: What is the connection between time-signature, beat, and tempo? What is the connection between time-signature, beat, and tempo? Mainly time-signature and tempo. Tempo is beats-per-minute I know, but does the time-signature

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Time signatures are primarily for notational purposes. Beat, tempo, and meter all describe a certain thing about the music, but the time signature is just how that's codified when it's written down.

As you know, Tempo is the frequency of the beat, and Beats are a kind of rhythmic emphasis that happens at regular intervals in most music. Meter is an important term--that tells us how each beat is subdivided (i.e. in two or in three) and how to group beats into larger chunks, also at regular intervals.

Time signatures, plus information about tempo, tells us how to get all of that information from notes written on the page. A quarter note alone has very little meaning until we contextualize it in terms of what note value the beat is assigned to and what the frequency of beats is.

You are correct in noticing that there is overlap between tempo markings and time signatures. That is because, while there are some conventions for how they usually relate, these conventions can be broken.

Consider 4/4 time with a tempo marking of q = 60 (bpm). This one is simple, there are sixty quarter notes per minute, and four quarter notes per measure. But what if the tempo marking was h (half note) = 60? There's no hard and fast rule that says the tempo marking has to be equivalent to the time signature (and in fact in 6/8, it rarely is), but in that case, there are still four quarter notes to each measure, it's just that the tempo is being given in how many half notes there are per minute.

Tempo is usually given with a note duration. If that duration is missing, then you can usually assume that it is referring to the bottom part of the time signature. But it is fairly rare to just see a number at the top of the page without any context. In the example you give, there is no note value assigned to the beat, so you would assume that it is referring to the quarter note for time signatures of 4/4 and 2/4.

For 6/8, things get a little bit hairy. This time signature always means that there are six eighth notes per measure, but not always does the eighth note get the beat! 6/8 is most commonly used to refer to a duple meter with a triple subdivision. This means that a dotted quarter note would get the beat, and each beat would be divided into three eighth notes. Most American folk songs fit this model (think about how you would write down "Pop Goes the Weasel"), and 6/8 time is used to avoid writing a triplet figure over every subdivided beat. 6/8 tempo markings are usually written as q. (dotted quarter) = x, but they occasionally will be e (eighth note) = x, signifying that you have six beats per measure.

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I wish in all my years studying music in grade school and High School I had this kind of explanation.

I was always given the explanation that 3/4 time means, three quarter notes per bar also known as a measure. And it means that a a quarter note is one beat.

All through my amateur musical career I thought, UH..SO What ? As with most aspiring young musicians I felt stupid for not understanding how time signature effect the sound or rhythm of music.

In fact, strictly speaking technically it doesn't. It's tempo and beat that does.
In itself time signature does not...not all by itself.

I've always suspected, but loath for any music instructor to ever tell me that.
"Time signatures are primarily for notational purposes." Right on!

It technically just states that the numerator designates how many beats per bar aka measure.

The denominator designates the note value that represents one beat.

3/4 time is 3 quarter note beats per bar. Quarter of What ! ? A whole note.
So what ?? How long is a bar/measure in real time? WTF?

What does that have to do with how the music sounds ?
Notes are relative too, to be sure but certain frequencies of sound vibrations correspond to specific notes.

This stuff is part of tradition. Music and in particular written music IS based on traditional convention on how it is represented on paper. But IMHO it is full of subjective and imperfect distillation into written form, developed more than a century ago. Not ALL of the sound is contained in the written sheet music....but most of it!

IMHO time signature baffles me on the rhythm and tempo of a piece of music. Tempo of course is at the discretion of the performer(s) as it should be. But I'm confused at what the time signature has to do with the sound, rhythm or tempo of the piece.

So....that being said, why is Dave Brubeck's unsquared dance (which I love) so interesting and why is that because of it's unusual 7/4 time signature.
The whole of Brubeck's album Time Out is about experimenting with time signatures.
It's all great and different and I'm told it's because of the unusual time signatures but I just don't get how the time signatures translate to my ear.

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Wikipedia entries on Tempo and Meter are informative on this topic.

A short answer is that tempo specifies the speed of playing, meter specifies how a score is logically divided into measures (bars) and beats, and that there are rules about how they interrelate but these rules have a bit of fuzziness:

For simple meters, the lower number of the time signature is the beat value, that corresponds to the BPM of the tempo.
For compound tuple meters (e.g. 6/8) the beat value is typically 3 times the lower number of the time signature, which is the same as the "dotted value of twice". In the case of 6/8, this is three eighths, or a dotted quarter.
If the tempo is noted with an explicit note value (e.g. Tempo: 120 = ?), then this trumps the above rules.
If tempo is specified by name (e.g. Allegro), the precise BPM is less significant than the ballpark BPM, which is fuzzily influenced by the type of piece and its cultural associations, but the above rules still apply.
If the tempo is significantly fast or slow, common sense may be applied that the BPM applies to half/double the above logic.
For complex meters (e.g. 2+2+2+3/8), the above rules presumably apply, but at that point, the musician probably has more context than any rules can provide.

In music, all rules are meant to be broken. For instance, John Cage's 4'33" is traditionally notated as having a tempo of 60 BPM, but is frequently performed at 1/3 this speed.

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What is the connection between time-signature, beat, and tempo? Mainly
time-signature and tempo. Tempo is beats-per-minute I know, but does
the time-signature affect it?

Time Signature tells us how many beats there is in a bar and what each beat consists of.

Tempo is an indication on how fast or slowly the peace is to be played

The beat is the amount of strong pulses a bar has.

2/4 time has two beats of crotchets. 9/8 time has three beats of dotted crotchets. 5/4 time has five groups of crotchets.

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For example, in a 120bpm tempo, are note durations of 4/4 time-signature different from 2/4 time-signature?

If in both cases the quarter note = 120 bpm, then no - the duration is the same.
Time Signatures such as 4/4 are used for notation. They do not describe the tempo, which is a physical (aural) phenomena. Tempo can be represented various ways: with words (adagio, allegro, etc) and a more precise bpm value. Bpm will always define what duration is given as the beat. For example:

quarter note = 120 bpm
half note = 60 bpm

You won't (shouldn't) ever see just "120 bpm". Too many assumptions can be made from this.

Time signatures are not used to convey tempo.
Tempo, is not used to convey time signatures.

Meter was described well by NReilingh.

Hope this helps!