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Everywhere I look, every resource I read, says that time signatures determine the "feel" and meter of the music. And then the person giving the explanation will show a piece of music as an example of a type of some meter (usually a piece of music that doesn't have readily accessible sheet music); but then someone will argue that it's actually so-and-so a meter, and then a big internet argument ensues.

The time signature is supposed to determine the "feel." But I need hardly say that two different pieces with the same time signature can have very different "feels." A lot of Shostakovitch "feels" unmetered (or maybe I have no ear for music).

People will insist that certain beats will be stronger with certain time signatures. Yet I can find many examples in music where a different beat gets the accent, and also other theories of music that disagree on which beat should be accented.

The first movement of Beethoven's Fifth Piano Concerto is common time. Yet the grand piano solo at the beginning is shoved into one measure, regardless of the time signature, and nothing bad seems to happen.

opening measure

I'm currently learning Beethoven's Eighth Piano Sonata, and till recently I misread it and thought it was in common time. Turns out it's not common time (4/4) but actually cut time (2/2). What's the difference? You still have four quarter notes per measure. How would it have felt different if it had been written in common time instead of cut?

And then you get Rachmaninoff, who will do weird things like changing the time signature for one measure only. In Op. 23 No. 5, measure 16 changes to 2/4 time for one measure, and then back to common time. In Op. 32 No. 12, measure 13 changes to 6/8 time for one measure, and then back to 12/8 for the rest of the piece (I'd post pictures, but the website won't let me).

I cannot for the life of me figure out what purpose these time signature changes serve (and neither can my piano teacher).

The only purpose of measures I can think of is if a conductor wants to say: "All right, let's take it from measure 42!"

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There's an important distinction between meter, which is a musical concept, and time signatures and measures, which are notation concepts. Notation must always be in service of the music, and to the reader. There are an infinite number of ways to notate a piece of music, but only a small subset of those ways will help the reader not want to gouge their eyes out.

I don't think you are arguing that meter doesn't exist, but as long as it does, notation needs some way of communicating that to the reader. Measure numbers and time signatures are the tools embedded in the western classical notation tradition that provides this. In some cases, the differences are overt readability and economy of INK (you're going to waste a lot of time writing flags by notating a piece of music that should be in 2/4 as 2/32), but in many cases, the differences can be subtle, and even rely on the cultural context of the existing body of notated music. (In my opinion, this often accounts for the difference between notating in 4/4 and 2/2.)

A couple of days ago, a jazz band I was in tried playing a piece of music notated in 7/8+3/4. Each measure contains 13 eighth notes. This time signature tells us what we can expect from the rest of the band if, for example, we are counting rests and need to know when to come in, or if we get off and need to get back with the rest of the band. It also tells us whether to interpret a note as a downbeat or a pickup, based on where in the measure it is located. Further examination of rest grouping and other notational features tells us that the 7/8 part of the measure is often grouped as 2+2+3, but sometimes it is grouped as 3+2+2 — information NOT contained within the time signature, but still relevant.

We tried playing the same piece of music notated as 4/4, 4/4, 5/4 (26 eighth notes) — wherein one sequence of those measure equaled two measures of the above. The "music" itself was identical, but all of a sudden, the notation was completely discongruous, and it was IMPOSSIBLE to understand where even the BEAT was relative to the notation. Sure, you can theoretically execute this notation in isolation much more easily than counting sevens, but when playing in an ensemble, every musician MUST be on the "same page" and be able to feel the beats and groupings as a single musical apparatus. Metric notation is HUGELY important in making that work.

And don't discount the importance of rehearsal efficiency. If we didn't care about rehearsals, we wouldn't need notation in the first place.

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Everywhere I look, every resource I read, says that time signatures determine the "feel" and meter of the music.

Either these materials are wrong, or you're interpreting them incorrectly. The time signature doesn't determine the meter, the music itself does that. A time signature is a device we use when notating music that makes it easy to see where in the meter each note falls. The composer/engraver should choose the time signature that matches up with the music in the cleanest way.

People will insist that certain beats will be stronger with certain time signatures.

This is also incorrect. Different genres of music accent beats differently, but we use the same set of time signatures to notate them out of simplicity. Again, time signatures do not determine anything about the music.

The first movement of Beethoven's Fifth Piano Concerto is common time. Yet the grand piano solo at the beginning is shoved into one measure, regardless of the time signature, and nothing bad seems to happen.

What you have here is a cadenza, a portion of the music that is more free-form, and which therefore doesn't fit into our notational systems as well.

Turns out it's not common time (4/4) but actually cut time (2/2). What's the difference?

Not much. 4/4 vs. 2/2 in particular is a fuzzy distinction for exactly the reasons you listed, and much of the time the choice between them is more about history and convention. Marches are almost always in cut time, for example.

Also note that you could always double or halve all of the note durations and adjust the bottom of the time signature. For example, take a piece notated in 4/4, cut everything in half (quarters turn into eighths, etc.), and change the time signature to 4/8. Basically the same thing, but less conventional. Musicians would look at it and probably assume that it was supposed to be played very fast, which may or may not be correct.

It's also very common in musical theater for the arranger to just write 4/4 (probably because it was the default in their software) when really the music would be better served to be notated in 2/2 because of its speed.

And then you get Rachmaninoff, who will do weird things like change the time signature for one measure only.

This is because the music is too complicated, and its rhythm changing so much that it doesn't fit neatly into one time signature. These one measure time changes are there to account for a rhythmic hiccup or extension.

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If you cherry pick examples of changing meters, additive meters, syncopation, etc, then of course you will come to the conclusion that time signatures do not convey the pulse of the music. But then you are looking at only half the picture. You need to look at both metrical and unmetrical music to really understand the point.

Start with examples that confirm meter. Probably the easiest way to do that is with Baroque dance suites, because one of the important aspects of the dance suite is for each dance of the suite to present contrasting meters. The allemande or gavotte will demonstrate duple meters of either 4/4 or 2/4. The sarabande and courante will demonstrate triple meter 3/4 (or 3/2), and the gigue will demonstrate compound 6/8 meter. Those are the standard types in most Baroque dance suites. But keep in mind the time signatures can vary. Sometimes old notation will simply give "3" for triple meter, or a gigue may be in compound 12/8 rather than 6/8, etc. etc.

Try Handel's suites as good examples. HWV 437 is the one with the famous sarabande.

After you spend some time with those example and get a feel for regular, metrical music and various common meters, then look at examples where rhythm and accent work contrary to meter. You can then analyze why the music isn't metrical.

Probably the most obvious reason some music isn't regular and metrical is simply because not all music is regular and metrical. Lot's of real folk music involves meter changes when it is notated. There are also lots of examples of rock music where a measure has a beat added or omitted. In many cases I think these "irregularities" are not perceptible unless you carefully count or notate the music. If the extra/omitted beats are at the end of phrases, they just feel like pauses of varying length ...after phrases that were otherwise regular.

Syncopation is the other obvious musical device that conflicts with meter. Something first must establish the sense of meter, then some rhythm is syncopated. Meaning the rhythm shifts the accent to notes other than the strong metrical notes. Beat one is the strongest metrical note. Accenting "up strokes" is another syncopated rhythm. Syncopation adds a lot of excitement to music.

Also, there is a lot of unmetered music. Medieval chant is a well known example. Often chant is described as having a "floating" or "drifting" feel. In part that is because it isn't metrical. A metaphor might help with understanding. If walking with your feet on the ground is the basic point of reference to a metrical experience, then floating would disconnect us from that metrical sense. So, unmetered chant isn't walking with its feet on the ground - so to speak - rather it floats.

You might look up "tyranny of the bar line" for more discussion about the stylistic associations of meter.

Basically, music can be metered or unmetered. It's a choice depending on musical intensions. Certainly if you want to dance, metrical music is typical. But also in styles where playing with the listener's expectations - surprise chord changes, recapitulation, etc. - meter will help set up those expectation, because meter provides fixed rhythmic points for certain expected changes.

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