Hi everyone,
In 15 tone equal temperament or 15ed2 / 15edo (15 equal divisions of the 2nd harmonic / the octave), there are three types of 7th, three types of 2nd, and two tritones. Thus, the three “extra” notes. Of course, a majority of the other notes have changed position from 12 tone equal temperament (or 12edo / 12ed2), aside from the root position augmented triad. This is because both 12ed2 and 15ed2 divide the octave into 3 equal parts (or 400¢) as 12 and 15 are both divisible by 3, and thus the augmented triad.
The lowest of these 7ths is a 960¢ interval, very close to the ~967¢ 7:4 harmonic 7th, which is an octave reduced 7th harmonic, thus the name. Because of its close proximity to a low harmonic, and perhaps the simplest ratio describing a 7th, there is a low amount of beating, and thus paired with a major third, this 7th can function as a root position chord as well as a dominant function chord. It can function as a dominant as well due to it still being a tritone (in this case a lesser tritone) from the major third.
It is debatable whether 7:4 or 9:5 is in fact the simpler ratio, due to the first being the 4th prime number (7) octave reduced (over the 1st prime / 2nd harmonic (2) twice), or the 2nd prime twice (3rd harmonic / 3) over the 3rd prime (5th harmonic / 5). In fact, simpler may not be an accurate term to describe the distinction, as both functional and harmonic relations are existing here.
The middle 7th in 15ed2 is a 1040¢ interval, quite close to the 11:6 neutral 7th ratio. Being a 4th (flat 4th) away from the minor 3rd, it can function as a minor 7th. And being a greater tritone, away from the major third, can also make a dominant function. As a melodic interval, its function as a part of the 11th harmonic is more prevalent.
The high 7th in 15ed2 is tuned to 1120¢ and essential functions normally as the major 7th.
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PS: naming is a challenge currently, which tuning systems often having multiple names. I prefer 15ed2 as the most technical, accurate, and general name. However, 15 tone equal temperament is most widely understood, followed by 15edo.
The “tritone” interval was named such as the sum of three whole tones. In 15ed2, the small whole tone (of 2 steps) used 3 times is the 4th (of 6 steps), and the large whole tone (of 3 steps) used 3 times is the 5th (of 9 steps). Hence the tritone name is erroneous here in this context. Unless, of course, you accept two different sized tones in combination to generate the tritone, in which case, both 15ed2 tritones may be generated. Such is an argument for the role of context in music, and particularly the large amounts of compositional uses of this type of contextual structure in microtonal / xenharmonic music.
Noah Dean Jordan 