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 it’s 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, I’m 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?
In major scale harmony, the vii chord is a diminished triad, or a m7b5 chord (1, b3, b5, b7) as a seventh chord. There are other places where those chords occur. For example, the vi and vii chords of melodic minor harmony.
You’re confusing diminished triads with diminished 7th chords (1, b3, b5, bb7). Dim 7th chords contain consecutive minor 3rd intervals and therefore have inversions that themselves can be root chords (e.g., Bdim7, Ddim7, Fdim7, G#dim7 are enharmonically inversions of each other).
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I am aware of the difference between diminished and half diminished chord
but I still do not understand where diminished chord follows?
Chord B dim.(b d f g#) is resolving on which chord? Cmaj?
D dim. ,F dim.,and G# dim. where they are going?
Bdim7 is enharmonic equivalent of Ddim7, Fdim7, and G#dim7.
You’re right that Bdim7 can resolve up to C or Cm, that’s a resolution up a half step, but it’s true for the other three roots/inversions/names as well:
Ddim7 can resolve up to Eb or Ebm
Fdim7 can resolve up to Gb or Gbm
G#dim7 can resolve up to A or Am
But since all of those four chords are actually the same, then:
Bdim7, Ddim7, Fdim7, or G#dim7 can ALL resolve to C, Cm, Eb, Ebm, Gb, Gbm, A, or Am.
The same previous sentence would be true if you transposed all 12 chord names up a half step, same relationships, and still true if you transposed up another half step. But if you transpose up three half steps, you actually just get the same group of chords.
It’s worth noting that diminished seventh chords can resolve in other ways too. For example, this progression in C:
Em7 D#dim7 Dm7 G7 Cdim7 Cmaj7, same duration on each chord.
In this case, the diminished seventh chords are not resolving in the same way.
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Another thing about the “four dim 7 chords in one” in harmonic minor is that when doing the stacked 3rds thing, the only degree that actually generates those four tones is the 7th. So, in A Harmonic Minor, that’d be G# dim 7. Chris Brooks’ “Neoclassical Speed Strategies” book talks about this stuff and how G#, B, D and F diminished seventh chords can each resolve to the tonic chord (Am) because of their connection to the E7b9 chord (E, G#, B, D, F). In other words, those 4 dim 7 chords can be treated like the dominant chord of the minor key.
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Today I found something like this about “demonished hordes”
"Diminished chords are very often used as links
between two diatonic chords. The most common use:
A. Ascending:
- raised first degree (#I ??)
- as a link between IMaj7 and IIm7
- raised second degree (# II ??)
- as a link between IIm7 and IIIm7
- raised fourth degree (# IV ??)
- as a link between IVMaj7 and V7
- raised fifth degree (#V ??)
- as a connector between V7 and VIm7.
B. descending:
- lowered third degree (bIII ??)
- as a link between IIIm7 and IIm7
- lowered sixth grade (bVI ??)
- as a link between VIm7 and V7.
C. With the same note as the basis:
- First lowered degree (I ??)
- as a transition chord between IMaj7i and IMaj7
- The fifth degree lowered (V ??)
- as a transition chord between V7 and V7".
We are dealing with three types of diminished chord use, and thus as:
I. a chord whose base rises by half a pitch up between two diatonic chords (points A1, A2, A3 and A4)
II. the basis of the chord connecting two diatonic chords drops by half a tone down (points B1 and B2)
III. auxiliary chord. The basis of the chord remains unchanged
when the chords change.
The above examples of solutions are almost the rule,
but we can also meet the so-called alternative solutions. Here they are:
? chord #I ?? instead of solving it on Iim7, it can solve on dominant of IMaj7, that is, the G chord with the fifth in the bass (C # ?? → G / D),
? chord # II ?? instead of solving on IIIm7,can solve on first one with third in the bass (D # ?? → C / E),
? chord # IV ?? instead of solving on V7, it can be solved on first with fifth in the bass (F # ?? → C / G),
? chord #V ?? instead of solving it on Vim7, it can solve on dominant inserted into IIm7 (G # ?? → A7),
? chord bIII ?? instead of solving on Iim7, it can be solved on the V7
with the fifth in the bass (Eb ?? → G / D),
? chord bVI ?? instead of solving on V7, it can be solved on first with the fifth in the bass (Ab ?? → C / G) (example 4)".
Ufff,It’s too difficult for me in theory, I would have to see it
Who has a profile on YT and can do a video about it? 
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This might be of use:
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Diminished chords can go up or down. In jazz they are used a lot to connect chords a whole step apart.
So In the key of C you have
I- C Maj 7
vii°/ii- C#°7
ii- D Minor 7
vii°/iii- D#°7
iii- E minor 7
vii°/IV- E°7
IV- F Maj 7
vii°/V- F#°7
V- G7
vii°/vi- G#°7
vi- A minor 7
vii°/vii- A#°7
vii°- B half diminished 7
Note that since diminished 7 chords are symmetrical (built in minor thirds) hat every note is actually a root note. So G#°7 = G#BDF. It’s the same chord as B(Cb)/D/F/Ab Diminished. Also these diminished chords are really just a rootless 7b9 chord. I’d you looked at G7b9- GBDFAb. You can see that there is a B°7 within the Chord. This is why B°7 can lead to C- as it has a dominant function.
Basically these secondary diminished chords are rootless 7b9 Secondary Dominants.
So vii°/ii is just a rootless VI7(b9/)ii.
A C# E G Bb.
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…in other words, we have… 13 diatonic chords?
…and now we can connect it with the modal interchange 
Oh no,not him,not again 
thanks anyway
You can find the same concept in detail in the work of Barry Harris
Title please any books etc he wrote?
The Barry Harris Harmonic Method for Guitar.
Which I hope you googled in the last 16 hours rather than waiting for me to tell you.
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Three Note Voicings and Beyond…great book that I’ve been working on over the past few months. Highly recommended. Good call.
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There are a few basic rules I was taught when studying jazz arranging that proved to be very useful when transferred on guitar, both in analyzing several stock voicings and coming up with new ones. Given that one already has a good grasp of the basic chord qualities-maj7, dom7, min7 and m7b5- at least in drop2 and all inversions, here’s how it works:
-In any voicing, substitute 9 for I. For dominant chords b9 and #9 can also be appropriate.
-In any voicing, substitute 11 for 3. However, if we want the 3rd to be included in the voicing, as in the case of minor chords or when the 11 is #, then use 11 for 5.
-In any voicing, use 13 for 5. Can be b13 in dominant chords.
Of course the above subs can be applied in combination. A wide variety of useful, common practice voicings can be generated this way.
A next logical step to take is see if certain shortcuts can be taken. Here’s an example:
Chord: C maj7.
Notes: C E G B
Sub 9 for 1: D E G B…now, that’s an Em7. So, for Cmaj9, I can play any Em7 voicing over C.
Another example, taking the previous a bit further:
Sub both 9 for 1 and #11 for 3: D F# G B…now, that’s a G maj7. So, for C maj7/9/#11 (C Lydian that is) I can play any Gmaj7 voicing over C.
You get the idea.
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