So it sounds as if I have proved that no-one needs analysis--or theory either! Not quite true. Ordinary listeners can enjoy music quite well without knowing the theory behind it and most performers can play music very well without knowing much about theory. There are also some interesting philosophical problems with the relationship between theory and composition that I talked about in this post.
But theory and analysis are still very useful and valuable activities. You can skate around on the surface of music forever without delving into it, but that does not mean that a deeper understanding is not desirable. The French horn player I was talking about before had to play individual lines in an orchestral texture and also oversee the French horn section, but the deeper understanding of the whole musical structure was more the responsibility of the conductor. If you are a solo pianist or guitarist, with responsibility for the whole musical texture, you really need to know more about it than just where to put your fingers and where to play loud and soft. So pianists (and even the occasional guitarist) often study the history and theory of music in much greater depth.
The music school I graduated from had courses in harmony and counterpoint in the first two years and in the last most students had a choice of 19th Century Theory and Analysis or 20th Century Theory and Analysis. Honors students had to take both. In counterpoint you study first modal counterpoint which covers music up to around 1600, then tonal counterpoint from then until 1750 (the death of Bach). Harmony, which means tonal harmony, is basically about music from 1700 to 1900, the so-called "common practice" period when nearly every composer and musician worked within the same understanding of the tonal structure of music. Late in the 19th century this structure got so burdened with deviations from the basic tonal structure that some composers decided to invent a new way of structuring music. The most important of these was Arnold Schoenberg who, in the 1920s, developed a way of composing, called 'serialism' that avoided tonal structure entirely. Since then music composition has diverged into a thousand different camps. Every composer is more or less expected to invent their own musical 'language'. There is no longer such a thing as a "common practice" which makes things very dicey indeed for audiences who usually have no idea what they are hearing or what to listen for.
Partly as a result of this, some composers have resorted to all sorts of non-musical means to reach out to the audience. These include elements of theater (even in chamber music), references to mysticism, inclusion of elements from folklore, world music, or popular music, writing music with text such as opera and other means.
That little history of music theory starts to answer the question of my title: how to analyze music. You should start by taking into account when the music was written. For example, here is the opening of a simple two-voice bicinium by Orlande de Lassus. There is no text and no instruments are specified so you could play it on any instrument or combination that can play the notes.
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You do need to know the basics of notation before you can begin such as what the clefs indicate. A set of five lines is called a staff and the top staff, the cantus, has a treble clef. This is a very stylized letter G that curls around the second line from the bottom telling you that this note is 'G'. The lower staff has what is called a 'C' clef. It is not really a stylized 'C', but the center of the bracket, pointing to the middle line, tells you that this note is 'C'. The other thing at the beginning is the flat sign which, on both staves, indicates that every time there is a 'B' note you play B flat instead. Now, since Lassus' dates are 1532 - 1594 we know that this falls in the counterpoint stage of music history. What does this mean? We should look at this music melodically and see if we can see relationships between the voices. Bearing in mind that the clefs are different, what we see here is that the Cantus has exactly the same notes as the first two measures of the Altus. The Altus starts with D, Bb, A then a scale from G to F and the Cantus has exactly the same notes (an octave higher). This is what is known as imitative counterpoint. The music is written so that the second voice comes in after the first voice, but with the same notes and the melody is written so that it will go together with itself harmoniously. There are lots of refinements and complications but if you understand this, then you understand the fundamentals of how music was structured in the 16th century.
Now, that wasn't so hard, was it? Here is a bicinium by Lassus played on two viols. It is not the same one as the example above, but it is structured the same way: the second voice imitates the first one measure later:
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