We do it all the time. An index is just indicative that there is a mapping (a function), usually from the integers. However we don't use the subscript notation when indexing by a continuum due to the discomfort you describe.
The point is that we need some way to deal with objects that are inherently infinite-dimensional.
Do you have any particular pieces in mind when you wrote this?
Bach is impressive, no doubt, but to each their own perhaps. I acknowledge that I have not received the appropriate training to fully appreciate the complexity in his works, so I wish I could hear what you do. To my ear, (and this isn't a novel opinion in the slightest), I think the Baroque era was more limited in expression due to the inherent limitations in the instruments and consequent styles at the time. Within those constraints, calling Bach an absolute titan of composition would be an understatement. But one wonders what he could have made without those constraints.
Bach's most approachable music might be his cello suites.
But also, I think there are two camps of fans of "classical music" (by which I mean music in the styles: Baroque, Classical, Romantic, Impressionist, etc). There are those who listen to the music, and those who play it.
For the most part, those who only listen to music often prefer Romantic and Impressionist styles. From the moody and dramatic to the gentle and contemplative, these styles are very approachable to the untrained ear.
But those who play an instrument (or sing in a choir) spend lots of time practicing and rehearsing and interpreting the music as it's written on the page. This extra time makes all of the little nuances of Baroque music truly come to life. The classic example is Bach's Crab Canon, which is a fine little piece of music... but once you realize that the whole thing is a palindrome, and you can actively appreciate how the same parts work in a forward and backward context, it becomes really interesting and pleasant.
So if Bach doesn't do it for you, and you play an instrument, try diving into playing it yourself.
I think that's true about Bach's instrumental music, but his big sacred works like his Passions and the Mass in B minor are as "romantic" as the Baroque period gets. Like OP, I think of these works as basically the pinnacle of human artistic achievement. They somehow have all the nuance and complexity you're referring to -- while also telling a deeply emotional story, and just being heart-wrenchingly beautiful even if you don't know the story.
I think Bach's lute music is the most approachable because it sounds the most modern like guitar music. Even though the baroque lute is an alien instrument visually to the average person today, the sound is closer to what people have grown up on.
The whole question though is like what is the best David Bowie album to start with multiplied by 100.
The catalog is just so immense, the sounds are just so varied that one person's favorite might completely be wrong for someone else.
I think the most relatable after thinking about it more is Stephanie Jones playing lute music on classical guitar.
Funnily enough, I actually play the cello and have enjoyed playing some of his cello suites in the past. Yes, I certainly admire the famous Suite I and it has an incredible mood to it.
I most enjoy playing music as a social affair rather than in isolation though. That may have a fair amount to do with my impression of composers from each era (Baroque is fine in a group, Classical can be unforgiving, Romantic is a lot of fun, etc.).
Looking at many of the responses here though (which have been wonderful), there are quite a few pieces from Bach that I was not aware of, or had forgotten about. He really was incredible.
I cut my teeth on Bach on Cello when I was 7. By the time I was in high school I could play all the instruments. I still don’t consider Bach to be the genius everyone says he was. He was a nepo baby with a big purse. His brothers, his family, all musicians of note for prominent figures of society. However, his leaning on his long history of music within the family helped polish his work as structured which helped sell it. Now, Jean-Babtiste Lully was a character…
If you don't like it, that's fine, I won't argue over taste. But your other descriptions of Bach's life deserve to be fact-checked.
> He was a nepo baby with a big purse. His brothers, his family, all musicians of note for prominent figures of society. However, his leaning on his long history of music within the family helped polish his work as structured which helped sell it.
This interpretation is not particularly historically accurate. Let's investigate:
> He was a nepo baby with a big purse.
Musicians of the baroque era weren't particularly wealthy or notable. Musical fame wouldn't come until the Classical era. And yes, music was his family trade, but that's how most trades went in that time. His parents both died before he turned ten, so he was mostly raised by his older brother. By all accounts they were not wealthy. So I think the term "nepo baby" is misleading, and "and "with a big purse" is simply incorrect.
> His brothers, his family, all musicians of note for prominent figures of society.
This is highly overexaggerated. JS Bach had two brothers who survived childhood, and neither was particularly "prominent." Most of his "notable family" were his children, especially CPE Bach.
> However, his leaning on his long history of music within the family helped polish his work as structured which helped sell it.
Bach's career was one of slow and steady growth. It doesn't appear that he leaned on his connections or family name much.
Bach did get some widespread acclaim by the end of his life, but mostly as an organist, not as a composer. His compositions were mostly discarded and ignored for a whole century until Felix Mendelssohn revived interest in his compositions. The cello suites, for example, were lost for nearly two hundred years, and only re-discovered in the 1920's.
He was known as an organist until the 18th century when someone decided to lump him in with the greats. His works were polished. Yes, he dedicated his life to music - but that’s also where his tenure started. Baroque style borrowing from others and making “commercial” music of his day. He was a nepo baby by our standards. His older brother that raised him wasn’t a Duke, but wasn’t poor either. He went to the best schools. They all borrowed from each other in this age.
He wasn't so "commercial" because he was doing more complex and countrapuntal music after it was falling out of fashion, and he never did an opera, which was all the rage.
His father had lots of children, 4 of which became musicians, of which JSB was the last child, the baby. Barbara Margaretha tried to take the family purse (having already been twice widowed). JSB was “orphaned” but his older brothers were adults. Let’s be real.
At the time, many people. Death stalked the land, children were lucky to reach adulthood, women were lucky to survive childbirth, and almost everyone experienced grief and bereavement.
It's all in his music - the manic passion of trying to master a craft against that background, a burning faith in a better future, against constant reminders of the horrors of the present.
It's not just four part counterpoint. There's a lot more going on.
Sure! When I think of why I love Bach, I often think of works where he demonstrates an ability to express often conflicting emotions at the same time. For example, in St. Mathew’s Passion, there’s a famous piece entitled “Mache Dich, Mein Herze” — it’s sung at a part where the followers of Christ are laying his body to rest, and somehow merges genuine despair with hope, representing the promise of resurrection. I think his ability to represent despair and hope at the same time is pretty extraordinary.
Karl Richter’s version is my personal favorite but there’s lots of different recordings. IMO Bach’s St Matthew Passion is the best piece of musical art, maybe art in general too idk.
There were a lot of these components in middle Germany at that time. Basically every reigning dynasty employed one, and there were a lot of those. They aren't famous now, but Bach wasn't famous at that time either. That he is famous now, is due to Mendelsohn.
Jan Dismas Zelenka wrote for the Saxon king, and many of his works were never released as a result.
Then, they burnt to ashes in 1945. The only extant copies were caught in the bombing of Dresden. We tend to think of "lost works" as something that happened in Antiquity. Nope.
I think this is the real long-term harm the Nazis have done. Sure killing and murdering people is very bad, but after a century it amounts all to the same. What you can't bring back is the whole culture they destroyed.
This is also one aspect why they absolutely hated the Jews. The Jewish culture emphasizes education and Christian were forbidden to take interests in the middle ages. So the Jews became the wealthy educated elite. They were the substrate for the German culture. So in some sense that hatred against the Jews was hatred against the educated and "the establishment".
Also the cities and cultures a lot of famous people, like philosophers and also later statesmen, essentially the countries elite, came from is now destroyed and doesn't belong to Germany any more.
The city where I live, Ostrava, had mostly Jewish intelligentsia, as it was otherwise an industrial city with most people working in mines and steel mills.
I don't think it ever came back, intellectually, from the Holocaust.
I'm not the GP but I can recommend Bach's Partita in D minor, said to have been composed after returning from travel to find that his wife had died and been buried in his absence.
Brahms said of it: "On one stave, for a small instrument, the man writes a whole world of the deepest thoughts and most powerful feelings. If I imagined that I could have created, even conceived the piece, I am quite certain that the excess of excitement and earth-shattering experience would have driven me out of my mind."
You should listen to Hilary Hahn's renditions of Bach's partitas and sonatas. She brings out the subtleties of Bach's composing beautifully, and the purity of his music is easy to appreciate in these solo pieces.
Don't think so. Her recordings of his violin concerts on the other hand are able to clearly show his genius due to the more complex orchestration and interplay between the different instruments.
I mean, I enjoy them too. Good mention. I had a feeling the parent wasn't a fan of the baroque counterpoint, and the violin concertos feature more of that.
I would highly recommend listening to any of his fugues. The great thing about them is you do not need any training to appreciate the complexity. All you need to do is listen to the starting melody.
That melody will repeat itself again and again, if you listen closely. It will harmonize with itself as more voices are added. It will be modulated into different keys and durations.
In a way, you can kind of think of Bach as the first electronic musician, in the sense that his works consist of "discrete tracks" that get layered on to each other. I'm sure there are youtube videos out there that demonstrate this visually.
which I think stands up just fine against pretty much any other classical piece baroque or not.
Personally I have a very big soft spot for his organ works, as I play (badly) some organ myself, and among those I don't see the trio sonatas recommended nearly often enough (here is a live recital of all of them, which is super impressive)
among those I probably enjoy the most the vivace of BWV 530. Other favorite pieces are the passacaglia and fugue https://www.youtube.com/watch?v=nVoFLM_BDgs the toccata adagio and fugue in C major https://www.youtube.com/watch?v=Klh9GiWMc9U (the adagio especially is super nice), but there's so many. Among organists I often come back to Helmut Walcha, and am always amazed at how he was able to learn everything just by listening, him being blind.
you can play it with an ensemble of random instruments.
you can play a single voicing all by itself.
all of it screams "musical". which, if you do play say, Tuba, or one of the larger instruments, is a godsend, as most of your lines in other pieces will bore you to death.
Nice to see the Zenph recording get some love. It's such a fascinating process they had to do. It's way better than the original Gould recordings with all his singing along.
> But one wonders what he could have made without those constraints.
I had a friend that said if Mozart/Bach/et al had access to modern music production equipment, they'd all write psytrance. But it is just another example of "take great talent from long ago and put them in modern day" comparisons.
This is only scratching the surface but I will present one of his most famous pieces to people who might ask why something like this is said. Keep in mind this was written 300 years ago. That's 300. fucking. years. ago. Think about how dated something from the 80s might sound. How modern does this sound? How completely universal is it's beauty? To me, this could have been written today and still sound fresh and beautiful.
This piece is still deeply moving despite centuries of tastes changing. This is only barely scratching the surface of Bach. As a musician, when I listen to other great musicians speak, they all speak about Bach as the best. Of course that's subjective, and there are no 'wrong' answers on who is your favorite, but when the feeling is so nearly unanimous amount people who are often, frankly, contrarian and counter culture it says something.
To be honest, I find most of these inconsistencies to be inconsequential for enjoying the film. The ones that really get to me though are the dramatic overestimates on the devastation caused by Chernobyl, and the effects of the radiation itself. Most of the effect of the film comes from this belief that the radiation really is that dangerous. When you know it isn't, it takes quite a bit away from the premise.
I just don't see how the US (possibly even the world) isn't heading for a colossal recession within the next year. The job market is gone. Everything is a recession indicator, we're basically in one right now. Once the AI hype dies, possibly even at the next Nvidia meeting, there's very little left.
There are many big events that could quickly solve the problem. First and foremost: Russia rejoining the global oil market by resolving the Russia-Ukraine war in some way. That would provide enormous impetus.
I think it really is important to stress that these are correlations only. Something I've noticed: those with poor attention spans or generally low engagement with deeper material tend to be attracted to SFV. Likewise, those in a state of depression or have ADHD can easily get into the quick satisfaction coming from SFV. It may exacerbate existing issues, but not necessarily be the cause.
Is it important to stress that though? This feels like a personal responsibility argument while also acknowledging that it disproportionately affects people who don't have a ton of control over their response to it.
I think that this is an opinion. For instance, one might think online gambling should be illegal, regardless of whether it causes addictive behavior or exacerbates some genetic predisposition to it.
LLMs are severely overhyped, have many problems, and I don't want them in my face anymore than the average person. But we're not in 2023 anymore. These kinds of comments just come off ignorant.
I dunno, I'm not fully anti-LLM, but almost every interaction I have with an LLM-augmented system still at some point involves it confidently asserting plainly false things, and I don't think the parent is that far off base.
Agreed, some days I code for 4-6 hours with agentic tools but 2025 or not I still can't stomach using any of the big three LLMs for all but the most superficial research questions (and I currently pay/get access to all three paid chatbots).
Even if they were right 9/10 (which is far from certain depending on the topic) and save me a minute or two compared to Google + skim/read-ing a couple websites, it's completely overshadowed by the 1/10 time they calmly and confidently lie about whether tool X supports feature Y and send me on a wild goose chase looking through docs for something that simply does not exist.
In my personal experience the most consistently unreliable questions are those that would be most directly useful for my work, and for my interests/hobbies I'd rather read a quality source myself. Because, well, I enjoy reading! So the value proposition for "LLM as Google/forum/Wikipedia replacement" is very, very weak for me.
There are two types of LLM defender; those who claim that it’ll be non-shit soon, just keep believing, and those who claim that it is already non-shit and the complainer is just stuck in year-1 where year is the current year.
Given that this has now been going on for a few years, both are wearing thin.
Like, I’m sorry, but the current crop of bullshit generators are not good. They’re just not. I’m not even convinced they’re improving at this point; if anything the output has become more stylistically offputting, and they’re still just as open to spouting complete nonsense.
This is an incredible model. But once again, we find an announcement for a new AI model with highly misleading graphs. That SA-Co Gold graph is particularly bad. Looks like I have another bad graph example for my introductory stats course...
I'm thinking this might have broader use than artistic appeal. From what I've heard, knot generation is a young but increasingly important topic in knot theory, since it can be used to generate data to train ML models on, and subsequently (hopefully) discover new algorithms for knot classification. See https://www.nature.com/articles/s41586-021-04086-x for example.
The point is that we need some way to deal with objects that are inherently infinite-dimensional.
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