note appears before a quarter-note, the grace- and quarter-notes would be played as if they were both 8th-notes on-beat.