After reading all this I’m off for a lie down. Lovely pictures! 
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Thanks to a question on the recent University Challenge final I now have a great way to remember that distinctive but angular and tricky to sing sounding Lydian Mode
It’s the Simpson’s Theme tune ( Starts C E F# A G E)
D’Oh!
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I reckon the reason Locrian doesn’t get used is because it is the only mode that doesn’t have a major dominant. The dominant to tonic interval or V - I cadence is so important to western music that it’s easy to see that a scale without that feature becomes an unattractive prospect for music makers.
And I think bands like Sabbath that used the flat fifth a lot, didn’t play in ‘chords in the key’ of Locrian
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That’s an interesting observation which I hadn’t noticed before. I didn’t really look past the fact of it having a very unstable tonic chord - diminished chords simply do not sound like home base, as a place of resolution, to my ears.
I work on resigning myself to the parallel naming convention, even though lines sharing a vertex cannot, by definition, be parallel unless vectoring 180º.
Relative modes on the other hand, are indeed parallel. Slip sliding parallel.
musings from my head. 
Edit add:
let me add I’ve struggled hopelessly at times over 6y or so to understand this stuff. It started to come together finally about a year ago with the help of more thorough on line site that explained it fully. Successively playing relative modes around a circle sounded just that - major scale. Parallel modes with an open drone note of A, E or D then allowed me to hear a real mode in context harmonically.
Your explanations are taking this to the next level still for me in terms of clarity with no ambiguity. So much on line leave out contextual detail entirely and muddy the waters terribly.
Great stuff, thanks for posting all this.
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Modes Part 4 - Modal Melodies: the sound of the modes?
It is time to involve the ears in our cognition.
We are going to hear seven parallel modes, all with root note C. First in a simple ascending and descending sequence. Then in a three-note sequence formed from playing three notes up, going back two, three notes up, going back two and so on. This inverts once the upper most note of the pattern is reached. These may still sound like the scale exercises / etudes I decried earlier when discussing modes in series. However, I hope that this time, a distinct difference in character and the unique flavour of each mode becomes just that bit more evident in the eras.
We are going to start with the Lydian mode - the mode found when the modal frame holding a seven-note cluster was rotated to its furthest point clockwise. We will work anticlockwise, ending with Locrian. The order will be:
Lydian
Ionian
Mixolydian
Dorian
Aeolian
Phrygian
Locrian
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The C Mixolydian Mode
—
C Mixolydian ascending and descending
C Mixolydian ascending and descending 3-in-a-line
The C Dorian Mode
—
C Dorian ascending and descending - audio file
C Dorian ascending and descending 3-in-a-line
The C Aeolian (natural minor) Mode
—
C Aeolian ascending and descending
C Aeolian ascending and descending 3-in-a-line
The C Phrygian Mode
—
C Phrygian ascending and descending
C Phrygian ascending and descending 3-in-a-line
The C Locrian Mode
—
C Locrian ascending and descending
C Locrian ascending and descending 3-in-a-line
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Modal Melodies
We have heard the modal scales played in sequence. Hopefully your ears are beginning to discern some differences between them. We now progress to some modal melodies.
I created a fairly simple melody. I was not looking to write the world’s greatest, most memorable melody for a worldwide chart hit. I merely wanted something that moved around the scale, using the notes.
The first example you will hear uses the Lydian mode with its sharp 4 (#4). This is often considered the ‘brightest’ mode. You will then hear how the melody changes as, one at a time to take it from mode to mode, one note is flattened. The order follows that above.
Lydian melody
Ionian melody
Mixolydian melody
Dorian melody
Aeolian melody
Phrygian melody
Locrian melody
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Now, just for a bit of silliness, and to illustrate one way that modes can be made to sound awkward and wrong, I present some variations on a tune. A well-known tune. This time going from Lydian to Locrian. Apologies if you had managed to put festive thoughts away for the year. 
Ding Dong Lydian
Ding Dong Ionian … the true scale for this tune.
Ding Dong Mixolydian
Ding Dong Dorian
Ding Dong Aeolian
Ding Dong Phrygian
Ding Dong Locrian
Crikey! That was something wasn’t it? Wow.
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We have now listened to seven variants of modal melody.
Each does sound different.
Perhaps in listening to each, starting at Lydian and moving towards Locrian, there is a sense of the mood of the melody changing. Ordering the modes from Lydian to Locrian has us moving anticlockwise around the Circle of Fifths, around the colour wheel. It also sees one new note lowered by a semitone (flattened) with each rotation. And many musical descriptions will describe this as moving from the brightest mode to the darkest mode. Somehow, it seems, flats being a sombre and foreboding character to the music.
Lydian – no flats, one sharp – bright.
Locrian – lots of flat notes – dark.
That is not really a meaningful why / how explanation, but I shall just place it there for the moment.
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What I would like to do next is to categorise our modes in to two groups of three and discard one altogether.
The two groups of three are the major types and the minor types.
Major types:
- Ionian (THE major scale);
- Lydian;
- Mixolydian.
Minor types:
- Aeolian (THE minor scale);
- Dorian;
- Phrygian.
Remember, major type scales have a major third and minor type scales have a minor third.
So much music makes use of Ionian and Aeolian that our ears are accustomed to hearing those sounds. There is probably value in hearing the same modal melodies side-by-side in order to hear the subtle differences, the nuances in sound a little more clearly.
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Here are the three major modes playing the melody already heard above in this order:
Ionian → Lydian → Ionian → Mixolydian all separated by four chimes of two C-notes an octave apart
Ionian sits in the middle of the three Major modes.
To move from Ionian to Lydian, one of the notes is raised by a semitone.
To move from Ionian to Mixlolydian, one of the notes is lowered by a semitone.
You will hear Ionian twice.
Ionian → Lydian → Ionian → Mixolydian
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Here are the three minor modes playing the melody already heard above in this order:
Aeolian → Dorian → Aeolian → Phrygian all separated by four chimes of two C-notes an octave apart.
You will hear Aeolian twice.
Aeolian sits in the middle of the three minor modes.
To move from Aeolian to Dorian, one of the notes is raised by a semitone.
To move from Aeolian to Phrygian, one of the notes is lowered by a semitone.
Aeolian → Dorian → Aeolian → Phrygian
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I have deliberately chosen to start with, then move away from, then move back to Ionian and Aeolian as those are – supposedly – the ones our ears are most accustomed to. This sequencing may allow you to hear the divergence from the commonly used major and minor modes.