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Jazz standards with a Barry Harris-inspired notation.

uppercase letter or uppercase roman number: major six of the given note or degree
lowercase letter or lowecase roman number: minor 6th opf the given note
prime symbol: an arbitrary diminished chord of the given chord (this will serve as a basis of the usual dominat chords)
double prime symbol: like the previous but one semi-tone higher (e.g. secondary dominant)
triple prime symbol: like the previous but two semi-tone higher (a very nice alternative cadence, especially for the ii minor7, see e.g. "I can't give you anything")

BH-notation	conventional
I	Imaj6
C	Cmaj6
iv	Fmin6
f	Dmin6
I'	IIdim or IVdim or bVdim or VIdim
C'	Ddim or Fdim or Abdim or Bdim
C"	Dbdim or Edim or Gdim or Bbdim
C"'	Cdim or Ebdim or Gbdim or Aim

superscripts generally mean extensions (or melody notes)
a somewhat special use of superscripts is to denote min7 chords (this is for convenience; min7 can be considered the 6th inversion of a maj6 chord in the BH-system). We will only use this for minor tunes (to avoid transposing the root)

BH-notation	conventional
i⁷	Imin7
c⁷	Cmin7
c^7,9	Cmin79

subscripts denote the (recommended) bass note (i.e. recommended inversion. No subscript usually means root in the bass (recommendation only).

BH-notation	conventional
I' ⁵ ₅	V7
C' ⁵ ₅	G7

further supeerscripts denote melody notes (also usually good extensions for comping). Melody often anticpatates the next chord, in such cases we write it to that chord.

BH-notation	conventional
C^5,3,7	Melody of Misty ('Look at me...')

here's how a major II-V-I looks like:

BH-notation	conventional
IV₆	Dmin7
I'⁵ ₅	G7
I⁷ ₁	Cmaj7 or C67

and here's the mino II-V-I:

BH-notation	conventional
iv₆	Dmin7b5
i'⁵ ₅	G7
i⁷ ₁	Cmin7

Don't be afraid when seeing the IV instead of the II. You will see that alternating between the I and the IV becomes a supper-common move anyway with this system

Yes, this is a minor ii-V-i

base note, kind, bass(+addons), melody notes relative to the base note

Improvise easily:

look only at the first two character. Only 3 main characters: e.g I,+ -> major 6, I,- -> minor 6 I,= -> minor 7 (inversion of major 6)
this usually changes much slower than in the fake/real books but still makes a lot of sense usually
consider touching melody notes

Comping:

same as for soloing

Bass:

look at the first three characters, the third is your root

Compoing without bass:

look at the first three characters, the third is your root

Solo performance

use all!

Major II-V-I

II chord

II,=: IImin7 (IV major sixth with 6th in the bass)

I,d,21 (arbitrary diminished with the 2th of the I in the bass and with the root of the I added)

V chord

I,d,5: (arbitrary diminished with the 5th of the I in the bass)

I chord

I,-- (minor 7th) note: I,-- is actually the same as III,+ and I,+ is the same as VI,-- but to clearly display the tonality of the song (minor/major), we will use I,-- whenever possible

minor II-V-I