When two notes are "tied" they are always at the same pitch. It's just a slightly different way of telling you "how long the note lasts.

Two eighth notes tied togeter are played just exactly as if a single quarter note had been written.

An obvious place for using the tie is when the first eighth note is at the end of a measure, and you want the note to last through an eighth note of the following measure. Many tunes also have a "basic beat, so if you would expect to play an eighth note - but it is sustained, the eighth note you "expect" is written explicitly and tied to a following note to tell you to "hold on to it."

By maintaining a "pattern" in the notation, it is much easier to read.

Two notes of different pitch joined by the same "arcish" symbol as the "tie" are a slur, and the meaning is that you go from the first note to the second "without an attack" on the second note. Generally, a short space is played between notes - that isn't notated, so that there is a distinct "stop one note before you start the next." The slur tells you that there shouldn't be a "gap." It does not mean you should "slide the pitch from one note into the next," which should be notated as a "glissando" if that's what the composer really wanted. In practice though, slurs are often played with a slight "bend-into" the second note.

In the time signature, the number on top tells you what how many beats are in a measure, and the note on the bottom tells you what "note value" gets one beat. Technically, it's not a fraction, but you can "rationalize" it as if it was. A tune written in 4-4 time can be played as though in 2-2 time, or even in 1-1. A tune writen in 3-4 time can be played as though in 6-8, and it will sound the same if the pattern of the "beats" is not changed.

By convention certain time signatures imply a "beat," telling you which "beats" of the measure are accented - played a little stronger; although a separate notation is really needed if you want this to be clear.

Tunes in 6-8 are probably most ambiguous. Marches are frequently written in 6-8 time, but always played two beats to a measure, with "three eighth notes" to the beat. (Maybe they should be noted as 1.5-2?) Polkas written as 6-8 sometimes are played 3 beats to the measure and sometimes 2 beats to the measure, depending on the "national tradition" being respected. The key to using the 6 numerator for the notation is that the "beat" to which the tune is played contains 6 "steps" per measure, either three "duplets" or two "triplets."

The tune called Carolan's Argument with His Landlady is a classic example of writing in an "unexpected" time signature so that it sounds like the "usual one." You may, in fact, find it rewritten "like the landlady wanted it" by those not "in" on the joke.

Common time (C) and "cut-common" (the slashed C) are both written identically to what you would show in 4-4 time. The "slash" means that you should play it "two beats to the measure," so it could have been written as 2-2 time. (It doesn't mean "play it really fast," as some seem to believe.)

A hundred years ago, a lot of stuff was notated with the half note getting one beat. The only real difficulty with that is that is that it gets hard to "count the flagbars" on all those hemi-demi-semi-quavers. Convention now is generally to use the eighth or quarter note for the "one-count" in notation, even when a different count is expected in playback.

The first requirement for written notation is that it accurately shows the notes to be played, and when and for how long each note continues to be played. The second "need" is that it be easy for the performer to read - and that's the main source for what may look like "unnatural" use of tied notes in place of the "whole note value" that could be substituted.

John