I recommend reading the Chord Progression page as an intro to this text. It’s import to move on from the Natural Major and Minor system and add the properties of the Harmonic Minor Scale to your knowledge base. The Harmonic Minor Scale is a self-contained system just like the Major Scale but is used as a supplement or a plugin to our main musical relationship the Major/minor Diatonic Scale rather than being used as the main system although there are examples of such compositions. Most of the time the Harmonic Minor is used to deviate from the Major/minor Scale momentary in the song.
Traditionally I like to organize my theoretical knowledge about music from the perspective of the Major Scale. This means that I usually treat the minor scale and minor compositions as derivatives of Major. In an E minor song, for example, E for me is always the 6th chord and Am is the 2nd and Bm the 3rd. I recognize that when you are playing a Minor composition you need to whole heatedly accentuate the Minor Root and the lead tones in the melody that is built on top of that particular chord progression. It follows that it makes perfect sense that you think about the Em chord as the Root, Am as the 4th and Bm as the 5th. It’s the same thing really.
Think of it as a numbers game. Either you start counting from The major root or start your formula on the minor side of the chordal relationship. It’s still the same system. This way of looking at Chord Progressions becomes relevant when we start plugging chords from Harmonic Minor into a songs that are in the Major key.
The most common Chord Substitution wherein Harmonic Minor is used as a suppliment is the Harmonization of the third as I like to call it. Usually when you see a Major(Dom 7th) chord instead of the typical minor(Min 7th) found in the third position of our Major Progression.
Lets consider my guitar lesson Hallellujah by Leonard Coen. How is the song composed? We see that the song is in the key C Major. This means that the chords used in the song are chords from the Chord Progression of C Major. The standard Major Chord Progression is always the same. The first chord is a Major; the second a minor; the third a minor; the fourth a Major; the fifth a Major; the 6th is minor (natural minor), and finally the seventh chord is diminished(used least). Do you remember the lyrics from the song where he sings out the Chord Progression formula? “well it goes like this the fourth the fifth the minor fall and the major lift”. The composer is explaining how he uses the standard progression, goes from the 4th chord of the basic progression to the fifth. The formula stays the same but the notes used might change depending on the key chosen.
Now if you look at the video you see that all the Chords used in the song conform to the typical Chord Progression except for one chord and that is the third (iii) or the E chord. The Chord progression formula tells us that this chord should by all accounts be a minor not a Major. What happens here is that we move momentarily from the standard chord progression into the Harmonic Minor System by plugin in the 5th chord of the Harmonic Minor Progression into the Third Position of the Major system. Look at this picture to clarify. It gets confusing when you are not sure from which end you are counting.
I like to call this common trait in both major and minor compositions harmonizing the third which in my mind means that you take the third chord from the Major chord progression (the 5th from the minor perspective) and substitute it with the 5th chord from the Harmonic minor chord progression. This means that you substitute an extended minor 7th chord or a minor triad from the traditional major scale system with the extended Dominant 7th or a major triad from the Harmonic Minor Scale Progression.