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For my understanding, the top notes D and F are slurs, the bottom notes G and C, what are they? Please help! Thanks!

While slurs and ties look pretty well the same, it's straightforward. Connecting two notes of different pitch, they're slurs; connecting two notes of the same pitch, they're ties. So, the bottom notes in each stave are held over, the B and D on top converge to C, and the F at the bottom drops to E.

We use those curved lines in music for three different purposes:

1. Extending duration (ties) - the notes involved must be the same pitch, and only two notes can be joined by a tie. If you want a duration that requires three notes to indicate, you need two ties, one for notes 1-2 and another for notes 2-3.




2. Indicating an articulation (slur) - for those instruments that can change pitch without a new attack (e.g. a clarinet changing fingering without tonguing) it means using a single attack produced both pitches. For instruments that require a new attack for each note, like the piano, it's an indication that the sounds should be connected as smoothly as possible.




3. Indicating a phrase ("legato line" or "phrase mark"). This is a little fuzzier... it indicates a smooth connection of the notes, saying they should be played in a legato manner - hence "legato line". It generally extends over a musical phrase - hence the term "phrase mark". Because there are two different interpretations, some composers have created additional symbols (like a dashed line to indicate a phrase that is not legato). Unlike the tie and slur, a single legato line or phrase mark connects more than two notes.

Some people will tell you that slurs connecting the same pitch are actually ties and do not warrant sounding the same note again. That is, of course, wrong. If there is any length-indicating articulatory mark on the first note (like a tenuto bar or a staccato dot), obviously the note has to be sounded differently. But there are also other cases, like the following extract from the Ciaconna of Bach's solo violin Partita 2 (this is in 3/4 time, execution a double stop on both G and D string followed by alternating uses of open A string and fingered D string): (Urtext). Here it is glaringly obvious that the notes connected with a slur and with the same pitch cannot possibly be intended as ties since they are spread across two different strings (and thus have to be sounded twice) and since playing them as ties would be completely out of character for the passage.

Even when playing this on the piano, you would most certainly sound the respective notes twice. So you should always look for all available clues before making the tie/slur decision. It is not as easy as "same pitch -> tie".

Those are double slurs used for joining chords. Ties would not have the same amount of curvature, and if you had different behavior intended for moving and staying notes, you'd use separate stems to indicate separate voicing. Of course, differences are subtle and it depends on who engraved those notes: it is pretty common for people using notation programs to use slurs for everything, so the differences are hard to rely upon. Here is some LilyPond code and output:

version "2.19.82"



global =

{ key f major

  omit time 4/4

}



<< { global <>^"Slurs" <g' b' d''>2^(_( <g' c''>4) r4

     <>^"Ties" <g' b' d''>2^(_~ <g' c''>4) r4

     <>^"Voiced Ties" << { <b' d''>2( c''4) } \ { g'2~ g'4 } >> r4

     <>^"Full voicing" << { <b' d''>2( c''4) } \ { g'2. } >> r4 }

     { global clef "bass" <c' f'>2^(_( <c' e'>4) r4

       <c' f'>2^(_~ <c' e'>4) r4

       << { f'2( e'4) } \ { c'2~4 } >> r4

       << { f'2( e'4) } \ { c'2. } >> r4 }

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