Tridecimal neutral seventh chord
Tridecimal neutral seventh chord is a chord consisting of three approximately equal neutral thirds which can be set to any kind of tridecimal neutral third interval.
Theory
By tempering certain commas, it's possible to make the chord equidistant (or partially with an equidistant triad), meaning that the two three neutral thirds that make up the chord are the same. Such tempering occurs among the buzurgismic chords, where 13/12 is tempered with 14/13.
However (at least in Eliora's experience), tridecimal neutral 7th chord sounds much more noble when the neutral third commas are left untempered, as it enables for a more precise 13/8 sound, and the slight asymmetry of one of the neutral triads that makes up the chord creates a tension that seeks to be resolved.
Inversions
The chord has three inversions as any seventh chord has, tridecimal neutral 6-5 chord (N65) , tridecimal neutral 4-3 chord (N43), and tridecimal neutral 4-2 chord (N42 or N2).
Since the 13 N7 chord has one slightly bigger N3 and the two others are smaller (16/13), or the 169/168 can go into 14/13, this gives a total of 12 possible chords that will have a mildly different sense depending on the position of larger or smaller neutral triad in each chord.
If the 13-N43 has 13/12 as the neutral second, it will have 4/3 between the 0th (root) and 2nd notes, and also 4/3 between 1st and 3rd notes. If instead the neutral second is equal to 16/13 - 14/13 - 16/13 in the second inversion, meaning that the perfect fourth that occurs between the root and the 2nd note, and between 1st and the upper notes will be 169/168 sharp of 4/3, creating tension that needs to be resolved.
353edo's Rectified Hebrew temperament's 19-tone scale sets the chord to be this way. This is used in Mercury Amalgam's instrumental piece Bottom Text.
13/8 N4-3 chord cadence
It is the progression 13/8-N43 - D7 - T53. In this case, 13/8 is the root of the N43 chord, meaning that the first note above the root is the tonic.
Such cadence is used in Mercury Amalgam songs Hemoclysm Totem and Bottom Text.
Example in 60edo:
42-0-7-25 - 35-55-10-25 - 0-20-35 (or 0-15-35) [using numbers]
G#2/5-C-C#2/5 - F - G-B-A-F - C-E-G (or C-Eb-G) [using 12edo and fractions]
As means of extending standard Western music theory
Tridecimal neutral seventh chord can be used alongside ordinary 12edo chords in resolutions, and it provides a unique sound that is both peculiar and consonant.
The best EDO for such play is 60edo, since it is the first multiple of 12 with such an excellent 13/8, which it inherits from 10edo. If one seeks tempering of 169/168, 24edo is the best choice for that, as it has only one kind of neutral triad and the tridecimal neutral 7th chord just becomes a "neutral 7th chord".