If we have C major, the diatonic chords are:
C, Dm, Em, F, G, Am, Bdim, C

If we turn each of these chords to diatonic 13th chord versions of themselves:
Cmaj13, Dmin13, Emin13, Fmaj13, G13, Amin13, Bdim13

The notes of these chords are now equivalent to the modes of each starting root.
C Ionian, D Dorian, E Phrygian, F Lydian, G Mixolydian, A Aeolian, B Locrian.

Here's a picture of what I mean from one of my other questions. So I was wondering if this is a known relationship in harmonic theory and are people using it. For example, using extended chords to modulate to modes and vice versa.

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    I don't quite understand the question, but it seems that you're confusing whether chords arise out of modes or modes arrive out of chords. The answer was originally that chords were natural consequences of several notes from a given mode sounding together, but more recently a mode could be derived from a specific chord or pitch collection construction (e.g. Scriabin's Mystic Chord: en.wikipedia.org/wiki/Mystic_chord) – LSM07 Feb 24 '19 at 16:22
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    Major 13th chords are a rarity. Dominant 13ths are commonplace. Usually with 13th chords, some of the thirds on the way are omitted. Just gets too busy. I think you're barking up the wrong tree. – Tim Feb 24 '19 at 17:22

You can see relationships between just about anything. The real question is: is the relation meaningful?

For example, there's a strong correlation between murder rates and ice cream sales. Will banning ice cream reduce the murder rate? Answer: of course not, because the relationship is indirect - it's hot weather that raises both.

That's what you've got here. Since all of the modes related to one scale have the same notes, all of the 13th chords formed with seven tones from the tertian series will have the same notes. The relationship between both the chords and the modes are dependent on our definitions of mode and chord.

The reason it's not meaningful is that chords with seven notes are rarely voiced with all seven, and there's certainly no requirement to always voice a chord in root position. Your Cmaj13 chord might be voiced as root-seventh-third-thirteenth, which would make it identical to first inversion voicings of Am(add9). And because notes are left out of chord voicings in practice, even if you've identified C as the root in the context of a progression you could play in C Lydian over it without a problem, since the particular voicing lacks any F note.

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