I'm working through The Jazz Piano Book by Mark Levine and this is a personal summary with some illustrations.
Intervals within an octave (simple/main intervals) come in pairs that add up to a full octave, except for the tritone, which divides the octave exactly in half and thus pairs up with itself.
| Semitones | Name | Semitones | Name |
|---|---|---|---|
| 1 | Minor second | 11 | Major seventh |
| 2 | Major second | 10 | Minor seventh |
| 3 | Minor third | 9 | Major sixth |
| 4 | Major third | 8 | Minor sixth |
| 5 | Perfect fourth | 7 | Perfect fifth |
| 6 | Tritone | 6 | Tritone |
This symmetry is illustrated nicely by the following diagram, where the red lines represent the intervals of interest relative to C.
Larger intervals (compound intervals) can be decomposed into octaves and a simple interval. The following pop up often in Jazz.
| Semitones | Compound interval | Simple interval |
|---|---|---|
| 13 | Minor ninth | Minor second |
| 14 | Major ninth | Major second |
| 15 | Minor tenth | Minor third |
| 16 | Major tenth | Major third |
| 17 | Perfect eleventh | Perfect fourth |
| 19 | Perfect twelfth | Perfect fifth |
| 20 | Minor thirteenth | Minor sixth |
| 21 | Major thirteenth | Major sixth |
The modes of the major scale are all constructed from the same sequence of seven intervals rotated one by one.
From these modes we can build chords with different qualities. Ionian mode is somewhat special because the conventional chord notation is based on it.
ii-V-I is the bread-and-butter chord progression in Jazz and we play it with the most basic voicing. We start with a root note. Picking notes from the Ionian, Dorian, or Mixolydian modes gives us major, minor, or dominant chords.
All we need is a 3rd and a 7th to distinguish between these three. The 5th is always the same and thus optional.
Play a major triad (in the second inversion for least dissonance) and the major second above the triad's root as new root. For example, F/G (read: F over G). The root of the triad becomes the 7th of the new chord, the triad's 3rd becomes the 9th, and finally its 5th acts as a suspended fourth.
If you follow this recipe, the new chord ends up without a 3rd and 5th. Traditionally, the 3rd has been left out in suspended chords to avoid the dissonance, but in Jazz anything goes, so you're free to add both of them at your discretion.
We start with a dominant 7th chord (or in fact a dominant 9th, which can be used interchangeably most of the time). This type of chord is built on the 5th degree of the major scale, so we are naturally in the Mixolydian mode. Now we find the 13th of this scale and use it as the root of the dominant 7th/9th chord. With this new root we can re-interprete the whole thing as a chord built from the Phrygian mode.
The Esus♭9 notation is based on the Ionian mode (major): there's a suspended 4th because the 3rd in Ionian mode would be G♯, ♭9 because the Ionian 9th would be F♯. A more succinct notation is G7/E or G9/E (read G7 over E, G9 over E).
You can always add the 5th, 6th/13th, or the 9th without changing the tonal qualities of the chord fundamentally. Depending on the context, we might add other notes.
| Minor 7th | Dominant 7th | Major 7th |
|---|---|---|
| 4 | ||
| ♭5 | ♯4 | |
| ♭6 | ♭13/♯5 | ♯5 |
| ♯7 | ||
| ♭9 | ||
| ♯9 | ||
| ♯11 |
Rotating a dominant 7th chord by 180°, such that the tritone interval within the chord remains unchanged, gives us the chord for tritone substitution.
The 3rd becomes the 7th and vice verse. We only need the new root (and the 5th if you want to play it).
To be continued...
Levine, M. (1989). The jazz piano book. Sher Music Co.