How to create 9th 11th 13th chords in order to
removing the least important degrees but
contain the essential harmonic quality of this chords?
Perfect 5th in the first place,what next??
PS. and how about 9th 11th 13th shell voicing?
Keep the 3rd and the 7th, then whatever extention you like above that.
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For shell voicings I would including the root, 3rd and then the 9th / 11th / 13th.
The book Three Note Voicings and Beyond is the most comprehensive book I know of written on the subject of shell chords. You should check it out if youāre interested in learning more.
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I think of it like:
basic seventh chord structure
1 3 5 7
fifth can often be omitted as itās not essential for defining chord quality
technically a 13th is, I guess, 1 3 5 7 9 11 13, but practically speaking I think of it more like:
9th:
1 3 (5) 7 9
11th:
1 3 (5) 7 11
13th:
1 3 (5) 7 13
And you can mix and match 9, 11, and 13 to taste or to get the desired sound given the melody or the progression in the tune.
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with the exception of the locrian scale?
b d f a + c or e or g?
forgetting about the locrian mode for a minute, but just thinking of the chord:
m7b5 - yes, it needs a b5 to be defined as m7b5 - however, to be simplistic for a second, it doesnāt sound bad to simply omit the b5, and often context will imply the b5 even if you donāt play it. For example if itās Cmaj7 Fmaj7 Bm7b5 E7 A7, if you omit the b5 on the Bm7b5 the progression still makes sense and itās not like the listener starts hearing an F# if you donāt play an F natural.
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also, I corrected a typo in my first post.
As stated by everyone else, the 5th is the first to go. Iāll kill the root note next. After that I think anything goes, you just gotta decide whatās important to you in that moment.
For me it ultimately depends on whatās happening in the melody, bassline and the general context of the chord. If Iām playing a jazz head with a Cmaj#11 chord but the #11 is already in the melody, then Iāll likely skip it in the chord.
Thereās a great video of Herbie Hancock where he says that he felt stuck and bored by his playing and Mile Davis told him ādonāt play the butter notesā which forced Herbie to think about what the ābutterā notes of a chord were (he decided they were the 3rd and 7th tones, the ones we consider essential!) and it completely changed his views on harmony and his playing. So sometimes you gotta break the rules of what āworksā and just try a bunch of other shit and see what works.
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My understanding is, even though the 2 and 9 are the same note, the two is contained in the bounds of the chord (between the 1 and 7) while the 9 is on top of it. This is why itās not just called a 2. You donāt necessarily have to remove one of the chord tones to play a tension although it might sound better and/or be easier to play
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Regarding the Herbie thing, yeah sometimes we just get bored with whatās obvious, and want to push ourselves and the listener.
Related, hereās a short pdf I made for my students with some (imo) important considerations for getting started with comping for jazz standards and some of the common variables: https://drive.google.com/open?id=1-944yuSdDVBTeWrM1uhziscE0nnOZtDz
An excerpt:
Some considerations about being āobviousā
- Clearly playing the root notes helps establish exactly what the chord is. Maybe you want to do this because you want the sound, and the harmony, to be very clear. Maybe you want the people you are playing with to be very aware of which chord you are playing, so they know where you are in the form. Maybe you donāt want to sound as āobviousā and want to trust that either the other players, or the audience, or both, donāt need such strong indications of the harmony, and can enjoy a looser interpretation of the chords
- Similar things can be said of other issues. For example, if there is no bass player, you can fulfill that role and play low notes. Or you can leave the low register āblankā and not feel the need to create a clear bass line. This is acceptable, it may simply be asking more of your audience, or of the people you are playing with.
- Thereās nothing wrong with ācompingā with āsingle noteā chords
- Similar to #3, depending on context, you may need to include less and less information about the original chord. If the people you are playing with know the tune very well, and everybody involved has a strong sense of the form of the tune and is keeping the same sense of time, we do not necessarily need to spell the harmony out in obvious ways. For example, depending on context, simply playing the 9th and 6th together of a minor seventh chord can be enough to support that chord even though your voicing has none of the original notes of the m7. This is the kind of thing that makes more sense the more you play with people, and will be more or less applicable depending on the style of comping you prefer, or that you feel supports the soloist best in that context
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I think its called the 9 because the chords are built in thirds (1 3 5 7 9 11 13), so even though the 2nd note its just right next the root if you count every 3 notes it ends up being the 9. you donāt have to play it above the octave though.
Thereās this master class that kurt rosenwinkel made, and he basically keeps the root with the 3rd and builds from there and then he does the same with the 5 and 7 (of course he uses this for chord melody stuff but it is useful as a template for making other voicing just add the other degrees)
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My favorite part about that story is that Hancock later came to believe that when he thought Miles had said ābutter notesā, Miles had actually said ābottom notesā. This is one of those cases of a mistake/misunderstanding or happy accident leading to a really cool yet unexpected result. Not everybody would have taken the heard advice ādonāt play the butter notesā in the cool direction Hancock did, but Hancock almost certainly took it somewhere cooler than he could have taken ādonāt play the bottom notesā. A true āchocolate and peanut butterā moment. 
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Interesting story. Glad someone investigated and got to the butter of that.
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Second degree is 2th without 7th (sus2) 2th become 9th with 7th.
By the way,6th is substitution for 7th.
and there is story about butter notes 
for whatever itās worth, I always thought this naming system was most logical, assuming a C root to make it easier to write:
1 2 5: Csus2
1 4 5: Csus4
1 2 3 5: Cadd9 or Cadd2
1 3 4 5: Cadd4
And then if we have a seventh, the 2s are 9s, the 4s are 11s.
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Cadd2
1 2 3 5 7 - Cadd9
The presence of the seventh makes it Cmaj9, not Cadd9.
Cadd9 is triad plus 9th or triad plus 2. Cmaj9 implies 7th. Whether you spell it 1 2 3 5 7 or 1 3 5 7 9, chord name is the same.
At least, Iām referring to terminology as it applies to jazz lead sheets. But Cadd9 has no 7th, nor does Cadd2.
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To everything above, I would also suggest this : take any 4 note 7th voicing you know and substitute 9(and alterations) for 1, 11th(or #11) for 3rd or 5th if you wish for 3rd to be included, and 13th(or b13) for 5th. This is how arrangers produce extended harmony when writing for 4 part sections-and it works great on guitar.
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I Agree, this is a great next step after becoming very familiar with seventh chords
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One more question,it is true that diminished chords
solves on chord by half tone higher?
B dim. go to Cmaj?
But diminished chord do not have inversion,only four root positions.
So,if diminished chords B solve on chord by half tone higher Cmaj
D dim. go to D#maj
F dim. go to F#maj
and G# dim. go to Amaj?