It all begins from the diatonic scale - the scale of the key.
Originally Posted by dublshot
DIATONIC CHORDS, MAJOR KEY
As you pile extensions (from a key scale) on to a triad, those extensions will form other chords.
Eg, for a C chord in C major, add the 7th, B: C-E-G-B. E-G-B = Em triad.
Add the 9th: D. Cmaj9 = C-E-G-B-D. E-G-B-D = Em7. G-B-D = G major.
Trouble is, not all 7 notes (1-3-5-7-9-11-13) can be added to every chord without some of them confusing the harmonic function of the chord.
Eg, if you add the 11th (F) to Cmaj7, you get a G7 chord G-B-D-F, with that distinctive tritone B-F. The F will also sound bad against the E lower down. G7 is the dominant chord, and C is the tonic - which is its opposite. So the chord is a mess.
So the F is considered an "avoid note". It's the one note in the scale that threatens to oppose the function of a Cmaj7 chord. IOW, it sounds bad! (But only if held over the chord, as if part of the chord - it can still be played in scale runs.)
All chords in a major key except one have at least one avoid note in their full set of diatonic extensions. Here's a chart (the extensions to the base chord contained by each arp are shown in brackets):
* 6ths (13ths) on a ii chord can sound fine, but are not often used as extensions; they make the chord too similar in sound to an extended V chord.
CHORD AVOID NOTES SUPERIMPOSED ARPS
I 11 iii (3-5-7), iii7 (3-5-7-9), V (5-7-9), vi (13-1-3)
ii 13* IV (3-5-7), vi (5-7-9), vi7 (5-7-9-11), I (7-9-11)
iii b9, b13 V (3-5-7)
IV - vi (3-5-7), vi7 (3-5-7-9), I (5-7-9), iii (7-9-#11), V (9-#11-13), ii (13-1-3)
V 11 vii (3-5-7), vii7 (3-5-7-9), ii (5-7-9), iii (13-1-3)
vi b13 I (3-5-7), iii (5-7-9), V (7-9-11)
vii b9, b13 ii (3-5-7)
Note that some of those superimposed arpeggios may omit important chord tones. Watch out for those that omit the 7th of the base chord - ie, the "13-1-3" options above. Over a maj7 in particular, the "1" is not a good note to use, without being able to resolve it down a half-step.
IOW, I disagree with All Your Bass. Arps up a 4th (down a 5th) are not a good idea - unless you are working over sus4 chords. Sus4 chords will not have maj7s, but commonly have b7s. They do not have avoid notes, but you may need to consider whether the missing 3rd will be major or minor, which depends on position in key.
Generally, sus4s are assumed to be major, because there's no need to omit a b3 when adding a 4/11 to a chord.
In jazz, sus4s are normally V chords, with b7s. As such, the following superimposed arps work:
V7sus4: ii (5-7-9), IV (7-9-11), IVmaj7 (7-9-11-13), vi (9-11-13).
You can even use an arp which contains the major 3 of the chord, if it's voiced high (above the 4/11), such as a Imaj7 (11-13-1-3) or iii (13-1-3).
DIATONIC CHORDS, MINOR KEY
"Diatonic" has a broader meaning in a minor key, because the 6th and 7th of the scale can be altered; so more varieties of chord are available. The ground rules are that the tonic chord is melodic minor (major 6th and 7th), the V chord is major (with two scale options), the vii chord is from harmonic minor (but uses the WH dim scale); but all other chords are harmonised from natural minor. That gives us this chart:
Chords ii, III, iv and VI are the same as for vii, I, ii and IV in the relative major.
CHORD AVOID NOTES SUPERIMPOSED ARPS
i 11* IIIaug (3-5-7), V (5-7-9)
ii b9, b13 ii (3-5-7)
III 11 iii (3-5-7), iii7 (3-5-7-9), V (5-7-9), vi (13-1-3)
iv 13 IV (3-5-7), vi (5-7-9), vi7 (5-7-9-11), I (7-9-11)
V7b9 - vii7 (3-5-7-b9)
V7alt - vim (b9-3-b13), vim6 (b9-3-b13-7), bviim (#9-#11-7), bviim7 (#9-#11-7-b9), bII7 (#11-7-b9-3), bII9 (#11-7-b9-3-b13)
VI - vi (3-5-7), vi7 (3-5-7-9), I (5-7-9), iii (7-9-#11), V (9-#11-13), ii (13-1-3)
vii - vdim7 (9-11-b13-maj7)**
* The 11th is not an avoid note in the way it is on a major chord, but it's very rarely used on a tonic minor chord.
**The vii7 chord - a full diminished 7th - will rarely have a superimposed arp, but a dim7 chord a half-step below (or whole step above) provides the other 4 notes from the WH dim scale.
If those roman numerals on V7alt throw you, here's examples for an E7 chord (key A minor):
E7alt - Arps: Fm, Fm6, Gm, Gm7, Bb7, Bb9
...in fact, any chord from the F melodic minor scale can work!
This is the charm of melodic modes for jazz players. There are no avoid notes in any of the modes - although they each have specific applications, not connected to the minor key scale degree they derive from (apart from I)
Naturally, music frequently contains chords from outside the current key, or diatonic chords that have been altered in some way.
However, ANY chord (in key-based music) - with one common exception - can be referred back to a role in a diatonic major or minor key, as listed above.
Eg, if you get an A7 chord in key of C major, you can treat it as a V7 from the key of either D major or D minor (perhaps depending on what chord follows it), and apply the relevant subs/arps.
The important exception is Lydian Dominant chords (7#11, 9#11, 13#11). These are diatonic to the 4th degree of melodic minor, but (as mentioned) they are not used that way in minor keys.
Where they will be used is as bII chords in minor or major keys, and sometimes as bVII chords in major keys.
The arps you can superimpose are related to those for 7alt chords.
So, for a C7#11 chord, you can use any arp from the G melodic minor scale. (Same as for F#7alt, the tritone sub for C7#11). These include the following (with the C7 chord extensions shown):
Gm (5-7-9), Bbaug (7-9-#11), D (9-#11-13), Em9b5 (3-5-7-9-#11), F#7#5 (#11-7-9-3)
This chord (these arps) will typically resolve to Bm (as will F#7alt), and sometimes B major or D major.
Which is a crucial point: Don't get too lost in these rarefied upper regions of chord tones! The way you get from chord to chord (melodic intervals and voice-leading) are more important than anything you do on one specific chord.
In music which spends a long time on a single chord (eg modal jazz), you are freer to explore various outside sounds (eg chromatic arps of various kinds), returning to the "inside" chord tones before any chord change.
Otherwise, you need to bear in mind the function of each chord (the role it's playing in the sequence), and not stray too far from the guide tones (3rds and 7ths) for too long. (Arps that contain the relevant 3 and 7 will always work, but too many with 9ths, 11ths and 13ths will end up disconnecting the lead line from the chords.)
In short, don't overcomplicate matters before you understand the basic rules!