Below is the the Harmonic Minor Scale and it's diatonic arpeggios.  This PDF is similar to those for the Major Scale and Melodic Minor Scale, so check out those posts if you haven't.  The PDF outlines the Harmonic Minor, it's seven modes and the diatonic arpeggios.

As with the other Scale PDFs, it might be worthwhile to start with the arpeggios for each mode.  Not only are they less notes to focus on, but they are lines in themselves that make succinct, harmonic 'sense' out of the modes.  Once you understand the chord tones for a mode, then you can 'flesh out' the rest of the scale around it.

For the novice jazzer, it is less useful to understand this material for compositional movements, than to just be aware of the arpeggios that are buried within this scale.  While the Harmonic Minor is a great choice for the Min-Maj7 chords and sounds in tunes like Nica's Dream and Nardis, the harmonic minor is probably most used in jazz for altered sounds -- using the Phrygian Dominant over a #11 chord or the Ultra-Locrian for tri-tone colors over a functional Dominant.  Just learning the harmonic scale will give you access to these sounds, but being able to pick out all the arpeggios within it can give you a richer starting point for your lines.

Another key use of the Harmonic Minor in jazz is the 'Barry Harris' approach.  Barry Harris has a whole cosmology of scales and tones that starts with chromatics, splits into diminished and ends with sixth chords.  The 'Barry Harris' scale -- the Major#5 -- is really the third mode of the Harmonic Minor in this PDF.  Check out his 'Sixth-to-Diminished' theory which is at the heart of his teachings.  If you act as if the Third Mode of the Harmonic Minor is the 'One' chord for Major tunes, you can develop a unique perspective into the tunes of the Swing and BeBop era.  Either way, if you are familiar with the arpeggios in the Harmonic Minor and Melodic Minor scales, you will be able to spot them when transcribing the lines of Charlie Parker and other boppers when they go 'outside' the harmony.

Below is a reference for learning the Melodic Minor Scale diatonically.  The Melodic Minor is sometimes referred to as the 'Jazz Minor' due to its applications to Jazz.  You may have gotten hip to the fact that the 7th mode of the Melodic Minor can be used serve up altered sounds over a functional Dominant (a V going to a I).  At some point I also realized that the fourth mode of the Melodic Minor can be used over #11 Dominants.  Learning the totality of the scale with diatonic context of all the arpeggios will give much greater command over these sounds.

While the Melodic Minor only differs from the Major scale by one note, I still find myself struggling to visualize it thoroughly over the fretboard, certainly more than Major scale.  My ability to visualize the Major scale was significantly bolstered by learning all the diatonic arpeggios in the scale, across all the scale 'boxes' where the modes sit with the root on the sixth string.  Learning arpeggios also significantly strengthened my ability to generate lines strongly outlining the harmony. 

You can focus on parts of the neck by breaking it up into 'r6 mode boxes'.  These scale 'boxes' I term with the 'r6' meaning 'root is on the 6th string.'  For instance, 'III-r6' represents the way the notes sit when you play the Phrygian scale with its root on the sixth string.  This lines up as a box, or chunk of the fretboard which is easier to concentrate on, especially at first.

Just like the the Major scale, there are seven notes in the Melodic Minor.  Each of these seven 'degrees' has a mode, chord, and arpeggio associated with it.  Modes are simply the same notes of a scale arranged with a different note serving as the root.  The arpeggio for each degree can be found by skipping every other note in the mode until the next octave is reached.  An arpeggio is basically the notes of a chord played sequentially.

Below are the scales and chords for each degree of the Melodic Minor Scale:

​The Melodic Minor scales are:
I  - Melodic Minor (R, 2, b3, 4, 5, 6, 7)
II - Phrygian #6 (R, b9, b3, 4, 5, 6, #6)
III - Lydian Augmented (R, 2, 3, #4, #5, 6, 7)
IV - Dominant #11 (R, 2, 3, #4, 5, 6, b7)
​V - Mixolydian b6 (R, 2, 3, 4, 5, b6, b7)
VI - Locrian #2 (R, 2, #2, 4, b5, #5, b7)
VII - Super Locrian (R, b9, b3, 3, b5, #5, b7)

The accompanying chords are:
I  - Min-Maj7 (R, b3, 5, 7)
II - Min #6 (R, b3, 5, b7)
III - Maj7 #5 (R, 3, #5, 7)
IV - Dom.7 #11 (R, 3, 5, b7)(#11)
​V - Dom.7 b13 (R,3, 5, b7)(#13)
VI - Min7b5 (R, b3, b5, b7)
VII -Min7b5 (R, b3, b5, b7)
​
I recommend starting with the Melodic Minor arpeggios in the middle section of the PDF.  Take four or five days and run over the arpeggios for one mode (probably start with Min/Maj7) in all the different r6 mode boxes.  Maybe put on a relevant backing track and definitely a metronome.  By the fifth day, you should really concentrate on connecting between the boxes and try to work horizontally or diagonally.

After you have the arpeggios, the scales are just the rest of the notes to flesh them out.  Start with the most useful scales for playing over Dominants, like the Lydian Augmented and Super Locrian.

Finally, learn to run all the arpeggios within a particular mode box.  This will reinforce both the scale pattern and the diatonic progression within the Melodic Minor.

While you are not going to necessarily find a lot of complex harmonic movements of the Melodic Minor in Jazz tunes, learning the diatonics of the Jazz Minor will give you greater facility at minor two-fives and the ability to super-impose these scales in other applications.

The world of harmony as explained by a Barry Harris creation story...

First there was nothing.  From nothing, God created the chromatic scale -- the world for all music to inhabit.

​The world of the living is 12 notes:
World = Chromatic Scale
C, C#, D, D#, E, F, G, G#, A, A#, B

The world is divided in half, creating Man and Woman. Each of them takes 6 tones of the world.
Man: C, D, E, G, A
Woman: C#, D#, E, G, A, B

There are three babies from these parents -- all diminished. Each diminished receives 2 tones from man and and 2 tones from woman whole-tone scales. The formula for diminished is 1, b3, #11, bb7(6).  The b3 and bb7 are 'chromosomes' from the opposite parent.  All diminished scales are inversions of one of these three sets of notes.

C, Eb, Gb, A
C#, E, G, Bb
D, F, Ab, Cb

The 'Man' and 'Woman' Parents (whole-tones) are the first major scales.  They have Major 3.
Children (diminished) are the first minor scales.  They have Minor 3.

So far we know there is 1 chromatic scale.
There are 2 whole tones scales.
There are 3 diminished scales.

Those 3 diminished babies (birthed by the two diminished scales) will now have babies.  The first baby is the dominant.  You get this baby by altering one "chromosome" aka, lowering one tone a half-step.  

Dominants (1,3,5,b7) - are diminished with one tone moved down a half-step.  Each diminished can have four baby dominants. 

Example, Cdim can have B7, D7, F7, Ab7 babies.

C, Eb, Gb, A becomes
B [C down half-step], D#(Eb), F#(Gb), A =  B7

C, Eb, Gb, A becomes
D, Gb, A, C = D7

C, Eb, Gb, A becomes
F, A, C, D#(Eb) = F7

C, Eb, Gb, A becomes
Ab, C, Eb, Gb = Ab7

The diminished 'scale' is created putting two of the three diminished arpeggios together -- all the chromosomes of a diminished, and all the tonics of the baby dominants.  Following the Root world of C, that would be:
C, D, Eb, F, Gb, Ab, A, B

This is known as the 'Whole-Half' Diminished scale which starts with a whole-tone jump followed by a half-tone jump.  This gives us a Major 9 and a Minor 3.

All the diminished children (birthed by the two Whole-Tone scales) have babies.  The first baby is the Dominant.  You get this baby by altering one "chromosome" aka, lowering one tone a half-step.  

Dominants (1,3,5,b7) - are diminished with one tone moved down a half-step.  Each diminished can have four baby dominants. 

Example, Cdim can have B7, D7, F7, Ab7 babies.

C, Eb, Gb, A becomes
B [C down half-step], D#(Eb), F#(Gb), A =  B7

C, Eb, Gb, A becomes
D, Gb, A, C = D7

C, Eb, Gb, A becomes
F, A, C, D#(Eb) = F7

C, Eb, Gb, A becomes
Ab, C, Eb, Gb = Ab7

If you raise a tone of a diminished a half-step and the baby you get is a Min6 (1, b3, 5, 6)

C, Eb, Gb, A becomes
Gb, A, Db[C up half-step], Eb =  Gbmin6 (Eb-7b5)

The Half-Diminished and Minor 6 arpeggios are inversions of each other.

Raise two tones of a diminished a half-step and the baby you get is a Maj6 (1, 3, 5, 6)

C, Eb, Gb, A becomes
A, C#[C up half-step], E[Eb up half-step], F#(Gb)  =  A6 (F#min7)

The Major 6 and Minor 7 arpeggios are inversions of each other.

Lower two different tones of a diminished a half-step and the baby you get is also a Maj 6 (1, 3, 5, 6)

C, Eb, Gb, A becomes
D[Eb down half-step], F#(Gb), A, B[C down half-step]  =  D6 (Bmin7)

From the Chromatic split you get the Whole-Tone.
From the Whole Tone's intermingling, you get the three diminished.
By moving one or two tones up or down from the diminished arpeggios, you get all the Major 6, Minor 7, Half-Diminished, and Dominant arpeggios.

From each diminished:
lower one tone a half-step = dominant
raise one tone a half-step = min6 (min7b5 inversion)
raise two different tones a half-step = Maj6 (min7 inversion)
lower two different tones a half-step = Maj6 (min7 inversion)

This material comes largely from this video and this video.

The Mixolydian mode is an 'uneven' scale - it only has 7 notes, while we ordinarily play in 4/4 time.  If you continue marching up or down the the Mixolydian scale tones in sequence, after 8 beats, your scale's chord tones are no longer hitting on the down beats -- they are flipped -- giving an awkward sound.

The solution is the 'Bebop Scale.'  The C7 bebop scale has extra 'passing tone' -- the B natural (the Major 7 of C).

C Mixolydian

C-D-E-F-G-A-Bb-C

1-2-3-4-5-6-b7-1

C 'Dominant Bebop'

C-D-E-F-G-A-Bb-B-C

1-2-3-4-5-6-b7-7-1

The 8 notes of the 'Bebop' scale ensures you always fall on the downbeat when you start on a chord tone, even if you march into the next octave.

The Minor modes (like Dorian) have the same problem when playing over a minor -- the chord tones fall off the downbeat if you do a run 8 beats or more.

The Jazz Minor (aka Melodic Minor) is essentially the Major scale with a flat 3rd.

C-D-Eb-F-G-A-B-C

1-2-b3-4-5-6-7-1

However, this scale only has 7 notes. You can turn the Jazz Minor into an 8-tone 'Bebop' scale to enhance the chord tones on downbeats after you enter the successive octave. There are two ideas of how to construct the Jazz Minor Bebop to get 8 tones -- either use the #5/b13 or the flat 7.

The first solution is adding the extra passing tone of a flat 13, also known as the sharp 5.

G 'Jazz Minor Bebop'

C-D-Eb-F-G-G#-A-B-C

1-2-b3-4-5-#5(b13)-13-7-1

descending, you spell the #5/b13 as Ab.  This bebop scale ensures the 'quality' of the Major-minor scale.

Another approach, is to construct the 'Jazz Minor Bebop' by incorporating both the Major and Minor 7.

C-D-Eb-F-G-A-Bb-B-C

1-2-b3-4-5-6-b7-7-1

In classical music, the minor scales are different ascending and descending.  In order to add the 'jazzy flavor' of a Major 3rd with a Minor chord, you can choose to use incorporate Major 3rd as the descending 'Bebop Minor' when doing downward runs.

'Jazz Minor Bebop Descend Variation'

C-Bb-A-G-F-E-Eb-D-C

1-b7-6-5-4-3-b3-2-1

or you can descend with the Dominant Bebop scale shown at the beginning.

You can any play of these above mentioned scales over a ii-V.  For instance, you can play the Jazz Minor ascending, and Bebop descending.

The beauty of these scales is that when you move into the next octave, you are still playing chord tones on the downbeats.

The videos linked above also talk about 'Broken Chords' which simply mean, phrasing chord tones, like an arpeggio.  Any arpeggio within the diatonic progression of a key can create a useful phrase in a solo.

The videos also briefly touch on half-step in bebop.  Half steps can be used to 'keep your footing' like in bebop scales, or be used as 'approach' tones in the key, or be used to help 'enclose' a chord tone.

I made this chart to highlight the Essential Chord Tones (ECTs) for each chord in a diatonic progression relative to the numbers in the Key scale (Ionian Major).  My shorthand for this concept is the "Diatonic Exchange Rate." 

When you play the root of the ii chord, you are just playing the second note of the Key scale.  The third of the ii chord is simply the 4 of the Key scale.  If you think of just these number for each chord in the progression, you can focus on the intervals of the Major scale instead of thinking about arpeggios relative to the Major scale.

I thought this might be helpful to come back to in light of chord solos.  In a chord solo, we have a melody, say on the 'B' string.  For each note we want to accompany with a chord, we have to find at least one ECT common to the chord for the rest of our fingers.  For a ii chord, we would really implicate a minor strongest if we have the flat 3rd of the ii.  Instead of finding the root of the ii and then finding its b3, I would rather just go directly to that note, which happens to simply be the 4 of whatever key I'm in. 

The chords are ordered in 4ths/5ths, common to progressions in Western music.  If you go bottom to top, you are moving in 4ths.

On the second page I grouped the ECTs in order of precedence.  Minors differentiate themselves from Major by virtue of the b3, where as Dominants differentiate by the b7.  You will see a pattern here: moving in 4ths, the third is the seven of the next chord in the sequence.

In the PDF below, I have diagrammed the arpeggios in each 'mode box' for diatonic chords with the 'diatonic exchange rate' pasted on each note.  I excluded the IVmaj7 on the first page for room, but it is diagrammed on page 5.  I also added some extra diagrams for altered dominants.

This chart shows the 'boxes' or scale patterns I typically visualize across the neck for a key.  Each box corresponds to a mode with the root on the sixth string.  If you don't know what a mode is, it is just the notes of a Major scale, but with the root on a different key.  For instance, Dorian is the major key scale if you use the second tone as the root.  Mixolydian is the mode if you use the fifth tone of the major scale as the root.

In the 'CAGED' method, each box also corresponds to an open chord fingering (and it's arpeggio).  For instance, the open C chord fits over the r6 Phrygian fingering (that starts on the 6th string).

I've also included the intervals to the I chord.  I used roman numerals for the ii and V to impress that these numbers are where the scales start for other chords in a diatonic (major) progression.* The ii-V-I is about the most important progression in jazz, so being able to visualize the relationship of these roots are key.  However, it extends to all the diatonic chords.  The root of the vi chord is always on the 6th tone.  You can see with the Aeolian mode (which starts on 6) that 'Aeolian' is a perfect scale to play over the vi chord.  Phrygian is the best mode to play over the iii chord, etc.

I also included the Locrian and Lydian modes, although I don't visualize them so much.  They are so close to the other fingerings, it is harder to distinguish which box you are in.  It is good to practice all modes to see how the sound, however. 

It is essential to know where the 7 is in the major scale however.  It is not only a 'guide tone' of a I7 chord, but it is the root of the vii chord - always a min7b5.  When you know all the intervallic relationships, you can quickly find a root based on the progression -- then position a chord and find the scalular notes or arpeggios.

*It is too unwieldy to use roman numerals throughout the chart because 'iv' and 'vi' get confusing.

I made this chart to begin creating visual "anchors" to the diatonic modes I am already familiar with.

I've been practicing the Harmonic Minor, which I know serves as the Altered Scale when its root is on the b9 of a Dominant.

Now I need "anchor" it to the modes I already am familiar with, so switching to the Altered Scale will be instantaneous.  I'm practicing switching between each section of the Altered scale with the corresponding mode in the diagram, measure to measure.  Then I'm practicing a ii - V7alt - I progression to get to the scale in time.

I basically visualize 5 "boxes" which correspond to five modes where the root is on the 6th string.  The Lydian and Locrian are so close to the Phrygian and Ionian respectively that I just picture the later boxes.

I use 'r6 Aeolian' to connote the shape of the Aeolian mode where it starts on the sixth string.  I'm using 'Vr4' to connote the root, arpeggio or scale for the V (Dominant) of the diatonic progression where it appears on the fourth string.

Thanks to the CAGED method, I know the shapes of all the major scale positions well.  But I'm still lagging visualizing a mode if the mode's root is not on the first or sixth string.  I made this chart to help me memorize the shapes for all the modes for all the roots.