Help with interpreting notation in Chopin's nocturne op. 9 no. 2
What are the following notations in Chopin's nocturne op. 9 no. 2:
Tilde sign with accidentals:
tr~~~
Striked note's flag:
Small notes. (I know what mean single small notes right before a normal note, but this is very long section written in small notes.)
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It's a turn where the note above is flat and the note below is natural.
It's a trill (though it is technically two trill marks)
It's a grace note.
It's most likely either optional notes or notes that belong to another part. Either way with the pedal marks above them they are probably extra ornamentation after playing the dotted whole note pattern with the fermata beforehand.
This is called a turn. The 'basic' version would be written without the accidentals, and the player would play the first note, then quickly play one tone (note) above, the main note again, a tone below, the main note, and the resolve one the final note. The accidentals clarify exactly which notes to "twiddle" to.
The turn can be either directly over a note, or between two notes. (It looks to me like this is intended to be between two notes.) Over a note, the whole value of the note would be used for the twiddle, between two notes, the 'twiddle' would be squeezed into the gap!
I'd play this example like this:
This is a trill. Quickly alternate between the marked note and a tone above, for the entire duration of the note.
An Acciaccatura (a.k.a. crushed note, grace note.) Play the small note as quickly as possible, just before the next main note, as a little lift/fall to the main note on the beat.
This notation I'm less familiar with, and not aware of it's specific name. (NReilingh provides a much better explanation in the comments.) Here, the small notes are to be played 'more freely' over the sustained chords underneath - not strictly in time. This youtube recording is a good example. (This part is around 3.40)
The "small notes" near the end of the piece is called a cadenza. It is meant to be played freely and without tempo, at the liberty of the performer. However, in the case of the Nocturne, it is often played very quickly and almost trill-like. All of the other questions have good answers to them already, so I will refrain from answering those.
The key to this section is the term "Senza tempo", which means "not in tempo", or in other words that this is completely free decoration. The rising arpeggio in the left hand can be taken leisurely, and the small notes can start when you feel like, and at the speed you feel like (usually pretty fast though). Really they are a variant form of trill, which Chopin has helpfully written out in full for you.
(Dom has already answered 1, 2, 3, but his comment on 4 suggests he is not familiar with the piece. I'm not sure how to add this answer to just part of the question, so feel free to copy it or whatever.)
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