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My question is mainly related to extended chord. Is there a general rule for using the 9th, 11th and 13th? I have seen in a II,V,I progression in the key of C this progression: Dm9,G13, C.

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There are two basic rules of thumb for chords.

  1. You always want your harmony to reflect your melody so if the melody you are harmonizing has a 9th, 11th, or 13th of the chord you are playing in it it would be easy to use it in a chord.
  2. Look for common tones and chromatic movement between chords in a progression that will help lead to other chords in your progression. For example in the progression you listed the notes in a Dm9 are D, F, A, C, and E, the notes of G13 are G, B, D, F, (A), (C), and E and the notes of C are C, E, and G. As you can see there are a lot of common tones between all the chords and you can even see the E is common in all chords and is the 9th in the Dm9 and the 13th in the G13.

Another thing to note is be careful with the use of the 11th chord. A dominant 11th (ex. G11) chord is rarely used because of the clash between the 11th and the 3rd. Typically when it is played the 3rd is omitted, but then the 11th is acting more like a suspension then a extension.

Besides that it is whatever you like. The rules are just guidelines. If it sounds good it is good.

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I recommend The Jazz Piano Book by Mark Levine for a fuller explanation of this, but there are some guidelines. In a Western Music context, you can add a 2 or 9 to just about any chord. A true 9th chord also includes the flatted 7th. That chord is most commonly used as a dominant, so a G9 in the key of C, A9 in the key of D, F9 in the key of Bb, etc. The Dominant or V chord is most typically followed by the Tonic, or I. 13th chords are used as Dominants the same way. In a blues setting, you can use a 9th (or 7th, or 13th) chord for all three of the main chords, I, IV and V (CFG, DGA, etc.).

Regular 11th chords are not as common as #11 chords. An example of an 11th chord would be an F major chord played over a G bass note. Some folks call that F/G, while jazzers sometimes call it a Gsus4. It is often seen where G is the Dominant (C is the Tonic), and sometimes resolves to a regular G chord or G7 chord. #11 chords are often used to spice up Tonic chords (I) and Sub-Dominant chords (IV). This is a very "off balance" sound, and comes up in jazz a good bit. A C#11 chord could be voiced as a D with a C bass, or D/C.

  • With the #11, one can slip from say, Ionian, into Lydian mode. It pays to be careful how the chord's written. C#11, or C #11. – Tim Oct 23 '14 at 8:03
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The first thing you need to consider is this: What kind of music are you playing?

The use of the extended chords differ from genre to genre. If for instance you are playing a Bach song, it would be hard to find an extended chord, unless the 9th,11th or 13th note exists from a previous chord.

In Jazz, generally 9th,11th and 13th chords are really common; they are especially found in dominant chords. To be really simple, you can use them pretty much every time you want.

If you look at my answer on another question (What is the secret of jazz phrases?) you'll see that extended chords are also used as this:

If the chord being played is G7 you can play:

  • Fmaj7 (the notes that make Fmaj7 are the tension notes of G7 -> the 7th, 9th, 11th, 13th of G7)
  • Am7 (same as above, but with the 9th, 11th, 13th and then G)
  • Dm7 (same as above, but with the 5th, 7th, 9th and 11th of G)
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One option is to Look at 9, 11 and 13's as extensions to the normal major or minor chords - first write your progression with standard major and minor triads. Then work out the key your progression is in (if the last chord is a fifth or fourth apart from your first chord then it is almost certainly in the key of your first chord). then add the 7ths to the chords, which will either be major or minor 7ths (7 or b7) work out which one by finding the one thats in the key (look at the notes in the scale). Now that your chords are 7th chords that means if you add a 9th, 11 or 13th to them they will be 9th, 11th or 13th chords, if you cant add these because theyvare not in the key, then you might be able to add #11, b9, #9 or b13, which will create equally jazzy chords. With extended chords, they usually sound better when you omit the fifth, so lastly do this. Extended Chords are like buildings if youve got the lower layers, then the top layers (11, #11, b9, 9, #9, b13, 13) can fit right on top of them.

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