Analysis in music is the practice, skill or process of examining how the music is put together. The needed skills are usually the ability to read musical notation and an understanding of harmony. The typical kind of analysis one does as a music undergraduate is to do harmonic analysis of things like Bach chorales and Mozart sonata movements. What you do is label each harmony with a Roman numeral that indicates its function. You also use superscript Arabic numerals to indicate the inversion of the chord. Let me do a phrase from a Bach chorale for you to show how it works.
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UPDATE: Just noticed a mistake. The chord on the second beat of measure 3 is not a I6, of course, but a I 4/2. But it is probably better to see the bass note as a passing note. It gets a bit complicated when you have simultaneous different passing notes. The E is on its way to the F#, the G is held over and the B is passing. So the "real" chord on that beat, shorn of decoration, is ACF#, a viiº6.
First, a couple of apologies. Finale, the music software I use, is perfectly capable of doing all the funny little symbols you use for harmonic analysis. But, alas, I have a Spanish language keyboard and that particular font does not quite work with it. So, for example, all the little Arabic numerals, which are supposed to be superscripts, come out subscripts. And I can't do either the symbol for diminished chords (all the vii chords in the example should actually be viiº chords) nor the figures for seventh chords. For example, the second beat of measure six should actually be a ii6/5 chord, but I can't get it to show. So bear with these minor omissions!
How harmonic analysis works is you first determine the key. In music from around 1600 until nearly the end of the 19th century, this is pretty easy. The piece above is in G major. With one sharp in the key signature your choices are only that or E minor. But it is not E minor because in that case there would be some D#s here and there. So, G major. This means that the triad on G, spelled GBD, is labeled "I". If the B is in the bass, then it becomes I6 to indicate first inversion. If we see the notes ACE, as in the second beat of measure six, then this is a chord built on the second degree of the scale. As it will always be a minor chord, we use lower case: ii. Again, if the C of the chord is in the bass, we indicate first inversion with a superscript "6". A chord with an added seventh in first inversion is indicated with two figures stacked 6/5 (which I wasn't able to do, sorry!). Why these numbers? If you look carefully, you will see that they simply record the intervals above the bass note. For this reason they are called "figured bass" numerals.
As you can see, harmonic analysis is a fairly mechanical process involving identifying the chords and their inversions. There can be a lot of subtleties to it, such as passing notes, suspensions and so forth. There are three basic harmonic functions: the tonic, the pre-dominant and the dominant. Harmonic analysis helps to identify these functions. You will find in all the music of the Classical era, and much of the Baroque and 19th century as well, that harmonic function is pretty clear. Every piece will end (and usually begin) with a tonic chord. Every phrase will have some sort of cadence, either full (as in the end of the above example) or half (as in measure 4 in the example) or deceptive (meaning that the cadence goes V to vi).
Phrase structure is another thing you can analyze and that is more complex than just harmonic analysis. You need harmonic analysis to analyze phrase structure. If you look at the above example, it is a phrase in two parts. The first is four measures ending in a half cadence. Then there are three measures ending with a full cadence. This makes the phrase irregular, because the standard would be eight measures. The second half is compressed. Composers often vary their phrases by both compression and expansion which is why we can find thirteen measure phrases sometimes.
On the next level we come to the overall form of the piece. Reading Wikipedia articles on things like Sibelius symphonies you might think that formal analysis is pretty much hit or miss, because they always say stuff like this theorist thinks it is a sonata, the other theorist thinks it is a rondo and the other guy just throws up his hands and says, "Dude, Sibelius was really jammin' on this one!" Kidding about the last one. But you do get the impression that formal analysis is pretty subjective, little more than a shot in the dark. I think this is mistaken. Formal analysis works pretty well in repertoire like Haydn, Mozart and Beethoven. It works less well with Sibelius or Shostakovich because I think that we haven't quite figured out what they were doing. It took a long time to figure out what Haydn, Mozart and Beethoven were doing and it is early days yet with 20th century composers.
Aesthetics in music is an entirely different kind of procedure and to understand the difference it helps to be familiar with what has been known for a couple of centuries as the "Is-Ought Problem" in philosophy. The basic idea, as laid out by the Scottish philosopher David Hume, is that there are two different kinds of statements: those dealing with relations between ideas (1+1=2) and those of matters of fact (today is the 29th of March). Unfortunately, this leaves a whole other category of statements, called "prescriptive" or "normative" statements, floating around without foundation, at least according to Hume. The reason being that you can't get from an "is" to an "ought" logically. Thus comes one of the fundamental problems in ethics. Aesthetics has very much the same problem, which is why some people see ethics and aesthetics as related. Incidentally, I wrote several posts about aesthetics and used David Hume as my starting point. Here is the first one.
Hume writes about aesthetics as an empiricist. One develops a sense of taste through experience and learns to distinguish high quality from low quality. Yes, there are differences in aesthetic quality that, while they are experienced subjectively, certainly have objective status. Bertrand Russell answered regarding the same problem of relativism in aesthetics by saying "I cannot see how to refute the arguments for the subjectivity of ethical values, but I find myself incapable of believing that all that is wrong with wanton cruelty is that I don't like it." In the aesthetics of music you could say that "I cannot see how to prove the existence of objective aesthetic value, but I am incapable of believing that the only difference in quality between Bach and Justin Bieber is that I like one and don't like the other."
Analysis and Aesthetics
So analysis is basically an objective process though there can be disagreements over details. In the case of modern composers, the basic fundamentals are still being fought over among theorists, which is why we see so much disagreement. But there is general agreement over, for example, the analysis of a Haydn string quartet because, in the last two hundred years, theoretical understanding of Classical style has become worked out to the point of being, more or less, objective truth. The objective truth of, say, the formal structure of a Sibelius symphony, is still being figured out. We can all look at the first measure and see, yes, that's an A quarter note, but the significance or function of it in the overall form is not so clear yet.
Aesthetics, on the other hand, has to do with good and bad. It, like ethics, is prescriptive or normative, that is, it involves a judgement as to quality. But there is no clear and easy route, as Hume says, from analysis to aesthetics. Analysis is about matters of fact and the relations between ideas. It deals with the matter of fact of things like G major and cadences. Analysis uses certain definitions as to what a key is or what meter is. These are true by definition. But the problem of trying to get from analysis to aesthetics is just like the problem of getting from "is" to "ought". Just because you can do something is no guarantee that you should do something. I can do something to harm a completely innocent person, like punch them in the nose. But I should not do so! Similarly, a composer could, perhaps, write a piece using every single possibility of invertible counterpoint. But this does not mean he should. It is the easiest thing in the world to write a piece of dry, boring, but correct counterpoint that no-one would want to listen to.
An aesthetically good piece of music is one that is charming, delightful, profound, moving, expressive, dynamic, energetic, languid, doleful, effervescent or a thousand other adjectives. Sometimes it can be several of these at once. Just exactly how a piece of music can do this, while another piece is dreary, boring, irritating, annoying, bullying and tiresome is the magic of music (and every composer's challenge). But I think that it is pretty easy to show, by demonstration, that some pieces have a high aesthetic quality and others have a low aesthetic quality. How we come to aesthetic judgements on music is a subtle process involving knowledge and experience and listening skills.
Well, that was long and I hope, not too tiresome! Now let's listen to some music. Here is Mitsuko Uchida playing the Piano Concerto No. 17 in G major by Mozart: