How should I understand time signature?
Now, I am not asking for the general answer given in band class ("The top number tells how many beats are in a measure; the bottom number tells what note gives the beat"). I'm more asking what it is and by what concept is it defined. My thoughts are:
A time signature is the "feel" of the song. For example, in most songs that are 4/4, there is a drum pattern. It starts with the bass, then the snare, then bass again and snare again. It creates a pulse; that pulse is called a "beat". If someone were to cut a song into pieces, which are, in a sense, equal, the way that makes the most sense is by dividing it into this pattern. Since the pattern consists of 4 parts, in this time, the song would have 4 beats per pattern. Other songs, however, have different patterns and those patterns are what define the measure, what gets the beat, and how many beats each measure has.
Is that correct?
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Time signatures are how we define the measures that we use to organize our music. Music is not always rigid - in fact sometimes very fluid. At the risk of getting too zen, your own concept of time relative to the music that you realize externally dictates how you perceive and apply time signatures in your music.
While most of us can agree on ranges of tempi, many of us would disagree how something should be notated. For example, a quick passage in 4/4 can be rewritten as a slower passage in 2/2. Changing our relative time perception changes our aesthetic interpretation, and therefore changes our aural experience.
Time signatures generally accomplish two things: they suggest a pattern of strong and weak pulses (and relative strong and weak pulses), and they define the notation for those pulses. Each "measure" contains one sequence of the pulse-emphasis pattern.
Simple meter
The most basic patterns are two-time (strong weak | strong weak) and three-time (strong weak weak | strong weak weak). A time signature with a 2 or 3 on top, suggests one of these patterns, unless otherwise noted (or notated). The lower number then just tells you which type of note corresponds to this basic pulse. Thus 2/2, 2/4, and 2/8 (or 3/2, 3/4, and 3/8) would all have the same pulse-emphasis pattern, just notated differently.
Compound meter
When you have an upper number that is a multiple of 2 or 3, then you get overlapping pulse-emphasis patterns. Thus in four-time, you have two strong-weak pairs in which the first is stronger than the second.
strong - weak - strong - weak
strong - weak
_____________________________________
strong - weak - semi-strong - weakest
An extensive discussion of this arose in the question Rhythmic/Metric Stress Patterns
Complex meter
When the upper number is not a multiple of two or three, then the pulse-emphasis pattern needs to be clarified. This can be done via the notation itself or via the time signature. Five-time can be notated as groupings of 2 and 3 beats, or 3 and 2 beats, or the time signature could be written as 3+2/X or 2+3/X.
Irregularly pulsed music
In John Adams's Phrygian Gates, the composer wants every note to be played with exactly equal emphasis. The score gives no time signature, and bar lines are only give to help the performer keep track of where they are in the piece.
György Ligeti, in the first of his Études pour piano, "Désordre", wants an irregular pulse pattern, both within each hand and between the two hands. To accomplish this, he uses a combination of accent markings and bar lines that don't align between the left-hand and right-hand staves.
Erik Satie eliminates both time signature and bar lines altogether in his "Gnossienne #1 ". Although the notation of the piece clearly suggests 4/4 time, the absence of bar lines indicates an equality of pulse, or that pulses may be applied at the performers discretion.
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