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When I see classical sheet music I don't see the chords named, were chords not used back then? And if they were, do the notes that are under a bass clef generally outline the shapes of chords?
I only see chords when I look up a classical song on YouTube and type "harmonic analysis" along with it, for example Harmonic Analysis: Beethoven Moonlight Sonata
Disclaimer: I omit a bunch of hedging about what I'm referring to when I say "classical music" below; think Bach or Brahms
It sounds like you might be coming from a pop/rock background and are familiar with, say, guitar tabs as a way of notating the structure of a song. Other answers have pointed out that:
Harmony and harmonic analysis is a really big subject but there are a couple things we can say about popular music, harmony, and notation that might be helpful:
First, the understanding of what a chord really is and what it does is really pretty different between popular music and classical music (think Brahms, say). Some of this is a little tangential to your question:
Getting back more directly to your question: the way that pop/rock guitarists label chords is in a couple important ways different from the way I would write chords when analyzing a piece by Brahms (say):
Guitarists (and also jazz musicians, where I think a lot of this language comes from) give labels to "chords" which in classical music we don't think of as chords. e.g. "sus" chords like a "C sus4" or whatever; in classical harmonic analysis we don't think of this as a chord; we'd call it a "C major chord" and then talk about a suspension, that is a non-chord tone.
In classical music these suspended notes tend to get resolved; they create tension by being out of place for a moment. The implication is that the listener hears a C major chord, and hears that there's a dissonance. In contrast, in pop music these chords don't need to be passing or resolve, they can just sit there and be an interesting color, effect, or maybe play with the vocal line in an interesting way.
There is a lot of language in classical music for different sorts of non-chord tones, e.g. passing tones, appoggiaturas, etc. If you analyze Bach's 4-part chorales you can see all of these. He'll even use idioms that involve a cascade of suspensions which pass through many chords in complex ways and where at no point can you take a vertical segment and find a pure "unadulterated" chord.
The point I'm trying to emphasize is that a harmonic analysis of a piece of classical music (that is how we'd label it with chords) is often about the tension between what we expect to be hearing as a listener and what notes are being heard, and generally ignore how the chord is voiced or omit some notes that are sounding. Sometimes there are multiple valid interpretations.
I'm sorry this is getting long. Related important concept that should be higher up: counterpoint, the way that multiple melody lines interact to suggest some harmonic structure, i.e. to form chords and do playful things within that chord structure (non-chord tones, above). This is what you see distilled in Bach's chorales say.
Other eras and types of pieces are much less heavy on counterpoint, e.g. a simple Mozart piano piece or a Schubert song accompaniment might have a left hand that strongly outlines the chord structure of the piece, playing arpeggios or the I and V of the chord. It sounds like you're most familiar with that type of writing (and that's what's common in pop music).
In the vast majority of classical music, the player is tasked with playing exactly the notes that the composer wrote. It's not very important for the player to understand the theory behind the piece, and a great number of classical players know little to no theory (at least until they reach conservatory, if they go that route) and don't suffer for it.
Let's consider the well known Mozart piano sonata in C:
And let's consider adding chord symbols to it:
Frankly, these chord symbols add nothing. So what if I know that the first measure is C? I still have to play the notes that Mozart wrote. The only thing they do is distract me.
Turn this around and ask "why are chord names, rather than the staff notations of the notes in the chords, so frequently used for written popular music and jazz but not for classical music?"
A typical amateur, around the campfire guitar player, for example, does not read music, but if you ask that person to play a "C chord" he or she knows where to put their fingers. Chord names are the most concise way of communicating the minimum information required.
The professional jazz musician, possibly using a fake book, knows many different ways to play any of the chords indicated by the cord name and how to use that knowledge to improvise on that harmonic structure. To this person the chords given are a suggestion, not something cast in stone.
Classical music is/was addressed to an population of musicians who are able to fluently read musical notation. It is also generally written with the idea that the exact melodic lines, harmony and instrumentation that the composer envisioned will be used. It is not 100% "cast in stone", since all music is interpreted, but the composer had a very detailed concept and requires a more detailed means of communicating it.
Chords were obviously used; one needs to only to look and listen to see and hear that.
But in addition to Matt's great answer, another reason is that such chord labeling doesn't always fit in the music of the past. There are excerpts of Bach's polyphonic music especially that are not at all conducive to such reductive analysis.
Furthermore, a lot of popular music today is chord-based, meaning that it's much more conducive to this type of labeling.
(If there's popular music written today that is as polyphonic as Bach's, I'd be curious to see how they chose to notate it.)
The way to notate chords and inversions in Baroque times was as figured bass. Since that notation is in reference to the bass line and since the harmonic content and the inversions tend to change faster than current-day chord changes and are notated more specifically regarding the intended inversion, interpretation of the half-improvised accompaniment was more suited to keyboard instruments rather than guitars, today's primary target of chord notation.
Figured bass more or less went down with Bach since he tended to spell out everything explicitly rather than relying on the improvisation skills of the keyboard players.
You've received some great answers already, but I wanted to point out more specifically how only looking at chords cannot sufficiently describe how to perform pieces.
Assuming that "classical" music refers to all music in the common practice period as well as the 20th century, chords don't suffice for describing music entirely. One can find many examples of this in the 20th century. A great example of this is Ligeti's Nouvelles Aventures.
Henry Cowell, in Dynamic Motion, calls for playing chords which use the entire arm of a pianist (around 33 seconds):
As a much more extreme example, what if there aren't any pitches whatsoever? One can also look at Reich's Clapping Music:
I'm surprised no one has mentioned tracker modules and tracker music yet.
https://en.wikipedia.org/wiki/Music_tracker
These types of programs were used to write the great TV series and classic video game tracks of the 1990's.
If you think the classical composers like Bach and Mozart were verbose in specifying every note (instead of oversimplified chords), tracker modules take this paradigm to a new level. A module defines a set of samples, and then the track itself invokes the samples when they are needed. Each sample could be associated with effects encoded in separate bytes as well, effectively dictating interpretation as well. The only problem was that different tracker module programs used different effect codes, which means that a module written for a specific program might be partially ruined if a different interpreter was used.
The chord paradigm fundamentally limits the composer into a small subset of possible music. From what I'm reading, the 2 main appeals of chord-based approaches are simplicity of composing and wide-open range of interpretation, both of which lend themselves readily to modern performing arts. This probably also accounts for why I've been seeing increasing complaints (from musicians) in various news articles over the years that modern music is becoming too unoriginal. In short, the chord approach has a low skill curve - you can make popular music quickly and easily but the "ceiling" on the music you can produce is rather low compared to classical or tracker-based approaches.
For those of you who say that music that doesn't fit in a chord framework is old-fashioned or excessively specified, I recommend listening to some of the 1990's classics by famous tracker module artists such as Alexander Brandon, Elwood, and Purple Motion. Something you will see a lot with the best of the tracker module artists is how they weave together at least 4 audio channels (percussion, bass, and 2 main instruments) - which arguably draws the line between modern "average" (percussion/bass/1 main instrument or voice) and "really good" composing. Weaving several main instruments or themes together was also done in the classical era (for example: 2-part inventions and 3-part sinfonias).
In the old days where you were limited to 4 audio channels (an artifact of the Commodore Amiga computer era), you couldn't even do a full chord parallel across audio channels because you'd eat 2 or 3 of them and leave no room for much else. Artists had to encode full chords into samples and invoke those. An example of this can be found in the tracker module "Space Debris" by an artist calling himself "Captain". If you watch the instrumentation in the Impulse Tracker version, you can see instruments named Chord1 Major and Chord1 Minor.
I'm curious to see what the classical composers would have come up with if they had access to modern digital audio workbench (DAW) or even tracker module software. I write my own tracker-style music in a modern DAW and it's hard enough composing with the complex 4-channel paradigm above. Writing the same on a character-cell tracker module program is definitely harder, and it takes even more skill to compose it on paper only with no computer.
