I'm trying to work my way through the Mark Levine Jazz Theory book but I'm scratching my head at the start of chapter 2 (Pg15). Maybe this book is too advanced for me and I need to pick something less Jazzy. A stave in the first example shows the notes F,Ab,Eb,G with the Chord F-7 written above it. I can't fit these notes into the chord, I would have expected to see F, Ab, C, Eb. In the next bar it names the chord as Bb7 with the notes below Bb,Ab,C,E,G. Again I would have expected to see Bb,D,F,Ab. Can somebody help and explain how or why these notes appear ?
In a book on chords we might expect the labelled chord to be played exactly as the chord symbol indicates, but this book is more about how a musician may approach the basic chord chart and embellish upon it creatively. Essentially, when a musician playing in a jazz context sees a chord, they will very often put their own substitute voicing on top that has a few more of the interesting extensions in it. In the Mark Levine book, he is concerned with showing the underlying chord progression as it may appear in a Realbook or leadsheet, and displaying it alongside what a musician may actually play. The chord charts as written in jazz lead sheets are fairly rough simplifications, giving the improvising musician a good clue as to what that chord is 'doing' in the piece of music, and leaving the space open for them to extend it in a way that suits them, and suits the stylistic sensibilities of the current group they are playing in.
In the example fig 2-1, we are seeing a ii V I progression in Eb.
F, Ab, Eb, and G are the intervals 1, b3, b7, 9, a minor9 chord, a common extension of the minor7 chord in the ii position. The 5th is omitted, but it doesn't effect the naming of the chord, the 5th is very often the first to be omitted from a voicing as it serves no strong purpose in defining the FUNCTION of the chord (here, at least).
Bb, Ab, C, D (not E), G are the intervals 1, b7, 9, 3, 13. A dominant13 chord, again, a common extension of a standard dominant7 chord in the V position. Note, you can generally add any upper extensions below the 'highest' extension you get. Ie in a 13 you can happily add all the extensions up to the 13th, there are some caveats and rules to that but perhaps thats not for right now, you'l come to them in time!
Edit, looking again at the example I would agree with Pied Piper that actually the Bb's 13th (the G) is better thought of as a suspension, in this context, as it is part of the melodic line on top and quickly resolves to F within the second measure!
Musicians playing jazz (depending on their personal choice and sound) will generally extend most chords in one way or another to get a more interesting or complex sounding harmony. Mark Levine is showing the underlying changes (the chord symbols) in such a way that their function can be easily seen, alongside a transcription of how someone may choose to actually play them.
This may seem a little complex for the learner, but it's not so bad. The underlying quality of the chord will be held in it's 3rds and 7ths, generally.
1st, 3rd and 7th - some kind of major7
1st, b3rd and b7th - some kind of minor7
1st, 3rd and b7th - some kind of dominant7
The extensions, in the example you asked about, are chosen from the other notes available from the scale, Ebmaj in this case. The ii chord naturally has a natural 9 so it has been added, the V chord has a natural 9 and 13 in the key of Ebmajor so they too have been added. These extensions are said to be diatonic to the key, ie. they are the ones we would expect.
Worth noting, playing extensions that are NOT from the key is very common, in order to create tension or other effects. But this is something to look at later perhaps, and IIRC in this book Mark Levine introduces the learner to these extensions as they come up.