This entry will bring to your attention the concept of numbering notes rather than letterising them, such as is the traditional method. The purpose of numbering the notes is to make it easier to play in new or different keys than you might usually prefer, to help with improvising and finally, to make it easier to play nice tasty 'jazz' chords. With this ideology in your mind, you will see the piano as numbers 1-13, with the chosen key's root as number 1, rather than having to memorise confusing letters which all overlap in different keys and cause confusion. It is so much easier to play Jazz (or any jazzy-sounding music) on the piano if you apply this concept.
Let's take C as our chosen key. Everyone is familiar with it and it doesn't involve black notes, should you be more of a beginner. More advanced players can simply transpose the concept into their favourite key (mine being Eb - E flat).
The idea is built upon the foundation that Jazz comes from major scales. Yes, minor scales and other complicated scales exist, but just work with major scales and acknowledge that other scales are simply obtained by modifying the underlying primary major scale.
In the major scale of C, we have C D E F G A B and C again for the octave; number 8. We number these from C, 1, up to B, 7. Begin to see these notes as numbers when you look at the piano. See the F as number 4, not F. From this, we can add that a standard chord 'triad' is made from 1 3 5, so C E G. If we continue this triad ascension, we would continue up to 13, which is an A, before the final triad returns us to C. For example, number 11 would also be F if we continued numbering from 1 - 13 (C -> A).
This might seem complicated, but don't let it be. We're simply numbering a major scale's notes from 1, it's root which could be any of the 12 keys, up to 13 - and why 13? Because chords are built on thirds, so if we ascend from 1 (C), we get C - E - G - B - D - F - A ... then C which would be 15, but that's impossible because C is 1 again.
From this, we can ascertain that 3 notes (9, 11 and 13) are beyond the octave. These notes are D F and A. If you notice the spaces from 1 - 7, you will be surprised to realise that they are indeed the notes D, F and A! How can this be? Well, it simply proves to you that a major scale provides all the notes required for major-type chords (i.e; not minor). Just like I said! So, with this triad ascension, we actually cover all the major scale notes (this applies to any key, not just C), but it just so happens that the extensions, D, F and A or better - 9, 11 and 13, are the gaps between 1-7! So in the end, all the major scale notes are used when ascending in major triads until 15, where we start again at 1, C.
Now, imagine that we choose the key of F. This gives us F, G, A, Bb, C, D, E and F (octave, number 8). Most of these notes are in the key of C, only the Bb is not - but that isn't important - don't get confused for no reason. The point is that most of the letters are shared between keys, but it's the number value of the physical lettered note which changes.
Let's select a random letter: D. This is number 6 in the key of F. It is also number 13 if we count from F as 1, then go up in triads, which gives: F A C E (then the extensions 9, 11 and 13) -> G, Bb and D. Again, these 3 notes are the spaces between the F A C E notes, once again using all 7 notes of the major scale.
I selected D. It is number 6 or 13 (more on the differences or when to call it 6 or 13 in a later entry) in the key of F. However (!), it is also number 2 or 9 in the key of C. What does this mean? Theoretically, it's not very exciting, but when playing, to know that you are aware of every possible number value of every note, depending on your key or chord of the moment, you have an array of sounds and opportunities at your fingertips (pun duly intended).
In my next entry, I'll give musical examples for you to go and try, really opening the world of Jazz sounds up to you; it's really very exciting to have such knowledge and authority over the piano. For now, however, I'll leave you with a simple example but not before informing you of the following: Learn all 12 major keys like the back of your hand. This sounds boring and school-like, but I'm not demanding rapidity and correct fingerwork, just learn them by heart so you can see them in your mind and immediately at the piano.
Now, the little example to give you a teaser of what is to come: Imagine you're playing a nice jazz piece. You are improvising a little over the melody or even completely improvising. How do you know what will work? What will sound good? What won't clash? What gives sound variety? It is this concept of note value awareness. You play one note against one chord, so for our example it is D. We play this against a C major chord. It has a particular sound, this 9th. It sounds open and gives way to many possible feelings. However, when we continue to play this D as we change to an F major chord, the physical note of D becomes a 6th or 13th (like I said, I'll explain in another entry when to call it a 6th or 13th) - but the point is, it sounds a particular way, this D against an F major chord.
This results in one physical note having two possible sounds and thus emotions tied to it. With this knowledge, imagine in all keys and all chords, you really have a world of sounds to play with, more often than not, without really having to move your fingers all over the place to aim for different sounds!
I'll let the dust settle on this before writing my next entry. Good luck!